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Page 1: Report on US ARMY AWARD DAAD19-00-1-0534 · −− DFT based DMT, optimum bit allocation Transmitted Power (dB) −. General DMT, optimum bit allocation #channels, #users = 2 (1,1)
Page 2: Report on US ARMY AWARD DAAD19-00-1-0534 · −− DFT based DMT, optimum bit allocation Transmitted Power (dB) −. General DMT, optimum bit allocation #channels, #users = 2 (1,1)

Report on US ARMY AWARD DAAD19-00-1-0534

Optimal Subband Coders for Cyclostationary Signals

PI: Soura Dasgupta

University of Iowa

The purpose of the award was to study the subband coding of cyclostationary signals.Accomplishement to date are listed below.

• We had proposed to study subband coding of cyclostationary signals with uniformfilter banks, under two bit rate constraints. (i) When the bit allocation among thesubband signals is static, and (ii) when the bit allocation is periodic but the bitbudget across the subband signals is constant at each instant of time. Problem (i)was solved before the award was received. We have solved (ii). In addition we havealso proposed a third criteria in which the bit budget is defined as a fixed average

over a period of the signal cycle. We show that though the optimizing filter bank isthe same for all three, and have characterized this filter bank, the new criteria leadsto a higher coding gain.

Figure 1 shows the coding gains under the last two schemes, for an input with 2-periodic statistics. The crosses are the second scheme and the circles the third.Observe that the last scheme leads to higher coding gains times better.

0 5 10 15 20 25 30 35 400

10

20

30

40

50

60

70

80Coding Gain vs # of channels: Subband coding of WSCS signal with period 2

o : (TDBA Scheme)

x : (PDBA Scheme)

# of channels

Cod

ing

Gai

n (in

dB

) as

ratio

of d

isto

rtion

Figure 1: Coding gain plots.

1

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• Realization of these optimizing filter banks requires the notion of compaction filters inthe case of stationary signals. We have introduced the concept of energy compactionfor cyclostationary signals as well. We have also generalized the notion of M -bandfilters to cyclostationary case. We have studied the theory and design of cyclosta-tionary energy compaction filters and showed how they relate to the optmizing filterbanks.

• We had proposed to study the use of nonuniform filter banks for subband coding forboth the stationary and cyclostationary signals. En route to such a study, we haveshowed that every biorthogonal dyadic filter bank has a tree structured decomposi-tion. This permits the optimization of such filter banks to be conducted in a treestructured framework, making the underlying task much simpler.

• A common occurrence of cyclostationarity is in Orthogonal Frequency Division Mul-tiplexed (OFDM) communications. We have shown that certain channel resourceallocation problems for OFDM systems are dual problems of subband coding. Wehave solved the optimum resource allocation problem for OFDM in the multiuser en-vironment. Specifically, we have considered in turn a variety of settings culminatingin one in which each user is assigned different number of subchannels and differentbit rates and is required to achieve differing symbol error rates and supports po-tentially different modulation schemes. Our goal is to select the input and outputblock transforms, the linear redundancy removal scheme at the receiver, the numberof bits/symbol assigned to each subchannel, and the subchannel assignment to eachuser, in order to achieve the QoS specifications under a zero ISI condition with min-imum transmitted power. We assume knowledge of the equalized channel and thesecond-order statistics of the noise at the receiver input.

(A) The optimum input/output block transforms are orthonormal.

(B) The selection of the optimum input/output transforms and redundancy removalschemes depends only on the channel/interference conditions, and does not de-pend on such service requirements as the required bit rates and symbol errorrates. Thus there is a conceptual separation between the selection of thesevariables, and the remaining tasks of bit loading and subchannel selection. Inpractical terms this considerably simplifies the optimzation problem.

Figure 2 compares the transmitting power of the DFT based DMT under no optimumbit allocation and optimum bit allocation with an optimum unitary transceiver. Weassume the equalized-channel to be C(z) = 1+0.5z−1, and a noise source v(n) whosepower spectral density is shown in fig. 1. We assume the DMT system supports twouser services. Both services employ QAM modulation schemes, and the target ratesfor the two users are 600 Kbps and 1 Mbps respectively. The (i, j) on the x-axis of theplot indicates that user 1 and 2 were respectively allocated i, j number of channels.

2

Page 4: Report on US ARMY AWARD DAAD19-00-1-0534 · −− DFT based DMT, optimum bit allocation Transmitted Power (dB) −. General DMT, optimum bit allocation #channels, #users = 2 (1,1)

The plot shows that there is a 10 dB saving in transmit power with our design overthe DFT based DMT under optimum bit allocation, and a 14 dB improvement overthe conventional DMT with no optimum bit allocation.

0 20 40 60 80 100 120 140 160 180 2000

1

2

3

4

5

6

7x 10

−14

Power spectral density of noise

−75

−70

−65

−60

−55

−50

−45Transmitted power under different schemes

__ DFT based DMT, no optimum bit allocation

−− DFT based DMT, optimum bit allocation

−. General DMT, optimum bit allocation

Tra

nsm

itte

d P

ow

er

(dB

)

#channels, #users = 2

(1,1) (2,3) (3,5) (4,7) (5,9) (6,11) (7,13) 2 5 8 11 14 17 20

Figure 2: Comparison of transmit power levels.

• For optimum coding over transmission channels it is imoportant that the transmitterbe aware of the channel it transmits over. While in recent years many algorithms havebeen proposed that assume such channel knowledge at the transmitter, no methodfor making exists for making the transmitter channel aware. We have proposed a newfeedback scheme that permits the transmitter to directly estimate the channel.

• In both subband coding and DMT bit loading is an important problem. Specifically,for an N -subchannel system in these problems reduce to general problem:

MinimizeP (b1, .., bN ) =N∑

k=1

φk(bk) (1)

subject to

Constraint :N∑

k=1

bk = B , bk ∈ {0 , 1 , ...B}, (2)

3

Page 5: Report on US ARMY AWARD DAAD19-00-1-0534 · −− DFT based DMT, optimum bit allocation Transmitted Power (dB) −. General DMT, optimum bit allocation #channels, #users = 2 (1,1)

where φk is a convex function nonnegative integers bk, and B is a positive integer. Insubband coding

φk(bk) = αk2−2bk (3)

where αk is determined by the signal variance in the k-th subchannel, and P (b1, .., bN )is the average distortion variance. In multicarrier systems

φk(bk) = αk2bk (4)

where αk reflect target symbol error rates (SER), and channel and interference condi-tions experienced in the the k-th subchannel, and P (b1, .., bN ) is the total transmittedpower. Higher values of αk reflects more adverse subchannel conditions and/or lowertarget SER; bk is the the number of bits assigned to each symbol in the cognizantsubchannel.

The complexity of most existing algorithms for such discrete bit loading grows withB. We have formulated a new bit loading scheme whose complexity is independent ofB and yet like existing schemes depends as O(N log N) in the number of subchannels.

A comparison of the performance of the algorithms of [2] and [1] and the proposedalgorithm with respect to the number of computations required is shown in the figures3 and 4, for the cases where N = 32 and N = 64, respectively. In implementing [1],which is a suboptimal algorithm, the maximum number of bits, B∗ that any channelcan be assigned is kept at B.

0 100 200 300 400 500 600 700 80010

2

103

104

Number of Channels = 32

Number of Bits to be allocated B

Num

ber o

f Com

puta

tions

Req

uire

d

: Proposed Algorithm: Campellos Algorithm: Krongold, Ramachandran Algorithm

Figure 3: Runtime comparisions of the three algorithms for N=32

Number of computations needed for each algorithm to converge to the optimal solu-tion was calculated by assuming that addition, subtraction, div, mod, multiplicationor division of two numbers would need one computation as would the logical compar-isons between two decimal numbers. The results show that the algorithm described

4

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0 100 200 300 400 500 600 700 80010

2

103

104

Number of Channels = 64

Number of Bits to be allocated B

Num

ber o

f Com

puta

tions

Req

uire

d

: Proposed Algorithm: Campellos Algorithm: Krongold, Ramachandran Algorithm

Figure 4: Runtime comparisions of the three algorithms for N=64

in [1] is linear with respect to B while the algorithm in [2] needs large number ofcomputations to converge as B grows. The number of computations needed for theproposed algorithm is independent of the change in B the minor variations whosesource is discussed in the paper referenced below. The improvement in performanceis very significant if B is large when compared to N .

Collaboration with DOD Labs: We are currently part of a team that is initiatingcollaboration with TACOM on research on digital humans. Many of the resource allocationideas implicit in this work are crucial to that project.

Below we list publications acknowledging this award.

List of Publications

1. A. Pandharipande and S. Dasgupta, “Subband Coding of Cyclostationary Signals”,IEEE Transactions on Circuits and Systems I, IEEE Transactions on Circuits and

Systems -I, pp 1884-1887, December 2002.

2. A. Pandharipande, S. Dasgupta and D. Kula, “Filter Banks for Optimum SubbandCoding of Cyclostationary Signals with Dynamic Bit Allocation”, in Proceedings of

ASILOMAR, November 2001.

3. A. Pandharipande and S. Dasgupta, “Optimal transceivers in DMT based multiusercommunications”, in Proceedings of ICASSP, 2001.

4. A. Pandharipande and S. Dasgupta, “Optimal DMT based transceivers in multiusercommunications”. To appear in IEEE Transactions on Communications.

5

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5. A. Pandharipande and S. Dasgupta, “Optimum compaction filters for cyclostationarysignals”, in Proceeedings of ICASSP, Orlando, Florida, May 2002. Journal versionunder preparation.

6. A. Pandharipande and S. Dasgupta, “Optimality in multiuser DMT, multiple de-scription coding and the subband coding of cyclostationary signals”, in Proceedings

of IFAC Workshop on Periodic Control, Como, Italy, August 2001.

7. A. Pandharipande and S. Dasgupta, “On Biorthogonal Nonuniform Filter Banks andTree Structures”, IEEE Transactions on Circuits and Systems -I, pp 1457-1467, Oc-tober 2002.

8. H. Xu, S. Dasgupta, and Z. Ding, “A Novel Channel Identification Method for FastWireless Communication Systems”, Submitted to IEEE Transactions on Communi-

cations.

9. A. Pandharipande and S. Dasgupta, “Subband Coding of Cyclostationary Signalswith Overdecimated Filter Banks”, in Proceedings of Eusipco, Toulouse, France,September 2002. Journal version in progress.

10. M. Vemulapalli, S. Dasgupta and A. Pandharipande, “A new algorithm for optimumbit loading”, Submitted to the International Conference on Communications. Journalversion awaiting Patent Application.

11. Pandharipande, A. and Dasgupta, S., “Channel and flow adaptive multiuser DMT”,in Proceedings of 35th Asilomar Conference on Signals, Systems and Computers,November, 2002.

12. Pandharipande, A. and Dasgupta, S., “Optimum multiuser DMT with unequal sub-channel assignments”, in Proceedings of ICC, May 2003, to appear.

13. H. Xu, S. Dasgupta, and Z. Ding, “An Improved Feedback Scheme for Dual ChannelIdentification in Wireless Communication Systems”, in Proceedings of ICASSP, HongKong, April 2003.

14. A. Pandharipande and S. Dasgupta, “Optimum biorthogonal DMT systems for multi-service communication”, in Proceedings of ICASSP, Hong Kong, April 2003.

15. A. Pandharipande and S. Dasgupta, “Optimum channel and multiflow-adaptive biorthog-onal DMT systems”, Submitted to IEEE Transcations on Signal Provessing.

16. Dasgupta, S. and Pandharipande, A., “Complete Characterization of Channel Resis-tant DMT with Cyclic Prefix”, IEEE Signal Processing Letters, May 2003.

List of Personnel apart from the PI

6

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1. A. Pandharipande. Ph.D. December 2002.

2. H. Xu. Ph.D. August 2003.

3. T. Tigrek. MS, March 2002.

4. X. Song. Current Ph. D. Student.

5. M. Vemulapalli. Current Ph. D. Student.

Patents

We are planning to file a patent application on the bitloading algorithm.

References

[1] J. Campello, “Practical bit loading for DMT”, IEEE International Conference on

Communications, pp 801-805, 1999.

[2] B. S. Krongold, K. Ramchandran and D. L. Jones, “An efficient algorithm for optimalmargin maximization in multicarrier communication systems”, IEEE Global Telecom-

munications Conference, pp 899-903, 1999.

7

Page 9: Report on US ARMY AWARD DAAD19-00-1-0534 · −− DFT based DMT, optimum bit allocation Transmitted Power (dB) −. General DMT, optimum bit allocation #channels, #users = 2 (1,1)

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SUBBAND CODING OF CYCLOSTATIONARY SIGNALS WITH OVERDECIMATEDFILTER BANKS

Soura Dasgupta, Ashish Pandharipande

Department of Electrical & Computer EngineeringThe University of Iowa

Iowa City, IA-52242, USA.Email:

�dasgupta, pashish � @engineering.uiowa.edu

ABSTRACTWe consider optimal orthonormal filter banks for subbandcoding of wide sense cyclostationary signals, with � -periodicsecond order statistics. An � -channel over decimated uni-form filter bank, with � -periodic analysis and synthesis fil-ters, is used as the subband coder. An average variance con-dition is used to measure the output distortion. We show thatfor at least three potential bit allocation strategies, the opti-mum filter bank is a principal component filter bank. Thisis in the same vein as our earlier results on subband codingwith maximally decimated filter banks.

1. INTRODUCTION

Wide sense cyclostationary (WSCS) signals arise in manyapplications, [1], [2]. We consider optimum orthonormalsubband coding of zero mean WSCS signals with � -periodicsecond order statistics, i.e. signals that obey for all ����� :� �� ��� ���� ������� � �� ������� ���� ��������� where

� � � denotesthe expectation operator.

The subband coder itself is an � -channel over decimateduniform filter bank (UFB), (see fig. 1), with!#" �$�and � -periodic linear analysis and synthesis filters, %'& ���)(*�and +�& ���)(,� , respectively. Each subband signal -.& ��� , isquantized at the � th instant, by a / & ��� bit quantizer, 0 & .Subject to bit rate and orthonormality constraints, we wishto allocate bits /1& �2� , and select, %3& ���)(*� and +�& �4�5(*� tominimize the average variance of 6�7 ���78 �� ��� .

Among many possible bit rate constraints one can adopt,three are of interest here. The first called static bit allocation(SBA) involves constant /1& ��� , and a bit rate constraint/9� :�;�<>=? &A@�B /C&ED�FG�IH (1.1)

Supported by ARO contract DAAD19-00-1-0534 and NSF grants ECS-9970105 and CCR-9973133.

The second and third, both assume � -periodic bit allocation:/C& �J�K�����L/C& ���5H (1.2)

In the second, the average bit rate over all the channels isconstant at each time instant, i.e. given / and all � ,/9� : ;�<>=? &A@�B / & ��� D FM�$H (1.3)

The third assumes a fixed average bit rate over periods oflength � : /9� N�O�QP <>=?R @�B ;�<>=? &A@�B / & ���5H (1.4)

Among these, (1.2) requires the least computation and (1.4)is the most general. On the other hand, (1.3) is preferred over(1.4) in applications, such as control over networks, wherethe bit rate constraint must be enforced at every time instant.

Subband coding under these three constraints, with max-imally decimated filter banks (i.e. �S� !

) has been studiedin [6] and [7]. These references show that, while the op-timum bit allocation schemes differ among (1.1 - 1.4), theoptimizing % & ���)(*� and + & ���)(*� can be chosen as the sameregardless of the allocation scheme. In fact a Principal Com-ponent Filter Bank (PCFB), represents the common optimiz-ing solution.

Recent studies, [4, 5] have established that the optimumUFB subband coder for Wide Sense Stationary (WSS) sig-nals is a PCFB, [3]. The principal contribution of this paperis to show that even on the over decimated case, optimalityis attained through PCFB’s, despite the differing bit alloca-tion constraints, reinforcing the universality of PCFB basedsolutions for problems such as these.

2. OPTIMUM BIT ALLOCATION

For any zero mean signal�7 ��� , define T�UV ���W� � � U ����� .

All subband signals - & ��� have � -periodic second order statis-tics. As in [4], [5], we assume that the quantizers are mod-eled by additive zero mean noise sources, independent of the

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� ��� � ��� � � � � ��� � � ���� ��� � � ��� �

������� ��� � ����� ���

� ������ �

���

������� ��������

......

......

� �

� �

� �����

� �

� �

� �����

Fig. 1. An � -channel over decimated filter bank as subbandcoder.- & �2� , with variances of the formT U�! ��� �#"%$ < U'& )( R * T U+, ���5� (2.5)

with " a distribution dependent constant. Note that under(1.2), T�U�! ��� are � -periodic.

Observe that the overall filter bank is! � -periodic. Let-. (*� and

-/ (*� be the transfer functions of! � -fold blocked

versions of the analysis and synthesis banks respectively. Inparticular,

-. (,� is �O�10 ! � and-/ (,� is

! �20��O� .A key difference between the over decimated and the maxi-mally decimated cases is that these transfer functions are nolonger square.

Define� & ����� �7 ! � 843 � , � & ��� � �� ! ��843 � and the

WSS vectors,

-�� ��� � � B �����5�CHCHCH � � B �65 8 � � N �)�1HCHCH�� �87 <>= �����)�HCHCH�� � 7 <>= ��� 8 � � N � �:9��-- ����� -GB �����)�1HCHCH���-MB �65Q8 � � N �5�CHCHCH � - ;�<>= ���2�5�H1HCH���- ;�<>= ��� 8 � � N � �:9�� (2.6)

with power spectral density (PSD) matrices ;=<V ?> � and ;@<+ ?> �respectively. Observe,

-- ��� � -. (*� -� ���5HWe assume ;A<V ?> � is known. We assume the perfect recon-struction and orthonormality conditions,-. (*� -.CB (*� �ED � -/�B (,� -/ (*�5� and

-/ (*��� -.CB (*�5H (2.7)

We propose to minimize the average variance of 6F �2���6�� ���78 �� �2� and under (1.3) and (2.7), obtainN�O� ; P <>=?R @�B T U G� ��� � N��� P <>=?R @�B ;�<>=?H @�B T U�!I ���

� "��� P <>=?R @�B ;�<>=?H @�B $ < U,& I ( R * T U+,I �2�

J "%$ < U'&� P <>=?R @�B :�;�<>=KH @�B T U+ I ����D =ML5;

(2.8)

with equality holding iff for each 3���� �)�$ < U,& N( R * T U+' �����O$ < U,& I ( R * T U+ I ���5H (2.9)

Likewise under (1.4), the lower bounded becomes

"%$ < U,&� : P <>=KR @�B ;�<>=KH @�B T U+ I ��� D =PL); P � (2.10)

with the bound met iff for each 3���� �)� = �)� U$

< U,& ( R Q * T U+' � = ���#$< U'& I ( R R * T U+ I � U �5H (2.11)

On the other hand under the static bit allocation strategy of(1.1), as shown in [6], the lower bounded becomes

"%$ < U,&� ;�<>=KH @�B : P <>=?R @�B T U+,I �2� D � (2.12)

with the bound met iff for each 3����$ < U,& : P <>=?R @�B T U+' ��� D �S$ < U,& I : P <>=?R @�B T U+ I ��� D H (2.13)

Observe, the optimum bit allocation scheme (2.11) is themost stringent among (2.9), (2.11) and (2.13).

Consequently UFB selection reduces to the followingproblem:

Problem 2.1 Consider the �O�T0 ! � system-. (,� with

WSS input vector-�7 ��� with given Hermitian PSD matrix

; <V U> � . Suppose-- ��� in (2.6) is the output of

-. (*� . For(1.3) (resp. (1.4)), (resp. (1.1)) find

-. (,� such that V = (resp.V U ) (resp. V�W ) is minimized subject to (2.7).

V = � P <>=?R @�B ;�<>=KH @�B T U+ I ��� � =PL);

(2.14)

V U � : P <>=KR @�B ;�<>=KH @�B T U+ I ��� D =PL); P (2.15)

V W � ;�<>=KH @�B : P <>=?R @�B T U+ I �2� D (2.16)

Observe all three of (2.14) - (2.16) are quite differentfrom one another. While V U is similar to the correspondingcost function in the WSS case, [5], V = and VXW are more com-plicated. Further while V U does not change by permuting thesubband variances, V = and VXW do. Indeed given a set of sub-band variances at different time instants we need consideronly the arrangements that lead to the minimum value of V = ,V W . Such optimal arrangements are characterized below.

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Optimum Arrangement for V = : Among the various per-mutations of T U+, �� � , ones that minimize V = obeys, [8]:

T U+�� � = � J T U+�� � U ��� ;�<>=K&�@� T U+, � = � � ;�<>=K&�@�� T U+, � U � (2.17)

For a $ -channel filter bank, � � $ , this requires that thelargest be paired with the smallest, the second largest withthe second smallest etc..

Optimum Arrangement for V W : Among the various per-mutations of T�U+, �� � , ones that minimize V = obeys, [6]: foreach � , one partial sum equals the sum of the � largest amongthe T�U+' �� � , another equals the sum of the next � largest, etc.

3. OPTIMUM SUBBAND CODER

We now characterize the optimum selection of-. (*� , by

introducing the notions of majorization and Schur concavity,[8].Definition 3.1 Consider two sequences

� ��� � &�� �&A@ = and� ��� � &�� �&A@ = with� & J � &�� = and � & J � &�� = . Then we say

that � majorizes�

, denoted as��� � , if the following holds

with equality at � ���R? &A@ = � & �R? &A@ = � & � N ��������H

Definition 3.2 Consider two sequences� ��� � & � H&A@ = and� ��� � &�� H&A@ = with

� & J � &�� = and � & J � &�� = . Then we saythat � weakly supermajorizes

�, denoted as

���! � , ifH? &A@ R � & J H? &A@ R � & � N � ����� H

We also have the following Fact from [8].

Fact 1 Consider any � ! 0'� !Hermitian matrix

/with

eigenvalues " = J " U J HCHCH J " P 7 , and an � ! 0 � !matrix #L�%$ / $ B , with the � ! 0 � !

matrix $ obeying$&$ B � D . Then the diagonal elements #�&(' & of # obey

�)# &(' & � ; 7&A@ = � �*" P 7 <>; 7 <>= �CHCH1H���" P 7 � H (3.18)

Further if! � � ,

�+# &(' &�� 7&A@ = � �," = �CHCHCH���" P 7 � H (3.19)

Definition 3.3 A real valued function - (*����- ( = �CHCH1HG�)( � �defined on a set .0/ / � is said to be Schur concave on . if��� � 1 �2. � - � � J - � �)H- is strictly Schur concave on . if strict inequality - � � "- � � holds when

�is not a permutation of � .

Further we note the following result from [8].

Theorem 3.1 Let - be a real-valued strictly Schur concavefunction defined and continuous on 3 as in Theorem 3.1.Then �4� � �5- � � J - � �5�with equality holding only if

�is a permutation of � .

We will now state a theorem that results in a test for strictSchur concavity. We denote

- (R* (*��� 6 - (*�6 ( R �7- ( &(' 8 * (*� � 6 U*- (*�6 ( & 6 ( 8 �

and

V = ������� � 6 V =6 T U+'I ��� � and V = �4� � �:9��;����� 6 U V =6 T U+'I ��� 6 T U+ � 9 � HTheorem 3.2 Let - (*� be a scalar real valued function de-fined and continuous on 3 �<� ( = �CHCHCHM�)(+� �>=�( = J HCHCH J( �?� , and twice differentiable on the interior of 3 . Then - (,�is strictly Schur concave on 3 iff: (i) - (

R* (,� is increasing

in � , and (ii)

- (R* (*����- ( R � = * (,�@� - (

R'R* (,�78A- ( R ' R � = * (*�8 - (

R� = ' R * (,�>�B- ( R � = ' R � = * (*�DCFE H

If only (i) holds then - (*� is only Schur concave.

It is known, that V U is strictly Schur concave, [8]. Wealso have the following lemma.

Lemma 3.1 The real valued function V = as defined in (2.14)is strictly Schur concave under (2.17).

Proof: Clearly V = is symmetric in its arguments T�U+ I ��� ,satisfying (i) of Theorem 3.2. Note that

V = ��������� N� G+H ;�<>=&A@�B T�U+ ���JI =PL);T U+,I �2� H (3.20)

If T�U+ I Q � = � J T�U+ I Q � = � , then under (2.17)

V = � = � � = �D�EV = � U ��� U �)�satisfying condition (ii).

To establish (iii), note that

V = ����� ���#V = 9��:���LK MHANPO Q RQPS!T RU (

R*WV Q�X NT RU I (

R*� M

H NYO Q RQYS T RU ( * V Q�X NT RU � ( * � (3.21)

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V = ����� �:9��:��� ������� �����

=1<>;; R MHANPO Q RQPS T RU (

R* V Q�X N T RU I ( R * � R if � ��9������ � ,=; R M H NPO Q RQPS T RU (

R* V QJX NT RU I (

R* T RU � (

R* if � ��9������� � ,E if �����9������ � ,

and hence, under (3.21)

V = ����� �5�4� � � 8 V = �4� � �:9��;��� 8 V = 9 �:�7�)�������� V = 9 �:�7�:9 �:��� CFE HFinally we note that under the pertinent optimum arrange-

ment, VXW is also Schur concave, but not in the strict sense.This follows from a slight variation of the fact that V U isstrictly Schur convcave, see also [6]. Note that

;@<+ ?> ��� -. ?> �M;A<V ?> � -. B U> �5H (3.22)

Now suppose the � !eigenvalues of ;=<V U> � , are

� -"2B ?> �)� -" = ?> �5�CHCH1H�� -" ; P <>= ?> � � �with

-" & ?> � J -" &�� = ?> � " E at all>

. Define the � ! 0 � !matrix whose columns are the unit eigenvectors correspond-ing to the smallest ��� eigenvalues of ;=<V ?> � . Observe

-� BC ?> � -� U> ��� D�HThen, because of Fact 1

� $��T U+ ��� � P <>= ' ;�<>=R @�B)' &A@�B �� �� U� B -"4& U> ��� > � 7 P <>=&A@ 7 P <>; P HNote the number of diagonal elements of ; <+ ?> � is less thanthe number of eigenvalues of ; <V U> � , as the overdecimatedcondition forces

-. (,� to be rectangular. Consequently, un-like [6] and [7], where maximal decimation forced a square-. (*� , weak super majorization, rather than majorization mustbe used.

We then have the following result.

Theorem 3.3 Consider Problem 2.1 and all quantities de-fined therein. Then optimality is attained if for a suitable fre-quency inavriant permutation matrix � ,

-. ?> ����� -� B ?> � .We note that for V U this solution is unique to within an

arbitrary permutation matrix � . For V = too this solution isunique to any permutation matrix � that enforces an opti-mum arrangement. This is so because both V = and V U arestrictly Schur concave. For V�W , on the other hand, even though� must enforce an optimum arrangement, the solution isby no means unique, as V W is not strictly Schur concave.Nonetheless it is intriguing that despite the difference be-tween the V & , a common

-.optimizes all three.

4. CONCLUSIONS

We have derived conditions for the optimal orthonormal sub-band coding of � -WSCS signals, using an over decimated� -channel uniform filter bank as subband coder with � -periodic filters three bit allocation schemes. As with the re-sults of [6], [7] an optimum filter bank in each case is thesame PCFB.

5. REFERENCES

[1] W. A. Gardner “Exploitation of spectral redundancy incyclostationary signals”, IEEE Signal Processing Mag-azine, pp 14-37, April 1991.

[2] G. Giannakis,“Cyclostationary Signal Analysis”,Chapter 17, CRC DSP Handbook.

[3] M. K. Tstatsanis and G. B. Giannakis, “Principal com-ponent filter banks for optimum multiresolution anal-ysis”, IEEE Transactions on Signal Processing , pp1766- 1777 August 1995.

[4] M. K. Mihcak, P. Moulin, M. Anitescu, K. Ramchan-dran, “Rate-distortion-optimal subband coding withoutperfect-reconstruction constraints”, IEEE Transactionson Signal Processing, pp 542-557, March 2001.

[5] P.P. Vaidyanathan, “Theory of optimal orthonormalsubband coders”, IEEE Transactions on Signal Pro-cessing, pp 1528-1543, June 1998.

[6] A. Pandharipande and S. Dasgupta, “Subband cod-ing of cyclostationary signals with static bit alloca-tion”, IEEE Signal Processing Letters, pp 284-286,Nov 1999.

[7] A. Pandharipande and S. Dasgupta, “Subband codingof cyclostationary signals”, Submitted to IEEE IEEETransactions on Circuits and Systems.

[8] A. W. Marshall and I. Olkin, Inequalities: Theoryof Majorization and its applications, Academic Press,1979.

Page 21: Report on US ARMY AWARD DAAD19-00-1-0534 · −− DFT based DMT, optimum bit allocation Transmitted Power (dB) −. General DMT, optimum bit allocation #channels, #users = 2 (1,1)

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OPTIMUM COMPACTION FILTERS FOR SUBBAND CODING OF WIDESENSECYCLOSTATIONARY SIGNALS

Ashish Pandharipande and Soura Dasgupta

Electrical and Computer Engineering Department, The University of Iowa, Iowa City, IA-52242, USA.Email: [email protected] and [email protected]

ABSTRACT

In this paper, we develop an optimum compaction filterbased design of filter banks for the subband coding of wide-sense cyclostationary (WSCS) signals. The design of theoptimal orthonormal filter bank is specified in terms of op-timal compaction filters. Each filter is designed by just theapriori knowledge of the cyclic autocorrelation of the inputWSCS signal. This design theory is developed by first pro-viding further insight to the optimal compaction filter designin the case of wide-sense stationary (WSS) signals.

1. INTRODUCTION

The energy compaction concept plays an important role insubband coding theory, and energy compaction filters findapplications in the design of orthonormal subband coders.Optimality of filter banks is in the sense of maximizing cod-ing gain, which is a measure of the distortion due to subbandquantization. The optimal orthonormal subband coding ofWSS and WSCS signals has been treated in [10], [11] and[1], [6] respectively. It has been shown [8], [11] that the op-timal orthonormal filter bank in the WSS case can be con-structed by designing the analysis filters one at a time bychoosing them to be optimal compaction filters for appro-priate power spectral densities (psds) derived from the in-put signal psd. Compaction filters thus are of interest dueto their connection with optimal subband coding and prin-cipal component filter banks. Compaction filters have beentreated in some detail for the WSS case in [8], [9], [11]. Thecompaction filter was formulated as an eigen problem in [8]and given a principal component approach. In [9], com-paction filters were derived by an energy analysis. Proper-ties of compaction filters have been further studied in [11].

In this paper, we develop the compaction filter conceptin the context of WSCS signals. Cyclostationarity is exhib-ited by some parameters for most manmade signals encoun-tered in communication, telemetry, radar, and sonar sys-

This work was supported by US Army contract, DAAAD19-00-1-0534, and NSF grants ECS-9970105 and CCR-9973133

tems, [2]. Considering these underlying periodicities andmodeling the random signals as cyclostationary can lead toimprovements in performance of signal processors.

A signal x(k) is WSCS with period M if for all k; l:

E [x(k)x�(l)] = E [x(k +M)x�(l +M)];

where E [�] denotes the expectation operator. The subbandcoder considered is an N -channel uniform maximally dec-imated filter bank depicted in fig. 1. We will consider a 2-channel (N = 2) subband coder in this paper to illustrate theideas. Here x(k) is WSCS with period M and at time k, Qi

is a bi(k)-bit quantizer. When x(k) is Wide Sense Station-ary (WSS) one selects the analysis filters Hi(k; z) and thesynthesis filters Fi(k; z) to be linear time invariant (LTI).Since x(k) here is WSCS with period M , we assume thatthese filters are Linear Periodically Time Varying (LPTV)with periodM . A linear filter with impulse response h(k; l)is called M -periodic, if for all k; l:

h(k; l) = h(k +M; l +M): (1.1)

The time index k in Hi(k; z) and Fi(k; z) recognizes theirlack of time invariance.

Given the autocorrelation of the WSCS signal x(k), thepsd matrix of theM -blocked version of x(k), which is WSS,can be found and will be assumed to be known. We are thusinterested in designing the filters Hi(k; z), Fi(k; z) by anenergy compaction approach. We shall suitably extend theidea of an optimum compaction filter for WSCS signals inthis paper. We show that the optimal orthonormal filter bankfor a WSCS input can be designed by choosing each analy-sis filter to be an optimum compaction filter for an appropri-ately ”peeled-off” signal, with the peeled-off signal derivedfrom the input signal. We first provide useful insight to thecompaction filter design for WSS signals. This analysis isthen paralleled to the case when the input to the filter bankis WSCS.

In Section 2, we recapitulate the optimum compactionfilter design for WSS signals. The results essentially pro-vide a different interpretation to the compaction process de-scribed in [11]. Section 3 then introduces the notion of an

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- H0- # N - Q0

- " N - F0 -

H1 # N Q1 " N F1

HN�1 # N QN�1 " N

-

-

-

-

-

-

-

- FN�1

?-

-

6-

6

x(k) x(k)

......

......

v0

v1

vN�1

w0

w1

wN�1

Fig. 1. An N -channel filter bank as subband coder

optimum compaction filter as applicable to WSCS inputs,and then describes the energy compaction approach to sub-band coder design when the input signals are WSCS. Sec-tion 4 sums the contributions of this paper.

Notation: For compactness, we use [x(0 : 1 : N � 1)]to denote the vector

[x(0); x(1); : : : ; x(N � 1)];

and [x(0 : �1 : N � 1)] to denote the vector

[x(0); x(�1); : : : ; x(�N + 1)]:

We use [X(z)]y to denote the transposed conjugate of thematrix [X(z�1)].

2. OPTIMUM COMPACTION FILTERS FOR WSSSIGNALS

Consider fig. 2. The filter H(z) is said to be an optimumcompaction filter for the zero-mean WSS input signal x(k)if it maximizes the output variance �2

vsubject to the con-

straint that jH(!)j2 is Nyquist-N , that is,

N�1Xk=0

jH(! �2�k

N)j2 = N; 8!:

- H - # N -x(k) s(k) v(k)

Fig. 2. Illustration for compaction filter

We now state a result which essentially describes theoptimum compaction filter design process for WSS signals.

Theorem 2.1 Consider fig. 1 with WSS x(k). Given thatH0(!) is an optimum compaction filter of the signal x(k).Then the analysis filter H1(!) is an optimum compactionfilter of x(1)(k) = x(k)� H0(!)p

Nx(k); : : : ; andHN�1(!) is

an optimum compaction filter of x(N�1)(k) = x(N�2)(k)�

HN�2(!)pN

x(N�2)(k).

Observe that Theorem 2.1 is a different interpretation tothe design methodology discussed in [11].

3. OPTIMUM COMPACTION FILTERS FOR WSCSSIGNALS

Define for i = 0; 1,

xi(k) = x(2k � i); si(k) = s(2k � i);

and denote the M blocked versions of x i(k); si(k); i = 0; 1as

~xi(k) = [x(2(Nk � (0 : 1 : N � 1))� i)]T ;~si(k) = [s(2(Nk � (0 : 1 : N � 1))� i)]T :

Call~x(k) = [~xT0 (k); ~x

T

1 (k)]T;

~s(k) = [~sT0 (k); ~sT1 (k)]

T:

Every LPTV system with period M can be interpretedas a multiple-input/multiple-output system LTI system withM inputs and M outputs. Let ~hmn(k; l) be the M �M im-pulse response matrix relating ~xn(k) and ~sm(k). Obviouslythis is an LTI system, and we can define

~Hmn(z) =Xk

~hmn(k)z�k:

Note that ~H00(z) and ~H01(z) respectively represent the LTIsystems relating the blocked even and odd samples of theinput of the M -periodic system H(k; z) to the blocked evensamples of the output of H(k; z).

Define the 2M � 1 vector

~v(k) = [v0(Nk�(0 : 1 : N�1)); v1(Nk�(N�1 : �1 : 0)]T :

Even when the analysis and synthesis filters are LPTV,the polyphase representation shown in fig. 3 still holds,[12]. The analysis and synthesis sides in fig. 1 are respec-tively replaced by the 2M � 2M LTI operators ~E(z), ~R(z).Note that the operator ~E(z) relates the 2M�1 vectors ~x(k)and ~v(k).

Perfect reconstructability reduces to the requirement that~R(z) = ~E�1(z), and orthonormality to the requirement thatfor all !

[E(!)]yE(!) = [R(!)]yR(!) = I:

Since S~x(!), the psd matrix of ~x(k), is positive definiteHermitian symmetric, it can be expressed as

S~x(!) = ~U(!)~�(!)h~U(!)

iy(3.2)

with ~U(!) unitary at all !, and

~�(!) = diag f~�0(!); � � � :~�2N�1(!)g (3.3)

obeying at all !

~�i(!) � ~�i+1(!) > 0: (3.4)

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Further, S~v(!), the psd matrix of ~v(k), obeys

S~v(!) = ~E(!)S~x(!)h~E(!)

iy: (3.5)

The canonical solution to the subband coding problemunder dynamic bit allocation and static bit allocation treatedin [1], [6] respectively is given by

~E(!) =h~U(!)

iy: (3.6)

In this case S~v(!) = ~�(!). The Hi(k; z) and Fi(k; z) canbe obtained by unblocking ~E(z) and ~E�1(z).

The main points of this Section are to define an opti-mum compaction process for WSCS signals that leads tothe design of the subband coder. To do so we first de-fine the M -periodic optimum compaction of an M -periodicWSCS process. The definition of optimum compaction ofa WSS signal given earlier involves a filter H(z) for whichH(z)H�(z�1) is Nyquist-2. For an LPTV system H de-fine the adjoint filter Ha whose impulse response ha(k; l)is related to the impulse response h(k; l) of H by

ha(k; l) = h

�(l; k): (3.7)

Observe that the adjoint of an LTI systems with transferH(z) has transfer function H

�(z�1). Thus the analog ofa system with transfer functionH(z)H �(z�1) in the LinearTime Varying (LTV) case, is the LTV system HH

a.

g+~w

~q

~v~x-

?--- ~E(z) ~R(z)

~x

Fig. 3. Blocked polyphase representation

We now give the appropriate definition of LTV Nyquist-2filters. When H(k; z) is M -periodic then HH

a is Nyquist-2 iff for all !,

h~H00(!); ~H01(!)

i264h~H00(!)

iyh~H01(!)

iy375 = I: (3.8)

We now define the optimum compaction process for WSCSsignals.

Definition 3.1 Consider fig. 2, with x(k) WSCS with pe-riod M , H LPTV with periodM , andN = 2. Then H is anoptimum compaction filter for x(k) if subject toHH

a beingNyquist-2, and for some index set fk0; k1; � � � ; kM�1g =f0; � � � ;M � 1g, it simultaneously maximizes the partialvariance sums

lXi=0

�2v(ki); (3.9)

for all 0 � l �M � 1.

Observe that this definition is targetted to accomodate thefact that v(k) is WSCS with period M . Consequently M

variance values must be considered. In the sequel, we willcall an optimum compaction filter canonical if in (3.9),

ki = i:

Note that even the canonical optimum compaction filter isnonunique. This is consistent with the fact that LTI opti-mum compaction filters for WSS processes are also nonunique,[11].

We now state the main results of this Section.

Theorem 3.1 Consider fig. 1 with N = 2, Hi(k; z) M -periodic, andx(k) WSCS with periodM . Then theH0(k; z)provided by the canonical solution (3.6) is an M -periodiccanonical optimum compaction filter of x(k).

Denote by Sx(!) the M �M psd matrix of the M � 1WSS vector x(k) obtained by blocking x(k). We can write

Sx(!) = V (!)�(!)[V (!)]y

with V (!) unitary for all !. Denote

V (!) =

�V (!

2) 0

0 V (!�2�2

)

�: (3.10)

Theorem 3.2 Given H0(k; z) is an optimum compactionfilter of the WSCS signal x(k). Then the filter H1(k; z) isan optimum compaction filter of the signal obtained by un-

blocking the 2M�1 vector ~x(1)(k) =hV (!)� H0(!)p

2

i~x(k),

whereH0(!), a 2M � 2M matrix, is the 2M blocked ver-sion of H0(k; z).

Theorem 3.1 together with 3.2 defines the optimum com-paction filter design for WSCS signals.

4. CONCLUSIONS

In this paper, we presented the optimum compaction filterdesign for the subband coding of WSCS signals. We firstgave a different interpretation to the compaction process inthe case of subband coding of WSS signals. The design pro-cedure was then extended to WSCS signals by consideringa 2-channel subband coder. We showed that the analysis fil-ters of the orthonormal filter bank can be designed sequen-tially by an optimal compaction process, with the optimumcompaction filters designed with the apriori knowledge ofjust the second order statistics of the input WSCS signal.

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5. REFERENCES

[1] S. Dasgupta, C. Schwarz, B.D.O. Anderson, “Op-timum subband coding of cyclostationary signals”,IEEE International Conference on Acoustics, Speechand Signal Processing - Proceedings, pp 1489-1492,Mar 15-Mar 19 1999.

[2] W. A. Gardner, “Exploitation of spectral redundancyin cyclostationary signals”, IEEE Signal Processing,pp 14-36, Apr 1991.

[3] R.A. Horn, C.R. Johnson, Matrix Analysis, CambridgeUniversity Press, 1999.

[4] P. Moulin, K.M. Mihcak, “Theory and design ofsignal-adapted FIR paraunitary filter banks”, IEEETransactions on Signal Processing, pp 920-929, Apr1998.

[5] S. Ohno, H. Sakai, “Optimization of filter banks usingcyclostationary spectral analysis”, IEEE Transactionson Signal Processing, pp 2718 -2725, Nov 1996.

[6] A. Pandharipande, S. Dasgupta, “Subband codingof cyclostationary signals with static bit allocation”,IEEE Signal Processing Letters, pp 284 -286, Nov1999.

[7] V. Sathe, P.P. Vaidyanathan, “Effects of multirate sys-tems on the statistical properties of random signals”,IEEE Transactions on Signal Processing, pp 131-146,Jan 1993.

[8] M.K. Tsatsanis, G.B. Giannakis, “Principal compo-nent filter banks for optimal multiresolution analysis”,IEEE Transactions on Signal Processing, pp 1766-1776, Aug 1995.

[9] M. Unser, “On the optimality of ideal filters for pyra-mid and wavelet signal approximation”, IEEE Trans-actions on Signal Processing, pp 3591-3596, Dec1993.

[10] P.P. Vaidyanathan, “Properties of optimal compactionfilters in subband coding”, IEEE Digital Signal Pro-cessing Workshop, pp 89-92, Sep 1-4 1996.

[11] P.P. Vaidyanathan, “Theory of optimal orthonormalsubband coders”, IEEE Transactions on Signal Pro-cessing, pp 1528-1543, Jun 1998.

[12] P.P. Vaidyanathan, S.K. Mitra, “Polyphase networks,block digital filtering, LPTV systems, and alias-freeQMF banks: a unified approach based on pseudocircu-lants”, IEEE Transactions on Acoustics, Speech, andSignal Processing, pp 381-391, Mar 1988.

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OPTIMAL TRANSCEIVERS FOR DMT BASED MULTIUSER COMMUNICATION

Ashish Pandharipande and Soura Dasgupta

Electrical and Computer Engineering, The University of Iowa, Iowa City, IA-52242, USA.Email: [email protected] and [email protected]

ABSTRACT

This paper considers discrete multitone modulation (DMT)for multiuser communications where different users are sup-ported by the same system. These users may have differingquality of service (QoS) requirements, as quantifiedby theirrespective bit rate and symbol error rate specifications. Ourgoal is to minimize the transmitted power given the QoSspecifications for the different users, subject to the knowl-edge of colored interference at the receiver input. In particu-lar we findan optimum bit loading scheme that distributes thebit rate transmitted across the various subchannels belong-ing to the different users, and subject to this bit allocation,determine an optimum transceiver.

1. INTRODUCTION

The discrete multitone modulation (DMT) channel cod-ing scheme has established itself as an effective high ratedata communication technique in both wired and wirelessenvironments and is used for example in ADSL and HDSL,[1]. We consider DMT in a multiuser environment. Thusthe DMT system studied here supports multiple users, withvarying quality of service (QoS) requirements, quantifiedbytheir respective bit rate and symbol error rate (SER) specifi-cations.

Specifically, consider the DMT system as in fig. 1 whichdepicts an

�-subchannel filterbank model of a DMT sys-

tem. We consider an overinterpolated ( ��� � ) filterbankas the transceiver. We assume that the channel ����� is FIRof lenth � (preequalization is assumed to have been done),and � ���� is additive colored noise with known spectrum.Thus for example � ���� could represent co-channel interfer-ence. Note, [8] provides models for cochannel interferencein a variety of settings. To mitigate intersymbol interference(ISI), a form of redundancy is incorporated by choosing��� ��� � . The transmitting filters, �������� , and the re-ceiving filters, �������� , are constrained to length � , and actas modulating and demodulating transforms respectively. In

This work was supported by US Army contract, DAAAD19-00-1-0534, and NSF grants ECS-9970105 and CCR-9973133

a DFT based DMT implementation [1], the IDFT and DFTare used as the modulating and demodulating transformsrespectively.

In this paper, as in [5] we will consider more generaltransformations leading to a generalized DMT system. Tocapture a multiuser environment,we assume that there are � -users each having been assigned

��� � subchannels. Furtherthe � -th user requires a bit rate of !� , and an SER of no morethan " � . Our goal is to select ��# and ��# , and distribute the bitrates among the various sub-channels to achieve the abovespecificationswith the minimum possible transmitted power.The problem addressed here thus directly generalizes thatin [5], which also addresses the same power minimizationissue, but assuming a single user subject to only one bitrate and SER constraint. The multiuser setting renders theoptimization problem highly nontrivial in comparison to thesingle user case. Further we show as much as 8 dB and 12dB savings in transmit power in our simulations with generalDMT systems over DFT based DMT systems with optimalbit allocation and no bit allocation respectively.

Related literature includes [4] which develops fast load-ing algorithms using table lookups and a fast Lagrange bi-section method for a single user setting. [7] considers asingle user optimization of the transceiver mutual informa-tion. [3], considers the optimum bit loading problem whentwo users are present.

Section 2, definesthe generalized DMT system and for-mulates a precise mathematical problem. Sections 3 and 4respectively consider the bit rate allocation and filterselec-tion problems. Section 5 gives simulations.

2. DMT BASED MULTIUSER SYSTEM MODEL

In this Section we give some preliminaries. Specifically,in Section 2.1, we recount the details of the generalizedDMT system provided in [5]. Section 2.2 provides a preciseoptimization problem.2.1. Polyphase representation of the DMT system

Consider the filterbank based DMT model in fig. 1. � ����is a zero mean wide sense stationary additive noise. As

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$ %'& $ (�)+*-,/. $%'& (102*-,/.%'& (4365 0 *-,/.

$ $$ $

77

8 *-,/. $ 9 $ :')+*-,/. $; < = &>? 3@5 0:A02*-,/. = &

:B3@5 0 *-,+. = &$$

$$$9 $$...

......

...

C *ED�.?F)? 0? 3@5 0

>?F)>? 0

Fig. 1. Filter bank based DMT model.

the filters � � ��� and � � ��� have lengths GH� , we maywrite the following polyphase decompositions: � � ����I�JLKAMON#EPRQ � M #�S #�T � , and �������U� JVKWMON#XPRQ � #�Y � T # , with constantS #�T Z and Y #�T Z . Definethe �\[ � matrix S with ]_^ -th ele-ment S #�T Z and the

� [`� matrix Y with elements Y #�T Z . Callthe constant matrices S and Y the transmitting and receivingmatrix repectively. Then with a and ba the vector of the sig-nals c1# and dce# , repectively, fg , the blocked version of �1���� ,one has the equivalent system in fig. 2. Here the pseudocir-culant matrix hI���� [9], is formed by the coefficientsof theFIR channel ������U��i Q � i N � MON �kj+j/jl� i/mn� M m . It obeys:hI����U�po h`q h�r���ts (2.1)

where huq is constant, ��[ � , and h�r���� is ��[v� . Notethe knowledge of the autocorrelation of � , yields the auto-correlation matrix of fg .

w x x w y{zX|2} w ~ w � � w; � ��� ����Fig. 2. Polyphase representation of the DMT system.

For DMT systems using zero padding, the transmittingand receiving matrices are respectively given byS ���u�������� Y ��� MRN o � � Q s (2.2)

where � is the� [ � unitary DFT matrix with � �L�_� T �L�N� ��� M Z���� � �� , � ��� � � � j/j+j � ���¡ 

, � Q is the� [¢�

submatrix of � having the first � columns of � , and � is the� [ � diagonal matrix with elements that are the�

-pointDFTs of the channel impulse response, [1]. We considermore general DMT systems that can lead to reduction insidelobes and better noise rejection properties of the filters.The transmitting matrix of such a general DMT is given byS ��� S Q� � (2.3)

where S Q is an arbitrary� [ � unitary matrix. The con-

dition for perfect reconstruction (PR) is given asY hI��� S �¤£ (2.4)

Using (2.1) and (2.3), the PR condition reduces toY h q S Q��¤£ (2.5)

Using singular value decomposition, h q can be written as

huq@� o¦¥ Q ¥ N s§ ¨ª© «¬ �`­ � �W®`¯ � ¥ Q ­ ®6¯ (2.6)

where ¥ and ® are respectively �°[u� and� [ � unitary

matrices whose columns are the eigenvectors of h q h q ¯ andh q ¯ h q . ­ is the� [ � diagonal matrix with diagonal

elements that are the singular values of h q .Using (2.6), one clear choice for Y satisfying (2.5) isY � S ¯Q ® ­ MRN ¥ ¯Q (2.7)

2.2. Problem definition

The optimum bit loading problem is to findthe best bit rateallocation scheme to minimize the transmit power, underdifferent bit rate and SER budgets of the users. The optimaltransceiver is then designed to minimize the power subjectto optimum bit loading.

Assume there are ± users, with each user being allocated� subbands (in fig. 1,� �²±³� and �´� �µ� � ). Let

the input power in the ^ -th subband of the � -th user be¶ �·+¸º¹ » . Due to PR, this is also the output signal power ¶ � �·+¸º¹ »in the ^ -th subband of the � -th user. Let the output noisepower in this subband be ¶ �¼½¸º¹ » , and ¾ Z�T � be the numberof bits allocated in this subchannel. Due to different QoSrequirements, we may have different bit rate constraints forthe users. The average number of bits for the � -th user is¾ª�¿� NÀ J À MRNZ�PRQ ¾ ZÁT � . However we need to account for thereduction in bit rate due to the zero padding. The average bitbudget for the � -th user is then !�¦� ÀK ¾ª�6� NK J À MONZ�PRQ ¾ Z�T � .

With a high bit rate assumption made on the modulationsystem, we have, [5], for the � -th user¶ �·+¸º¹ » �¤i/�F �2à ¸º¹ » ¶ �¼½¸º¹ »where the constant i/� depends on the SER "n� . We seek tominimize the average transmission power given byÄ �  � ÅÆ� P N

À MRNÆZ�POQ ¶ �·+¸º¹ » (2.8)

�  � ÅÆ� P NÀ MRNÆZ�POQ i/�F �2à ¸º¹ » ¶ �¼ ¸º¹ » (2.9)

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subject to the bit rate budgets

� �  � À MRNÆZ�PRQ ¾�ZÁT � � ���   � j/j+j � ± � (2.10)

and the PR requirement (2.7).

3. OPTIMUM BIT ALLOCATION

The problem of minimizing (2.9) under the set of constraints(2.10) is a constrained optimization problem. Using theAM-GM

Ninequality and (2.10),Ä �  � ÅÆ� P N

À MRNÆZ�POQ i/�F �2à ¸º¹ » ¶ �¼ ¸º¹ » (3.11)

Ç �� ÅÆ� P N i+���À MONÈZ�PRQ  �2à ¸º¹ » ¶ �¼½¸º¹ » N�É À (3.12)

� i± ÅÆ� P N i/���ºÂ � KËÊ »À MONÈZ�PRQ ¶ �¼Ì¸�¹ » N�É À (3.13)

with equality holding iff for all ^ � � :

¾�ZÁT � � � � � �  Â ��ÍÏÎÐ�F� À MRNÈZ�PRQ ¶ �¼½¸º¹ » N�É À �  Â ��ÍÏÎÐ�F� ¶ �¼½¸º¹ » j (3.14)

This is the optimum bit allocation strategy. The optimaltransceiver design is to findmatrices Y � S so as to minimizeÑ � ÅÆ� P N ��Ò �

À MONÈZ�PRQ{Ó Z�T � NÔÉ À (3.15)

where Ò � �¤i �  � K{Ê » Ó ZÁT � � ¶ �¼ ¸�¹ » j (3.16)

Observe, if one chooses �¢�� , and ÒÕ���ÖÒ for all � , then(3.15) reduces to the optimization function considered in[2], for the subband coding of cyclostationary signals.

Optimal arrangement: Observe, [2, 6], that given a set ofpositive numbers ×ÏØÏ�ÚÙ � �� P N , ØÏ� Ç Ø+� � N the minimum amongall possible

J Ø �2Û Ø � ¸ isJ � � P N Ø � Ø+� � M � � N . Thus among the

various permutations of Ó Z�T � , any that minimizes (3.15) musthave the following property:

Ó ��T �ªÜ Ç ÓÚÝ T �2Þàß�ÒÕ�ªÜ À MRNÈZláPO� Ó ZÁT �ªÜ�GâÒ��2ÞÀ MONÈZláP Ý Ó Z�T �2Þ (3.17)

and ÒÕ� Ç Ò Ý ßÀ MONÈZ�PRQ{Ó Z�T �ÖG

À MONÈZ�PRQËÓ Z�T Ý j (3.18)0The arithmetic mean (AM) of a set of positive numbers is greater than

or equal to their geometric mean (GM), with equality iff all the numbersare equal.

4. OPTIMUM TRANSCEIVER DESIGN

In this Section we address the problem of filterselection tominimize (3.15). This reduces to selecting a unitary matrixS Q . Given that the matrices in (2.6) are known, (2.7) fixesY . Observe that the situation in fig. 3 prevails, and ã �¼ , theautocorrelation of fä , is known. Further the autocorrelationmatrix of ä is given byã ¼ � S ¯Q ã �¼ S Q � (4.19)

and that Ó Z�T � in (3.15), are simply the diagonal elements ofã ¼ .We need a few results from the theory of majorization

that will be used in solving the optimization problem at hand.We will firstintroduce the notion of majorization and Schurconcavity [6].Definition4.1 Consider two sequences ck�H×+c # Ù Ý#XP N andå �æ× å # Ù Ý#EP N with c # Ç c # � N and å # Ç å # � N . Then we saythat å majorizes c , denoted as cvç å , if the following holdswith equality at ���k��Æ #EP N c # G �Æ #EP N å # �   Gk�IGè� jDefinition4.2 A real valued function é����U�LéÕ��� N � j/j+j � � Ý definedon a set êìëâã Ý is said to be Schur concave on ê ifcvç å Íl�íê ß é���c Ç éÕ� å jé is strictly Schur concave on ê if strict inequality éÕ��c A�éÕ� å holds when c is not a permutation of å .

We will now state a theorem that results in a test for strictSchur concavity. We denote

é z � } ����U�ïî éÕ����î � � Ó �Oð é z #�T Z } ����U�ïî � éÕ���î �Ï# î �ªZ jTheorem 4.1 Let éÕ���� be a scalar real valued function de-finedand continuous on ñ , and twice differentiable on theinterior of ñ . Then éÕ���� is strictly Schur concave on ñ iff

(i) é is symmetric in its arguments,

(ii) é z � } ���� is increasing in � , and

(iii) é z � } ����`�pé z � � N } ����`ßòé z � T � } ��� � é z � T � � N } ���� �é z � � N T � } ��� � é z � � N T � � N } ���'ó � .Theorem 4.2 If � is an ��[I� hermitian matrix with diag-onal elements ô N � j/j+j � ô Ý and eigenvalues õ N � j+j/j � õ Ý , thenôöçkõ on ã Ý .

To connect the results from majorization theory devel-oped to our optimization problem, we state the followinglemma.

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x x w ÷1ø ù Ü w �Õúûw w¬ úû�� �ü �ü üFig. 3. Receiver block diagram.

Theorem 4.3 The real valued scalar functionÑ

as definedin (3.15) under the optimality conditions (3.17-3.18) is strictlySchur concave.

In particular as the search of S Q is restricted to unitarymatrices, if one chooses S Q to be ý a matrix of orthonormaleigenvectors of ã �¼ , then ã ¼ is a diagonal matrix containingthe eigenvalues of ã �¼ . Note that diagonal elements of ã ¼are ¶ �¼½¸º¹ » . Thus from theorem 4.2, this choice of S Q yieldsa sequence of ¶ �¼½¸º¹ » that majorizes all other achievable se-quences. Consequently if arranged optimally, Theorem 4.3holds, that such a sequence will minimize (3.15). It remainssimply to arrange the eigenvalues of ã �¼ among the ¶ �¼½¸º¹ » ,through exhaustive search if need be, so that an arrange-ment that minimizes (3.15) is obtained. Thus for a suitablepermuation matrix, þ , the optimizing S Q isS Q��¤þ@ý j (4.20)

5. SIMULATION RESULTS

In this section, we compare the trasmitting power of theDFT based DMT under no bit allocation and optimum bitallocation with our optimum transceiver. We assume thechannel to be �����'�  ÿ� � j � � MON , and a noise source �1����whose power spectral density is shown in fig. 4. The plotshows that there is an 8 dB saving in transmit power withour design over the DFT based DMT under optimum bitallocation, and a 12 dB improvement over the conventionalDMT with no optimum bit allocation. We however note thatthere may exist noise environments where the DFT basedDMT performs as well as our optimal design.

6. CONCLUSIONS

In this paper, we have presented an optimum bit allocationstrategy and transceiver design for minimizing the transmitpower when different users have varied QoS requirements.Simulations confirmthe efficacy of our results.

7. REFERENCES

[1] J.S. Chow, J.C. Tu, J.M. Cioffi, “A discrete multi-tone transceiver system for HDSL applications", IEEEJournal on Selected Areas in Communications, pp 895-908, Aug 1991.

0 20 40 60 80 100 120 140 160 180 2000

1

2

3

4

5

6

7x 10

−13 Power spectral density of noise

1 2 3 4 5 6 7 8 9 10−70

−60

−50

−40

−30

−20

−10

0Transmitted power under different DMT and bit allocation schemes

__ DFT based DMT, no bit allocation−− DFT based DMT, optimum bit allocation−. General DMT, optimum bit allocation

Tran

smitt

ed P

ower

(dB)

#channels/user, #users = 2

Fig. 4.

[2] S. Dasgupta, C. Schwarz and B.D.O. Anderson, “Opti-mal subband coding of cyclostationary signals", IEEEInternational Conference on Acoustics, Speech, andSignal Processing, pp 1489-1492, Mar 1999.

[3] L.M.C. Hoo, J. Tellado, J.M. Cioffi, “Discrete dualQoS loading algorithms for multicarrier systems ",IEEE International Conference on Communications,pp 796-800, 1999.

[4] B.S. Krongold, K. Ramchandran, D.L. Jones, “Compu-tationally efficientoptimal power allocation algorithmsfor multicarrier communication systems", IEEE Trans-actions on Communications, pp 23-27, Jan 2000.

[5] Y. P. Lin, S.M. Phoong, “Perfect discrete multitonemodulation with optimal transceivers", IEEE Transac-tions on Signal Processing, pp 1702 -1711, June 2000.

[6] A. W. Marshall and I. Olkin, Inequalities: Theoryof Majorization and its applications, Academic Press,1979.

[7] A. Scaglione, S. Barbarossa, G.B. Giannakis, “Filter -bank transceivers optimizing information rate in blocktransmissions over dispersive channels", IEEE Trans-actions on Information Theory, pp 1019-1032, Apr1999.

[8] T. Starr, J.M. Cioffi,P. Silverman, Understanding Dig-ital Subscriber Line Technology, Prentice Hall, 1999.

[9] P.P. Vaidyanathan, Multirate Systems and Filter Banks,Prentice Hall, 1992.

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1

A Novel Channel Identi�cation Method for

Wireless Communication Systems

Honghui Xu1, Soura Dasgupta1, and Zhi Ding2

1Electrical & Computer Engineering Department

The University of Iowa

Iowa City, IA-52242, USA.

2Department of Electrical & Computer Engineering

University of California

Davis, CA 95616, USA.

Emails: hoxu, [email protected], [email protected]

Keywords: Identi�cation, Equalization, Feedback, Wireless, Training

1Work supported in part by NSF grants ECS-9970105 and CCR-9973133.2Work supported in part by NSF grant CCR-9996206

August 1, 2002 DRAFT

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Abstract

We present a novel dual channel identi�cation approach for mobile wireless communication systems.

Unlike traditional channel estimationmethods that rely on training symbols, we propose a bent-pipe feedback

mechanism which requires the mobile station (MS) to send portions of its received signal back to the Base

Station (BS) for wireless channel identi�cation. Using a �lter-bank decomposition concept, we introduce an

e�ective algorithm that can identify both the forward and the reverse channels based only on this feedback

information. This new method permits transfer of computational burden from the MS to the resource rich

BS and leads to signi�cant savings in bandwidth consuming training signals.

I. Introduction

We propose a new approach to the estimation and compensation of forward link channels in

mobile wireless communication systems that centers on a novel bent pipe feedbackmechanism.

In principle, this feedback mechanism enables Base Stations (BS) to simultaneously estimate

both the Forward Link Channel (FLC) from the BS to a Mobile (MS) and the Reverse Link

Channel (RLC) from the MS to the BS, without any training signals or resorting to blind

estimation techniques. While practical realities temper these theoretical expectations, as we

will demonstrate in this paper, our techniques bring with them certain signi�cant advantages.

Our paper is motivated by the strong surge in mobile wireless communication systems.

The rapidly expanding arena of wireless services including mobile computing and broad-

band multimedia would require much higher wireless capacity and higher data rates than

is currently needed, over the often unreliable wireless medium. Indeed, unlike their wire-

line counterparts, wireless communication links are highly susceptible to channel variations,

particularly in mobile environments. Three speci�c advantages motivate our approach.

(A) Adaptive Coding

The high variability of wireless links poses a serious challenge to assuring quality of service

(QoS) to various traÆc ows under harsh and dynamically distortive channel conditions.

To address QoS needs, future wireless communication systems must respond to potentially

rapid channel changes in an e�ective and timely manner. Currently, FLC estimation and

equalization responsibilities are assigned almost entirely to the MS [1], [2], [3]. At the same

time, to improve adaptivity to channel conditions, several researchers have proposed a num-

ber of schemes that require at least a partial awareness of the FLC at the BS. Thus, Paulraj

et. al., [6] demonstrate substantially improved performance through the use of adaptive

2

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space time coding that assumes that the BS has partial knowledge of the FLC. Goeckel et.

al. [10] propose other forms of adaptive coding again relying on similar information. Bit

loading in OFDM systems similarly assumes a channel aware BS, [8], as indeed does the

power minimization scheme of [9]. Similarly precoding techniques [11] can be signi�cantly

improved should the BS have partial knowledge of the FLC. Our bent pipe feedback tech-

nique naturally apprises the BS of the prevailing FLC conditions that are needed by these

schemes.

(B) Shared FLC compensation

As noted earlier, the tasks of FLC compensation and estimation are often assigned to the MS.

It is recognized that most future wireless cellular services will be characterized by an FLC

that supports higher data rates than does the RLC. For a given channel, higher data rate

induces longer delay spread and more severe ISI. In other words the discrete time baseband

model is of a higher order, and its equalization and compensation more onerous. At the

same time it is the BS that houses greater computational reserves, even though it is assigned

the less burdensome task of estimating and compensating the RLC. It is thus desirable to

shift at least a part of the FLC compensation and estimation burden to the more resource

rich BS. The feedback mechanism we propose permits a better utilization of this resource

disparity, by transferring much of the FLC estimation and compensation tasks to network

nodes with greater resources.

More speci�cally, our method in principle permits the BS to estimate and pre-compensate

the wireless channel. In practice, because of roundtrip delays and channel variations that

occur within the resolution of such delays, the BS will only partially compensate the dynamic

channel, and the MSmust take part in combatting the residual ISI. Nonetheless, this partially

compensated channel will induce reduced levels of ISI, whose removal would consequently

impose far less computational burden on the MS.

(C) Reduced Training

Channel estimation at the MS is typically assisted by the frequent transmission of training

data. As the FLC supports higher data rates, and is consequently described by a higher order

discrete time model, it requires longer training sequences, transmitted more frequently. This

is obviously at the expense of the all too precious forward link bandwidth. In GSM for

3

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example roughly one sixth of the transmission time is devoted to training signals.

As noted in (B), due to roundtrip delays our scheme permits the the BS to only partially

compensate the FLC, and the MS must combat the residual ISI with the assistance of some

training. Nonetheless, this partially compensated channel su�ers from reduced levels of ISI.

As we demonstrate through simulations in a later section, the number of training symbols

needed for the estimation and compensation of the partially compensated FLC is signi�cantly

lower, with consequent savings in the bandwidth allocated to FLC training. This advantage

is buttressed by the fact that no training is needed on the RLC, and that instead the time

slot normally used for RLC training can be used for the feedback data.

It should be noted that 2G CDMA systems and emerging 3G systems do employ some

simple feedback. Such feedback, however, is often limited to an estimated power loss param-

eter that enables the BS to compensate multipath distortion via power-control. Although

power-control or higher SNR can improve MS performance, particularly against at fading

channels, channel distortion as a result of multipath fading cannot be eÆciently compensated

by mere increase of transmission power. Other proposals for channel information feedback

are in [4]-[6], which involve feeding back channel parameter estimates to the BS at appropri-

ate instants of time. These proposals continue the practice of assigning the sole responsibility

of channel estimation to the resource challenged MS, and do not reduce the training burden

on the FLC.

Unlike the conventional feedback of channel estimates in [4], [5], our new approach only

requires that the MS feed back to the BS a portion of the received signal, over the time slot

conventionally reserved for RLC training, in epochs where either it normally transmits or in

epochs where it detects a performance degradation. Clearly, this permits the BS to estimate

the Roundtrip Channel (RTC).

However, the key novelty of our approach lies in the following discovery: By feeding back

only a portion, rather than the entire received signal, one empowers the BS to identify both

the FLC and the RLC from the roundtrip feedback signal alone. This novel channel feedback

does not require high speed reverse links and can naturally accommodate asymmetric data

link structures. Furthermore, no additional training signals are necessary for estimating the

RLC at the BS. As will be explained further, this scheme requires no greater overhead than

4

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those associated with [4]-[6], while at the same time having the fundamental advantage of

shifting substantial processing burden from the MS to BS, and reduced levels of training

data.

�� - -

����

�?

-

- - -

� j

high rate 1=T1

?6

?�

Wireless propagation MS TransceiverBS Transceiver

Data

Data

MSDemod.FLC

RLC

Transmitter

M

plexerBS

low rate 1=T2

1=T1

1=T2

Demod.

channelestimator

demulti-Transmitter

L

Fig. 1. A wireless communication system with signal feed back.

Our paper is structured as follows. First, the basic principle of roundtrip feedback is pre-

sented in Section II. Its implementation issues are addressed and practical caveats analyzed.

Conditions under which RLC and FLC can be obtained from the RTC are in Section III.

An algorithm to estimate the roundtrip dynamics is formulated in Section IV. Section V

provides the unravelling algorithm. Simulations are in Section VI.

II. Bentpipe Feedback

The feedback scheme we use is depicted in Fig. 1. Speci�cally in this �gure the FLC and

RLC respectively, operate at the rates 1=T1 and 1=T2, with T1 < T2 and T1=T2 = L=M

for integers L and M that need not be coprime. The sampled data at the FLC output is

converted to the RLC rate by the decimator/interpolator combination depicted in the �gure.

This is interlaced in the stead of normal RLC training data, with the data that the MS needs

to transmit through the RLC in its normal course of operation, and fed back to the MS.

ii- - - 6?i?

v(n)? -?h(n) M L

x(n)g(n)

y(n)

u(n)w1(n) w2(n)

Fig. 2. The equivalent digital expression of the system (noise and interference considered)

Fig. 1 can be transformed into Fig. 2, where x(n) is the digital data sequence transmitted

5

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by BS at time nT1 at the rate 1=T1, y(n) is the data sequence received by BS at time nT2

after sampling at the rate 1=T2, and h(n) and g(n) are the FIR impulse response of the

FLC and RLC respectively. Further, w1(n) and w2(n) are the noise sequences at the FLC

and RLC outputs, and u(n) models the interference caused by the normal RLC data due to

imperfect synchronization. Throughout we make the following standing assumption.

Assumption 1: The signals x(n), u(n), w1(n), w2(n) are zero mean, white and mutually

uncorrelated.

It is clear that given that the BS is aware of the data it has transmitted, under assumption

1 it can estimate the RTC. Using ideas from [12], we show that under mild assumptions

on the FLC and RLC, the BS can in fact directly unravel the FLC and RLC from the

RTC, without any training in either FLC or RLC. This ability to separate RLC and FLC

from RTC estimate can be seen as a consequence of the rate changing mechanism that

separates the two channels. Rate changers are time varying systems, and consequently, even

when w1(n) = w2(n) = u(n) = 0, the LTI operators h(n) and g(n) cannot be arbitrarily

interchanged.

We now pose and answer two questions surrounding the practicality of this approach. In

doing so at appropriate places we discuss the data and computational overheads associated

with this scheme relative to those of the feedback schemes mentioned in [4]-[6].

(i) What about roundtrip delay? Over reasonable distances the roundtrip delay is in

tens of microseconds. Thus, for example, over a roundtrip distance of 5 km, this delay is

about 16.67 �s. Over such time spans, the environment as seen by the mobile unit undergoes

little change. This is underscored by the fact that in GSM each data frame has a duration

of 557 �s, and training occurs only once per data frame. Thus the channel variation within

the resolution of this delay occurs mainly because of Doppler e�ect. Still, even with Doppler

e�ect on high speed mobiles, the channel characteristics are unlikely to change drastically

within such a short time interval. As a case in point, a vehicle traveling at 100km/hr su�ers

a maximum Doppler shift of 50 Hz at cellular band. The channel variation thus endured will

not be enough to prevent the transmitter from substantially compensating the FLC. Thus

the residual ISI that must be equalized at the receiver will be signi�cantly milder leading to

the need for much shorter training sequences on the FLC. Given that no training is needed

6

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on the RLC, and that feedback data occupies the RLC training slot used in conventional

communication, this implies substantial savings in the bandwidth devoted to the overall

training. Simulations presented later support this contention. Since the BS has greater

computational reserves, this would also e�ect a bene�cial transfer of computational burden

to such a resource rich BS.

We should also note that the feedback schemes suggested in [4]-[6] would su�er the same

latency e�ects manifested by the roundtrip delay. Thus while the addition of the other ad-

vantages of our scheme are conspicuously absent in the approaches of [4]-[6] the e�ectiveness

of adaptive coding noted in the introduction should be comparable in both instances.

(ii) What about battery life? Overly frequent feedback transmissions may deplete power

resources at the receiver. In epochs where the receiver also transmits, the training data

currently sent can be substituted by the feedback data, as RLC training is no longer needed.

During silent uplink episodes, feedback can be restricted to epochs where the receiver detects

a high level of packet loss due to obsolete channel estimates, in much the same way as

existing proposals for channel estimate feedback require. Thus, the associated overhead is

again comparable to that in [4]-[6].

Taking a more long range view, whereas battery technology continuously improves, with

ever lengthening battery life, bandwidth resources will remain scarce. Given the bandwidth

savings reduced training brings about, one can expect bent pipe feedback to gain an increas-

ing advantage in this trade-o�.

III. Conditions for estimating RLC/FLC from RTC

Borrowing ideas from [12] we show that it is possible in principle to unravel the FLC and

RLC dynamics from the RTC. We consider the case without noise and user interference at

�rst. The case with noise and interference will be treated later.

Consider the polyphase representation of the FLC and RLC transfer functions H(z) and

G(z) [13], given below.

H(z) =M�1Xi=0

Hi(zM )z�i (1)

G(z) =L�1Xi=0

Gi(zL)z�(L�1�i): (2)

7

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In the sequel Hi(z) and Gi(z) will be called the polyphase components of H(z) and G(z),

respectively. Then with

xi(n) = x(nM � i) and yi(n) = y(nL+ L� 1� i); (3)

we can redraw Fig. 2 as Fig. 3.

Thus, in principle the knowledge of x(i) and y(i) estimates the rank one matrix transfer

function

F (z) =

2666666664

G0(z)

G1(z)...

GL�1(z)

3777777775�H0(z) H1(z) � � � HM�1(z)

�: (4)

- -

-

-

-

-

-

?

-

?

- -

?

?

-

? ?

?

?

?6

z�1

6

6-

-

M

M

M

L

L

L

H1

H0 G0

G1

GL�1HM�1

x0

x1

xM�1

y0

yL�1

y1

x

y

...... ...

...

z�1

z�1

z�1

z�1

z�1

Fig. 3. The polyphase representation of the system.

Now we will need one of the following two assumptions (both need not be satis�ed).

Assumption 2: The greatest common divisor (gcd) of the set of polynomials Hi(z) is a

pure delay z�d (d integer). Further their maximum order is known.

Assumption 3: The gcd of set of the set of polynomialsGi(z) is a pure delay z�d (d integer).

Further their maximum order is known.

Assumption 2 for example is quite common in fractionally spaced blind equalization, [15],

[16]. Further pure delay common factors in Hi(z) and Gi(z) result from delays in H(z) and

G(z), respectively, though the converse need not be true. In particular only delays of M

(respectively L) or larger result in the Hi(z) (respectively Gi(z)) having a pure delay as a

common factor. Then we have the following Theorem.

8

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Theorem 1: Suppose one or both of the two assumptions 2 or 3 hold. Then the matrix in

(4) gives H(z) and G(z) to within a common nonzero scaling factor and a delay as long as

H(z)G(z) 6= 0.

Proof: Suppose assumption 2 holds. Observe, that one has available G0(z)Hi(z),

i 2 f0; � � � ;M � 1g. Then the gcd of G0(z)Hi(z), i 2 f0; � � � ;M � 1g is �z�dG0(z) for some

scalar � and integer d. Thus by dividing by this gcd one obtains the Hi(z) to within a scalar

and delay, and hence also the Gi(z). The relation to assumption 3 can be similarly deduced.

Should however, theHi(z) andGi(z) have respective common factors 1��z�1 and 1��z�1,

� 6= �, then these can be exchanged between the Gi(z) and Hi(z) in (4) without changing

the input output relationship exempli�ed by (4). Consequently the Gi(z) and Hi(z) cannot

be separately extracted. This of course does not show how the H(z), G(z) can be determined

to within a scaling factor. The sequel provides such algorithms. Note, [12] does not have

these algorithms.

IV. Estimating the roundtrip dynamics

In this section, we describe how the roundtrip dynamics can be estimated at the BS using

the feedback information. To this end, in Section IV-A we provide a z-domain description

of the relation between x(n) and y(n). Section IV-B explains how this description leads to

an estimation algorithm.

A. An input output relationship

Consider now Fig. 2 with u(k) the interference caused by the normal RLC data due to

imprecise synchronization. Adopt the standard notation of representing the z-transform of

signals represented by small letters such as a(n), by capital letters, e.g. A(z).

De�ne,

w2k(n)def= w2(nL+ L� k � 1); 0 � k � L� 1 and uk(n)

def= u(nL� k); 0 � k � L� 1:

Then, [13]

Y (z) =L�1Xk=0

z�(L�1�k)Yk (zL) and X (z ) =

M�1Xk=0

z�kXk (zM ):

9

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Likewise,

W2(z) =L�1Xk=0

z�(L�1�k)W2k(zL) and U(z) =

L�1Xk=0

z�kUk (zL):

Thus we can rewrite

L�1Xk=0

z�(L�1�k)Yk(zL) =

L�1Xk=0

z�(L�1�k)Gk (zL)V (zL) +

L�1Xk=0

z�(L�1�k)W2k(zL)

+L�1Xk=0

L�1Xj=0

z�(L�1�(k�j))Gk (zL)Uj (z

L) (5)

Equating coeÆcients on both sides of (5) one obtains:

0BBBB@

Y0(z)...

YL�1(z)

1CCCCA =

0BBBB@

G0(z)...

GL�1(z)

1CCCCA V (z) + bG(z)

0BBBB@

U0(z)...

UL�1(z)

1CCCCA+

0BBBB@

W20(z)...

W2(L�1)(z)

1CCCCA (6)

where

bG(z) =

0BBBBBBBB@

G0(z) G1(z) � � � GL�1(z)

G1(z) G2(z) � � � z�1G0(z)...

...

GL�1(z) z�1G0(z) � � � z�1GL�2(z)

1CCCCCCCCA

(7)

Observe,

V (z) =M�1Xj=0

Xj(z)Hj(z) +W10(z) (8)

where

w10(n) = w1(Mn): (9)

Thus one has0BBBB@

Y0(z)...

YL�1(z)

1CCCCA = F (z)

0BBBB@

X0(z)...

XM�1(z)

1CCCCA+ bG(z)

0BBBB@

U0(z)...

UL�1(z)

1CCCCA+

0BBBB@

W20(z)...

W2(L�1)(z)

1CCCCA (10)

where F (z) is given by (4).

10

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B. Estimation of RTC

Roundtrip dynamics estimation involves the estimation of F (z) from (10). The key thing

to note about the estimation of F (z) is that under Assumption 1, x(k) is uncorrelated with

w10(n), vi(n) and w2i(n). Thus one can express (10) as

0BBBB@

Y0(z)...

YL�1(z)

1CCCCA = F (z)

0BBBB@

X0(z)...

XM�1(z)

1CCCCA+ �(z) (11)

where (n) is uncorrelated with x(n).

Suppose the order of H(z) is lH and the order of G(z) is lG. In (1), Hi(z) =P

n hi(n)z�n,

where

hi(n) = h(Mn+ i) 0 � i �M � 1; 0 � (Mn+ i) � lH (12)

Similarly, in (2), Gi(z) =P

n gi(n)z�n, where

gi(n) = g(Ln+ L� 1� i) 0 � i � L� 1; 0 � (Ln+ L� 1� i) � lG (13)

By padding zeros, let the length of the sequence of coeÆcients of Hi(z) be lh + 1 and that

of Gi(z) be lg + 1, where lh and lg are the maximum orders of polynomial set Hi(z) and

polynomial set Gi(z) respectiveley. They are related to lH and lG as follows:

lh + 1 = d(lH + 1)=Me

lg + 1 = d(lG + 1)=Le

where dxe stands for the smallest integer that is greater than or equal to x.

Let lf = lg + lh and de�ne

Fij(z) = Gi(z)Hj(z) =lfXk=0

fij(k)z�k (14)

For some integer N to be speci�ed in a later section, the Toeplitz �ltering matrix of Fij(z)

is de�ned as

TN (Fij) =

0BBBB@

fij(0) � � � fij(lf ) 0

. . .. . .

. . .

0 fij(0) � � � fij(lf )

1CCCCA (15)

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TN (Fij) is an N � (lf +N) matrix.

Denote

Xi(k) = [xi(k); � � � ; xi(k � lf �N + 1)]T ; 0 � i �M � 1 (16)

X (k) = [X T0 (k); � � � ;X

TM�1(k)]

T (17)

Yi(k) = [yi(k); � � � ; yi(k �N + 1)]T (18)

By aligningM matrices TN (Fij), we get the block Toeplitz matrix

Fi(N) = [TN (Fi0); � � � ; TN (Fi(M�1))] (19)

Hence, equation (10) can be expressed as

Yi(k) = Fi(N)X (k) + �(k) 0 � i � L; (20)

where �(k) is uncorrelated with X (k). The inverse of RXX = E(XX T ) exists because of

assumption 1. Then we have

Fi(N) = RYiXR�1XX

(21)

where RYiX= E(YiX

T ) and E(�) denotes the expectation operation. Observe, the estimation

of Fi(N) for each i in f0; 1; � � �L � 1g provides F (z) accomplishing the estimation of the

roundtrip dynamics.

V. Unraveling FLC and RLC from the estimated RTC

We now demonstrate how the estimate of Fi(N) obtained in the previous section can be

used to separate the FLC and RLC dynamics. For simplicity sections V-A and V-B provide

unraveling algorithms under assumptions 3 and 2 respectively, with the added assumption

that the cognizant polyphase components do not even share delays. Recall that this does

not necessarily imply the absence of delays in the pertinent channels but rather that delays

are smaller than M or L depending on whether the channel in question is the FLC or RLC.

Methods of accomodating common delay factors among the polyphase components are in

section V-C.

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A. Algorithm when the Gi(z) are coprime and do not share a delay

Similar to the de�nition of TN (Fij) in equation (15), we can de�ne TN (Hi) and TN (Gi)

as the Toeplitz �ltering matrix associated with Hi(z) and Gi(z) respectively. TN (Hi) has a

dimension N � (lh +N) and TN (Gi)'s dimension is N � (lg +N). De�ne the LN � (N + lg)

block Toeplitz matrix,

G =

2666666664

TN (G0)

TN (G1)...

TN (GL�1)

3777777775

(22)

We note �rst the following well known fact, [16]:

Fact 1: If the Gi(z) have no common factors, not even delays, then for a suÆciently large

N speci�ed in [16], G has full column rank. Further, the left null space of G provides Gi(z)

to within a scaling constant.

In the sequel choose N to be the smallest integer for which G has full column rank. Now

de�ne the (lg +N)�M(lh +N) matrix

H =

�Tlg+N(H0) Tlg+N(H1) � � � Tlg+N (HM�1)

�: (23)

Note that the following relation holds

TN (Fij) = TN (Hj)Tlh+N (Gi) = TN (Gi)Tlg+N(Hj) (24)

Consequently, one obtains:

F = GH (25)

De�ne

F = [F0(N)T ; � � � ;FL�1(N)T ]T (26)

and recall that section IV provides F .

It is evident that H is full row rank. Therefore, the column vectors of F spans the same

subspace de�ned by the column vectors of G, referred to in the sequel as the signal subspace.

More importantly the left null space of F is identical to the left null space of G. Fact 1 then

sets up the direct estimation of G(z) from the left null space of the estimate of F provided

in the previous section.

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Allowing for estimation errors because of �nite data records, call F the estimate of F

provided by the algorithm in the previous section. Denote the error matrix as N , i.e.

F = GH+N :

Hence we have the following result after performing a singular value decomposition (SVD)

of F :

F = GH+N = (Vs Vn)

0B@ �s 0

0 �n

1CA0B@ WH

s

WHn

1CA (27)

where (�)H stands for the Hermitian transpose, the column vectors in Vs span the signal

subspace, and the column vectors in Vn span the left null space of F and hence of G. The

dimension of Vs and Vn are LN � (lg +N) and LN � (LN � lg �N).

If there is no estimation error, the i-th column pi of Vn satis�es

pHi G = 0 (28)

where pi = [pi(0); � � � ; pi(LN � 1)]T . In practice, when estimation errors exist, (28) can be

solved in the least square sense, i.e. by minimizing the following quadratic form

q(g) =LN�lg�N�1X

i=0

jpHi Gj2; (29)

where

g = [g0(0); � � � ; g0(lg); � � � ; gL�1(0); � � � ; gL�1(lg)]: (30)

In order to solve for g in equation (29), we follow the method in [16]. Similar to the

de�nition of TN (Fij), we de�ne the �ltering matrix Tlg+1(Pij) associated with Pij(z), 0 � i �

LN � lg �N � 1; 0 � j � L� 1, where

Pij(z) =N�1Xk=0

pi(jN + k)z�k (31)

The dimension of Tlg+1(Pij) is (lg + 1)� (lg +N). De�ne

Pi = [Tlg+1(Pi0)T ; � � � ; Tlg+1(Pi(L�1))

T ]T (32)

Then equation (29) can be transformed into

q(g) = gPgH ; where P =

LN�lg�N�1Xi=1

PiPHi (33)

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The solution of g is the smallest eigenvalue of matrix P . Note that g consists of all the

coeÆcients of G(z). Thus the RLC is identi�ed up to a multiplicative constant. From the

estimated G(z), we can construct G as in (22). From

H = Gy

F (34)

where (�)y indicates pseudoinverse, we can also identify the FLC up to a multiplicative

constant.

B. Algorithm when the Hi(z) are coprime and do not share a delay

The foregoing algorithm deals with the case when assumption 3 is satis�ed together with

the stronger condition that the polyphase components Gi(z) do not share any delays. In this

case, we identify G(z) �rst, and then identify H(z) from the estimated G(z). Now suppose

that assumption 2 is satis�ed together with the condition that the polyphase components

Hi(z) do not share any delays.

De�ne for some integer ~N

~H =

2666666664

T ~N (H0)

T ~N (H1)...

T ~N (HM�1)

3777777775: (35)

Under assumption 2 and the lack of common delay factors among the Hi(z) there exists an

~N for which ~H has full column rank. Choose

~Fi( ~N) = [T ~N (F0i); � � � ; T ~N (F(L�1)i)]

~F = [F0( ~N)T ; � � � ;FM�1( ~N )T ]T

and

~G =

�Tlh+ ~N(G0) Tlh+ ~N(G1) � � � Tlh+ ~N(GL�1)

�:

Then (24) leads to:

~F = ~H ~G;

and an algorithm very similar to that in Section V-A accomplishes the desired unraveling.

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C. Treating common delay factors

We now consider the situation where the pertinent polyphase components may have com-

mon delay factors. We will explain the underlying ideas with Assumption 3 in force. Similar

ideas apply to the case where Assumption 2 holds.

Suppose that Gi(z) = z�l �Gi(z) where the �Gi(z) have no common factors, i.e. z�l is the

gcd of the Gi(z). Suppose Hi(z) = z�ni �Hi(z). Then because of the upper triangular Toeplitz

nature of TN (Gi) and TN (Hi), and because of (24), TN (Fij) has l+ nj columns that are zero

vectors. These zero columns also appear in F . The matrix �F obtained by removing these

zero columns can be expressed as

�F = �G �H

where �G in particular is

�G =

2666666664

TN ( �G0)

TN ( �G1)...

TN ( �GL�1)

3777777775

(36)

and as the �Gi(z) have no common factors has full rank and has identical null space as �F .

Consequently the algorithm in section V-A working with �F rather than F suÆces to estimate

G(z) and H(z) to within a scalar and delay ambiguity.

VI. Simulations

We present two simulation examples, the �rst to illustrate the basic performance of the

algorithm, and the second to illustrate the reduction of training levels needed on the FLC

when the channel parameters change with time. Before presenting these simulations we make

precise our notions of SNR and SIR (caused by the interference e�ect of poorly synchronized

RLC data.)

Denote noise, interference and signal at the output of RLC as w(n), uo(n) and z(n)

respectively. We can see y(n) = z(n) + w(n) + uo(n). SNR and SIR are de�ned as

SNR =E(z2(n))

E(w2(n))(37)

SIR =E(z2(n))

E(u2o(n))(38)

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If the variance of x(n),w1(n),w2(n) and u(n) are �2x,�2w1, �2w2

and �2u respectively, and if the

signals are wide sense stationary (WSS), by a derivation given in the appendix, we have

SNR =�2xPL�1

i=0

PM�1j=0

Plfk=0 f

2ij(k)

�2w1

PlGk=0 g

2(k) + L�2w2

(39)

SIR =�2xPL�1

i=0

PM�1j=0

Plfk=0 f

2ij(k)

L�2uPlG

k=0 g2(k)

(40)

where fij(k) is de�ned in (14) and lG is the order of RLC.

Simulation example I: The FLC and RLC are generated from two delayed raised-cosine

pulse C(t; �), where � is the roll-o� factor. C(t) is limited in 8T for FLC and in 6T for

RLC, where T is the symbol interval.

FLC = 0:1C(t; 0:25) + 0:8C(t� T=2; 0:25) (41)

RLC = 0:5C(t; 0:10)� 0:7C(t� T=3; 0:10) (42)

Their corresponding discrete time channel coeÆcients are

FLC = [0:0129;�0:0326; 0:0693;�0:1485; 0:6019; 0:5019;�0:1485; 0:0693;�0:0326] (43)

RLC = [0:0521;�0:0786; 0:1423;�0:0783;�0:2882; 0:1128;�0:0677] (44)

We use down sampling factorM = 3 and upsampling factor L = 2. Hence lh = 3 and lg = 4.

Noise signal w1(n) and w2(n) are zero mean and have the same variance. The input signal is

i.i.d BPSK. We focus on estimating and compensating the FLC in the interference free case.

To quantify the quality of channel estimation, we de�ne the normalized root-mean-square

error(NRMSE) as

NRMSE =1

k h k2

vuut 1

Mt

MtXi=1

k h(i) � h k22

(45)

where Mt is the number of Monte Carlo runs; h is the real channel and h(i) is the estimate

in the i-th run. Fig. 5 shows NRMSE versus total SNR, where total SNR is de�ned in (39).

After FLC and RLC are estimated, a pre-equalizer is built for the FLC and a post-equalizer

for the RLC. The length of the zero-forcing equalizer is three times the corresponding real

channel length. The equalizer SNR is SNR at the output of the equalized system. That is,

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for FLC SNR is that at the output of the channel, but for RLC SNR is that at the output

of the post-equalizer. BER versus equalizer SNR is displayed in Fig.6.

To evaluate the e�ect of iterference due to uplink data, Fig.7 shows the BER versus SIR

at �xed SNR.

1 2 3 4 5 6 7 8 9−0.4

−0.2

0

0.2

0.4

0.6

0.8FLCRLC

Fig. 4. The channel impulse responses of FLC and RLC.

−5 0 5 10 15 20 25 30 3510

−2

10−1

100

Total SNR

NR

MSE

FLCRLC

Fig. 5. NRMSE versus total SNR without interference. 100 Monte Carlo runs. 500 symbols for each run.

Simulation example II: Reduced training for a time varying channel:

We use COST-207 Typical Urban(TU) [7] model with 100 echo paths, BPSK data and

maximum Doppler frequency 55Hz. We assume the channels to be quasistatic, i.e. time-

invariant in one frame and time-variant from frame to frame. Each frame lasts 400�second

(cf 557 �second for GSM) and the roundtrip delay is 16.67�second. The receive �lters for

FLC and RLC are raised cosine functions with roll-o� factors 0.2 and 0.1 respectively. The

18

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−5 0 5 10 15 2010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Equalizer SNR

BER

FLCRLC

Fig. 6. BER versus equalizer SNR without interference. Channels are estimated at 20dB with 500 symbols

in each run. 100 Monte Carlo runs.

0 2 4 6 8 10 12 14 16 18 2010

−6

10−5

10−4

10−3

10−2

10−1

100

SIR

BER

FLCRLC

Fig. 7. BER versus SIR with 20dB total SNR and 25dB equalizer SNR. 500 symbols in each run. 100 Monte

Carlo runs.

FLC sustains a data rate of 1 Mbps, and the RLC supports 0.667 Mbps.

Two situations are compared:

(a) Training aided equalization of FLC at the MS and of the RLC at the BS, with no

feedback.

(b) No training on the RLC, but instead sending feedback data of the same length as the

RLC training data in (a). A precompensator, obtained using the scheme of this paper is used

on the FLC and is augmented by a post-equalizer estimated at the receiver using reduced

training.

Both methods use the same input signal power and noise power. Figure 8 shows nb=nt

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versus input SNR for both methods to achieve the same BER. Here nt is the length of the FLC

training sequence used in (a) and nb the length of training used for partially compensated

channel estimation in (b), so that the same FLC BER is obtained in both cases.

As is evident from the simulation that at 18db SNR oyr algorithm requires only 32% train-

ing length on the FLC than the conventional training based FLC compensation. Translated

to a GSM setting instead of devoting 1/6th of transmission time for training only 1/19th is

needed to achieve the same performance in this time varying setting.

Given that the feedback data on the RLC replaces and has the same length as the training

data in (a), this represents substantial savings in bandwidth.

2 4 6 8 10 12 14 16 180.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

Input SNR(dB)

n b/nt

Fig. 8. nb=nt versus SNR at the same BER.

VII. Conclusion

We have proposed a new feedback scheme that permits FLC and RLC channel estimation

from the BS, in principle without the use of training signals or blind estimation methods. In

practice it leads to signi�cant savings in the bandwidth consumed by training signals, and

the transfer of some of the computational burden currently shouldered by the MS, to the

more resource rich BS. The feasibility of these ideas have been demonstrated through both

20

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theory and simulations. The possibility of unraveling FLC and RLC from the information

of RTC is a core novelty of the paper.

Appendix

Similar to equation (3), we de�ne

zi(n) = z(nL + L� 1� i); 0 � i � L� 1 (46)

So E(z2(n)) =PL�1

i=0 E(z2i (n))=L. And we know zi(n) =

PM�1j=0

Plfk=0 fij(k)xj(n� k)

E(z2i (n)) = E[M�1Xj=0

lfXk=0

fij(k)xj(n� k)M�1Xp=0

lfXq=0

fip(q)xp(n� q)] (47)

From assumption 1, we know E(xj(n�k)xp(n�q)) = Æ(p�j)Æ(k�q)�2x,where Æ is Kronecker's

delta. Then,

E(z2i (n)) = �2x

M�1Xj=0

lfXk=0

f2ij(k) (48)

E(z2(n)) = �2x

L�1Xi=0

M�1Xj=0

lfXk=0

f2ij(k)=L (49)

From equation(10) and (8), we can see w10(n) will pass through Gi(z) for all 0 � i � L� 1.

Note that the variance of w1(n) is equal to w10(n), and w1(n) and w2(n) are uncorrelated

and zero mean. Similar to the derivation of E(z2i (n)), we have

E(w2(n)) = �2w1

L�1Xi=0

lgXk=0

g2i (k)=L+ �2w2(50)

And we knowPL�1

i=0

Plgk=0 g

2i (k) =

PlGk=0 g

2(k). From (49) and (50), we get the expression for

SNR in (39). It's easily seen that the interference component at the output of RLC is the

signal by passing u(n) through channel G(z). So we have

E(u2o(n)) = �2u

lGXi=0

g2(i) (51)

References

[1] T. Krauss and M. D. Zoltowski, \Multiuser Second-Order Statistics Based Blind Channel Identi�cation for Using

a Linear Parameterization of the Channel Matrix," IEEE Trans. on Signal Processing, vol. 48, no. 9, September

2000, pp. 2473-2486.

[2] Y. Zhang, K. Yang, and M. Amin, "Adaptive array processing for multipath fading mitigation," IEEE Transac-

tions on Antennas and Propagation, vol.49, no.4, pp.505-516, April 2001.

21

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[3] S. Kayhan and M. Amin, "The application of spatial evolutionary spectrum to direction �nding and blind source

separation," IEEE Transactions on Signal Processing, April 2000.

[4] J.H. Winters, \Switched diversity with feedback for DPSK mobile radio stations", IEEE Transactions on Ve-

hicular Technology, pp 134-150, 1983.

[5] G. Raleigh and J. CioÆ, \Spatio-Temporal Coding for Wireless Communications", IEEE Transactions on Com-

munications, COM-46(3): 357-366, March 1996.

[6] R. W. Heath, Jr. and A. Paulraj, \A Simple Scheme for Transmit Diversity using Partial Channel Feedback,"

Proc. of 32nd Asilomar Conf. on Signals, Systems and Computers, pp.1073-8, Paci�c Grove, CA, 1998.

[7] P.Hoeher,\A statistical discrete-time model for the WSSUS multipath channel", IEEE Trans. on Vehicular

Technology, VT-41(4):461-468,April 1992.

[8] L.M.C. Hoo, J. Tellado, J.M. CioÆ, \Discrete dual QoS loading algorithms for multicarrier systems", IEEE

International Conference on Communications, pp 796-800, 1999.

[9] J. Zhang, A. Sayeed and B. Van Veen, \Optimal transceiver design for selective wireless broadcast with channel

sttae estimation", in Proceedings of ICASSP, May 2002, Orlando, Fla.

[10] D. L. Goeckel, \Adaptive Coding for Fading Channels using Outdated Channel Estimates", IEEE Transactions

on Communications, Vol. 47, pp. 844-855, June 1999.

[11] X.-G. Xia, \New precoding for intersymbol interference cancellation using nonmaximally decimated Multirate

Filterbanks with ideal FIR equalizers", IEEE Trans. Signal Processing, 45:2431-2441, Oct. 1997.

[12] G. Williamson, S. Dasgupta and M. Fu, "Adaptive Multirate, Multistage Filters", Proc. of ICASSP, 1996.

[13] P. P. Vaidyanathan, Multirate Systems and Filter Banks, Prentice Hall, 1992.

[14] A. J. Goldsmith and P. P. Varaiya, \Capacity of fading channels with channel side information", IEEE Trans.

on Information Theory, 43(6):1986-1993, November 1997.

[15] L. Tong, G. Xu and T. Kailath, \Blind identi�cation and equalization based on second-order statistics: A

time-domain approach", IEEE Trans. Information Theory, vol. 40 no. 2, pp. 340-350, March 1994.

[16] E. Moulines, P. Duhamel, J.-F. Cardoso and S. Mayrargue, \Subspace methods for the blind identi�cation of

multichannel FIR �lters", IEEE Trans. Signal Processing, vol. 43 no. 2, pp. 516-525, Feb. 1995.

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Channel and flow adaptive multiuser DMT

Soura Dasgupta, Ashish Pandharipande

Department of Electrical and Computer Engineering, The University of Iowa, Iowa City, IA, USA.Email:

�dasgupta, pashish � @engineering.uiowa.edu

ABSTRACT

This paper considers the design of biorthogonal DMT mul-ticarrier transceiver systems supporting multiple services.The supported user services may have differing quality ofservice (QoS) requirements, quantified in this paper by bitrate and symbol error rate specifications. Our goal is tominimize the transmitted power given the QoS specifica-tions for the different users, subject to the knowledge ofcolored interference at the receiver input of the DMT sys-tem. In particular we find an optimum bit loading schemethat distributes the bit rate transmitted across the varioussubchannels belonging to the different users, and subject tothis bit allocation, determine an optimum transceiver. Thiswork differs from our prior work [6] where orthonormaltransceivers were considered.

1. Introduction

Future broadband communication systems will be expectedto deliver multiple services, such as voice, data, video, withmultiple-stream support. Because delivery of these streamswill be under differing requirements such as informationrate and error performance, allocation of critical resourceslike power would have a significant impact on the overallperformance of the communication system. Discrete Mul-titone (DMT) transmission involves a channel coding tech-nique to achieve reliable, high data rate communications insuch systems. It is a current standard in various wirelineapplications like ADSL, VDSL, [10], and in the form of Or-thogonal frequency division multiplexing (OFDM) has beenproposed for fixed wireless standards like IEEE 802.11a,[12]. This paper considers transceiver optimization for suchmulticarrier transmission systems operating in a multiuserenvironment.

More specifically, we assume that a single DMT sys-tem supports � users, each having its own QoS specificationquantified by its bit rate and symbol error rate (SER). The�

-th user requires a bit rate of ��� , and an SER of no morethan ��� . An equal number of subchannels are assigned to

Supported by ARO grant DAAD19-00-1-0534 and NSF grants ECS-9970105 and CCR-9973133.

each user. As proposed in several recent papers, [7], [6],[9], we consider general DMT transceivers which are moregeneral than the traditional DFT based systems in that theinput and output transforms are general block transforms.We consider biorthogonal systems employing zero paddingredundancy with the redundancy removal at the receiver be-ing a general linear operation. Our goal is to select the inputand output block transforms � , � (see fig. 1), the lin-ear operation reflecting redundancy removal, the number ofbits/symbol assigned to each subchannel, and the subchan-nels assigned to each user to achieve the QoS specifications,under a zero intersymbol interference (ISI) condition withthe minimum possible transmitted power. We assume thatthe channel and equalizer are known and so is the interfer-ence autocorrelation.

We thus generalize our earlier result reported in [6], wherethe same optimization problem was considered with an or-thonormal transceiver under the assumption that each useris assigned the same number of subchannels. We shall see inthe following sections how the extension to [6], consideredhere nontrivially modifies the optimization problem.

Figure 1 depicts the DMT communications system un-der consideration. An incoming data stream is convertedinto � parallel data streams of lower rate. An � -pointblock transformation , of these streams of data is fol-lowed by a parallel-to-serial conversion, prior to transmis-sion through the communication channel. An equalizer isemployed to shorten the dispersive effects of the transmis-sion channel. The equalized channel ������� is assumed to beFIR of length � . For an FIR equalized channel of length � ,extra redundancy of length � in the form of zero paddingis added at the channel input to infuse resistance to chan-nel induced ISI. At the channel output, one performs insuccession the operations of redundancy removal, serial-to-parallel conversion, and the application of an inverse blocktransform, �� .

Past treatment of optimum resource allocation, [1], [2],[3], has been restricted mostly to bit loading and power allo-cation algorithms. Some authors have studied the optimumtransceiver design in the single user case, [7], [9]. While[7] was concerned with optimizing the transmitted power,[9] focussed on the maximization of the mutual informa-

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tion between the transmitted and received signals. In [5] theauthors consider the problem considered here for the sin-gle user case of ��� � , and with orthonormality conditionenforced. In [7] the single user case is considered with or-thogonality removed. Reference [7] shows that in the singleuser case biorthogonality leads to no improvement in thetransmitted power. Likewise a major conclusion of this pa-per is to show that even in the multiuser environment withpotentially asymmetric subchannel allocations, optimal per-formance is acheived by orthonormal transformations.

��

��� �

� �Noise

+Interference

+

...

...

������� ���������

...

InverseTransform� � S/P

RemoveRedundancy

Transform� ���

�� P/S � Pad

zeros� Channel

��Equalizer

...

Fig. 1. DMT communication system.

2. Formulation

In this Section we give some preliminaries. Specifically, inSection 2.1, we recount the details of the generalized DMTsystem, along the lines of [7]. Section 2.2 provides a precisedescription of the optimization problem.

2.1. Preliminaries

Barring [7], most papers assume that is unitary, i.e.

�� ����� (2.1)

In the biorthogonal case considered here we relax (2.1) andsimply assume that and � can be arbitrary nonsingular����� matrices. Denote the blocks of � input and outputsymbols respectively by � ��� ��� � ! ��� �#"%$&$%$&"'!)(�*,+ ��� �.-0/ ,and 1� �2� �3�4� 1! �2� �5"%$&$%$%" 1!)(�*6+ �2� �7-8/ . With 9 ��� � , the noiseand interference effect at the output of the equalizer, denote: �2� �;�<� 9 �>=�� �#"?9 �>=��A@ � �5"%$&$%$%"?9 �>=��B@C=4D � �.-0/ , as the= -fold blocked version of 9 �2� � , with =�� �E@ � . Thenone can show, [5] that with FAG an =H� � constant ma-trix characterized by the � order FIR equalized channel, and� + , an �I�C= matrix, representing the linear redundancyremoval operation, the blocked input-output relation of thesystem is given by

1� �2� �J� � � + FBG &� ��� �K@ �� � + : �2� �5� (2.2)

We impose the perfect reconstruction (PR) condition,i.e., in the absence of noise/interference, 1� �2� ���L� ��� � forall � . In other words,

� �K+%F G �M��" (2.3)

and the DMT system has no ISI. To obtain a more usefulcharacterization of PR, consider the singular value decom-position of FAG

F G �ONJP QAR PSUTWV �P �ON R P V �P (2.4)

where N P and V P are respectively =O�X= and �Y� � unitarymatrices and

R P is a �Z� � real, positive definite diagonalmatrix. Then, because of (2.3), given , the class of all� � + enforcing PR is completely characterized by

� [� *,+ � (2.5)

and

� + � V P R *6+P]\ � ( ^`_ N �P " (2.6)

where ^ is any arbitrary �a� � matrix. In the sequel it willbe useful to partition NbP as NbPc�d� N Nb+e- , where N is=f� � and N + is =f� � .

Note, as V P , NJP andR P are supplied by the channel,

the only quantities that need to be found to determine thetransceiver completely are � and ^ .

2.2. Problem formulation

As mentioned earlier the � subchannels are distributed amongthe � users with each of the users being allocated gH��ih � subchannels. Thus consider disjoint subsets j �lkm S "%�&�%�5" �nD �o with p j � pq�igsr � , and j �ut j�v[��wx" �zy�|{ .Subchannel assignment to the

�-th user constitutes deter-

mining j � . We assume that the { -th subchannel of the�

-thuser is assigned } v?~ � bits per symbol. To meet the bit ratespecification for the

�-th user one requires that�=��v5���� }?v?~ � � � � � (2.7)

Let the input power in the { -th subchannel of the�

-th userbe �6��&��� � . Assume that �6��.��� � is the noise power in this sub-channel. Under high SNR most modulation schemes, [8],require that to achieve a given SER the required SNR is pro-portional to �q� ��� � . More precisely,

� ��&��� � �i� � � � ��� � � ��.��� � " (2.8)

where the constant � � depends on the desired SER, � � , forthe

�-th user. For example, for QAM, � � � +� � � *6+ �&� �� �.-0� .

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Under this framework, the transmitted power for the biorthog-onal DMT system is given by��� � �

���� + �v5����� � ��%��� � � �� - v7v (2.9)

� ����� + �v5����� ���� � ��� � � ��7�>� � � � - v7v � (2.10)

Define ��� denoting the known autocorrelation matrix ofthe noise vector : ��� � , and� � � � �� � � and �� � � + � � � �+ � (2.11)

Then � ��.��� � are the diagonal elements of � � , the autocorre-lation matrix of the receiver output noise vector � �2� � . Thus,because of (2.5), (2.10) can be rewritten as� � � � �b� �

���� + �v5���� � �� � ��� � � � * � � *,+ - v7v � � � � � - v7v �(2.12)

Thus the optimization problem becomes: Given ��� , g ,��� , � � , minimize (2.12) subject to (2.7) by selecting } v?~ �(bit loading), selection of j � (subchannel assignment), � (transformation selection) and because of (2.6), ^ (redun-dancy removal selection).

We show that there is a conceptual separation betweenthe three selections, i.e. the optimizing ^ is determined ex-clusively by � � , provided by the knowledge of the interfer-ence and equalizer characteristics; � is determined en-tirely by ^ and the channel characteristics; j � are deter-mined entirely by � � , in turn provided by � + and � , andthe bit allocations are determined once the above quanti-ties are found. Further as noted in the introduction, we willshow that without loss of generality, the optimizing � , are unitary.

3. Optimum selections

In this section we consider the selection of the various vari-ables.

3.1. Optimum Bit Loading

From the Arithmetic Mean-Geometric Mean (AM-GM) in-equality that states that the Arithmetic Mean, exceeds theGeometric mean, with equality if all samples are equal, wehave that for a given choice of j � and � , under (2.7),��� � � ����� /� ����� + ��� �� ����� ���v5����� � � * � � *,+ - v7v � � � � � - v7v��� +����

with equality iff for all�

and ".{� � ��� � � � * � � *6+ - v7v � � �� � � - v7v � � �"! � � � � * � � *6+ -$#%# � � �� � � -&#'#?�(3.13)

This is in turn equivalent to the optimum bit loading rule:

}?v?~ � � = � �g D)(%*,+ � - � ��.��� � � � 5- v7v�/. v5����� � �� �>� � � � - v7v � +��0��1 �

(3.14)Note that

� � �2� / is much more complicated than its spe-cializations, � � � , studied in [5]. Thus, under optimum bitloading the remaining variables must be selected to mini-mize

� � �2� / . Observe, that while the choice of these othervariables impacts the selection of }ev?~ � , � � �2� / itself is inde-pendent of } v?~ � . This underscores the fact that the remainingvariables can be selected regardless of the precise values of}?v ~ � obtained through (3.14).

3.2. Selection of � , j � and ^Assume for the moment that ^ and hence � + has been se-lected and that the resulting positive definite Hermitian � has the SVD: �� � N R � N � (3.15)

withR � m43 "%$&$%$5" 3 (�*6+ o real, diagonal and N unitary.

The goal is to select �� and j � to minimize� ����� / .

For convenience we first work with the minimization of5 � � �J� (�*6+� # � 76 # � � � � � - #'# � � * � � *6+ - #%# (3.16)

given positive 6 # . Note5 � �� � has the form of

� �.

It is noteworthy that in [5], the �� � � N � , with�

a permutation matrix miminimizes�8� * �2� / . If � is re-

stricted to be unitary, then [11] shows that this choice of� also minimizes (3.16). Consider, however, the examplewhere 6 # � � , � � � and � � diag

m9 " ��o . Then ob-serve that

5 �>� �;� � S but with

� � �: � Q � �� D � T Q � h : ; SS � T5 � ���A�=< . Thus in general � � N � does not minimize(3.16). However, we will show in the sequel that it doesminimize

� ����� / .The following result shows that the search space of �

can be restricted to a particular form.

Lemma 3.1 For some unitary V , (3.16) is minimized by

��[� V R *,+>� � N � (3.17)

and (3.16) becomes5 � � �J� (�*,+� # � 76 # \ V R V � _ �#'# (3.18)

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Denote � � � � � � ��� ���0� and �)vz� \ V R V � _ v7v . Thenunder optimum bit loading it suffices to restrict the searchof � to (3.17) and to seek to minimize under unitary V :���� � �

���� + � � �v5������ � � �0�� (3.19)

with j �� defining the optimal arrangement of the sequenceof � � .

Before proceeding, we need a few results from the the-ory of majorization, [4].

Definition 3.1 Consider two sequences ! � m ! # o��# � + and� � m � # o �# � + with ! # � ! # + and � # � � # + . Then we saythat � majorizes ! , denoted as !�� � , if

�� # � + ! #� �

� # � + � #holds for

� � � , with equality at� � � . We say that� weakly supermajorizes ! , denoted !���� � , if � �# � v ! # �

� �# � v � # " � C{� � .Fact 1 If � is an �M� � Hermitian matrix with diagonalelements �C� m � # o��# � + and eigenvalues

3 � m 3 # o��# � + , then��� 3

.

Definition 3.2 A real valued function � �����J��� � � + "%�%�&�%" � � �defined on a set � k � � is said to be Schur concave on �if !�� � on ����� �2! � � � � � �#�� is strictly Schur concave on � if strict inequality � �2! � r� � � � holds when ! is not a permutation of � . Further if!�� � � then also � ��! �3r�� � � � .

We will now state a theorem that results in a test forstrict Schur concavity. We denote � � �"! � � �J�$#�% �'& !# & � .

Lemma 3.2 Let � � � � be a scalar real valued function de-fined and continuous on (`� m ��� + "&�%�&�%"�� � �*) � + � �&�%� �� � o , and twice differentiable on the interior of ( . Then � �����is Schur concave on ( if �+� �"! � � � is increasing in

�.

The following Lemma provides an important propertyof j �� the optimum arrangement of the subchannels.

Lemma 3.3 Consider for g � � and 6 ",�J",- # " } v r S ,. � � 6 �u*6+�

��� - � � � ��� @ �/��u*,+�0 � } 0 � � �0�

with - # � - #' + and }�# � }�# + . Suppose for some e"�{- # rs}?v and

1 .1 - # r 1 .

1 } v � (3.20)

Then

2 � � 6 �u*,+���� &~ �43� # - � � } v � � ��� @ �/�

�u*,+�0 � %~ 0 3� v } 0 � - # � � �0��5 . �

Further1 . h 1 - # 1 . h 1 - #' + and

1 . h 1 } # 1 . h 1 } #' + .Thus from Lemma 3.3, any optimum arrangement for (3.19)requires that for all

� �Br6� 0 �1 � ��1 � �

5 1 � ��1 � 0 �

Under this condition,� ��

is Schur concave. Thus from Fact1, as m \ V R V � _ #%# o (�*6+# � � m43 "&�%�%�%" 3 ( *,+ o "the choice of V as a permutation matrix that enforces an op-timum arrangement of subchannels, minimizes (3.19). Thusto within a permutation matrix

�, under optimum bit allo-

cation, one can choose as an optimizing � � � R *6+�� � N � .Now note that for any diagonal nonsingular matrix 7 ,5 � � �;� 5 �87 � �#"and that for some diagonal matrix 9R ,

� � � R *6+�� � N � � 9R *6+�� � � N � �Thus as in [5], the minimizing � � � N � with

�enforcing

the optimum arrangement. Under these conditions

� � * � � *6+ - v7v � � �� � � - v7vand indeed � ��.��� � are the eigenvalues of � .

Thus, regardless of ^ the best � is a Karhunen-LoeveTransform of � , and the � ��.��� � equal the eigenvalues of � .From the comment on supermajorization made at the endof Definition 3.2, it follows that the optimizing ^ must besuch that the set of resulting eigenvalues of � weakly su-permajorize all possible sets of attainable eigenvalues. Theoptimizing ^ can then be shown to be given by, [6],

^ �<D N � � � N +;: N �+ � � N +=< *,+ � (3.21)

4. Simulation results

In this section, we compare the transmitting power of theDFT based DMT under no bit allocation and optimum bitallocation with an optimum unitary transceiver. We assumethe equalized channel to be ������� � � @ S �?> � *,+ , and anoise source 9 ��� � whose power spectral density is shownin fig. 2. We assume the DMT system supports three userservices, where each user is allocated an equal number ofsubchannels and with the same SER requirement and mod-ulation scheme for each user. The plot in fig. 2 compares

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the transmit power levels as the number of subchannels al-lotted to the users is varied. The plot shows that there isa 10 dB saving in transmit power with our design over theDFT based DMT under optimum bit allocation, and a 14 dBimprovement over the conventional DMT with no optimumbit allocation.

0 20 40 60 80 100 120 140 160 180 2000

1

2

3

4

5

6

7x 10

−14 Power spectral density of noise

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6−60

−50

−40

−30

−20

−10

0Transmitted power under different DMT and bit allocation schemes

__ DFT based DMT, no bit allocation

−− DFT based DMT, optimum bit allocation

−. General DMT, optimum bit allocation

Tra

nsm

itte

d P

ow

er (

dB

)

#channels/user, #users = 3

Fig. 2. Comparison of transmit power levels.

5. Conclusions

In this paper, an optimum bit allocation strategy and designof a general biorthogonal DMT multicarrier transceiver sys-tem employing zero padding redundancy were presented,for minimizing the transmit power when different users withvaried QoS requirements are supported and are assigned po-tentially different number of subchannels. We showed thatno gains in transmit power can be obtained by consider-ing biorthogonal transceivers over orthogonal transceivers.These results also show that the optimum transceiver de-pends only on the channel and interference conditions andnot on the QoS requirements. Indeed to within a permuta-tion of subchannels, the optimum transceiver obtained hereis identical to that obtained in [5]. Equally should the chan-nel/interference remain invariant after the initial connectionis established, then only bit loading and subchannel selec-tion need be updated in response to changing traffic needs.

6. References

[1] L.M.C. Hoo, J. Tellado and J.M. Cioffi, “Discrete dualQoS loading algorithms for multicarrier systems ”,IEEE International Conference on Communications,pp 796-800, 1999.

[2] D. Kivanc and H. Liu, “Subcarrier allocation andpower control for OFDMA”, IEEE Journal on Se-lected Areas in Communications, To appear.

[3] B. S. Krongold, K. Ramchandran and D. L. Jones,“Computationally Efficient Optimal Power Alloca-tion Algorithms for Multicarrier Communication Sys-tems”, IEEE Transactions on Communications, pp 23-27, Jan 2000.

[4] A. W. Marshall and I. Olkin, Inequalities: Theory ofMajorization and its applications, Academic Press,1979.

[5] Y.-P. Lin and S.-M. Phoong, “Perfect discrete mul-titone modulation with optimal transceivers”, IEEETransactions on Signal Processing, pp 1702 -1711,June 2000.

[6] A. Pandharipande and S. Dasgupta, “Optimaltransceivers for DMT based multiuser communica-tion”, IEEE International Conference on Acoustics,Speech, and Signal Processing, pp 2369 -2372,2001. Also IEEE Transactions on Communications,accepted subject to modifications.

[7] Y.-P. Lin and S.-M. Phoong, “Optimal ISI-freeDMT transceivers for distorted channels with colorednoise”, IEEE Transactions on Signal Processing, pp2702 -2712, Nov 2001.

[8] J.G. Proakis, Digital Communications, McGraw-Hill,Third Edition, 1995.

[9] A. Scaglione, S. Barbarossa and G.B. Giannakis, “Fil-terbank transceivers optimizing information rate inblock transmissions over dispersive channels”, IEEETransactions on Information Theory, pp 1019-1032,Apr 1999.

[10] T. Starr, J.M. Cioffi and P. Silverman, Understand-ing Digital Subscriber Line Technology, Prentice Hall,1998.

[11] S. Akkarakaran and P.P. Vaidyanathan, “Filterbank op-timization with convex objectives and the optimalityof principal component forms”, IEEE Transactions onSignal Processing, pp 100-114, January 2001.

[12] R. Van Nee and R. Prasad, OFDM for Wireless Multi-media Communications, Artech House, 2000.

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OPTIMUM BIORTHOGONAL DMT SYSTEMS FOR MULTI-SERVICE COMMUNICATION

Soura Dasgupta and Ashish Pandharipande

Department of Electrical and Computer Engineering, The University of Iowa, Iowa City, USA.Email:

�dasgupta, pashish � @engineering.uiowa.edu

ABSTRACT

This paper considers the design of biorthogonal DMT multicarriertransceiver systems supporting multiple services. The supporteduser services may have differing quality of service (QoS) require-ments, quantified in this paper by bit rate and symbol error ratespecifications. To reflect their service priorities, different users onthe system can be potentially assigned different number of sub-channels. Our goal is to minimize the transmitted power given theQoS specifications for the different users, subject to the knowledgeof colored interference at the receiver input of the DMT system. Inparticular we find an optimum bit loading scheme that distributesthe bit rate transmitted across the various subchannels belongingto the different users, and subject to this bit allocation, determinean optimum transceiver.

1. INTRODUCTION

Future broadband communication systems will be expected to de-liver multiple services, such as voice, data, video, with multiple-stream support. Because delivery of these streams will be underdiffering requirements such as information rate and error perfor-mance, allocation of critical resources like power would have asignificant impact on the overall capacity of the communicationsystem. Discrete multitone (DMT) is a channel coding techniqueto achieve reliable, high data rate communications in such systems.It is a current standard in various wireline applications like ADSL,VDSL, [9], and in the form of Orthogonal frequency division mul-tiplexing (OFDM) has been proposed for fixed wireless standardslike WLAN’s, [11]. This paper considers transceiver optimizationfor such multicarrier transmission systems operating in a multiuserenvironment.

More specifically, we assume that a single DMT system sup-ports � users, each having its own QoS specification quantified byits bit rate and symbol error rate (SER). The � -th user is assumedto have been assigned ��� subchannels, and requires a bit rate of� � , and an SER of no more than �� . The number of subchannelsassigned to each user is fixed a priori according to some priori-ties determined by the user service. As proposed in several recentpapers, [4], [6], [8], [10], we consider general DMT transceiverswhich are more general than the traditional DFT based systems inthat the input and output transforms are general block transforms.We consider biorthogonal systems employing zero padding redun-dancy with the redundancy removal at the receiver being a generallinear operation. Our goal is to select transforms �� , �� (see fig.1), and the linear operation reflecting redundancy removal, and

Supported in part by US Army contract, DAAAD19-00-1-0534, andNSF grants ECS-9970105 and CCR-9973133.

assign bits/symbol to each subchannel, to achieve the QoS spec-ifications, under a zero intersymbol interference (ISI) conditionwith the minimum possible transmitted power. We assume that thechannel and equalizer are known and so is the interference auto-correlation.

We thus generalize our earlier result reported in [6], where thesame optimization problem was considered with an orthonormaltransceiver under the assumption that each user is assigned thesame number of subchannels. We shall see in the following sec-tions how the extension to [6] considered here modifies the opti-mization problem. Further, the asymmetric subchannel assignmentconsidered here is by contrast more realistic as service prioritiesmay cause certain users to receive greater number of subchannelsthan others. For example, one may assign more subchannels tovideo services than to audio services.

Figure 1 depicts the broad contours of a DMT communicationssystem. An incoming data stream is converted into � -parallel datastreams of lower rate. An � -point block transformation of thesestreams of data is followed by a parallel-to-serial conversion, priorto transmission through the communication channel. An equalizeris employed to shorten the dispersive effects of the transmissionchannel. The equalized channel ������� is assumed to be FIR oflength � . For an FIR equalized channel of length � , extra redun-dancy of length � in the form of zero padding is added at the chan-nel input to infuse resistance to channel induced ISI. At the channeloutput, one performs in succession the operations of redundancyremoval, serial-to-parallel conversion, and the application of an in-verse block transform.

Past treatment of optimum resource allocation, [1], [2], [3],has been restricted mostly to bit loading and power allocation al-gorithms. Some authors have studied the optimum transceiver de-sign in the single user case, [4], [8], [10]. While [4] was concernedwith optimizing the transmitted power, [8] focussed on the max-imization of the mutual information between the transmitted andreceived signals. Essentially, [4] considers the same optimizationas presented here but under the assumption that only a single useris supported by the system, i.e. ����� . In comparison with [4],the multiuser environment considered in this paper renders the op-timization problem highly non-trivial as shall be seen.

In Section 2, we give a description of the DMT system andthe optimization problem under consideration. Section 3 presentsour results on optimum transceiver selection and the optimum bitrate allocation strategy that needs to be adopted. Section 4 presentssome simulation results showing improvements in transmitted powerlevels obtained with our optimum design. Section 5 concludes.

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��

��� �

� �Noise

+Interference

+

...

...

� �������

� �������

...

InverseTransform� � S/P

RemoveRedundancy

Transform� ���

�� P/S � Pad

zeros� Channel

�Equalizer...

Fig. 1. DMT communication system.

2. MULTIUSER DMT SYSTEM MODEL

2.1. Preliminaries

Define in fig. 1, � and �� as the nonsingular ��� � trans-mitter and receiver transform matrices respectively. We assume abiorthogonal system, i.e.

� � ������ (2.1)

Thus, the input block transformation is a general nonsingular trans-formation, and the output transformation is its inverse. Denote thecollection of � input and output symbols respectively by � � � � �� � � � � ����������� � ����� � � ��� � , and !� � � � � � !� ��� � ����������� !� ���� � ����� � .With " � ��� , the noise and interference effect at the output of theequalizer, denote # � � � � � " �%$ � ���&" �%$ �(' � �����������)" �%$ �*'+$-,� ��� � , as the $ -fold blocked version of " � � � , with $ � �.' � .Then the system in fig. 1 has the equivalent description of fig. 2.Here / ���� is the $ -fold blocked version of the channel-equalizercombination ������ �10 �2'30 � � �� ' ����� '40�5� � 5 . One can showthat / ���� �76 /*8 /:9 ����<; (2.2)

where /:8 is an $7� � constant matrix, and /:9 ���� is an $7� �matrix.

The addition of zero padding redundancy to an � -block vec-tor is equivalent to premultiplication by=�>@? �BADC �E 5GF �IH � (2.3)

Denote by � , a suitable �J�K$ matrix, representing the linearredundancy removal operation. Then the input-output relation ofthe system in fig. 2 is given by!� � � � � � � / ����� = >@? � � � � �' � � # � ��� (2.4)

� � � /:8 ��� � � �' �� � # � � � � (2.5)

We impose the perfect reconstruction (PR) condition, i.e., inthe absence of noise/interference, !� � � � �L� � � � for all � . In otherwords, �� � /*8� � � C � (2.6)

and the DMT system has no ISI. To obtain a more useful charac-terization of PR, consider the singular value decomposition of / 8defined in (2.2):/ 8 �NMPOQASR OE HUT:VO �NM � R O T:VO (2.7)

W XZY � []\ ^�_ ` \ ^G_ a \b^G_ c[]\ ^G_edfg\ ^G_�h h hX�i�j Y� � k k klGmon � p \ q�_ � +

Fig. 2. Block representation of DMT communications system.

where M O and T O are respectively $r�s$ and �t� � unitary matri-ces whose columns are the eigenvectors of / 8 / 8 V and / 8 V / 8 .R O is the �u� � real, positive definite diagonal matrix with diag-onal elements that are the singular values of /v8 . Then, because of(2.6), given � , the class of all � � enforcing PR is completelycharacterized by (2.1) and

� � T O R ��O 6 C � w ; M VO � (2.8)

where w is any arbitrary �D� � matrix. In the sequel it will beuseful to partition MPO as MPO � � M � M � � , where M � is $t� � andM � is $x� � .

Note, as T O , MyO and R O are supplied by the channel, the onlyquantities that need to be found to determine the transceiver com-pletely are � and w .

2.2. Problem formulation

As mentioned earlier the � subchannels are distributed amongthe � users with the � -th user allocated ��� subchannels. Thus con-sider disjoint subsets

= ��z1{ E � ����� � �|, �~} with � = ��� � � � , and= �:� =�� ����� ������ . Subchannel assignment ro the � -th userconstitutes determining

= � . We assume that the � -th subchannel ofthe � -th user is assigned � ��� � bits per symbol. To meet the bit ratespecification for the � -th user one requires that

�$��������� � ��� � � � � � (2.9)

Let the input power in the � -th subchannel of the � -th user be�������� � . Assume that �������� � is the noise power in this subchannel.Under high SNR most modulation schemes, [7], require that toachieve a given SER the required SNR is proportional to �g� ��� � .More precisely, � ������ � ��� �g� � ��� � � ������ � � (2.10)

where the constant � � depends on the desired SER for the � -thuser, �� . For example, for QAM, � � � �� � � ��� ��� �  ��� � . Underthis framework, the transmitted power for the biorthogonal DMTsystem is given by¡P¢ � £� ��¤ � �������� � ������ � � V� ��� ��� (2.11)

� £� ��¤ � �������� � � � � ��� � � �� ��� � � V� � � ��� � (2.12)

The minimization of the transmission power involves optimalselection of � ��� � (bit loading), selection of

= � (subchannel assign-ment), � (transformation selection) and w (redundancy removalselection).

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3. OPTIMUM SELECTIONS

With � � , ��� and ��� denoting the autocorrelation matrices of thenoise vectors � � � ����� � � � and # � ��� respectively,

� � � � � � V� and � � � � � � V� � (3.13)

Note that �y�� ��� � in (2.12) are the diagonal elements of � � . Thus(2.12) can be rewritten as¡P¢ � �� � � £� ��¤ � ����~��� � ��� � ��� � � � V� ���� � ��� � ����� V� � ��� � (3.14)

First consider the problem of determining optimum �� , orequivalently chossing � . Observe that the problem of choosing � , under (2.1), minimizing (3.14) has the following form:

Problem 3.1 With � � � positive definite Hermitian ��� and� � � nonsingular � , � � E, determine � to minimize

� � � � ������ ¤ � � � ����� V� ��� � � V� ���� ��� (3.15)

Now note that for any diagonal nonsingular matrix � ,� � � ���� ��� �� � . This means that in finding a minimizing � to (3.15),

one can restrict the search to an � for which the diagonal ele-ments of � V� ���� are ����� . The following Lemma considers theequivalent constrained optimization of Problem 3.1.

Lemma 3.1 With �|� � positive definite Hermitian matrix � � ,� � � nonsingular � , consider the minimization of����� ¤ � � ����� V� ��� (3.16)

such that for all ��� { E � ����� � �., �~} and some � E,� � V� ���� � � � ���� � (3.17)

Then the minimizing � obeys for some real, positive definite, di-agonal � , � ������ V� � � �� V� � ��� � (3.18)

In fact one minimizing � for (3.15) obeys

� � � � V� � � � V� � � C �The following result shows a particular � minimizing (3.15).

Lemma 3.2 Let the SVD of positive definite Hermitian � � be

� � �NM R � M V (3.19)

with R real, diagonal and M unitary. Then for some unitary T ,(3.15) is minimized by

� � T R ���� � M V (3.20)

and (3.15) becomes

� � � � ������� ¤ � 6 T R T V ; �� (3.21)

Before proceeding, we need a few results from the theory ofmajorization, [5].

Definition 3.1 Consider two sequences� � { � �}�� ¤ � and �{� ! } � ¤ � with

� #" � �$ � and %#"& !�$ � . Then we say that majorizes

�, denoted as

�(' , if ) � ¤ � � +* ) � ¤ � holdsfor � * � * � , with equality at � � � . We say that weaklysupermajorizes

�, denoted

�,'�- , if ) � ¤ � � �".) � ¤ � !&� � *� * � .

Fact 1 If / is an �<� � Hermitian matrix with diagonal elements0 �N{ 0 �} � ¤ � and eigenvalues 1 �N{213�} � ¤ � , then0 ' 1 .

Definition 3.2 A real valued function 4 ���� �54 ��� � � ����� � � � � de-fined on a set 6Jz7� � is said to be Schur concave on 6 if�8' on 6:9;4 � � �+"<4 �= � . 4 is strictly Schur concave on6 if strict inequality 4 � � ���>4 �= � holds when

�is not a permuta-

tion of . Further if�#' - then also 4 � � ���?4 �= � .

We will now state a theorem that results in a test for strictSchur concavity. We denote 4�@ �BA ���� �DC2E @GF AC F

� .

Lemma 3.3 Let 4 ���� be a scalar real valued function defined andcontinuous on H ��{ ��� � � ����� � � � �JI � � " ����� " � � } , and twicedifferentiable on the interior of H . Then 4 ���� is Schur concave onH if 4K@ �BA ����� is increasing in � .

We now return to the minimization of (3.14) with ��� havingthe form (3.19). Because of Lemma 3.2, with � � T R ����� � M V ,for any choice of

= � and � ��� � , and subject to T being unitary¡ ¢ " LNMGOPQPKR ¤TS £� ��¤ � � � ������ � � � ��� � � 6 T R T:V ; ��� � �" LNMGOPQP R ¤TS £� ��¤ � � ���!UWV � � � �YX���~��� � 6 T R T:V ; ��� � � � � �

Denote Z ���L� ��� UWV � � � � and [ � � 6 T R T V ; ��� . Then¡P¢ " ¡]\¢ � LNMGOPKPQR ¤^S £� ��¤ � Z � X�����`_� [ �� (3.22)

with= \� defining the optimal arrangement of the sequence of [ � .

The following Lemma characterizes such optimum arrangements.

Lemma 3.4 Consider for integers a��cbd"r� ,

e � �=]f��X��¤ � [ � � � � f ' �gZih

���Xj ¤ � � j � � � h

with s�cZs�k[l �)� � � E. Suppose for some �)�e�[ �4� � and m

em [l

�nme

m � � � (3.23)

Then o � �=qp f����¤ � � �%r¤ [ � � � � � ��� f ' �gZsp h

���j ¤ � � j r¤ � � j � [ � ��� h]t e .

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Hence if there exist pairs [ �)� � such that (3.23) holds, violatingthe Schur concavity condition in Lemma 3.3, the value of

ecan be

reduced by interchanging [ �)� � . From Lemma 3.4, an optimum

arrangement for (3.22) requires that for all [ �J� [ j , C �_�

C ��t C �

_�C ��� .

Under this condition,¡ \¢

is Schur concave. Thus from Fact 1,as {P6 T R T V ; � } ���� ¤ � ' {21 � � ����� � 1 ����� } , the choice T � Cminimizes (3.22). Thus the optimizing � � R ����� � M V , underoptimum bit allocation.

The optimum bit loading is obtained by using the fact that thearithmetic-mean of the � � numbers {�� � ��� � � �� ��� � � V� � � ��� } ���~� �in (2.12) is greater than their geometric mean, with equality iff

� ��� � � $ � �� � , ��� � A �������� � � V� ��� ���

� p ������� � �� ��� � � V� � � ��� � ��� � � H �Thus under the above optimum bit allocation, and because of (3.13)and Fact 1, the optimum � (from (2.1)) is a matrix of eigenvec-tors of ��� , and

� V� ��� ��� � ��� ��� � V� � ��� are the eigenvalues of� � . Thus even under the relaxed conditions on � , the best poweris achieved with orthogonal � . Further, the optimizing w must besuch that the set of resulting eigenvalues of ��� weakly superma-jorize all possible sets of attainable eigenvalues. The optimizing wcan then be shown to be given by, [6],w � , M V� � � M ��� M V� � � M �� ��� � (3.24)

4. SIMULATION RESULTS

In this section, we compare the transmitting power of the DFTbased DMT under no bit allocation and optimum bit allocationwith an optimum unitary transceiver. We assume the equalizedchannel to be ������ � �y' E � � � ��

, and a noise source " � � � whosepower spectral density is shown in fig. 3. We assume the DMT sys-tem supports two user services. The �=�)�e�� on the x-axis of the plotindicates that user 1 and 2 were respectively allocated �)�e� num-ber of channels. The plot shows that there is a 10 dB saving intransmit power with our design over the DFT based DMT underoptimum bit allocation, and a 14 dB improvement over the con-ventional DMT with no optimum bit allocation.

5. CONCLUSIONS

In this paper, an optimum bit allocation strategy and design ofa general biorthogonal DMT multicarrier transceiver system em-ploying zero padding redundancy were presented, for minimizingthe transmit power when different users with varied QoS require-ments are supported and are assigned potentially different numberof subchannels. We showed that no gains in transmit power can beobtained by considering biorthogonal transceivers - the optimumpower is achieved through an orthogonal transceiver. Simulationsdemonstrated potential improvements in performance over DFTbased DMT systems.

6. REFERENCES

[1] L.M.C. Hoo, J. Tellado and J.M. Cioffi, “Discrete dual QoSloading algorithms for multicarrier systems ”, IEEE Interna-tional Conference on Communications, pp 796-800, 1999.

0 20 40 60 80 100 120 140 160 180 2000

1

2

3

4

5

6

7x 10

−14

Power spectral density of noise

−75

−70

−65

−60

−55

−50

−45Transmitted power under different schemes

__ DFT based DMT, no optimum bit allocation

−− DFT based DMT, optimum bit allocation

−. General DMT, optimum bit allocation

Tra

nsm

itte

d P

ow

er

(dB

)

#channels, #users = 2

(1,1) (2,3) (3,5) (4,7) (5,9) (6,11) (7,13) 2 5 8 11 14 17 20

Fig. 3. Comparison of transmit power levels.

[2] D. Kivanc and H. Liu, “Subcarrier allocation and power con-trol for OFDMA”, IEEE Journal on Selected Areas in Com-munications, To appear.

[3] B. S. Krongold, K. Ramchandran and D. L. Jones, “Compu-tationally Efficient Optimal Power Allocation Algorithms forMulticarrier Communication Systems”, IEEE Transactionson Communications, pp 23-27, Jan 2000.

[4] Y.-P. Lin and S.-M. Phoong, “Optimal ISI-free DMTtransceivers for distorted channels with colored noise”, IEEETransactions on Signal Processing, pp 2702 -2712, Nov2001.

[5] A. W. Marshall and I. Olkin, Inequalities: Theory of Ma-jorization and its applications, Academic Press, 1979.

[6] A. Pandharipande and S. Dasgupta, “Optimal transceivers forDMT based multiuser communication”, IEEE InternationalConference on Acoustics, Speech, and Signal Processing, pp2369 -2372, 2001. Also submitted to IEEE Transactions onCommunications.

[7] J.G. Proakis, Digital Communications, McGraw-Hill, ThirdEdition, 1995.

[8] A. Scaglione, S. Barbarossa and G.B. Giannakis, “Filterbanktransceivers optimizing information rate in block transmis-sions over dispersive channels”, IEEE Transactions on In-formation Theory, pp 1019-1032, Apr 1999.

[9] T. Starr, J.M. Cioffi and P. Silverman, Understanding DigitalSubscriber Line Technology, Prentice Hall, 1998.

[10] P.P. Vaidyanathan, Y.-P. Lin, S. Akkarakaran and S.-M.Phoong, “Discrete multitone modulation with principal com-ponent filter banks”, IEEE Transactions on Circuits and Sys-tems I: Fundamental Theory and Applications, pp 1397 -1412, Oct 2002.

[11] R. Van Nee and R. Prasad, OFDM for Wireless MultimediaCommunications, Artech House, 2000.

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AN IMPROVED FEEDBACK SCHEME FOR DUAL CHANNEL IDENTIFICATION INWIRELESS COMMUNICATION SYSTEMS

Honghui Xu�, Soura Dasgupta

�, and Zhi Ding

�Department of Electrical & Computer Engineering

�Department of Electrical & Computer Engineering

The University of Iowa University of CaliforniaIowa City, IA-52242, USA. Davis, CA 95616, USA.

hoxu, [email protected] [email protected]

ABSTRACT

In an earlier paper we had presented a novel dual channel iden-tification approach for mobile wireless communication systems.Unlike traditional channel estimation methods that rely on train-ing symbols, this approach used a bent-pipe feedback mechanismrequiring the mobile station (MS) to send portions of its receivedsignal back to the Base Station (BS) for wireless channel identifi-cation. Using a filter-bank decomposition concept, we introducedan effective algorithm for identifying both the forward and the re-verse channels based only on this feedback information. This newmethod permits transfer of computational burden from the MS tothe resource rich BS and leads to significant savings in bandwidthconsuming training signals. This paper proposes a more informa-tive feedback method leading to significant performance improve-ment over our earlier scheme.

1. INTRODUCTION

Two important tasks in mobile wireless communications systemsare channel estimation, and compensation aided by frequent trans-mission of training signals. In most future cellular systems theforward link, carrying data from Base Stations (BS) to a Mobile(MS), will support higher data rates than the reverse link. Con-sequently, the estimation and compensation of the Forward LinkChannel (FLC) requires more resources and longer training se-quences than that of the Reverse Link Channel (RLC). Equally,the current practice is to assign the compensation and estimationof the FLC entirely to the MS, which generally has less computa-tional reserves than the BS.

To permit the resource rich BS to share in the compensationof the FLC, and to reduce the bandwidth consuming training ofthe FLC, in [2] we proposed a new approach to the estimation andcompensation of the FLC in mobile wireless communication sys-tems using a novel bent pipe feedback mechanism. In principle,this feedback mechanism enables the BS to estimate and compen-sate both the FLC the RLC, without any training signals on eitherlink or resort to blind estimation techniques. While practical reali-ties temper these expectations, as we demonstrated in [2], this ideahas significant advantages.

Specifically, the approach of [2] requires that the MS feed backto the BS a portion of the received signal, over the time slot con-ventionally reserved for RLC training. Clearly, this permits the BS

Supported by NSF grants ECS-9970105, CCR-9973133 and 999620.

to estimate the Roundtrip Channel (RTC). However, the key nov-elty of our approach lies in the following discovery: By feedingback only a portion, rather than the entire received signal, oneempowers the BS to identify both the FLC and the RLC from theroundtrip feedback signal alone. This novel channel feedback doesnot require high speed reverse links and naturally accommodatesasymmetric data link structures, and structures where the RLC andFLC have different carrier frequencies. Furthermore, no additionaltraining signals are necessary for estimating the RLC at the BS,though some training for synchronization will still be needed.

As the BS will miss changes in the FLC that occur withinfeedback latency, the MS must estimate and compensate the resid-ual ISI in the channel dynamics the BS cannot compensate. Overreasonable distances and mobile speeds these changes are modestenough to make the partially precompensated FLC dynamics rel-atively mild. Thus, a 5km roundtrip causes a feedback delay of16.67 ��� , a time span over which the FLC undergoes little change.This is underscored by the fact that in GSM each data frame has aduration of 557 ��� , and training occurs only once per data frame.Thus the channel variation within the resolution of this delay oc-curs mainly because of Doppler effect. Yet a vehicle traveling at100km/hr suffers a maximum Doppler shift of 55 Hz in the cel-lular band; a shift not large enough to cause drastic changes inthe FLC characteristics over latencies of tens of microseconds.Thus the residual ISI that must be equalized at the receiver willbe significantly milder leading to the need for much shorter train-ing sequences on the FLC. Given that no training for estimation isneeded on the RLC, and that feedback data occupies the RLC train-ing slot used in conventional communication, this implies substan-tial savings in the bandwidth devoted to the overall training, and asiginficant transfer of the FLC compensation/estimation burden tothe BS. Simulations presented in [2] support this contention. Thisscheme also permits the use of such adaptive coding at the BS ashas been advocated by several authors [3]- [6].

The scheme of [2] is preliminary in nature. One of its disad-vantages is that it fails to make as efficient a use of the feedbackslot as is desirable. In particular a large fraction of the feedbackslot carries zero samples that do not contain useful informationabout the FLC. The key contribution of this paper is to formulate amore informative feedback scheme that carries more informationabout the FLC leading to improved FLC estimation.

2. THE FEEDBACK SCHEME

For the most part in this paper we assume that the ratio of the datarates supported on the FLC and RLC is ��� , with ��� . Later

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we will comment on how to accomodate the case of � � .In fig. 1 ������� and ����� are the discrete time baseband modelsof the FLC and RLC respectively, � ������� are the noise sequenceat their output, ������� and ������� are respectively the data sequencetransmitted and received by the BS. The samples ��������� , receivedat the MS are rate converted by the � -branch rate convertor, [1]that generates samples for every � samples at its input, i.e.effects a rate conversion by a factor of � � . The sequence ���������is retransmitted over the RLC over the slots usually reserved forRLC training: ������� models the interference caused by the normalRLC data because of imperfect synchronization.

In this arrangement ��� � � . Effectively, over samplelengths of , �!������� contains � out of every � samples of � � ����� ,and has in addition #"#� zeros. The scheme in [2] uses only thetop branch of this arrangement, i.e. has �%$'& . Consequently in[2] out of every -symbol feedback slot only one sample containsthe data received at the MS, with the remaining (")& symbolsbeing zero samples. Thus in [2] the available feedback slot is underutilized as far as information exchange is considered. This causesimportant information to be unnecessarily discarded, reducing theability to track time variations, resulting in larger residual ISI in theFLC compensatedon the basis of the estimate at the BS. As we willdemonstrate in Section 4, the more sophisticated rate convertorwith �*$� , leads to improved performance.

Consider the � and fold type I and II polyphase decomposi-tions of ������� and ������ respectively, i.e. �#�����+$),.-0/ ��214365 �7��� - �8� / �and ������9$:,<; / ��2143>= � ��� ; �8� /�? ; / � / �8@ . Then, [1], absent noiseand interference, Fig. 1 can be transformed into Fig. 2, whereA ����� and B����� are respectively, left and right pseudocirculant ma-trices given by

A �����+$CDDE = 3F���� = � ����� GHGHG =JI / � ����= ������ = � ����� GHGHG = I �����

......

......= ; / ������K� / � = 3 �����LGHGHGM� / � = I / � �����

N!OOP (1)

and B����� =CDDE 5 3������ 5 � ����� GQGRG 5 -S/ � ������ / � 5 -0/ � ����� 5 3������ GQGRG 5 -S/ �T�����......

......� / � 5 -S/ IVU � �����K� / � 5 -S/ IVU �WGQGRG 5 -S/ I

N OOP6X (2)

Define 5 �����Y$)Z 5 3F�����7[HG\GHGH[ 5 -0/ � ������] (3)= �����+$)Z = 3������7[HG\GHGQ[ = ; / � ������]2^ X (4)

Define in Fig.2 _���`��a$bZ c � �����7[RGQGHGH[�c ; ������] ^ and d>��`��e$Z f � �����7[RGQGHG\[gf - �����] ^ where fh�7����� and c�������� are the � -transformof �i�R����� and �j�R����� . Then we have the following relation_���`��k$ A �����8B�����8d>��`�� X (5)

Since f � ����� and c � ����� are known to BS, the round trip chan-nelA �����8B����� can be estimated. The question is, under what con-

ditions can one extract ������� and ������ fromA �����8B����� . Clearly

the best one can hope for is to estimateA ����� and B����� to within a

scaling constant and common delays among the 5 ������� and = �7����� .In the case of [2], such an extraction is possible if either (not nec-essarily both) of the following conditions apply.

������� l �

l �������� � � ����� m

nnn

...

��������j�� � � / �� / �� / �� / �� /

�� / �� � ������i�����������

...mm

o oo...

mmm

m m pmpnnn

ppp

pplI

Fig. 1. System model of improved scheme: Rate changer with �branches. q

rrr

sss n

nmm q

� t �t �qvut

w+x �w+x�

w+x �wyx �wyx�

w x �...

...

q

mn

mm p

pppp

pp p mmt{zp p

|~} wy��(} wy�...

� �� ����

Fig. 2. Polyphase Representation.

Assumption 1 The greatest common divisor (gcd) of the set ofpolynomials = ������� is a pure delay � /�� ( � integer). Further theirmaximum order ��� is known.

Assumption 2 The gcd of set of the set of polynomials 5 �R���� isa pure delay � /�� ( � integer). Further their maximum order ��� isknown.

To see why, observe that in the setting of [2], i.e. �%$<& , (5)is replaced by _���`���$ = ���� 5 �����8d>��`�� . Thus the rank-1 matrix= ����� 5 ����� can be estimated. Observe the � -th row of = ����� 5 ���� issimply, =�� �����7Z 5 3������7[HG\GHGQ[ 5 -S/ � ������] . Under Assumption 2, thegcd of the elements of this row provides to within a delay and scal-ing, =�� ����� and hence also 5 �R����� and ������� . Similar unraveling ispossible should Assumption 1 hold.

Observe in the setting of this paper the rank-1 matrix = ���� 5 �����is not directly available. Yet in the next two sections, we show thatunder either Assumption 1 or 2, ������ and ������ can be obtainedto within a scaling and delay from the the roundtrip dynamics cap-tured by

A �����8B����� .3. PROOF OF IDENTIFIABILITY

In this section we show that 5 ����� is identifiable to within a scalingconstant, from

A �����8B����� , when � � '�<� , and assumption2 holds with the common delay � among the 5 � equalling zero.In section 3.3, we discuss the case where this common delay isnonzero. The knowledge of 5 ���� provides ������ . A similar resultcan be formulated when assumption 1 holds or when �� � ���or when :$ � �b� . Thus even the case of �$ � canbe captured. In each case the selection of � ensures that somereceived signal is discarded and �T���������$ � � ���h"#�i� .3.1. Definitions and notations

For an ���S� polynomial matrix �9�����+$�,<��2143 �������8� /4� , where� is the degree of �9����� , define the � �������4���e�8� generalized

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Sylvester matrix of �9���� as

��� ��� �y$ CE �������LGHGQG ����� �. . .. . .

. . .������� GRGQG ����� �NP X (6)

For the ��� � � matrix �'$)Z �0�����7[RGQGHGQ[��0��� "�&T��] , define� � - ����+$� / �� �2143 �S�����8� / � X (7)

where �0����� and� �� - ���9� have dimension �e� � . Note that� � - ���� is a function of � .

Define an � � � polynomial matrix - ����� as

- ����y$���� ?�-S/ �8@��F� � -0/ �� / � � ��� ? -0/ �8@�� (8)

where � -S/ � denotes � ��"�&T�+��� �M"�&T� identity matrix; � � ���denotes the �~� � zero matrix.

3.2. Identifiability

With �e�����+$ A �����8B����� ,��� ��� ^ �+$ ��� ��B ^ � � ��� U � U � � A ^ � (9)

Whenever ������8������ �$�� , ��� � A ^ � and��� ��B�� are full rank

for all intgers � ��� . Hence� � ��� ^ � and

� � ��B ^ � have iden-tical left nullspaces. Thus the knowledge of �a����� provides theleft nullspace of

��� ��B ^ � . The following theorem shows that theleft null space of

��� ��B ^ � under assumption 2 provides �#����� towithin a scaling.

Theorem 1 Suppose 5 ����� and B ���� are defined in (3) and (2)respectively and assumption 2 is in force, with � $ � and � ����� . Then for any integer � ��� � � �#� "#& , � � ��B ^ � has anontrivial left null space. Suppose � is a matrix whose rows spanthe left nullspace of

��� ��B ^ � . Let! �����+$)Z � � �� - ���9���8^y[HGQGHG\[" I / �- � � �� - ������8^�] X (10)

Consider an � -dimensional nonzero polynomial row vector #5 �����with degree ��� . Then� �T� #5 � � ��� � ! �+$$� iff #5 ����y$$% 5 ����� (11)

where % is a nonzero constant.

Thus indeed the left null space of� � � � ! � , constructed from the

left nullspace of��� ��� ^ � , provides 5 ����� to a scaling.

3.3. A Subspace Algorithm

Assume assumption 2 holds with �$$� . Suppose the signals ������� ,������� , � � ����� and �k������� are zero mean, white and mutually uncor-related. In view of the noise free model of (5), and the knowledgeof � ����� and ������� at the BS, a standard least squares scheme pro-vides an estimate of

��� ��� ^ � and hence its SVD:��� ���9^Y���'&M$ ��(*)+( � � ��, ) �� , � � ��-/.)- .� � (12)

Find � whose rows span the left nullspace of�*� ��� ^ � , where��� �a��� �)�%")& , and construct

! ����� defined in (10). Be-cause of the assumption of lack of correlation between ������� andthe noise and interference, � is provided by ( � . Then solving for� �T� #5 � as the eigenvector corresponding to the smallest eigenvalueof� � � � ! � � � � � ! � . , where �8G � . indicates transpose conjugate,

provides 5 ����� . Since� �10 U � ��B�� has full row rank one finds �����

also to a scaling constant, using� �� A �+$ � �T��� �\� � ��0 U �T��B����32 XIf 5 ������� have a common delay then this manifests in certain columnsof� � ��� ^ � being zero. Then applying the above procedure on the

matrix with these zero columns removed provides ������� and ������to within a scaling and a delay.

4. SIMULATIONS

We present two simulation examples. The first example shows thebasic performance of the scheme in this paper relative to that of[2]. The second example illustrates the reduction of training levelsneeded on the FLC when channel parameters change with time.

4.1. Simulation IIn the simulation, FLC and RLC are generated from two delayedraised-cosine pulse 4h�65R[�7y� , where 7 is the roll-off factor. 4h�65R[�7y�is limited in 8T for FLC and in 6T for RLC. The FLC and RLChave the respective analog models: � X 8 4h�657[3� X 9;: � �<� X = 4h�65v"> � 9 [?� X 9@: � and RLC $$� X : 4h�657[3� X &����T�A� X B 4h�65" > � 8 [�� X &���� . Weuse downsampling factor � $ 8 and upsampling factor '$ 9and ��$ . Noises � � ����� and �k������� are zero mean and havethe same variance. The input signal ������� is i.i.d BPSK. To ob-tain a performance measure of channel estimation, we define thenormalized root-mean-square error (NRMSE) as

NRMSE $ &CEDFC �HGIIJ &�'K -ML� �21 � C #D ? � @ " D'C � � (13)

where �'K is the number of Monte Carlo runs;D

is the actual chan-nel and #D ? � @ is the � estimation. In our simulation �NK $ &��@� .In each run 600 symbols are used. We call the scheme in [2] oldscheme and the scheme in this paper new scheme. Fig. 3 showsNRMSE versus input SNR, where input SNR is defined as

Input SNR $)&��PORQ@S &�� E ��� � ������� � E ��� �� ������� (14)

A zero-forcing preequalizer is constructed for FLC and a postequalizer for the RLC, on the basis of the FLC and RLC estimates.Both are housed at the BS. Equalizer SNR for the FLC, is com-puted using the signal power at the FLC input. For RLC, it usesthe signal power at the equalizer output. BER versus equalizerSNR is displayed in Fig. 4, where the input symbol is BPSK sig-nal. Both figures show the performance improvement due to themore informative feedback of this paper.

4.2. Simulation II: Reduced training for a time varying chan-nelWe use COST-207 Typical Urban(TU) [7] model with 100 echopaths, BPSK data and maximum Doppler frequency 55Hz. Weassume the channels to be quasistatic, i.e., time-invariant in oneframe and time-variant from frame to frame. The receive filters

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0 5 10 15 20 25 30 3510

−2

10−1

100

Input SNR(dB)

NR

MSE

Old,FLCOld,RLCNew,FLCNew,RLC

Fig. 3. NRMSE versus input SNR for both schemes

2 4 6 8 10 12 14 1610

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Equalizer SNR(dB)

BER

Old,FLCOld,RLCNew,FLCNew,RLC

Fig. 4. BER versus equalizer SNR for both schemes.

for FLC and RLC are raised cosine functions with roll-off factors0.2 and 0.1 respectively. The FLC sustains a data rate of 1 Mbps,and the RLC supports 0.667 Mbps. We use downsampling factor� $ 8 and upsampling factor ($ 9 . The schemes with ��$'&and �*$ 9 are called old scheme and new scheme respectively. Inboth cases we compare two settings:

(a) Training aided equalization of FLC at the MS and of theRLC at the BS, with no feedback.

(b) No training on the RLC, but instead sending feedback dataof the same length as the RLC training data in (a). A prec-ompensator, obtained by the new scheme or the old schemeis used on the FLC and is augmented by a post-equalizerestimated at the receiver using reduced training.

Methods in (a) and (b) use the same signal power at the FLCinput. Fig. 5 shows ��� ��� K versus input SNR for methods in (a)and (b) to achieve the same BER. Here ��K is the length of theFLC training sequence used in (a) and ��� the length of trainingused in (b), so that the same FLC BER is obtained in both cases.As is evident in Fig. 5, in order to achieve the same BER as theconventional method, the new scheme needs less training data and

thus saves more bandwidth than the old scheme when input SNR �B dB. By way of further comparison we note that at 18dB SNR,while the training sequence length in [2] is 30% of (a), the lengthfor the new scheme is only 15%, i.e. half that needed by [2].

0 2 4 6 8 10 12 14 16 180

0.2

0.4

0.6

0.8

1

1.2

1.4

Input SNR(dB)

n b/nc

New schemeOld scheme

Fig. 5. ��� ��� K versus SNR at the same BER.

5. CONCLUSION

In this paper, a bent pipe multi-branch feedback scheme is usedto estimate FLC and RLC from RTC. By exploiting the propertiesof the nullspace of pseudocirculant matrices, the identifiability re-sult and unravelling method are derived for this improved scheme.Since this improved scheme uses more information from FLC, weget performance improvement compared to the scheme in [2].

6. REFERENCES

[1] P. P. Vaidyanathan, Multirate Systems and Filter Banks, Pren-tice Hall, 1992.

[2] H. Xu, S. Dasgupta and Z. Ding, “A novel channel iden-tification method for fast wireless communication systemscommunications”, Proc. of ICC, vol.8, pp. 2443 -2448, 2001.Submitted to IEEE Transactions on Communications.

[3] R. W. Heath, Jr. and A. Paulraj, “A Simple Scheme forTransmit Diversity using Partial Channel Feedback,” Proc.of 32nd Asilomar Conf. on Signals, Systems and Computers,pp.1073-8, Pacific Grove, CA, 1998.

[4] L.M.C. Hoo, J. Tellado, J.M. Cioffi, “Discrete dual QoSloading algorithms for multicarrier systems”, IEEE Interna-tional Conference on Communications, pp 796-800, 1999.

[5] J. Zhang, A. Sayeed and B. Van Veen, “Optimal transceiverdesign for selective wireless broadcast with channel sttae es-timation”, in Proceedings of ICASSP, May 2002, Orlando,Fla.

[6] D. L. Goeckel, “Adaptive Coding for Fading Channels usingOutdated Channel Estimates”, IEEE Transactions on Com-munications, Vol. 47, pp. 844-855, June 1999.

[7] P.Hoeher, “A statistical discrete-time model for the WSSUSmultipath channel”, IEEE Trans. on Vehicular Technology,VT-41(4):461-468, April 1992.

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A NEW ALGORITHM FOR OPTIMUM BIT LOADING

Manish Vemulapalli1, Soura Dasgupta1 , and Ashish Pandharipande2

1Department of Electrical & Computer Engineering 2Department of Electrical & Computer EngineeringThe University of Iowa University of Florida

Iowa City, IA-52242, USA. Gainesville, FL 32611-6130, USA.mvemulap, [email protected] [email protected]

ABSTRACT

In this paper we present an efficient bitloading algorithm that ap-plies to both subband coding and multicarrier communication. Thegoal is to effect an optimal distribution of positive integer bit val-ues among various subchannels to achieve a minimum distortionerror variance for subband coding and transmitted power for multi-carrier communications. Existing algorithms in the literature growwith the total number of bits that must be distributed. The noveltyof our algorithm lies in the fact that its complexity is independentof the total number of bits to be allocated.

1. INTRODUCTION

An important problem in both subband coding and multicarriercommunications is bitloading. Specifically, for an N -subchannelsystem in these problems reduce to general problem of finding bk

to

Minimize: P (b1, .., bN ) =

N∑

k=1

φk(bk) (1)

Subject to :N∑

k=1

bk = B , bk ∈ {0 , 1 , ...B}, (2)

where φk is a convex function, and B is a positive integer. Insubband coding

φk(bk) = αk2−2bk (3)

where αk is determined by the signal variance in the k-th subchan-nel, [1] and P (b1, .., bN ) is the average distortion variance, and bk

is the bits assigned to the k-th subchannel. Further αk increaseswith increasing signal variance. In multicarrier systems

φk(bk) = αk2bk (4)

where αk reflect target performance, and channel and interfer-ence conditions experienced in the k-th subchannel, [11], [12] andP (b1, .., bN ) is the total transmitted power. Higher values of αk

reflects more adverse subchannel conditions and/or stringent per-formance goals; bk is the the number of bits assigned to each sym-bol in the cognizant subchannel.

It is recognized that for general convex functions φk(·), theabove constrained minimization grows in complexity with the size

Supported by ARO contract DAAD19-00-1-0534 and NSF grantsECS-9970105, ECS-0225432 and CCR-9973133. Pandharipande was withthe Department of Electrical and Computer Engineering, the University ofIowa, Iowa City, IA-52242, USA when this work was partially completed.Not for general circulation. Patent application to be filed shortly.

of B. Since B can be large, it is important to formulate algorithmsfor which the complexity bound is independent of B. The princi-ple contribution of this paper is to show that if one works with thespecial case of (4), then such an algorithm is indeed feasible. Thealgorithm we provide in this paper though formulated in the con-text of (4), can be trivially modified to accomodate both (3), andindeed a general function

φk(bk) = αkaηbk (5)

To place this work in context we note the presence of severalbit loading algorithms in the literature. These include, [3], [4], [6],[8], [10]. The two most advanced and recent are [10] and [3]. Thecomplexity of [10] grows as O(N log(N)) with the number ofsubchannels, but linearly with B. On the other hand [3] providesa suboptimal solution with complexity O(N). Strictly speakingits complexity does not grow with B, as it restricts the maximumnumber of bits to be assigned to any subchannel to some B∗. In-stead the complexity grows with B∗. The assumption of small B∗

is certainly problematic in subband coding, and even in communi-cations settings when certain subchannels experience deep fades.In such a case efficiency may demand that large number bits beassigned to subchannels with more favorable conditions. A stillanother contributor to the complexity of [3] is the dynamic rangeof αi, which again comes into play in the presence of deep fades.All other algorithms have run times that increase with B.

By contrast, we provide an exact solution to (1, 2), under (4),whose complexity has an upper bound that is determined only by Nand is in fact O(N log N). The role of B is only to induce cyclicfluctuations in the precise number of computations, and neither Bnor the dynamic range of αk, affects the upper bound of the runtime.

The paper is organized as follows. Section 2 recaps a resultfrom [13], that is specialized in this paper to formulate the algo-rithm given in section 3. The complexity and proof of correctnessare provided in Sections 4 and 5, respectively. Section 6 givessimulations comparing the run time of this algorithm with those of[10] and [3].

2. A GENERAL RESULT

We now present a general result from [13] that solves (1), (2) forarbitrary convex φk(·). This result is specialized to the cases of(3) and (4) in subsequent sections. Denote for k = 1, ..., N, x =1, ..., B,

δk(x) = φk(x) − φk(x − 1). (6)

The φk’s being convex, it follows that

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δk(1) < δk(2) < ... < δk(B), ∀k. (7)

Let S denote the set of smallest B elements of

τ = {δk(x) : k = 1, ..., N, x = 1, ..., B}

The following lemma from [13], gives an optimum solution to (1),(2).

Lemma 1 The optimal solution b∗ = [b∗1, ..., b

∗N ]T to problem

(1), (2), is defined as follows

b∗k =

{0 : δk(1) /∈ SB : δk(B) ∈ Sy : δk(y) ∈ S, δk(y + 1) /∈ S

In essence this lemma provides a conceptual framework forsolving (1), (2). Specifically, construct S, and for each k, deter-mine the largest integer argument bk for which δk(bk) is in S. Forgeneral convex functions φk the complexity of all known solutionsgrows with B. In the rest of the paper we show by example of (4)that the result when specialized to (4) leads to an algorithm thatdoes not depend on B. A trivial modification of this algorithm canbe formulated for (3), and indeed (5) and is omitted.

3. PROPOSED LOADING ALGORITHM

In the special case of (4), one finds that,

δk(x) = αk2x−1. (8)

The first step of the algorithm requires ordering the αi, andcan be accomplished in O(N log N) steps. Henceforth assumewithout sacrificing generality that:

α1 ≤ α2 ≤ .. ≤ αN . (9)

Define the sequence:

li = dlog2(αi

α1

)e, i = 1, 2, ..., N (10)

with lN+1 = ∞, where dae is the smallest integer greater thanor equal to a. The significance of the integers li is explained byLemma 2

Lemma 2 With li defined in (10),

α12li−1 < αi ≤ α12

li ,

i.e.δ1(li) < δi(1) ≤ δ1(li + 1).

Proof: From (10) we have li = dlog2(αi

α1)e. The definition of

the ceiling function gives us the following result,

li − 1 < log2(αi

α1

) ≤ li.

Hence the result.

Then the proposed algorithm for solving (1), (2) under (4) isgiven below. It assumes that the ordering implicit in (9), has al-ready occurred, and assigns bi bits to the i-th subchannel.

Proposed algorithmStep-1: Find the smallest k such that

Rk =

k−1∑

i=1

(lk − li) ≥ B (11)

Thenbi = 0 ∀i ∈ {k, k + 1, · · · , N}. (12)

Step-2: Find∆ = B − Rk−1 (13)

r = ∆ mod (k − 1) (14)

q = ∆div(k − 1) (15)

Step-3: Find the r smallest elements of the set

{δ1(lk−1 − l1), δ2(lk−1 − l2), · · · , δk−1(0)}. (16)

In particular, with ljisuch that with lji

∈ {1, 2, · · · , k − 1},

δji(lk−1 − lji

) ≤ δji+1(lk−1 − lji+1

), (17)

callJ = {j1, j2, ..., jr} . (18)

If r = 0, J is empty.Step-4: For all i ∈ {1, 2, · · · , k − 1},

bji=

{lk−1 − li + q + 1 if i ∈ J ,lk−1 − li + q else.

(19)

4. COMPLEXITY

Observe that the complexity inplicit in achieving (9) is O(N log N).Determination of k so that (11) holds requires at most 2N opera-tions, regardless of B. Indeed one has, with

ρ1 = 0

ρn = ρn−1 + ln,

Rn = (n − 1)ln − ρn−1.

The only impact that B has in the complexity of determining k isthat for sufficiently small B, k < N and the number of computa-tions is further reduced to 2(k − 1).

Determining the ranking manifest in (17) is detrmined only byr and k, and is

O(r log(k − 1)) ≤ O((N − 1) log(N − 1)).

Determination of r requires 2 operations, independent of B. Bdoes affect the precise value of r, which however is no greaterthan N − 1.

Thus the overall complexity, is bounded by O(N log(N)),with B playing no role in the determination of this bound. Theonly effect that B has on the overall complexity is to cause fluc-tuations in the precise number of operations, within a range that isindependent of B. To recap, these fluctuations occur when:

• For small B, k < N , and finding k requires only 2(k − 1)operations.

• As B changes r fluctuates between 0 and N − 1, and thenumber of operations required to determine the smallest relements of the set in (16) changes.

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5. PROOF FOR CORRECTNESS

We now show that the algorithm in section 3 does indeed solve (1),(2), under (4). In view of Lemma 1 it suffices to show that the set

S∗ = {δ1(1), · · · , δ1(b1), δ2(1), · · · , δ2(b2), . . . , δk−1(bk−1)},(20)

is such thatS∗ = S,

defined in section 2. This in turn requires the demonstration of thefollowing facts.

(A) |S∗| = |S| = B, where |·| represents the cardinality of itsargument.

(B) For all i, j ∈ {1, 2, · · · , N},

δi(bi+1) ≥ δj(bj).

The first theorem proves (A).

Theorem 1 With bi defined in (11-19), |S∗| = B.

Proof: Since bi = 0 for all i ∈ {k, k + 1, · · · , N}, we need toshow that

k−1∑

i=1

bi = B.

From (11-19) we have that

k−1∑

i=1

bi =∑

i∈J

bi +∑

i∈{{1,···,k−1}−J}

bi

= r(q + 1) + (k − 1 − r)q +

k−1∑

i=1

(lk−1 − li)

= ∆ + Rk−1

= B.

To prove (B) we need an additional Lemma.

Lemma 3 With li, k and q as in (10-15),

q

{≤ lk − lk−1 if r = 0< lk − lk−1 if r 6= 0

Proof: From (11-15)

(k − 1)q + r ≤ Rk − Rk−1

=

k∑

i=1

(lk − li) −

k−1∑

i=1

(lk−1 − li)

= (k − 1)(lk − lk−1).

Hence the result.

We now prove (B) for the case where r = 0.

Theorem 2 Consider (10-19). Suppose r = 0. Then (B) aboveholds.

Proof: For all i ∈ {2, · · · , k − 1}, from Lemma 2, we have:

δi(bi) = αi2lk−1−li+q−1 ≤ α12

lk−1+q−1 = δ1(b1), (21)

as l1 = 0. Thus δ1(b1) is the largest member of S∗ in (20).From Lemma 2, for all i ∈ {1, · · · , k − 1},

δi(bi + 1) = αi2lk−1−li+q > α12

lk−1+q−1 = δ1(b1). (22)

Further, as (12) holds, we have from Lemmas 2 and 3 that for alli ∈ {k, k + 1, · · · , N},

δ1(b1) = α12lk−1+q−1 ≤ α12

lk−1 < αk = δk(1). (23)

In view of (21), (22) and (23), prove the result.

Finally we prove (B) for the case where r 6= 0.

Theorem 3 Consider (10-19). Suppose r 6= 0. Then (B) aboveholds.

Proof: With the indices ji defined in (17), we first show that

δjr ≥ δi(bi) ∀i ∈ {1, · · · , k − 1}. (24)

In view of (17) this is clearly true for i ∈ J . Now consider p ∈{{1, · · · , k − 1} − J}. Because of (19) and Lemma 2,

δp(bp) = αp2lk−1−lp+q−1

≤ α12lk−1+q−1

< αj12lk−1−lj1

+q

= δj1(bj1

)

≤ δjr (bjr ),

where the last inequality follows from (17).For all i ∈ {{1, · · · , k − 1} − J}, (17, 18) demonstrate that

δi(bi + 1) ≥ δjr (bjr ). (25)

Further, from Lemma 2 for all i ∈ J ,

δi(bi + 1) = αi2lk−1−li+q+1 > α12

lk−1+q ≥ δjr (bjr ).

Then the result is proved by observing from Lemma 3 that

δjr (bjr ) = αjr2lk−1−ljr+q

≤ αjr2lk−1−ljr−1

≤ α12lk−1

< αk = δk(1).

6. SIMULATIONS

A comparison of the performance of the algorithms of [10] and [3]and the proposed algorithm with respect to the number of compu-tations required is shown in the figures 1 and 2, for the cases whereN = 32 and N = 64, respectively. In implementing [3], which isa suboptimal algorithm, the maximum number of bits, B∗ that anychannel can be assigned is kept at B.

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0 100 200 300 400 500 600 700 80010

2

103

104

Number of Channels = 32

Number of Bits to be allocated B

Num

ber o

f Com

puta

tions

Req

uire

d

: Proposed Algorithm: Campellos Algorithm: Krongold, Ramachandran Algorithm

Fig. 1. Runtime comparisions of the three algorithms for N=32

0 100 200 300 400 500 600 700 80010

2

103

104

Number of Channels = 64

Number of Bits to be allocated B

Num

ber o

f Com

puta

tions

Req

uire

d

: Proposed Algorithm: Campellos Algorithm: Krongold, Ramachandran Algorithm

Fig. 2. Runtime comparisions of the three algorithms for N=64

Number of computations needed for each algorithm to con-verge to the optimal solution was calculated by assuming that addi-tion, subtraction, div, mod, multiplication or division of two num-bers would need one computation as would the logical compar-isons between two decimal numbers. The results show that thealgorithm described in [3] is linear with respect to B while the al-gorithm in [10] needs large number of computations to convergeas B grows. The number of computations needed for the proposedalgorithm is independent of the change in B the minor variationsseen are attributed to the facts that for small B, k in (11) is small,reducing the number of computations slightly, and cyclic fluctua-tions induced by the variation in r (see (14)) between 0 and N −1.the sorting algorithm whose convergence depends on the input vec-tor) and the difference in the runtimes becomes very significant forlarge B. Further the proposed algorithm out performs that of [3],even when B is small, and even in [3] B = B∗ will be chosen.This is largely because of the fact that the run time in [3] growswith the dynamic range of αk.

7. CONCLUSIONS

We presented an optimum bit loading algorithm with a run timeof O(N log N) which is more efficient than the ones existing inthe literature, in that its complexity does not depend on the totalnumber of bits to be allocated. The improvement in performanceis very significant if B is large when compared to N .

8. REFERENCES

[1] S. Akkarakaran and P.P. Vaidyanathan, “Discrete multi-tone communication with principal component filter banks”,IEEE International Conference on Communications, pp 901-905, 2001.

[2] A.N. Barreto and S. Furrer, “Adaptive bit loading for wirelessOFDM systems”, IEEE International Symposium on Per-sonal, Indoor and Mobile Radio Communications, pp 88-92,2001.

[3] J. Campello, “Practical bit loading for DMT”, IEEE Interna-tional Conference on Communications, pp 801-805, 1999.

[4] P.S. Chow, J.M. Cioffi and J.A.C. Bingham, “A practical dis-crete multitone transceiver loading algorithm for data trans-mission over spectrally shaped channels”, IEEE Transac-tions on Communications, pp 773-775, 1995.

[5] T.H. Cormen, C.E. Leiserson, R.L. Rivest and C. Stein, Intro-duction to Algorithms, MIT Press and McGraw-Hill, SecondEdition, 2001.

[6] R.F.H. Fischer and J.B. Huber, “A new loading algorithm fordiscrete multitone transmission”, IEEE Global Telecommu-nications Conference,pp 724-728, 1996.

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[8] D. Hughes-Hartogs, “Ensemble Modem Structure for Im-perfect Transmission Media”, US Patent No.4679227(July1987), 4731816 (March 1988), 4833706 (May1989).

[9] N. Katoh, T. Ibaraki and H. Mine, “A polynomial time algo-rithm for the resource allocation problem with a convex ob-jective function”, Journal of Operational Research Society,pp 449-455, Vol 30, 1979.

[10] B. S. Krongold, K. Ramchandran and D. L. Jones, “An ef-ficient algorithm for optimal margin maximization in multi-carrier communication systems”, IEEE Global Telecommu-nications Conference, pp 899-903, 1999.

[11] Y.-P. Lin and S.-M. Phoong, “Perfect discrete multitone mod-ulation with optimal transceivers”, IEEE Transactions onSignal Processing, pp 1702-1711, June 2000.

[12] A. Pandharipande and S. Dasgupta, “Optimal transceivers forDMT based multiuser communication”, IEEE InternationalConference on Acoustics, Speech, and Signal Processing, pp2369 -2372, 2001. Also IEEE Transactions on Communica-tions, accepted for publication.

[13] Toshihide Ibaraki and Naoki Katoh, Resource AllocationProblems: algorithmic approaches, MIT press, Cambridge,Massachusetts, 1988.