joint modeling of the multipath radio channel and user-access method 1) communications department...

Joint Modeling of the Multipath Radio Channel and

User-Access Method

1) Communications DepartmentTechnical University of Cluj-Napoca, Romania

2)Electronics & Telecommunications Department University of Aveiro, Portugal

Zs. A. Polgar1), V. Bota1), M. Varga1), A. Gameiro2)

2

Outline

• Mathematical modeling of the multipath Rayleigh faded radio channel.

• OFDMA type multiple access techniques – short presentation.

• Theoretical computation of the received SINR p.d.f. for a specific OFDMA multiple access technique.

• Evaluation of the received SINR p.d.f. in the considered OFDMA multiple access based on measured/simulated data.

• Markov chain modeling of the channel SINR states in the considered OFDMA multiple access scheme.

• Further studies and developments.

3

• evaluation of the data transmission performances on a radio channel affected by frequency and time selectivity, requires the derivation of the channel state p.d.f., i.e. the p.d.f. of the instantaneous total SINR (total SINR = background SNR + SIR).

• the stated problem is a difficult one and pure analytical solutions are possible only for some special cases, e.g. the case of a radio channel characterized by multipath propagation with uniform power delay profile and Rayleigh fading [Alouini - 3] – in this particular case the global SINR of the channel will be characterized by a

distribution, while the SINR on each propagation path is characterized by an exponential distribution.

• another possible mathematical characterization of the entire transmission chain, very useful especially for packet based transmissions, can be made with Markov chains– beside the probability distribution of the average received SINR, it allows the

computation of instantaneous SINR for small time intervals.

Mathematical modeling of the multipath

Rayleigh faded radio channel.

4

r1

r2 r3

r2(t) 2

r3(t) 3

r4(t) 4

r1(t) 1

ag

ar1 ar2 ar3

sin(t)

sout(t)

mean attenuation (free space propagation

log-normal fading)

relative delay (relative to

reference path)

relative attenuations (relative to

reference path)

multipath propagation

Rayleigh fading (i – rms. of the i-th path)

exp1

p

BNSNR.av

BN2

SNR.inst

2

r

0r;0

r0;

2

rexp

r

rp

02

0

2

22

2

2

2

t1Lt2

tt

i

L

1iit

exp!1L

1p

0;:)pathsL(profiledelaypoweruniform

t

Fig. 1 Multipath Rayleigh faded radio channel model

(1)

(2)

(3)

(4)

(5)

(6)

(7)



5



• Possible solutions proposed in literature:– An infinite series for the computation of the c.d.f. and p.d.f. of sums of random

variables (particular case of Rayleigh variables) [Beaulieu - 14]• A number of several thousands of terms are required to obtain a good approximation of

the p.d.f. of sums of random variables.• It considers only the sum of real random variables – the phase variations induced by

multipath propagation and random phase variations induced by the small scale fading are not considered.

– Computation of the p.d.f of sums of random vectors, with identically and arbitrarily distributed lengths and uniformly distributed phases [Abdi - 13]

• Integration of the multipath propagation situation is required (phase distributed uniformly but with imposed mean value) .

• The p.d.f. obtained can be expressed as a definite integral including Bessel functions – the obtained p.d.f. of the vector length can be expressed by Hankel transform H0a, where J0 is the zero order Bessel fct and Exy is the expectation of the combined vector

• The previous p.d.f. can be decomposed as a a series of Laguere polynomial, easier to compute than the definite integral.

n

1ii0AAa0

00 AJE;HadaJaa.f.d.p

N1(8)

6



– Closed-form upper-bound for the distribution of the weighted sum of Rayleigh variates [Karagiannidis - 16]

• An upper-bound is defined for the sum of Rayleigh distributed variables. • The phase variations due to the multipath propagation and small scale fading are not

considered.• The proposed upper-bound relies on G Meijer’s functions – very difficult to compute in

the general case.

– Computing the distribution of sums of random sine waves and of Rayleigh-distributed random variables by saddle-point integration [Helstrom - 13]

• The c.d.f. of a sum of independent random variables can be computed according to the following integral on a curve, where h(z) is the moment generating function of the v variable:

• Saddle point integration is proposed to compute the previous integral; the saddle point which has to be found as well the moment generating function.

• Random phases and Rayleigh distributed amplitudes are considered separately.

• Conclusion: no simple closed form solutions are available for the stated problem.

C

z zlnvzzhlnz;j2

dzev.f.d.c (9)

7



• The proposed solution:

– It considers a multipath channel with N propagation paths characterized by gains a i and delays i (i - index of the path). The level on each path is x i=Aai, where A is the amplitude of a test sine with frequency fl; the level distribution on the propagation paths is given by distributions pi(x).

– The probability to obtain a level r at the output of the channel, if the phase distributions are not considered, is given by:

(10)

(11)

(12)

– The function Ql(r,x1,x2…,xN-1, 1,2,…, N-1), expresses the occurrence probability of a level xN on the last path, N, that would make the value of the entire

received signal equal r, when the signals on the other N-1 paths have levels x1 – xN-1;

)r.(f.d.pdxdxdxdx,,,x,x,x,rQxpxpxpxprp0

10

230

1N1N211N21l1N0

332211fl_ch 1N

0;pp

0;p

0;0

,,,,x,x,x,x,rQ

2lN1lN

1lN1N3211N321l

21N

1t1Ntt

21N

1t1Ntt2,1l f2sinxrf2cosx

8

– The channel instantaneous SINR on frequency fl can be obtained by dividing the received signal power level on the considered frequency to the noise + interferences power at receiver’s input.

o Since the noise power is constant for a short time interval and over a small frequency bandwidth, we may assume that the SINR p.d.f. is also expressed by (10), (11), (12).

– The channel instantaneous SINR on frequency fl can be obtained by dividing the received signal power level on the considered frequency to the noise + interferences power at receiver’s input.

– If the phase distributions, pi() on different propagation paths are considered, the probability of the level r at the output of the channel is expressed by:

(13)

– The presented relations are related to one discreet frequency (subcarrier);

– If the small scale fading is Rayleigh distributed then pi(x) are Rayleigh distributions and pi() are uniform distributions.

– The integrals related to the amplitude distributions can be computed on a

smaller, finite interval, according to the dynamic of the pi(x) distributions



l

iii121N121N1N3211N321l

2

0

2

0

2

0 0 01N1N22

0111N21fl_ch

f2';ddddxdxdx',',',',x,,x,x,x,rQ

xpxpxpppprp1N21

9

• the OFDMA type multi-user access is based on a frequency-time signal pattern that can be adjusted to different propagation scenarios and which is able to cope with the frequency selectivity generated by the multipath propagation and the time variability generated by the user motion. the mentioned signal pattern, called bin, chunk or burst, is proposed by several

existing and future high bit rate transmission systems - candidate systems for the future 4G mobile communication networks.

Examples of existing or future mobile communication systems based on OFDMA multiple access:

-the WINNER project - tends to propose radio interface technologies and system concepts for the future 4G mobile communication systems.

-the IEEE 802.16.x (x=a,…,e) standards (WiMAX) specify an OFDMA type multi-user access technique (called SOFDMA – Scalable OFDMA)

- the FLASH-OFDM high speed mobile transmission technology proposed by Flarion Technologies.

within these chunks, non-coded or coded QAM modulations are used adaptively, based on channel estimation or prediction, performed by the mobile station, using a number of scattered pilot sub-carriers.

OFDMA type multiple access techniques.

Short presentation.

10

OFDMA type multiple access techniques.

Short presentation.

chunk duration

time-frequency chunks

chunk bandwidth

n OFDM sub-

carriers

sub-carrier separation

m OFDM symbols

OFDM symbol period

chunk structure

Fig. 2 OFDMA multi-user access principle and chunk structure.

Tchunk

Bchunk

time

freq.

Fig. 3 Chunk allocation according to BFP algorithm- chunks are allocated by a scheduler using channel measure- ments performed by the mobile and channel prediction performed by the base station in the whole frequency band;- chunks are allocated in such a manner to increase the data amount transmitted by the BS;- one or several chunks are allocated to one user according to the channel and service characteristics;

Tchunkchannel coherence time

Bchunkchannel

coherence bandwidth

the chunk-allocation to different users in the downlink is performed based on the Best Frequency Position (BFP) method - each chunk is allocated to the user that predicts the best channel parameters (SINR) in the frequency band of that chunk – or on a fast frequency hopping (FH) method.

11

• Computation of the joint mobile channel-multiple access SINR p.d.f.– Frequency Hopping user-chunk allocation

• The probability of the rated level to lie between a pair of thresholds Tk(dB) and Tk+1(dB), wkFH, (Jk and Jk+1 - corresponding the linear values) where modulation configuration k should be used;

• The selection of a modulation configuration in a chunk m is made according to the average of the rated levels received on the Cu sub-carriers of that chunk, Pam.

• For FH bin-allocation method, the probability of a bin m to be assigned to one user is

Pm=1/Bu,Bu being the number of available chunks; Nu is the no. of payload subcarriers

Theoretical computation of the received SINR p.d.f. for the considered OFDMA multiple access technique.

(14)

k 1u

k

u

(mC q)u

k 1u

lk

JB

kFH k k 1 m am m mm 1 J

C

am m f m m u u uq 1u u

JN

kFH k k 1 f m ml 1u J

w J d J P P d d(d )

1 1P d P d ; P ; N B C

C B

1w J d J P d d(d )

N

12

• Computation of the joint mobile channel-multiple access SINR p.d.f.

– Best Frequency Position user-chunk allocation

– The probability, wkBFP, that the maximum rated level lies between thresholds Jk, Jk+1, where modulation configuration k should be used, requires the computation of the probability of the average rated level of chunk m to lie in this domain, Pkm, multiplied with the probability that the average rated levels of all other chunks, Qm to be smaller than the average rated level of bin m, it is expressed by:

(15)

Theoretical computation of the received SINR p.d.f. for the considered OFDMA multiple access technique.

u

k 1 uu

k

u

(mC q)u

m

B

kBFP k k 1 km mm 1

J BB

am m mj m mm 1 j 1J

j m

C

am m f mq 1u

dmj m aj m0

w J d J P Q

P d Q (d )d(d )

1P d P d ;

C

Q (d ) P y dy ; d average SINR in chunk m

13

Theoretical computation of the received SINR p.d.f. for the considered OFDMA multiple access technique – FH-allocation.

1 2 3 4 5 6 7 80

0.05

0.1

0.15

0.2

0.25 PFH

Constellation No.

Const. No 1 2 3 4 5 6 7 8

PFH -computed 0.0745 0.1228 0.1341 0.2362 0.2066 0.1488 0.0663 0.0107

NFH /10000 - simulated

0.0734 0.1215 0.1338 0.2381 0.2062 0.1515 0.0581 0.0174

Table 2 Measured and computed probabilities to employ the specified non-coded configurations (SINR =16 dB, FH method)

non-coded

nb SINR thr.(dB)

1 -

2 8.3

3 13.2

4 16.2

5 20.2

6 23.6

7 26.6

8 29.8

Table 1 Non-coded configurations and corresponding SINR thresholds

Fig 4 Measured and computed probabilities to employ the specified non-coded configurations (SINR of reference path=16 dB); FH chunk allocation method

Simulated

Computed

14

1 2 3 4 5 6 7 80

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45 POA

Constellation No.Fig 5 Measured and computed probabilities to employ the specified non-coded configurations (SINR of reference path=16 dB); BFP chunk allocation method

Theoretical computation of the received SINR p.d.f. for the considered OFDMA multiple access technique – BFP allocation.

Simulated

Computed

Const. No 1 2 3 4 5 6 7 8

POA – computed 0.0012 0.0013 0.0076 0.0756 0.2910 0.3936 0.2158 0.0139

NOA /10000 - simulated 0.0050 0.0025 0.0065 0.0675 0.2785 0.4165 0.2105 0.0130

Table 4 Measured and computed probabilities to employ the specified non-coded configurations (SINR =16 dB, BFP method)

Test parameters

Carrier freq. fc1.9GHz

OFDM symb. freq. fs

10kHz

OFDM guardInterv. G

11s

Chunk size (no. subc. no.

OFDM symb.)

206=120;108 payload symb.

Power delay profile: 4 paths

User speed

a1=3dB;1=200nsa2=3dB;2=400nsa3=3dB;3=600ns

120km/hTable 3 Test conditions related to fig. 4 and fig. 5

15

• the approach presented before requires a large amount of computation and should be performed for every SINR value; for the BFP allocation, it should also be performed for every cell-carrier loading (Lc).

• an approximate method to compute the p.d.f. of the received signal level (and the SINR at the receiver), for any desired average SINR0 of the first arrived path consists of three steps:– compute, simulate or measure the probabilities of the receiver’s SINR to lay between an imposed set of thresholds Tk, for a given SINR0 of the first arrived path. Computation can be done using the method described above;

– find an interpolation function f(x) that approximates the distribution of the SINR on the channel, fulfilling the conditions imposed by step a.

– translate and scale f(x) around the desired SINR of the first arrived wave, SINR0.

• this method requires a smaller amount of computation and ensures a good accuracy of the obtained p.d.f.

• as an example, the WP5 Macro channel model (18 propagation paths) for a given average SINR0 =16 dB of the first path and Lc (cell load) = 2%, 50%, 75%, 100% was considered; both the BFP and FH chunk allocation methods were considered.

Evaluation of the received SINR p.d.f. based on measured/simulated data.

16


• if f(x) is the function which interpolates the SINR p.d.f. and wk is the probability of the current SINR to lay within the k-th domain, the interpolation function should fulfill the conditions:

(Jk - linear values corresponding to logarithmic values Tk; J1 – min., JS – max.)

• The interpolation function f(x) should also be positive and it should have only one maximum across the whole range of x variable considered;

• To simplify the computation of f(x), the number of SINR domains is reduced by suppressing those who exhibit a wk probability below an imposed value, e.g. wk < 1∙10-3, and adjusting the lowest and highest thresholds, to Tkm and TkM.

k 1

k

J

k S 1 1J

w f (x)dx; k 1, ,S; a. f (J ) f (J ) 0; b.

(16)

• The f(x) was chosen to be a polynomial function of order S=kM-km+1, (17.a); Using the probabilities wk of the SINR to lay within the k-th interval, the coefficients of f(x) are computed using (17.b ) - (17.c).

M m M m

m M

k k 1 k k 1i i 1 i 1i

i k k k k 1 k m M 1i 0 i 0

cf (x) c x ; a. f (J ) f (J ) 0; c. w (J J ); k k k ; b.

i 1

(17)

17

Tk [dB] T1=-2 T2=8.3 T3=13.2 T4=16.2 T5=20.2 T6=23.6

Lc=2% 0 0 0 2∙10-5 1.4∙10-3 3.9∙10-2

Lc=50% 0 0 0 5∙10-5 3.2∙10-3 5.5∙10-2

Lc=75% 0 0 0 8∙10-5 8.5 10-3 7.3∙10-2

Lc=100% 0 0 3.1∙10-4 6.8∙10-3 2.6∙10-2 8.8∙10-2

Lc=x% 5∙10-4 1.1∙10-2 3.7∙10-2 1.47∙10-1 2.38∙10-1 2.67∙10-1

Tk [dB] T7=26.6 T8=29.8 T9=33 T10=36.2 T11=39.4

Lc=2% 3.4∙10-1 5.3∙10-1 8.7∙10-2 7.8∙10-4 0

Lc=50% 3.5∙10-1 5.0∙10-1 8.8∙10-2 7.9∙10-4 0

Lc=75% 3.4∙10-1 4.8∙10-1 8.7∙10-2 8.2∙10-4 0

Lc=100% 3.1∙10-1 4.8∙10-1 8.9∙10-2 1.0∙10-3 0

Lc=x% 2.21∙10-1 7∙10-2 4.5∙10-3 2∙10-5 0

• the SINR domains and the associated probabilities corresponding to BFP and FH chunk allocation are presented in in table 5;

Tab. 5 SINR domains and associated probabilities for BFP and FH chunk allocation; SINR0=16dB

Lc=x% corresponds to the FH chunk allocation

a0ref0refa SNRSNRc;xcfcf (18)


•Scaling the f(x) function for other desired SINR0 (SINR0-a) of the reference path, (fa(x)):

18

15 20 25 30 35 400

0.01

0.02

0.03

0.04

0.05

0.06

SINR (dB)

cell load < 2%

cell load = 50%

cell load = 75%

cell load = 100%

Fig. 6 Interpolation functions of the channel SINR p.d.f. for cell carrier load Lc=2%, 50%, 75%, 100% and BFP chunk allocation.

p.d.f. (SINR)

5 10 15 20 25 300

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

SINR (dB)

p.d.f. (SINR)

Fig. 7 Interpolation functions of the channel SINR p.d.f. for FH chunk allocation.

• both the analytical and the approximate method, ensure good accuracies of the wk SINR state probabilities.

• the wk SINR state probabilities delivered by the approximate method differ with less than 0.3% from the ones obtained by computer simulations.

• the approximate method provides the probabilities wk, with a slightly greater error,

but requires significantly less computation than the analytical approach.


19

• the BFP allocation method ensures high state probabilities only for a limited number of channel states, 2 maximum 3, the rest of the states being employed quite seldom.

the SINR average value of these states is 10-14 dB higher than the SINR level of the firstly arrived path - this is because the BFP method ensures (with high probability) the chunk with the highest SINR for each user, taking advantage of the frequency diversity of the multipath Rayleigh channel in a very efficient way.

• with the FH method the cell-carrier load does no longer affect the p.d.f., because the user chunks are allocated independently according to a pseudo random sequence.

the FH allocation method employs with significant probabilities wk, more channel states, 5 or 6, because it does not place the user on the particular chunk that ensures the best SINR.

the average SINRs of these states are smaller, with even 8 dB, or greater, with up to 14 dB, than the SINR of the firstly arrived wave, SINR0.

the most employed states ensure an average SINR higher with 0-10 dB than SINR0.

the FH chunk allocation method takes advantage of the channel frequency diversity in a less efficient manner - it performs an averaging over the

whole available frequency bandwidth.


20

• The p.d.f. of the total SINR is independent of the mobile speed.• the total SINR p.d.f. shows only the average probabilities of the channel’s SINR to lie between each pair of thresholds.

• employment of the p.d.f. only allows the computation of average performances of a transmission scheme, over a relatively long time interval.

• If the transmission analysis requires the probability distributions of the studied parameters, distributions depending on the mobile speed, a Markov-chain modeling of the channel and multiuser-access method should be developed.

• Different chains should be derived for various cell-carrier loadings and mobile speeds.• Some Markov-chains obtained by computer simulations, for some significant situations

related to the considered OFDMA access scheme, are presented.• The computer simulations required to derive these models with acceptable accuracy are

far less time consuming than the complete evaluation by simulations of the desired performances of a transmission system.

• Out of the set of possible non-coded QAM modulations only the first 8 are used adaptively in each chunk.

• the complete Markov-chain assigned to the channel SINR states has 8 possible states, one assigned to each SINR domain.

• the Markov chains can be simplified by dropping some states with low or very low associated probabilities, i.e. smaller than 0.01, to allow a simpler and still accurate performance evaluation.

Markov chain modeling of the channel SINR states in the considered OFDMA multiple access scheme.

21


St.6 p(6)

St.8 p(8)

p(6-6) p(8-8)

St.7 p(7)

p(7-7)

p(7-6)

p(6-7) p(8-7)

p(7-8)

p(8-6)

p(6-8)

Fig 8 Markov chains which characterize the radio channel in the case of BPF user chunk allocation, x % load of the cell carrier and y km/h mobile speed

100% load of the cell carrier; speed 4km/h and 120km/h

2% and 50% load of the cell carrier; speed 4km/h and 120km/h

St.6 p(6)

St.8 p(8)

p(6-6) p(8-8)

St.7 p(7)

p(7-7)

p(7-6)

p(6-7) p(8-7)

p(7-8)

p(8-6)

p(6-8)

St.5 p(5)

p(5-5)

p(8-5) p(6-5)

p(5-6)

p(5-8)

p(5-7)

p(7-5)

Initial state

Initial stateFinal state; speed

Tab.a BFP allocation, Lc=2% State probab.

6 7 8

6 120 km/h 0.0839 0.0554 0.0269 0.03775

4 km/k 0.9713 0.0018 0.00077

7 120 km/h 0.464 0.3972 0.1128 0.33888

4 km/k 0.0158 0.9807 0.0095

8 120 km/h 0.4484 0.545 0.3563 0.62305

4 km/k 0.0127 0.0174 0.9897Tab.6 SINR state transition probabilities. BFP chunk allocation, Lc=2%, speed 4km/h and 120km/h, SINR0=16dB

Tab.7 SINR state transitions for BFP chunk allocation, Lc=2%, speed 120km/h, SINR0=16dB, 105 transmitted chunks

Initial state 4 5 6 7 8

Final state

4

5

6

7

8

22


• The presented Markov chains can be used to compute two important parameters of a mobile transmission system, namely the packet error rate and the packet average length in bits, a packet of data being composed of an imposed number of chunks.

each branch is labeled with a variable X used to count the number of chunks in the packet, and with a product of the state transition probability and the correct chunk probability of the branch originating from that state; the correct chunk probability is computed based on the average SINR, QAM modulation corresponding to the considered state and chunk parameters; the average SINR of a given state is computed based on the previously established SINR’s p.d.f.; for each possible combination of initial and final states – i.e. the states corresponding to the first and last chunk in the packet - the transfer function of the graph (between the two considered states) is computed using the Masson formula [17]. finally only the terms with an imposed power L (L=packet length) are considered.

•To compute the average length of a packet, each branch in the graph will be labeled, beside the previously mentioned X variable, the state transition probability and with a term having the expression: Yi, i being the number of bits/symbol of the QAM modulation used in the branch originating state.

the obtained graph transfer functions are composed of terms having the following expression: pYtXz. From each transfer function, the terms XL are extracted , and finally the average packet length is computed.

23


Fig. 9 p.d.f. of the no. of bits/symbol in the case of BFP allocation of the user chunk for different loads of the cell carrier and different speed values of the mobile

5 5.5 6 6.5 7 7.5 80

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

no. bits/symb.

Lc=2%; 4km/h

Lc=50%; 4km/h

Lc=100%; 4km/h

Lc=2%; 120km/h

Lc=50%; 120km/h

Lc=100%; 120km/h

pdf(no. bits/symb.)

24

1. Study of methods which allow the accurate and fast computation of the p.d.f. of a sum of independent Rayleigh random variables – e.g. an infinite-series approach for the computation of c.d.f. and p.d.f. or saddle-point integration approach.

• Alternative approach: study of methods that give closed forms of upper/lower bounds of a sum of independent Rayleigh random variables.

2. Study/computation of the c.d.f./p.d.f. of Rayleigh distributed vectors.3. Computation of the joint multipath-Rayleigh channel – multiple access p.d.f.4. Theoretical Markov modeling of the multipath-Rayleigh channel

• Possible solution: - mathematical modeling of the level transition rate and (based on this) computation of the SINR-state transition rate;

- time dimension is required - leading to a time domain modeling of the multipath-Rayleigh process should be performed; - decomposition of the multipath-Rayleigh process as sum of

weighted sine signals with p.d.f. restrictions, might be a possible solution;

5. Joint Markov modeling of the multipath-Rayleigh channel and multiple access technique.

Further studies and developments.

25

References[1] Th. S. Rappaport, Wireless Communications, New Jersey:Prentice Hall PTR, 2001.

[2] B. Sklar,“Rayleigh fading channels in mobile digital communication systems. Part I: Characterization,” IEEE Comm. Mag., July 1997.

[3] M-S. Alouini, Analytical Tools for the Performance Evaluation of Wireless Communication Systems, lecture notes Communication&Coding Theory for Wireless Channels, http://www.iet.ntnu.no/projects/beats/Documents/parts1and2.pdf , NTNU, Trondheim, Norway, Oct. 2002.

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[8] V. Bota, Transmisiuni de date, Risoprint, Cluj Napoca, 2004.

[9] V.Bota, M. Varga, Zs. Polgar; “Performancesof the LDPC-Coded Adaptive Modulation Schemes in Multi-Carrier Transmissions” Proc. of COST 289 Seminar, July 7-9, 2004, Budapest, Hungary.

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[13] A. Abdi, H. Hashemi, S. Nader-Esfahani, “On the PDF of the Sum of Random Vectors”, http://web.njit.edu/~abdi/srv.pdf

[14] N. C. Beaulieu, “An Infinite Series for the Computation of the Complementary Probability Distribution Function of a Sum of Independent Random Variables and Its Application to the Sum of Rayleigh Random Variables”, IEEE Trans. On Communications, Vol. 38, No. 9, September 1990.

[15] C. W. Helstrom, “Cpomputing the Distribution of Sums of Random Sine Waves and of Rayleigh-Distributed Random Variables by Saddle-Point Integration”, IEEE Trans. On Communications, Vol. 45, No. 11, November 1997.

[16] G. K. Karagiannidis, T. A. Tsiftsis, N. C. Sagias, “A Closed-Form Upper-Bound for the Distribution of the Weighted Sum of Rayleigh Variates”, IEEE Trans. Letters, Vol. 9, No. 7, July 2005.

[17] S. Lin, D. Costello, Error Control Coding. Fundamentals and Applications, Prentice Hall, 1983.

http://www.iet.ntnu.no/projects/beats/Documents/parts1and2.pdf

http://cost289.ee.hacettepe.edu.tr/publications/COST289_MCM6_Bota.pdf



http://web.njit.edu/~abdi/srv.pdf

26

Annex – demonstration of relations (10) – (12)

I

QR

2Q

2IRRRQI

i

N

1iii

N

1ii

N

1iiii

X

Xtana;XXX;ft2cosXft2sinXft2cosX

ft2sinf2sinaAft2cosf2cosaA

f2sinft2sinf2cosft2cosaAtr

ft2cosAts

N

1i

N

1ijjiji

N

1i

2i

N

1i

N

1ijjiji

N

1ii

22i

N

1i

N

1ijjiji

N

1ii

22i

2R

2N

1iii

2N

1iii

2R

N

1iiiQ

N

1iiiIii

f2cosxx2x

f2sinf2sinxx2f2sinx

f2cosf2cosxx2f2cosxX

f2sinxf2cosxX

f2sinxX;f2cosxX;aAx

(17)

(18)

• the transmitted and received signal, s(t) and r(t)

• alternative expression of received vector length

27

Annex – demonstration of relations (10) – (12)

0;pp

0;p

0;0

,,,x,x,x,rQ

f2cosxrf2cosx

f2cosxr4

0rf2cosxx2x

f2cosx2xx

2N1N

1N1N211N21

21N

1iNii

21N

1iNii2,1

21N

1iNii

2

1N

1i

21N

1ijNiji

1N

1i

2i

1N

1iNiiN

2N

(19)

• the probability of signal level on the N-th path, conditioned by the received signal and the signals on other levels

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