EQUALIZATION FOR DIGI. .(U) MISSION RESEARCH CORP SANTA

UNCLSSIIEDBARBARA CA B ESAWYJER 01 NOV 83 MRC-R-8@i DNA-63i9T

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MICROCOPY RESOLUTION TEST CHARTNATIONAL BUREAU OF STANDARDS-1963-A

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AD-A157 809DNA 6319T

A BRIEF INVESTIGATION OF ADAPTIVEDECISION FEEDBACK EQUALIZATION FORDIGITAL HF LINKS EMPLOYING PSKKMODULATION

B.E. SawyerMission Research CorporationP.O. Drawer 719Santa Barbara, CA 93102-0719

1 November 1983

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A BRIEF INVESTIGATION OF ADAPTIVE DECISION T0c RotFEEDBACK EQUALIZATION FOR DIGITAL HF LINKS TcnclRpr

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19 K EY WOR S (C'!, -i, e -n rev r- , -de if necessary and identify by block number)

PSK Receiver MitigationAdaptive EqualizationDecision Feedback EqualizationHF Communications

2) ABSTPACT '( wirn , re.t . r ir if nruc ary and identi f by block number)

Adaptive decision feedback equalization is one of several mitigation tech-niques that can be employed to improve the performance of narrowband HF modemsoperating under time and frequency selective channel conditions. The decisionfeedback equalizer (DFE) is described and several adaptation algorithms arederived. Computer simulations of BPSK HF modems were used to compare link per-formance (in terms of bit error rate) with and without employment of an adap-tive DFE. Also the performance was quantified versus channel disturbanceparameters for a specific 2400 baud mode.

DD AN73 1473 FOITION OF I NOV 65 IS OBSOLFTE UNCLASSIFIED

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TABLE OF CONTENTS

Section Page

LIST OF ILLUSTRATIONS 3

INTRODUCTION 5

2 ADAPTIVE DECISION FEEDBACK EQUALIZATION 7

2.1 BASEBAND CHANNEL MODEL 8

2.2 THE T/N FRACTIONALLY SPACED DFE 11

2.3 STOCHASTIC GRADIENT ALGORITHM 13

2.4 LEAST SQUARES ADAPTATION AND THEWIENER HOPF EQUATIONS 17

2.5 THE KALMAN ADAPTATION ALGORITHM 20

2.6 THE FAST KALMAN ADAPTATION ALGORITHM 222.6.1 Forward Prediction 232.6.2 Backward Prediction 262.6.3 Extended State DFE Adaptation 282.6.4 Derivation of the Algorithm 28

3 SIMULATION DESCRIPTION 36

3.1 TRANSMITTED SIGNAL GENERATOR 36

3.2 CHANNEL MODELS 383.2.1 Channel Model No. 1: A Simple Delay

Spread Model 403.2.2 Channel Model No. 2: A Simple Delay

and Doppler Spread Model 43

3.3 RECEIVER MODELS 44

3.3.1 Receiver Model No. 1: An IdealizedConventional BPSK Receiver Model 46

3.3.2 Receiver Model No. 2: A DFE-AssistedBPSK Receiver Model 46

3.4 ERROR STATISTICS ACCUMULATION 48

l~

TABLE OF CONTENTS (Continued)

Section Page

4 RESULTS AND CONCLUSIONS 50

4.1 PERFORMANCE OF RECEIVER MODELS NO. I AND NO. 2IN THE CONTEXT OF CHANNEL MODEL NO. 1. 50

4.2 PERFORMANCE OF RECEIVER MODEL NO. 2 IN THECONTEXT OF CHANNEL MODEL NO. 2. 55

4.3 CONCLUSIONS 58

REFERENCES 61

Accession For

NTIS GRA&IDTIC TABUnannounced (JJustification

77 By.

Distribution/

Availability CodesAvail and/or

Dist Special

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LIST OF ILLUSTRATIONS

Figure Page

I. Block diagram of discrete baseband channel model. 10

2. Block diagram of T/N fractionally-spaceddecision feedback equalizer. 12

3. Block diagram of digital PSK HF link simulation. 37

4a. Representation of telemetry format. 39

4b. Signal flow diagram illustrating the generation ofthe channel sample sequence {dkl. 39

5. Delay power spectrum of Channel Model No. 1. 42

6. Delay power spectrum of Channel Model No. 2. 45

7. Block diagram of Receiver Model No. 1. 47

8. Block diagram of Receiver Model No. 2. 49

q. Bit error rate versus EISI/Es for ReceiverModel No. 1 with Es/N o equal to 6 dB. 52

10. Bit error rate versus EISI/Es for both receivermodels with Es/N o equal to 6 dB and 10 dB. 54

11. Bit error rate versus C/No for Receiver Model No. 2for doppler bandwidth of 4 hertz. 57

12. Bit error rate versus C/No for Receiver Model No. 2for various doppler bandwidths. 59

13. Bit error rate versus doppler bandwidth for variousvalues of C/N0 . 60

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SECTION 1

INTRODUCTION

Adaptive equalization is one of several mitigation schemes that

can be designed into HF communication systems to combat the effects of

frequency selective fading. Other possible schemes include maximum like-

lihood sequence estimation and various spread spectrum techniques. The

restriction on the scope of this work to adaptive equalization should not

be interpreted as a disparaging reflection upon the potential benefits of

the other possibilities, but merely represents the limitations caused by

time and resource constraints.

The use of adaptive equalization does not increase the bandwidth

of the transmitted signal. Also, for the narrowband modulation type of

interest here, PSK, long, deep fades can occur which span the modulation

bandwidth. Thus one would envision an adaptive equalizer to be-used in

the context of HF transceivers having the capability to scan the HF band

and to select and maintain a good carrier frequency automatically.

The particular type of equalizer addressed in this report is

called a decision feedback equalizer (DFE), so named because decisions on

the value of previously detected channel symbols are fed back to the

equalizer to he incorporated into its internal state vector.

Section 2 of this report is meant to be a complete introduction

to adaptive decision feedback equalization. By collecting the pertinent

contents of several journal articles into this section the reader is freed

from sorting through several sets of notation. Also all derivations in

5

Section 2 are generalized for fractionally-spaced sampling, coherent

demodulation and time-variant channels, while the journal articles tend to

address special cases. Of the numerous articles on adaptive equalization

to be found in the open literature, three in particular are widely refer-

enced, historically important, and pertinent to this report: Austin's

derivation of the decision feedback equalizer from decision-theoretic

principles (Reference 1); Gersho's treatment of the stochastic gradient

algorithm in the context of linear equalization (Reference 2) which is

easily extended to decision feedback equalization; and Godard's applica-

tion of Kalman filtering principles to equalizer adaptation (Reference

3). In addition to these, References 4, 5, and 6 develop the fast Kalman

adaptation algorithm. The derivation of the fast Kalman algorithm in

Section 2 closely follows the treatment of transversal filter equalizer

adaptation found in Reference 6, but extends the derivation to decision

feedback equalizer adaptation using the mathematical tools provided by

Reference 4.

The results obtained during this investigation were generated

with a Monte Carlo computer simulation of an HF link. The computer simu-

lation is described in Section 3, with primary emphasis given to the

channel and receiver models.

Section 4 discusses the results of a simulation-aided analysis

of HF PSK modems. First, the performance of conventional and

equalizer-assisted modems are compared for a simple time-invariant HF

channel model. Then an equalizer-assisted modem is analyzed for a simple

time-varying frequency-selective HF channel model.

%7'"'6

SECTION 2

ADAPTIVE DECISION FEEDBACK EQUALIZATION

A decision feedback equalizer (DFE) is a digital signal proces-

sor that can enhance the performance of several types of digital modems

operating under frequency selective channel conditions. This discussion

is generalized to include T/N sampling, phase coherent demodulation and

random channel variability. T/N sampling means the receiver samples the

analog channel output every T/N seconds where T is the minimum signaling

interval of the modulated waveform. Phase coherent demodulation means the

demodulator uses phase as well as amplitude information, which implies the

DFE must process complex numerical data. Random channel variability means

the channel transfer function is time-variant and is unknown a priori to

the receiver, which implies the DFE must be able to continuously adapt its

internal parameters to match the channel.

Two adaptation techniques commonly discussed in the technical

literature are the fast Kalman and the stochastic gradient algorithms.

Both of the algorithms have larger implementation requirements than the

OFE itself. The fast Kalman algorithm requires more hardware and/or

digital processing than that required to implement the stochastic gradient

algorithm. However, the fast Kalman algorithm converges faster than the

stochastic gradient algorithm.

We shall first discuss a baseband channel representation that is

appropriate for several digital modulation schemes, and then describe the

T/N fractionally-spaced DFE. The remainder of this report formulates

several adaptation techniques starting with the stochastic gradient

algorithm. Then various least square adaptation algorithms shall be

derived, culminating with the fast Kalman algorithm.

7

2.1 BASEBAND CHANNEL MODEL

It is convenient to model the modulation, channel propagation

and denodulation processes with a time-discrete complex baseband (or

analytic signal) representation. Such a representation of the modulated

waveform is the sequence of complex samples {dkl. The transmitted wave-

form represented by the discrete sequence {dk} is of the form

[Real(dk) cos(wct) + Imag(dk ) sin(wct) ] rect (t/T s - k)

k=O

where wc is the cdrrier frequency, Ts is the sampling interval, and the

rect function is defined as

( 1 -1/2 < X - 1/2rect (X) =

I 0 otherwise .

Several quadrature amplitude modulation (QAM) and M-ary phase

shift keying (PSK) modulation types can be represented with this discrete

hasehand formulation by appropriately selecting the alphabet for d . For

example, limiting dk to the set {1,-1} would correspond to BPSK modu-

lation. Also, T/N spaced sampling is represented by only changing dk

every N samples, in which case T = T/N. We shall use the cor4,ntion that' S

the receiver generates a detected symhol when k is an integer multiple of

N.

The time-discrete analytic signal representation of the channel

at the kth sample time is the set of complex samples {hml, the sampled-

data response of the channel to an isolated unit sample. More specifical-

ly, suppose that dn is one and dk is zero for all k not equal to n. Then

the discrete channel output (ignorinq noise) at the kth sample time is

0

k -n-Mhk The integer I corresponds to the propagation delay and is used

here as a convenience so that we can require hm eqlual to zero for in < 0,

since the channel is causal and hence M must be non-negative. It is

further assumned that the channel has finite leoory, and we introduce L,

called the channel meiiory duration, which is the sallest integer such

that hk is zero for all m > L. For our purposes the values hk in the

rdnge 0 < m < L completely specify the intersymbol interference character-

istics )f the channel and are used to form the vector h

T [h h' h2 ... hLk k k

The underbar denotes a column vector and superscript T denotes

vector transpose. The transpose form is used for typographical simplicity

be)cause it allows the coluimn vectors to be presented in a horizontal

orientation. Next qe define a vector of the mnost recent L+1 channel

syrmbo I S.

k [dk dk-i dk_2...d k L ] (1)

The kth received sample constituting the output of the discrete

-hannel, denoted Uk, is the convolution of the unit sample response with a

sequence of inputs, corrupted by additive noise.

h d + n k (2)k "= dk + k kh dk- m k

n=0

r, is the coilpl-x white gaussian noise comqponent received during

the kth sariplin. interval. 4 block diagram of the tapped delay line

implenentation of the has'-hand channel model is shown in Figure 1.

1) A linear forward predictor of the next N+1 samples to enter

.1k is derived. This forward predictor uses _k, the same Kalman

gain vector of an LS algorithm adapting a DFE.

2) A linear backward predictor of the last N+1 samples to leave

xk is derived. This backward predictor also uses 2k.

3) LS adaptation is applied to an extended state DFE to develop

the mathematical structure between the forward and backward

predictors.

4) It is shown that the forward and backward predictors can

work in tandem to efficiently calculate _gk(recursively.

2.6.1 Forward Prediction

Define a vector &k composed of the N+1 elements entering the

,tate vector at the kth sample time when k/N is an integer.

FT A u .. U dkDN (39)k k-i ... k-N+1 k-D-N3

The linear prediction of ._k+N is FkN xJk where Fk is an (N+1)

x (Kl+1+K2) rectangular matrix of forward prediction coefficients. The

forward prediction error vector is defined as

,f A 1 +N - F x .(40)

A comparison of this equation with Equation (21) shows a

similarity between this forward predictor and the DFE. To calculate the

optimal value of Fk, the scalar error LS cost function of Equation (8) is

generalized for vector errors.

23

,. .-. , ."- .-"--.'-. ,,- -," .' , [.-. " :J ,': ' --,. '.. ,,'' - , -"' .-." . - .-L " 1" i.. '',,N-11- .>.-

The sequence of operations performed by this algorithm after the

kth sample time (when k/N is an integer) is

4 : T k-N -4 ( 34 )

[33,34] k 4-k/(x + - ) (35)

[30,34, Tk = [TkN"kk (36)

35] k

[21] Ek = dk-D - £k-N k (37)

[23] N + k c'k" (38)

The implementation complexity of this algorithm can be dominated

by the first step which requires (Kl+I+K 2)2 multiplications. The follow-

ing algorithm is equivalent to this one, but is computationally superior

because it manipulates smaller matrices.

2.6 THE FAST KALMAN ADAPTATION ALGORITHM

None of the adaptation algorithms discussed thus far take into

consideration any deterministic relationship between .1k and XRkN. From

the definition of jk in Equation (5), all but N+1 of the elements in ? k-N

also reside in XR. The fast Kalman algorithm attains its computational

efficiency by exploiting this fact.

The derivation of the fast Kalman Algorithm will proceed as

fol lows:

22

Since X + 4 A-1 is a scalar and Ak is invertible,

[28] A- I - 1 N k-Nt ± k-j (29)Sk+ A AkN

Define T = A- and notice that Tk like Ak is hermitian sincek kkk

T T T * ** *TTk Ak = (AkTk) = I=(ATk =Tk AK = TK AK

T T *

and postmultiplication by the inverse of Ak implies that Tk equals Tk.

Substituting Tk into Equation (29) yields the following formula.

[29 Tk-N k T Tk-N (30)k -N

Now consider 9.k .

[191 _k =A <1 =Tk _ (31)

303111 TkN -k TkN k

T k-N -k + x T 3

[321 T k-N Xk (33)9-1 k - - .*--

+ k TkN - k

21

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'-" ''- "w -'; J' .'" "; '." > .''- " '"- '.' ," ,'''.r" .- '..',. .W".f - , . -. ,-.- . , • " -

2.5 THE KALMAN ADAPTATION ALGORITHM

Fi rst we shall deri ve a recursive calculation of A k from

which a recursive formula for 9k will easily follow. This algorithm

bypasses the necessity of niatix inversion, and as a result is more effi-

cient than the one discussed immediately above.

It is easy to verify the following equation.

k-N-kA k- A ' k-N--

[x + A-N~ 1 (24)

[24] -1 x 1 -1 X A

A' Ak [ +~ A-N ]

(25)

[25] 1 = [I + A-1 k '~ [I -A~~.A+~ A-j 1 (4 11

(26)

Postmultiply by A-N and premultiply by Ak

[26] 1 =[A k [A-k 1

A 1 x (x + * - A1-i (1k-N n* AkN k-N A ] (7

[13,27] 1 A~ [A-' - -1A-1 A-] (28)kA- AkN ~ A+ <Ak-N x)k*-

29)

The recursive calculation of Sk is now derived.

[10,14] A k - = I A k-N 4-N + 4 dk-D (15)

[13,15] Ak k (A k -kk --k-N + dk-D (16)

[16] Ak Sk = Ak k-N I 4 (dk-D - k-N -)k (17)

Premultiply by A- 1

[17] k : -k- + k (dk-D - 4k-N , ) * (18)

where - (19)

The sequence of operations of this recursive algorithm after the

kth sample time is

[13] Ak = A k-N + - (20)

Ek = dk-D - -- N k (21)

[19] k = A kI k (22)

[18,21] -k = -k-N + Jk Ek (23)

Notice that the calculation of Ak and _k is greatly simplified

and that the calculation of the quantity yk has been bypassed altogether.

However, Ak must be inverted to calculate k and the dimension of Ak is

KI+I+K often a large number. The calculation of 9k can also be done1 2't

recursively, and algorithms that do so are discussed in the next two

sections.

19

[8 n-k/N . kN n-k/N *n 0 n- / nN -N k = nN dnN-D ) (9)n:O n=O

The quantity on the right is a column vector and the quantity

premultiplying Ck on the left is a positive-definite hermitian square

matrix. Some of the forthcoming derivations are tedious. To simplify the

presentations, the numbers of preceding equations justifying a new

expression are listed in brackets to the left, as was done above. Equa-

tion (9) can be restated as

[9] Ak k (10)

where

k/N n k/N[9,10] Ak = X - N - n (11)n=O -nN !-nN

k/N *

[9,10] k = X X -nN dn (12)n=O - N D

The recursive nature of Equations (11) and (12) is apparent and

can he written as

[il] A k X XAkN k+ (13)

[12] 4 4 k-N + k dk-D (14)

A new coefficient vector is calculated after each equalizer

operation. Thus at the kth sample time the quantities AkN, kN and C k-N

are available so that Ak, yk and Sk may be calculated recursively.

18

2.4 LEAST SQUARES ADAPTATION AND THE WIENER-HOPF EQUATIONS

The stochastic gradient algorithm discussed above was based upon

the least mean square (LMS) cost function. Now we turn our attention to

algorithms that minimize the least squares (LS) cost function given by

k/N 2LS cost function = X nl/N dND-~-~n

n=O I nN D - _C XnN

This cost function is the geometrically weighted sum of the squared

magnitude of all equalizer output errors if the equalizer coefficient

vector had always been equal to ck. After the kth equalizer output has

been generated, the least squares adaptation algorithms generate the value

of c that minimizes the LS cost function. As done previously, the

discrete independent temporal index k increments with each received sample

and hence increases by N with each equalizer operation. All quantities

discussed henceforth are defined only when k/N is an integer. Thus, the

previous value of quantities that change only after each equalizer opera-

tion are denoted with the subscript k-N instead of k-i.

For time invariant channels the selection of c k should depend as

'uch on previous data as current data, in which case the constant para-

meter N appedrin] in the cost function should be unity. X is incorporated

into the LS cost function to allow for time varying channels. A real

positive value of A slightly smaller than unity causes received data to he

less influential as it becones older.

The opti mal value of a is found by setting the gradient of the

!S cost function with respect to C k to zero which yields the discrete

Wiener-HIop f equiations.

17

.........

Algorithm No. 1. An adaptive TIN fractionally-spaced DFE (stochasticgradient algorithm.

Step Number Operation Comment

1 k 0 Initialization

xN +o

c_N + 0

uk

+ S k-N + k- +a k-D-N Shift N new channel

samples and previousI decision sample intoUkLN+ 1- state vector

3 - Equalization

~~~k + -k-N + a( k DY

4 .+ (d a-DYk Adaptation

5 k k + N Transition to nextset of N inputsamples

6 Go back to step 2

C

4'J

Each term in this equation can be readily derived.

dk D = 0

I * * * *--N 2 2 -4 - -N .4y - k

Id k-D -k-N -k + dk-D (Sk-N I} = 2 dkD -k

Therefore

l V12 = -2 (dkD - = -2

Then the stochastic gradient adaptation rule becomes:

k kN + A Ck * (7)

To state the operation of a T/N fractionally-spaced DFE using

the stochastic gradient adaptation algorithm we define a square matrix and

a column vector both of dimension K1 +I+K2 and a rectangular matrix having

dimensions (Kl+I+K2 ) x N

N 1 [-sS i__ I:; A

_ K -1 L J

T X,< (K+Z)

th element

a [ 0...o1 0 ... o]

All unlabeled partitions contain zero elements only. I denotes the iden-

tity matrix with dimension specified by the subscript. The adaptive

equalizer operation is given by Algorithm 1.

15

This shall be done using a numerical search technique based upon the

method of steepest descent called the stochastic gradient algorithm. The

stochastic gradient iteration rule is

- '2

k+SkN- 112 2 .~ (6)

jEkj12 denotes the approximation of the gradient of lEki1 with

respect to CJkN where the slight dependence of L_-N on dkD and xk is

ignored. A is a real scalar controlling algorithm performance, and the

factor 1/2 is introduced here for later notational convenience. The use

of the term "gradient" conforms to the nomenclature of common engineering

practice and the technical literature. However, such usage is mathemati-

cally imprecise because the gradient of any real-valued function with

respect to a complex valued vector either does not exist or is identically

'. ' zero. To eliminate confusion the reader must interpret the gradient of a

real-valued function with respect to a complex-valued vector as the sum of

1) the gradient of the real-valued function with respect to the real part

of the complex vector and 2) the product of the square root of -1 and the

gradient of the real-valued function with respect to the imaginary part of

the complex vector.

The gradient of Ik 12 at -N points in the direction of maximumincrease of ICk1 2 . The stochastic gradient algorithm attempts to minimize

Ik1 2 by perturbing 4k-N in the direction approximately opposite that of

this gradient after each equalizer operation. To derive V C k12, first

expand Vk 1.

VI k V = { (dkD -£k-N --k (dk-D - 1k)N -- *

41"" - ~V {dk-D ck N - + dk-D (-£-k-N -k] J

414

not have a priori knowledge of dkDN, in which case the decision symbol

dk-D-N is based upon the equalizer output Yk-N. However, to simplify our

consideration of the adaptation algorithms we shall assume that knowledge

of dkDN is available to the receiver. The complex tap weights of the

feedforward section are chosen to coherently combine all the signal:..

components of the received data arising from the transmission of dkD and

can be viewed as a transversal filter qualitatively matched to the

discrete channel unit sample response. This linear combination of the

received data symbols will also have signal components arising from

transmitted channel symbols other than dkD, half of which were transmit-

ted prior to and half after dkD. Presumably, those transmitted prior to

dkD have already been detected correctly. Hence the associated inter-

symbol interference (ISI) components in the weighted sum of the received

data are deterministically related to the previous decision symbols. The

coefficients of the feedback section, then, are selected to cancel these

components. For the T/N fractionally-spaced DFE, L/N previously detected

symbols are contributing ISI components to the weighted sum of the

feedforward taps, and hence K, should be set equal to L and K2 should be

set approximately equal to L/N.

2.3 STOCHASTIC GRADIENT ALGORITHM

Define the error in the kth equalizer output as Ek = dk-D - Yk,

where the availability of dkD is assumed. One would like to iterate c

to eventually drive F-k to zero, but Ek cannot completely diminish due to

the random noise component in Yk. Failing that, it is desirable and

mathematically convenient to have the adaptation algorithm try to minimize

the least mean square (LMS) cost function.

LMS cost function = 21

13

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2.2 THE T/N FRACTIONALLY SP-CED OFE

According to the definitions given above, a particular transmit-

ted channel symbol dk L influences the currently received channel symbol

uk and previously received symbols uk_ 1 , "k-2 ..... Uk-L. An equalizer

ideally uses all of these received symbols to assist the receiver in

generating the decision symbol dk-L, the receiver's estimate of dk-L. As

mentioned earlier, the values of dk change at most every N samples, and

the receiver generates a decision symbol only when k/N is an integer. To

do this the equalizer actually processes the currently received symbol and

the K1 previously received symbols, where K, is chosen to be greater than

or equal to L. A OFE processes K2 past decision symbols also. After the

kth sample time the decision dkD is made, and the integer 0 must he in

the range L < D < K, for proper equalizer operation. Yk denotes the OFE

output generated after the kth sample and is calculated by

Yk M k (* denotes conjugate transpose) (3)

* A kK I kK1+1 1 0 K2(4where k: [ c ... c c cl ... c k ] (4)

c-L k k . ck k ~k k

and T A d d d ](5)a[uk 'k-l"Uk-Kl -D-N k-D-2N. .k--NK 2

The mechanization of this calculation is shown in Figure 2. k is called

the state vector and akN the coefficient vector. The elements of a -N

are the tap weights shown in the block diagram. For this coherent

haseband imnplementation of the equalizer, these coefficients are complex

scalars.

The block diagram in,' -ates the DFE can he impli e,: nted as two

tapped delay lines. The one on the left is called the feeH,)frward section

and the one on the right the feedhack section. IJsually thc receiver doesii

• ... . . .-.- - - .. : .. .. ; . . . - ,... ... .:-, ,........ .-. ', ,'. ., .. (. 7 .<

4-)

C~j _iz.9-L

CD,

k/N (4forward prediction x ( n-k/ x -FX(1LS cost function n=O -n-n k-n

Each column of Fk contains all the coefficients used to predict

a single element within _k-N- The Wiener-Hopf equations yielding the

optimal Fk are found by setting to zero the gradient of the trace of the

LS cost function with respect to each column of Fk. The result is

k/N / k/N[41] ( xN )Fk = n -nm nN (42)

n=0 n =0

or

f[11,42] A F =V (43)

k k k

where

V, f k/N n-k/N f*[42,43] V k x x nN -nN+N - V k-N + !k+N (44)

Repeating the manipulations of Equations (15) through (19) using

FkN and ._k+N instead of k-N and dkD yields

[18] Fk = FkN + gk(-_k+N - F kN4 (45)

[40,45] Fk = F + (46)

Now we define the minimum value of the LS cost function given byf

Equation (41) as Ek, which is found by substituting Equation (43) into

Equation (41) and simplifying.

24

[11,42, E k R_ V-k Fk (47)43,44]

where

fA k/N n-k/N f *Rk =0 -nN+N -RN+N ' k-N + -k+N -k+N (48)

n=O

A recursive calculation of Ek is now derived.

[44,469 - )** (49)47,46, Ek Rk-N +-4+N -k+N k (IkN +-k4-k+N Fk-N+4kfk) (9

f f f*[40,46, Ek = (RkNVkNkN) + (J_(+FkNxk)(fk+Fk_Nlk)

491f.

Vk-N A -fk k - 4+N 4 Fk (50)

[40,44, - XE-+_ x(F.-F) F N-47,50] k k.N +N -Nk + -N4

Sf*(V- -k+N) 9-k (51)

[46 ,51] E - X= E k _N (-4 *) + * f*

•*f*Ef +N + R -f'k +-k F-- (53)k~ (52)

[52 Ek+ = R-N

[52] f X E= f- (fk+ Fk -V f* f (53)

[43,46] A (F (54)

4k- Vkf "Fk-N (55)

f*

[19,55] k k =AVk Vk Fk-N k (56)

25

* -S...%

Since is a scalar,

[53,56] Ef = X Ek + (1-k k'f (57)

k k

q -One more identity is needed for the forward predictor.

[40,56] f -f*k+N + - Vk - (58)

Define as follows.

f(59)

[58,59] -k -k+N - Vk k (60)

2.6.2 Backward Prediction

Define the vector_.Rk composed of the N+1 elements leaving thestate vector at the kth sample time (when k/N is an integer) as follows.

,k [U-Kl1 Uk-K-2 ... Uk-KI-N dk-D-NK2-N] (61)

%*

The linear backward prediction of a based upon xk is BkN X k

where Bk is an (N+1) x (Kl+I+K 2) dimensional matrix of backward prediction

coefficients. The backward prediction error vector is defined as

S-k k Bk-N k (62)

and the backward prediction LS cost function is

26

) ,./. ' , . ., .. , ** .

k/N . )Nbackward prediction X X -BnBkX (63)

LS cost function n=O

Denote the optimal value of this cost function as Eb and define

the following two matrices.

b A k/N * b *Vk =nn=0 NN X V k + Pk (64)

k/Nk ~ n-k/N *bR k -PN- nN X R kN + 2k A (65)

n=O

The derivation of the backward predictor is identical to that of

the forward predictor. Because the definitions given by Equations (61)

through (65) are consistent with those of the forward predictor, replace-f f f b f

ment of the quantities Fk, -k' Vk , Ek, Rk, -k+N and _ , by Bk, 4k Vk, EfkRbRk' and b', respectively, into Equations (43) through (60) produces the

mathematical framework of the backward predictor. The expressions needed

for subsequence derivations are stated below.

[43] A B = b (66)k k k

[46] Bk = Bk-N + k bk (67)

* h b b* B (68)[47] Ek =Rk k k

biA b* (9[60] V _q (69): - k k (9

27

"A_ _

2.6.3 Extended State DFE Adaptation

From Equations (39) and (61) the elements of _ and _j are never

contained within the same state vector. To establish the interrelation-

ship of Eb and Vb with Ek and V, an extended DFE is considered. The

state vector of this extended OFE is defined below.

-T AS[u U ... u d d ... d ]-k k- k-Ki-N k-D-N k-D-2N k-D-NK2-N (70)

Since this extended DFE is just the original DFE with the

parameter Ki increased by N and the parameter K2 increased by 1, Equations

(8) through (38) apply here directly. Only the following two expressions

related to the extended DFE will be needed however.

k/N X n-k/N- -*71[II] Ak n__ .nk/ nN nN (71)

n=O - -

[19] Ak 9k (72)

where 9k i the extended Kalman gain vector.

2.6.4 Derivation of the Algorithm

To relate Ak and 1k to Ak and X two (Kl+K 2+N+2) dimensional

square matrices are introduced.

A I ,K + I _;

Qf __1+ Qb

L I KJL1

It is easy to verify that Of and Qb have the following proper-

ties.

J~ 2R

Tf Q 1 (73)

b Q-1 (74)

Qf 1ZI (75)

~bQ k=~ (76)

Next the extended Kalman gain vector is expressed in terms of

the forward predictor variables.

[71 Qf Ak T k/N

[71 QA n-k/N Q ~-rN IN QfT (77)

k/N

[75,77] Qf Ak Q = n n- L-NI _ -NN (78)

xnN-N

R f VkN

[1144T k-N (79)

48,78] k f : IN Ak-N9

[72] Qf Akk =Qf (80)

29

T -I[73,75, - (QfAkQf) Qf .( : - (81)

79,80] V f*T 0 " k-N -qk-N=fk ~I- 1(82)

[19,791 Q fk Qf [ _] - -k-N

,qk--N

[43,47, Q A Q [ - - - - -](83)f9 k f 0F_

By Equation (59) the left side of Equation (83) can be postmul-

tiplied by (EkN) f - k-N while the right side is postmultiplied bySE. N )-1 C _k -Vf *NAkN

(f 4&I -V f*

[83,59] Q k Qf L-E _I N _0-- -- - ] (84)

By inspection the right sides of equations (81), (82) and (84)

sum to zero. Setting the sum of the left sides to zero and premultiplyingT

by the inverse of Qf Ak Qf yields

[81,82, Q k= - (85)84]f

30

6 m ," . W:." ,'. " -.- ".-, - , , , - .. . -W '-'• .. . . , . .

A similar derivation is now performed to express the extended

Kalman gain vector in terms of the backward predictor variables.

[71] Q A QT = k N n-k/N -NN Q T (86)n 0 b n=Ob-n n b

k/N nNT k n-k/N r(87)

[86,76] Q A = LbN i __Nn:O LPn N()

AV b[11,65, Qb A T = vk k (88)66,87] kb Lb* R8bk Rbk'

[72] QbAk-9k kQb (89)

[7476, (QbAkQb) Qb k = (90)88,89] Vk -b-

[981 QAQT : -k A (91)[19,88] Qb Ak b Lj

k

[66,68, Q Ak (92)88] b I Rb_ V b*B Ek

k k Bk

31

F-B (Eb)-1 0[69,92] Qb k Qb F - k - k - - (93)

bk by (E-'bJ

(E b 1[90,91, = K k.. (94)Qb q - - -I--

93] (E b)-1bkj

Define a Kj+1+K2 column vector and an N+1 length column

vector Hk as

Qb (95)

Notice that 2 and Ek can be generated by rearranging the

elements of the partitioned column vector shown on the right side of Equa-

tion (85). A comparison of Equations (94) and (95) leads to the following

expression.

[94,95] 9 = 9 + Bk 4 (96)

A recursive expression for Bk using and can now be

derived.

[67,96] Bk = BkN + + Bkk) (97)

32

W.V6"M

[97] Bk( - b ) = B (98)

We assume I - A Pj is invertible.

[98] Bk (BkN + .4b)(I - 1k' (99)

The sequence of operations performed by this algorithm after the

kth sample time (when k/N is an integer) is

[40] fk-N = Fk-2N -A-N

[46] Fk-N = Fk-2N + -k-N -k-N

[59] =-N (1 -k-Lk-N -k-N

ff , *

[57,59] Ek N = Ek-2N + X*-N -fk-N

I - ( f -1f

[85,94] L QT b ..- -Q-- --=~k- Q _Ek-N ) !4 -N

[62] k - k

[99] Bk (Bk N + *)(I - - *)-1

33

[96] gk = gk + 'k k

[21] E: dkD -- k-N -

[23] -= k

The operation of a T/N fractionally-spaced DFE using the fast

Kalman adaptation algorithm is shown in Algorithm 2. Experience has shown

that the direct implementation of algorithms such as this can exhibit

numerical instability. Often adherence to the IEEE floating point stan-

dard results in a stable implementation. Alternately, the algorithm can

be restructured in the form of the so-called square root filter to improve

numerical stability.

34

• "-" "" "-" -"-- ," -,-?'-'. . • '.'.. '. '.; "-.'-+-.-. . , ., . ... ..-.,-.NI... ... wi;.. . ..

Algorithm No. 2. An adpative T/2-spaced DFE (fast Kalman algorithm).

StepNumber Operation Comment

1 k . 0 Initialization

+ 4-N + 2_

Fk-2N B-kN + U

Ek_2N + 61 (6 a small positive realnumber)

* Forward predictor2 f- k-FZk -N error

3 Fk-N + Fk-2N+ 4k-N fk-N Forward predictor

:4-N ( ' k-N--N fk-N -SkN kN is a realscalar

5 Ek + AN -N Ekf is hermitiank -N k -Ek2N+ -NkNk

' K-; : k-N Multiplication by

6 Qb Qf - F - -N- - Qb Qf merely records

qk k-olN(Ek.,) 4-N_ column vector

u k

1 S -4-N + P Uk- +

a dk-DoN Shift N new channel, . samples and previous

decision sample intoLUkN+1i state vector

8 - 4kk+~ ) Backward predictorerror

9 Bk " (Bk + N JI~ - kv]b Backward predictorB+ k-N + 1 coefficients

10 2k - + Bk Pk Kalman gain vector

11 Yk + k-N k Equalization

12 Ek + dk- - Yk Equalizer error

13 c* - + - E k Update equalizer

coefficients

14 k * k + N Transition o nextset of N inputsamples

15 go to Step 2

35

SECTION 3

SIMULATION DESCRIPTION

This section describes the Monte Carlo computer simulation used

to evaluate the adaptive equalizer algorithms discussed in thr orevious

section. The simulation is partitioned into four components as shown in

Figure 3. This simulation structure intentionally resembles the signal

flow diagram of a one-way HF link. The transmitted waveform is first syn-

thesized into a sampled-data representation, then corrupted by the propa-

gation medium, and finally processed within the receiver. The transmitted

and received digital data streams are compared to accumulate a symbol

error rate statistic. Each of the four components shown in Figure 3 are

described in this section.

3.1 TRANSMITTED SIGNAL GENERATOR

This component of the computer simulation generates the sample

sequence {dkj discussed in Section 2.1. The scope of this report is

limited to binary PSK modulation, so all channel samples are binary.

Further the information bearing symbols (also referred to as bits) are

assumed to be random, independent and uncorrelated. A pseudo-random

number generator implemented in software generates a stream of random ones

and zeroes constituting the information syriools.

Recall from Section 2 that N identical and consecutive samples

are (jonpratpe for each channel symbol, and that k is the sample, not

symbol, index. Also recall that. for BPSK modulation the elements of

36

... - ,.' ' • "

LUJ

C)~)

~LL.-4-

LLL

E

0

LL.

(/1 <c

37

The BPSK receiver that does employ adaptive equalization is

Receiver Model No. 2, described in Section 3.3.2. The equalizer

parameters used here are the following:

LTS = 42

LTB = 640

N =2

Ki= 20

K2 = 10

j 1 x 10- 3 after reception of an information symbolA 4 x 10- 3 after reception of a training symbol

The two simulated links were exercised for Es/N o equal to

6 and 10 dB. The signal-to-noise ratio for the DFE-assisted modem was

actually reduced by 0.2948 dB to allow for the seven percent training

sequence overhead. Thus the two modem types are compared on the basis of

equal transmitted energy per information symbol.

Where Eis diminishes, that is EIsI/E s is -- dB, the average

",it error rate for Receiver Model No. I is given by

P= _ Erfc (/Es/No)e 2 5

Jhere Erfc(-) is the familiar complementary error function. Evaluation

of this expression for E s/N equaling 6 and 10 dB yields 2.388 x 10- 3

ind 3.8756 x 10-, respectively.

The average bit error rate for Receiver Model No. 1 with Es

eqlualing 6 dB is plotted versus EIsI/E s in Figure 9. As EIS I diminishes

the curve appears to approach th'e predicted value of 2.388 x 10_ . The

SECTION 4

RESULTS AND CONCLUSIONS

This section presents two sets of results that were generated

during this investigati,,)n. The first set is a comparison of the perform-

ance of an idealized conventional BPSK receiver to that of a DFE-assisted

BPSK receiver in the context of the time-invariant, frequency-selective

channel model described in Section 3.2.1. These results are presented to

illustrate the problem of intersymbol interference and show that adaptive

equalization is a potential mitigation technique.

The second set of results show the DFE-assisted modem perform-

ance in the context of the time-varying, frequency-selective channel model

discussed in Section 3.2.2. These results give an indication of how this

modem type would perform over a range of doppler and delay spreads

exhibited by disturbed HF propagation paths.

4.1 PERFORMANCE OF RECEIVER MODELS NO. 1 AND NO. 2IN THE CONTEXT OF CHANNEL MODEL NO. 1.

The Monte Carlo computer simulation was exercised to investigate

the performance of the two BPSK modems described in Section 3.3 operating

under the time-invariant, frequency-selective channel conditions discussed

in Section 3.2.1.

The BPSK receiver not employing adaptive equalization is

Receiver Model No. 1, described in Section 3.3.1. It does not use a

training sequence and is bit synchronized and phase locked onto the signal

energy propagation mode.

50

,-. .w''S, ,.',-.'_,, '_,.w" .,- '.. .... -. ' " -- • -.-. . ''.' . . "-,. . '. . .-"- -". .- "- . . ".". .. - .. - " -

( --

C,)

LLL

494

the simulation. Instead the decision rule offset delay parameter D is

selected so that all signal energy associated with dkD is in the feed

forward section of the DFE when Yk is generated.

A block diagram of the model is shown in Figure 8. It differs

from Receiver Model No. 1 in three ways: 1) a DFE has replaced the

N-sample accumulation function; 2) the parameter D is larger than zero;

and 3) the decision symbol dk-D is fed back to the DFE function.

DFE adaptation can be implemented using either Algorithm 1 of

Section 2.3 or Algorithm 2 of Section 2.6, with one modification. The

variable dkD of Step 4 of Algorithm 1 and Step 12 of Algorithm 2 must

be replaced by dkD. The decision rule for dk D for Receiver Model

No. I differs from Receiver Model No. 2 because of the incorporation of

training sequence symbols into the telemetry format. The rule is given

by:

+1 Real {Yki > 0 and dk D was generated from an

information symbol

dk-D = -1 Real {Yk} < 0 and dk-D was generated from an

information symbol

dk-D dk-D was generated from a training symbol

3.4 ERROR STATISTICS ACCUMULATION

This part of the simulation simply compares the sequence {dkl

with {dkI to form an estimate of the average symbol error rate.- The

error rate estimate is calculated by dividing the number of incorrectly

received information symbols hy the total number of received information

symbols. Training sequence symbols are only used to adapt the DFE and are

not used in the calculation of the error rate estimate.

L/)

LUJ

L.-

3.3.1 Receiver Model No. 1: An Idealized ConventionalBPSK Receiver Model

This receiver model is only considered in the context of the

simply delay spread channel model described in Section 3.2.1. The

receiver is assumed to be synchronized and phase-locked onto the mode

propagating signal energy. Since the signal energy mode is represented by

hk, the decision variable sequence is denoted {dk}. The delay parameter

D equals zero because dk can be generated immediately after the

kth sampling interval.

This receiver model is represented by Figure 7. A value for

Yk is generated (for each k that is an integer multiple of N) by

simply adding uk , uk_, ..... ukN+ i . The binary decision variable dk is

then formed by the following rule:

k 1 Real {ykI > 0dk Real {y }< 0

k

where again dk is only calculated when k is an integer multiple of

N. By definition, the channel input samples dk, dk-1, ... , dkN+ are

equal, being identically generated from the same channel input symbol as

discussed in Section 3.1.

3.3.2 Receiver Model No.2: A DFE-Assisted BPSK Receiver

This receiver model is used for both channel models discussed in

Section 3.2. The OFE explicitly provides the phase tracking and "fine"

bit synchronization functions that are assumed to be perfect in Receiver

Model No. 1. In an operational receiver a "coarse" bit synchronization

capability would be required, but this is not implemented explicitly in

46

DISCRETE CHANNEL UNIT PULSE RESPONSE

DELAY

0 3T/N 9T/N 13T/N

DELAY (seconds)

Figure 6. Delay power spectrum of Channel Model No. 2.

45

B = Doppler bandwidth (hertz)

C = average carrier power (watts)

p = constant selected so that C = E{h dk} (note

that p equals 1/2 for N less than four)

For the purposes of this report, this channel model will bespecified by the four parameters C/No0, B, N and T. C is arbitrarilychosen to be unity without loss of generality. Since the parameter

C/No has units of watts/joule or hertz rather than being a unitless

ratio, the logarithmic form of the C/No specification has units of dB-Hz

(decibels relative to a one hertz bandwidth). The delay spectrum for the

channel model is illustrated by Figure 6.

This channel model shall not be employed to investigate conven-

tional BPSK modems, such as Receiver Model No. 1, which can only lock onto

one of the four modes. Therefore it is not appropriate to designate one

mode as the propagation path for signal energy with which all other modes

interfere. Rather the received energy from each mode of Channel Model

No. 2 is considered signal energy even though the four modes interfere

destructively with one another on the average.

3.3 RECEIVER MODELS

A receiver model generates the decision variable sequence {dk-D}

(for all k that are integer multiples of N) by processing the channel

model outpu- symbol sequence {Uk}.

Two receiver models have been employed for this investigation -

one that models an idealized conventional BPSK receiver and another thatmodels an adaptive DFE assisted receiver.

44

.1

--

model is inadequate if the modes exhibit different doppler frequency

shifts. Such channels are said to have a doppler spread and can be repre-

sented by a channel model that allows a distinct phase rotation rate for

each mode. One such channel model is described below.

3.3.2 Channel Model No. 2: A Simple Delay and Doppler Spread Model

This model can also be represented using the notation of

Section 2.

L = 13

exp {0.375 * 27kBT/N} for m = 0

/Cexp 10.125 *21TkBT/N} for m =3

mh k = CP exp {O.250 * 2nkBT/NI for m = 9

0/c exp 10.625 * 2fkBT/N} for m 13

0 otherwise

ENnkn k1 = NN/T k j

I k j

where k, N, nk, No retain the same definitions as in Section 3.3.1

and

43

I- ,*

5;-.

DISCRL E CHANNEL UNIT PULSE RESPONSE

DELAY o--

ELU S

I--

4-44 EISI

II li ill l I I I I Ill III II

- .-. - 20 SAMPLES

Figure 5. Delay power spectrum of Channel Model No. 1.

".4

42

4..A, ,'

N = ratio of sampling rate to modulation rate

nk = complex zero-mean, additive white Gaussian component

of Equation 2 of Section 2. Here the quadrature real

and imaginary components of nk are uncorrelated and

have the same variance

N = one-side power spectral density level of each quadrature0

component of the additive white noise preceding sampler.

For the purposes of this report, this channel model will -be

specified by the three parameters Es/N 0 , Es/EIs I and N. Es and T are

arbitrarily chosen to be unity without loss of generality.

This channel model can be visualized using Figure 5 which shows

a two mode energy delay spectrum resulting from the transmission of an

impulse. The earliest (left most) spike corresponds to the mode propa-

gating signal energy and the other spike corresponds to the mode propa-

gating intersymhol 4nterference energy. The association of one mode with

the signal and the other with the intersymbol interference is motivated by

how this channel model might affect a conventional PSK receiver. Assuming

the delay between the two modes exceeds a channel symbol interval (i.e., N

is less than 20) the matched filter of a conventional PSK demodulor cannot

combine energy from both modes that arises from the same channel symbol.

This channel model approximates only a limited range of HF prop-

agation possibilities. Three examples might he simultaneous reception

froin: 1) a groundwave and a skywave propagation path; 2) a single hop and

a double hop path; and 3) two single hop paths reflecting off distinct

ionospheric layers. In the context of adaptive equalization, the above

41

effects. Reference 16 reviews the current state of the art in HF propaga-

tion analysis in the context of nuclear phenomenology.

Since the current understanding of HF propagation disturbances

is incomplete, the choice of a channel model is somewhat arbitrary. Heretwo simple, non-statistical channel models shall he employed, which have

the salient characteristics needed for this investigation, but are not

general representations of HF channels. The first is time invariant,

exhibiting delay spread but not doppler spread. The second exhibits both

delay and doppler spreads.

3.2.1 Channel Model No. 1: A Simple Delay Spread Model

Using the notation of Section 2, this channel model can be

described by the following set of parameter values.

L = 20

m=,hk A /T m = 20 for all k

0 otherwise

" NON/T k= j

Ein n* IkjI0 k j

i, where

k = sample index

E = received signal energy per modulation interval

E z received intersymbol interference energy per

modulation interval

40

-~~~~~ is....

~40

LLLJ

4-s

<c CL

40 LeLj4

'4- L)LJa

CP

a

4-s

L.) V CA

t 4-s

EL CD

LL ca D.2- c

4-U)

p~' , L

I--

-i-

I- IU -J Ln

- '39

{dkl can be taken from the set ji,-I}. The convention used in the simu-

lation is as follows: the binary channel symbol 0 is mapped into N con-

sequetive elements of {dk} having the value +1; and channel symbol 1 is

similarly mapped into N consecutive elements having the value -1.

In addition to the correspondence with information symbols, some

elements of Idk} may correspond to training sequence symbols. A train-

ing sequence is a block of channel symbols, known a priori to the re-

ceiver, which are periodically injected into the transmitted data stream.

. Training sequence symbols are used within the receiver to initialize and

maintain the adaptation algorithm of the equalizer, as will be discussed

in Section 3.3.

The telemetry format of the simulated HF link is shown in

Figure 4a. The symbol stream is partitioned into telemetry blocks of

LTB symbols in duration. Each telemetry block starts with the same

sequence of LTS training symbols (represented hy cross-hatching in the

figure), followed by a LTB-LTS length segment from the information

symbol stream. The multiplexing of the training sequences into the data

stream is represented by Figure 4b. The generated sequence {dkl is

taken as input by the channel models.

3,2 CHANNEL MODELS

Much experimental and analytic work has been done to charac-

terize HF channels (e.g., References 7-15), but universally acceptedmodels of HF propagation disturbances have yet to be developed. In

natural environments, a wide range of channel characteristics, such as

absorption, delay spread and doppler spread, have been observed and theK most severe effects tend to occur on auroral propagation paths. An impor-

tant class of artificially induced effects are those reculting from high

altitude nuclear bursts. Such effects are not presently well understood,

but are estimated to be at least as severe as naturally occurring auroral

38

0-12

THEORETICAL LIMIT

-3 ~ AS E I /E SAPPROACHES -

10IS s

-12 -10 -8 -6 -4 -2 0 2Ratio of E,, to E, (d81)

Figure 9. Bit error rate versus EISI/Es for Receiver Model No. 1 withE5/N0 equal to 6 dB.

52

bit error rate increases with the level of intersymbol interfere as would

be expected. The severity with which this error rate increases accounts

for the rarity with which this modem type is employed in HF communication

links.

Two approaches for improvilg the average bit error rate were

explored: increasing the signal-to-noise ratio; and substituting Receiver

Model No. 2 for Receiver Model No. 1. Figure 10 presents a comparison of

the results of these approaches with the results of Figure 9.

Consider first the difference in error rate characteristics when

Es/N o is increased from 6 dB to 10 dB. As previously mentioned, this

would result in a decrease in average bit error rate from 2.388 x 10- 3 to63.8756 x 10- for EISI equal to zero. Figure 10 shows, though, that as

E /E is increased, the 10 dB error rate characteristic rapidlysi

approaches the same degraded performance level experienced for 6 dB.

Apparently as EIsI/E is increased, the significance of the additive white

Gaussian noise on the error mechanism diminishes in comparison to the

significance of intersymbol interference. Therefore increasing the

transmitted carrier power, which increases bothEIsI and F s but leaves

E IsI/E unaltered, does not effectively combat the effects of high levelsof intersymbol interference.

Now consider the difference in error rates with Es/N o held

V at 6 dB when Receiver Model No. I is replaced with Receiver Model No. 2

and the attendant change in telemetry format to accommodate a training

sequence. When EIsI/Es is very small, Receiver No. 2 suffers an effective

0.4 dB loss relative to the asymptotic error rate exhibited by Receiver

No. 1, hut as EIsI/E 5 increases the error rate for Receiver No. 2decreases, rather than increases. The 0.4 dB loss in the absence of

intersymbol interference is due in part to the 0.?948 dB training sequence

overhead mentioned earlier, and in part to the random misadjustment of the

53

;.. -

.~10

10- 2

S 0 ~E E6 VB ECIEt- MODEL NO.

THEORETICAL LIMIT FOR RECEIVER

3o MODEL NO. 1 AS E 5 /E5 APPROACHES

10-*-12 -10 -8 -6 -4 -2 0 2

Ratio of E,3, to Es (dB)

* Figure 10. Bit error rate versus EISI/E5 for both receiver models withEs/No equal to 6 dB and 10 .

54

,%

DFE coefficients caused by the influence of additive white Gaussian noise

on the adaptation algorithm. The decrease in average bit error rate with

increasing EIsI/Es results from the DFE using the intersymbol interference

energy as additional signal energy.

4.2 PERFORMANCE OF RECEIVER MODEL NO. 2 IN THE CONTEXTOF CHANNEL MODEL NO. 2.

The Monte Carlo simulation was exercised to investigate the

performance of the DFE-assisted BPSK modem described in Section 3.3.2

under the time and frequency selective channel conditions discussed in

Section 3.2.1. Channel Model No. 1 is time-invariant, allowing the

presentation of the results in Section 4.1 to be in a form that is

independent of the data rate.

Here Channel Model No. 2 is time-varying, and it is convenient

to consider one specific information symbol rate, which was arbitrarily

chosen to be 2400 baud. The DFE parameters used to generate all the

results presented in the section are as follows:

LTS :48

LITB : 96

N 2

' K1 : 8

K2 :16

j2 x 10- 1 after reception of an information symbol

8 x 10- after reception of a training symbol

T = 1/4800 secopds

55

Receiver Model No. 1 is not considered here because it does not

have a phase tracking capability. The investigation of Receiver Model

No. 1 in the context of any time-varying channel model would be pointless.

The stochastic gradient algorithm discussed in Section 2.3 was

used within Receiver Model No. 2 to adapt the DFE coefficients. This

algorithm has an inherent phase tracking capability which is similar to

that of a first order phase lock loop. Since each mode has a fixed

doppler offset, the adaptation algorithm must track a phase ramp. This

results in a lag error for each DFE coefficient, and the adaptation

algorithm could have been redesigned as a second order phase tracking loop

to reduce these lag errors. However, by not specifically optimizing the

receiver for Channel Model No. 2, the results obtained here are indicative

of the performance of this modem type for other channels of comparable

doppler spread but exhibiting more random phase fluctuations.

The results take the form of average bit error rates plotted

versus either the average carrier power-to-noise spectral density level

C/No of the doppler bandwidth B.

Figure 11 shows the bit error rate versus C/No for B fixed at

four hertz. As expected, the error rate decreases monotonically with

increasing C/No . However, the error rate characteristic flattens for

values of C/No above 46 dB-Hz. When C/No is below this value most

detected bit errors are caused by the AWGN component of the channel model

output. As C/No is increased the AWGN component diminishes in compari-

son to the residual intersymbol interference which the adaptive-DFE does

not correct. Further increasing C/No does not appreciably improve the

error rate because the effects of intersymbol interference are not

reduced.

56

-. 40

C~C9

.4.,0

CL

I, L.

C4J

cnDTAO j;f

57U

Figure 12 shows the error rate versus C/No characteristic for

several coppler bandwidth values. Figure 13 shows the same data plotted

in the form of error rate versus B characteristic curves for several

values of C/NO . The DFE coefficients misadjustment due to adaptation

lag increases with B, and the resulting effects upon the error rates are

shown in both of these figures.

4.3 CONCLUSIONS

The adaptive DFE is an effective solution to intersymbol inter-

ference if the channel impulse response is constant or very slowly

changing. For severe levels of intersymbol interference, the inclusion of

adaptive equalization into the link design was shown in Section 4 to be

much more beneficial than increasing the received signal-to-noise ratio.

Thf effectiveness of the adaptive DFE can be limited by the

rapidity of channel fluctuations, parameterized in Section 4.2 by the

doppler bandwidth B. The specific 2400 baud modem considered in that

section exhibited usable channel capacity for values of B up to 4 Hz. For

C/No equal to 44 dB-Hz and B equal to 4 Hz, the measured channel symbol

error rate was about 0.04. To make such a modem useful for practical

applications one would employ some form of channel symbol error control.

Augmenting this modem with rate 1/2, constraint length 7 convolutional

coding, interleaving and Viterbi decoding would produce a 1200 band data

link with an expected error rate of less than 10" . Employing rate 1/4 or

rate 1/8 codes instead of the rate 1/2 code would yield 600 and 300 baud

links with much lower error rates.

58

il :-, ... ... .:. -.-. ., . .. ;-: - . - .,. ., . . . .

00

i~DeJ o o e' o Irco

.- I~~4 1- 4 L.

L

N Ul

590

I - I , I- r I I I I

00

-00

EAN

00 (U

L

600

REFERENCES

1. Austin, M. E., Decision-Feedback Equalization for Digital Communica-tion Over Dispersive Channels, MIT Lincoln Laboratory Technical Report437, August 1967.

2. Gersho, A., "Adaptive Equalization of Highly Dispersive Channels forData Transmission," Bell Systems Technical Journal, Vol 48, No. 1,pp. 55-70, January 1969.

3. Godard, D., "Channel Equalization Using a Kalman Filter for Fast DataTransmission," IBM Journal of Research and Development, pp. 561-273,May 1974.

4. Falconer, D. D. and L. Ljung, "Application of Fast Kalman Estimationto Adaptive Equalization," IEEE Transactions on Communications,COM-26, No. 10, pp. 1439-46, October 1978.

5. Ljung, L., M. Morf and D. D. Falconer, "Fast Calculation of GainMatrices for Recursive Estimation Schemes," Bell Systems TechnicalJournal of Control, Vol. 27, No. 1, pp. 1-19, January 1978.

6. Mueller, M. S. "Least-Squares Algorithms for Adaptive Equalizers,"Bell Systems Technical Journal, Vol. 60, No. 8, pp. 1905-1925, October1981.

7. Grisdale, G. L., J. G. Morris and D. S. Palmer, "Fading of LongDistance Radio Signals and a Comparison of Space and PolarizationDiversity Reception in the 6-18 M c/s Range," Proc. IEE, Vol. 104,Part B., pp 39-51, January 1957.

8. Sifford, B. M., etc., HF Time and Frequency Dispersion Effects -Experimental Validation of an FSK Error-Rate Model, SRI, AD-618330,March 1965.

9. Goldberg, B., "300 KHz-30 MHz MF/HF," IEEE Trans. Commu. Tech., Vol.COM-14, pp 767-784, December 1966.

10. Watterson, C. C., J. R. Juroshek, and W. D. Bensema, "ExperimentalConfirmation of an HF Channel Model," IEEE Trans. Commun. Technol.COM-18, 792-803, 1970.

11. Rice, D. W., "Phase Characteristics of lonospherically PropagatedRadio Waves," Nature London, 244 (136), 86-88, 1973.

61

... .~T-~>~~ *~*p*.*~* -~-.- ~:X

12. Goldberg, B., Communication Channels, Characterization and Behavior,IEEE Press, New York 1975.

13. Bello, P. A., L. W. Pickering, G. E. Peo, and K. P. Jauniskis,Wideband HF Channel Measurements Studies, Project 47, CNR Inc.,Needham, Mass., March 1979.

14. Wagner, L. S., J. A. Goldstein and E. A. Chapman, Wideband HF ChannelProber: System Description," NRL Report 8622, March 1983.

15. Findings from the DNA High Altitude Nuclear Effects Summer Study (HFCommittee), June 1982.

16. Gutsche, S. L., and M. R. Frolli, Estimates of HF/VHF SignalCharacteristics in Nuclear AGW and Plume Scatter Environments,MRC-R-810, January 1984.

62

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