IEEE ICC 2014 - Wireless Communications Symposium

Modified Probabilistic Data Association AlgorithmsMilutin Pajovic*M.I.T. - W.H.O.I.

Cambridge, MA, 02139Email: milutin@mit.edu

Kyeong Jin Kim, Toshiaki Koike-Akino, Philip OrlikMitsubishi Electric Research Laboratories (MERL)

Cambridge, MA, 02139Email: {kkim.koike.porlik}@merl.com

Abstract-Probabilistic Data Association (PDA) algorithm hasshown promising performance in symbol detection and inter­ference cancellation in different communication schemes. Thispaper proposes new algorithms that build on PDA and introducemodifications in the way the symbol being detected is treated.While PDA models this symbol as a discrete sample from aconstellation, PDA with symbol uncertainty (SU-PDA) views itas a sum of a deterministic symbol and random noise, while theGaussian PDA (G-PDA) models it as a random variable witheither a single Gaussian or Gaussian mixture distribution. Theproposed algorithms are tested via computer simulations on bothsimulated and experimentally measured channels. The perfor­mance study reveals that the SU-PDA and G-PDA outperform theconventional PDA with the performance gain ranging from fewdBs on measured channel with block fading up to and exceeding10 dB on the simulated channel with fast fading.

Index Terms-probabilistic data association algorithm, largeMIMO, machine-to-machine communications, precoding, symboldetection.

I. INTRODUCTION

Machine-to-machine (M2M) communication systems areintended to enable machines to exchange short command andcontrol messages over wireless links. The main design goalis to achieve fast and highly reliable transmission of shortmessages over wireless channels with small and relativelysimple devices employing a small number of antennas (prefer­ably one or two) [1]. One of the major application areas ofM2M modems are factories wherein the automated productionprocesses would benefit if machines communicated with eachother wirelessly.

We consider a single-input single-output (SISO) commu­nication system with block transmission and detection as acandidate for M2M modems. The type of messages and therequired low latency call for short block sizes (e.g. < 100symbols).

A successful design of M2M modem with pseudo-randomphase precoding (PRPP) and Likelihood Ascent Search (LAS)detection has been reported in [2]. This scheme performsextremely well in practice when block size is at least 400symbols. However, our goal is to reduce the block size withoutlosing much of the performance.

Maximum likelihood detection (MLD) is the optimal sym­bol detection scheme. However, the complexity of the MLDgrows exponentially with the block size. Therefore, a variety ofsuboptimal detection algorithms with polynomial complexity

* M. Pajovic completed this work while he was an intern at MERL.

have been developed. Among the most known are PDA [3],QRD-M [4], LAS [5], Reactive Tabu Search (RTS) [6], andalgorithms based on graphical models [7].

The PDA algorithm was originally developed for targettracking and has gained interest in the communications com­munity. As such, the PDA has been applied for multiuserdetection in code division multiple access (CDMA) sys­tems [8], [9]. The application of PDA for turbo equalizationis reported in [10]. The symbol detection in multiple inputmultiple output (MIMO) systems using PDA algorithm ispresented in [11]. PDA has been combined with the decisionfeedback equalization (DFE) for detection in MIMO systemover a frequency selective channel [12]. A bit-by-bit detectionof a higher order quadrature amplitude modulation (QAM)using the PDA algorithm is given in [13]. A new symbolordering scheme for PDA detection suitable for flat fadingchannel is proposed in [14].

This paper proposes new algorithms based on PDA whichoutperform the conventional PDA for short block sizes «100 symbols) at lower bit error rates (BERs) of our interestin the M2M communication system. Given the equivalencebetween the SISO system with block transmission and largeMIMO [2], the proposed algorithms are equally applicable forsymbol detection in large MIMO.

Computer simulations have been conducted for both thesimulated and experimentally measured indoor channels.These tests show that the proposed algorithms outperformthe conventional PDA with the gain ranging from few dBs(i.e., one order of magnitude of BER) on the experimentallymeasured channel with larger blocks, up to and exceeding 10dB (i.e. more than two orders of magnitude of BER) on asimulated fast fading channel with shorter blocks.

Throughout the paper, boldface uppercase letters denotematrices and boldface lowercase letters denote vectors. An op­erator (.)T denotes transpose and (.)H complex-conjugation,i.e., Hermitian. Unless otherwise specified, all vectors areassumed to be column vectors.

II. SYSTEM MODEL

A block diagram of the SISO communication system withblock transmission and detection is shown in Fig. 1 anddescribed in this section.

The information to be transmitted is represented with com­plex symbols. The symbol transmitted at discrete time n isX n E X, where X is a finite dimensional symbol alphabet

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Fig. 1. Block diagram of a SISO communication system with block trans­mission and detection.

(5)z =x+w,

where z = G -ly . Note that the noise w is circularlysymmetric Gaussian, i.e., wrveN(O, R) with covariance R =(G H G)-l.

In essence, the PDA algorithm iteratively calculates theposterior probability of each symbol X n , conditioned on thereceived signal z,

rely on the additional specifics of the SISO system, they areequally applicable for symbol detection in other communica­tion schemes driven by (3).

III. PROBABILISTIC DATA ASSOCIATION ALGORITHMThe conventional PDA algorithm is outlined in this section.

For the sake of easy exposition and without loss of generality,model (3) is after multiplication with G -1 expressed as

transmitterfront-endprocessing

pprecoderp

x

receIverfront-endprocessing

makeblocks

X11

\virelesschannel

source ofsymbols

where H E ceNxN models distortions caused by the wirelesschannel. The signal-to-noise ratio is SNR and the additivenoise is circularly symmetric zero mean uncorrelated Gaussianprocess, i.e., vrveN(O, I).

The relation between the received signal y and transmittedblock x is obtained by substituting (1) into (2) such that

is the effective channel matrix. Throughout this paper, weassume the receiver perfectly knows the effective channelmatrix G and focus on symbol detection.

Although derived for a SISO communication system withblock detection, expression (3) models the input-output re­lationship in other communication systems as well. Sincethe algorithms proposed in this paper do not in particular

(i.e., constellation). Without loss of generality, we assume thatsymbol X n can take any value from X with equal probability.The consecutive symbols are grouped into blocks of size Nso that the transmitted block is x E ceNx 1.

The data block x is processed by a precoder. The precoderessentially spreads the energy of each symbol X n into Nsignaling intervals, giving rise to time diversity in the caseof a fast fading channel. Formally, the precoder is describedby a precoding matrix P E ceNxN and its output is given by

On the contrary, if the channel experiences very slow or blockfading, the precoding does not lead to diversity. Formally, inthis case we use the identity precoder, i.e., P = I.

The precoded block p is modulated onto a carrier andtransmitted over a wireless channel. The signal received onthe receiver antenna is filtered, demodulated, converted intothe baseband and sampled. The received baseband signal indiscrete time is y E ceN x 1and is related with the precodedsignal p through a linear model, (9)

(8)

(7)

w = LXiei +wi¥-n

J.-t n = E[wlz] = LE[xilz]ei'i¥-n

~n = cov(w, wlz) = Lvar(xilz)eief + R. (10)i¥-n

and

Pn(a) = TID [xn = a I z], a E X, n = 1,2, ... , N. (6)

In addition, the PDA assumes that the posteriors of all symbolsexcept X n computed thus far are the true posteriors. Thus, themean J.-t n and covariance ~n of the approximating Gaussiandistribution are easily obtained by evaluating the mean E[Xilz]and variance var(Xi Iz) of a discrete random variable withknown distribution.

Having approximated the statistics of the interference plusnoise in (7), the distribution of the received signal z condi­tioned on X n = a is complex Gaussian,

pzlxn (z I a) = eN(z; aen + J.-tn , ~n), a E X. (11)

I To simplify the exposition, we assume that Gaussian distributions through­out the paper are circularly symmetric. Algorithm development in the realdomain as well as the complex domain with non-circular symmetry isanalogous.

z = xnen + LXiei + w,i¥-n

where {ei}i=l is the standard basis (i.e., ei contains 1 inentry i and zeros elsewhere). The main idea behind the PDAalgorithm is to treat the contribution from all symbols exceptX n to the received signal as interference and approximate thesum of the interference and noise with a Gaussian distribution.That is, when detecting X n , the interference plus noise in (7)

is approximately eN(J.-tn, ~n)l. The mean J.-t n and covariancematrix ~n are matched to the mean and covariance of W, i.e.,

The algorithm operates as follows. Suppose it is at iterationi and X n is the desired symbol. Then the received signal zin (5) is expressed as

(3)

(1)

(2)

(4)

p=Px.

y=Gx+v,

G = VSNRHP

y=VSNRHp+v,

where

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( ) _ pzlxn (z I a) cy (12)Pn a - '""'" ( I !)' a E .A, ,L.Ja'E'XPzlxn z a

Therefore, the posterior Pn of a symbol X n is updated viaBayes' rule

(21)

(19)

( )_ Pz(z;xn=a)

Pn a - ~ ( . _ _ ')' a E X,a'E'X pz Z, X n - a

N

~n = cov(w, wlz) = 2:var(xilz)eief + R.i=1

and

where the summation in the denominator is over the finitealphabet X.

A pseudo-code description of how the SU-PDA algorithmevaluates the posteriors of binary phase shift keying (BPSK)modulated symbols is given in Algorithm 1. Note that theconditional mean (18) and variance (19) of symbol X n takingvalues from X = {+1, -I} are, respectively, 2p~) - 1 and4p~) (1 - p~)), where p~) is the estimate of TID [xn = 1 I z] atiteration i.

Note that the mean vector J..L n is the same as in theconventional PDA (9). On the other hand, the summation in theexpression for covariance matrix (19) includes contributionsfrom all symbols, as opposed to the conventional PDA (10).

Given that interference plus noise w is Gaussian distributed,the probability distribution of the received signal z, parame­terized by Xn , is

is,

The SU-PDA updates posterior Pn of X n by normalizing theparameterized distributions with respect to parameter x. That(13)

(14)

Xn = argmaxPn(a).aE'X

where the summation in the denominator is over alphabet X.Note that in (12) we exploit the assumption that symbols haveuniform prior.

The PDA algorithm then continues and updates the posteriorof the next symbol following some prespecified/adaptive or­dering scheme. After a certain number of iterations or untilconvergence is established, it outputs the estimates of theposteriors Pn. Hard estimates xn of the corresponding symbolsare obtained using the maximum a posteriori probability(MAP) rule,

IV. PROPOSED ALGORITHMS

This section presents new algorithms based on PDA. Thecommon feature of the conventional PDA and the proposedalgorithms is that they use the same model (5) (or (3)) and inferposterior distribution Pn of transmitted symbol X n , conditionedon received signal z (or y).

A. PDA with Symbol Uncertainty (SU-PDA)

Suppose we are detecting symbol X n . The main idea behindthe SU-PDA algorithm is to model X n as

where xn is a deterministic, unknown variable from finitealphabet X and xn is a zero mean random variable whosevariance is equal to the variance of X n conditioned on z andevaluated from the current estimate of Pn,

(15)

Intuitively, xn captures the uncertainty in the current knowl­edge about X n .

Substituting (14) into (7) yields

z = xnen + xnen + 2: Xiei + w. (16)i¥-n

Algorithm 1 SU-PDA detection of BPSK symbolsRequire: received signal z

Ensure: PI,P2,··· ,PNI .. 1· (0) (0)nltIa lze PI , ,PNfor i = 1,2, , I do

for n = 1 to N do- '""'" (2 (i-I) 1)J..L n - L.Jk¥-n Pk - ek

~ - '""'" 4 (i-I) (1 (i-I)) T~n - L.Jk Pk - Pk ek e k

(i) _ eN(z;en+JLn,~n)

Pn - eN(z;en+JLrl'~n)+eN(z;-en+JLn,~n)end for

end for

The contribution to the received signal z from the symbolsother than X n is treated as interference, while xn en is viewedas an additional noise term. Overall, the distribution of thesum of the interference and noise

is approximated with complex Gaussian distribution with themean J..L n and covariance ~n. As in the conventional PDA,the current estimates of posteriors {Pi}~1 are used to eval­uate the mean and covariance of the approximating Gaussiandistribution such that

B. PDA with Gaussian Approximation (G-PDA)

G-PDA starts with model (5) and updates posterior Pn. Asin the original PDA, the contribution of other symbols to thereceived signal z is viewed as interference. The distribution ofthe interference plus noise w is approximated with complexGaussian distribution with the mean vector J..L n and covariancematrix ~n, evaluated as in (9) and (10).

In the first step of the G-PDA algorithm, the expected valueof the interference is subtracted from the received signal.Since the noise w has zero mean, the expected value of theinterference is J..L n and the obtained interference-free signal is

W = xnen + 2:xiei + wi¥-n

J..L n = E[wlz] = 2: E [xilz]ei'i¥-n

(17)

(18)z=z-J..Ln · (22)

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The interference-free signal z is using (5) modeled as

Note that qrveN(O, ~n), where ~n is evaluated using (10).In the second step of G-PDA, the signal z in (23) is

processed with the minimum mean square error (MMSE) filterin order to estimate symbol X n . The impulse response of theMMSE filter is given in vector form by

(34)

After substituting (27) and (29) into (32), the mean andvariance of the a posteriori Gaussian distribution are given by

I e~~;;1 (z - J-Ln) 12 1f-L n = 1 T~-1 and an = T -1 . (33)

+en n en 1 +en~n en

Finally, the posterior Pn (a) is updated by integrating Gaus­sian distribution eN(f-L~, a';) over region Z(a) in the complexplane which is defined as the set of points which are closer inthe Euclidean sense to the constellation point x = a than toany other constellation point from X. Formally, we have

(23)

(24)q = L::xnen - J-Ln + w.i#n

where q is the equivalent noise, given by

Finally, substituting (23) into (27), yields a simple modelfor 20

The output from the MMSE filter is scaled and sufficientstatistic 20 is given by

where

Algorithm 2 G-PDA with single Gaussian approximationRequire: received signal z

Ensure: Pl,P2,··· ,PNI .. 1· (0) (0)nltIa lze PI , ,PNfor i = 1,2, , I do

for n = 1 to N do- ~ (2 (i-I) 1)J-L n - ~k#n Pk - ek~ 4 (i-l)( (i-I)) T

~n = ~k#n Pk 1 - Pk eke kI (1 T~-I) -1 T~-1 ( )f-Ln = + en~n en en~n z - J-L n12 (1 T~-I)-1an = +en~n en

p(i) = 0.5 + 0.5erf(~)n /2;;2iV~Un

end forend for

Z(a) = {z Illz - all < liz - a'ii, a' EX}. (35)

For BPSK modulated symbols where X = {+1, -I},Z(l) = {z E ~ I z > O} and thus the posterior Pn(l) isupdated with the probability that a random variable distributedaccording to (31) is positive. Hence,

Pn(xn = 1) = lP' [N(JL~,a;;) > OJ = ~ + ~erf ( ~),2 2 V 2a;'

(36)where the error function is defined as

2 [X 2

erf(x) = y'1r io e- tdt. (37)

A pseudo-code description of how the G-PDA algorithmwith a single Gaussian approximation updates and evaluatesthe posteriors of BPSK symbols is given in Algorithm 2.

(29)

(27)

(28)

(26)

(25)

(30)PXn(X) =eN(x;O,l).

where qrveN(O, a 2 ). The variance a 2 is evaluated as

2 1a =--~-T~-1en~n en

20 = X n + q,

In the final step, symbol X n is detected from 20 usingmodel (28). It can be confirmed that if X n is modeled asa uniform discrete random variable over alphabet X, theconventional PDA algorithm is obtained. Instead, dependingon how the distribution of X n is modeled, we arrive to twoversions of G-PDA.

1) Modeling X n with a single Gaussian: In this version, wemodel X n as a single Gaussian random variable whose meanand variance match the mean and variance of X n . Since X n isa point from a finite constellation, without loss of generality,we have

Substituting (25) into (26) and using the matrix inversionlemma for the inverse of the rank one update of a matrix yields

T~-1- en~n-Zo = T -1 z.

en~n en

Hence, given that both X n and q in (28) are Gaussiandistributed scalars, the posterior of X n conditioned on 20 (andtherefore on z because 20 is a sufficient statistics) is alsoGaussian, i.e.,

2) Modeling X n with a Gaussian Mixture: In this versionof the G-PDA, we assume the prior on X n is a Gaussianmixture whose components have means taking values from thealphabet X, the same variance a5 and equal weights. Formally,

where the mean and variance are, respectively, given by(38)

I Zof-Ln = ---2 andl+a

12 a 2a =---

n 1 + a 2 ·(32)

PXn(x) = I~I L::eN(x;a,a6),aEX

where a5 is a parameter that can be tuned. The posterior of X nis using the Bayes' rule determined to be a Gaussian mixture.

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As an example, we consider the BPSK modulation whereX = { +1, -I}. The distribution of X n conditioned on 20 (andthus on z) is using the Bayes' rule given by

Pxnlz(xlz) = WI eN(x; J11, a /2) + W2eN(X; J12, a /2

), (39)

where20a5 ± a2 12 a2a5

/I. - and a = (40)fA'1,2 - a2 + a5 a2 + a5'

while the weights are

WI 2 ex: exp (- ~zo2=f 1)~)) where WI + W2 = 1. (41), 2 a + ao

The version of the G-PDA with Gaussian mixture which doesnot update the weights WI and W2 i.e., keeps them equal isreferred to as G-PDA with partial update.

The posterior Pn (a), a E X is updated by integrating theGaussian mixture Pxnlz(xlz) as in (34) over region Z(a),defined in (35). For the BPSK modulation, we obtain

Pn(1) = 0.5 (1 + wlerf (V~~'2) + w2erf (V~:'2) ) .(42)

A pseudo-code description of how the G-PDA algorithmwith a Gaussian mixture updates and evaluates the posteriorsof BPSK symbols is given in Algorithm 3.

Algorithm 3 G-PDA with a Gaussian mixtureRequire: received signal z, variance a5Ensure: Pl,P2,··· ,PN

I .. 1· (0) (0)nlt1a lze PI , , PNfor i = 1,2, , I do

for n = 1 to N do- '"" (2 (i-I) 1)J-L n - Lk:f-n Pk - ek

~ - '"" 4 (i-I) (1 (i-I)) T~n - Lk:f-n Pk - Pk ek e k- (T~-I) -1 T~-1 ( )Zo = en~n en en~n z - J-L n

2 _ (T -1 ) -1 . _ zoa6 ±a2• 12 a 2 a6a - en~n en ,J11 2 - ~+2 ,a - 2+ 2, a ao a ao

( (zo =f 1)2) h - 1Wl,2 ex: exp - 2(a 2 +a5) ,were WI + W2 -

p~) = 0.5 (1 + wlerf (v'~~'2) + w2erf (v'~;'2))end for

end for

Once the posterior Pn of a symbol Xn is updated, the SU­PDA and G-PDA update the posterior of the next symbolfrom some ordering scheme and after a certain number ofiterations or after it has converged, the algorithm outputs thefinal estimates of the posteriors and hard estimates of thetransmitted symbols obtained using the MAP rule (13).

V. VALIDATION RESULTS

This section presents the performance results of the SU­PDA and G-PDA algorithms. The first part of this sectionpresents simulation results on fast fading channel. The secondpart presents the results obtained from simulating the proposedalgorithms using the channel impulse responses measured inan indoor commercial setting.

Fig. 2. Block size N = 16, QPSK modulation, DFT precoder, five-tapchannel with fast fading, ten iterations. G-PDA with partial update is G-PDAwith Gaussian mixture and no update on mixture weights.

A. Simulation on Fast Fading Channel

The BER performance of the communication system whichuses the proposed algorithms for symbol detection is evaluatedusing the Monte-Carlo simulations.

Equally likely quadrature phase shift keying (QPSK) sym­bols are grouped into blocks of size Nand precoded with thediscrete Fourier transform (DFT) precoder, whose

[P] - _1_ -j ~ (n-l)(m-l) - 1 2 N (43)n m - l7\Te , n, m - , , ... , ., yN

The obtained symbols are transmitted over a channel mod­eled as a linear time-varying filter which has five taps. Thecoefficients corresponding to a single tap exhibit fast fadingsuch that fdts = 1, where fd is the Doppler and t s is thesymbol duration.

We assume the receiver perfectly tracks the channel anddetects the transmitted symbols using the proposed algorithms.For reference, we also consider the MMSE receiver and theconventional PDA algorithm.

The BER performance of the described communicationsystem employing the MMSE, PDA, SU-PDA, G-PDA withsingle Gaussian approximation, G-PDA with Gaussian mixtureand G-PDA with partial update detection is shown in Fig. 2for N = 16 and Fig. 3 for N = 32. The BER performanceversus SNR for block size N = 64 and MMSE, original PDAand SU-PDA receivers is shown in Fig. 4.

As can be seen from the presented performance plots, theconventional PDA suffers from the error floor. On the contrary,the symbol detectors based on the PDA modifications donot exhibit an error floor in the region of considered SNR'sand the performance improvement is significant at SNR'scorresponding to the BER's of our interest. On the other hand,at relatively lower SNR's the original PDA has slightly betterperformance and the crossover point appears between 11-14dB and moves towards higher SNR's as N increases.

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Fig. 3. Block size N = 32, QPSK modulation, DFT precoder, five-tapchannel with fast fading, ten iterations. G-PDA with partial update is G-PDAwith Gaussian mixture and no update on mixture weights.

Fig. 4. Block size N = 64, QPSK modulation, DFT precoder, five-tapchannel with fast fading, ten iterations.

The comparison between different PDA modifications re­veals that all versions of the G-PDA outperform the SU-PDAfor N = 16 and N = 32. Among the G-PDA versions, the onewhich uses the Gaussian mixture approximation of distributionof the symbol being detected has a slightly better performancethan the one with a single Gaussian approximation.

B. Simulation on Experimentally Measured Channel

The BER performance of the PDA algorithm and proposedmodifications SU-PDA and G-PDA when used to detect thesymbols received from the real, experimentally measuredchannel is presented in this part.

The channel transfer function is experimentally measuredin an indoor environment. The realizations of the channelimpulse response at 2.4 GHz band and of 5 MHz bandwidthare calculated from the channel transfer functions. One suchrealization corresponding to the non-line of sight (NLOS)

Fig. 5. One realization of a measured channel impulse response.

channel with transmitter-receiver range of 50 m is shownin Fig. 5. For simulation results presented in this sectionwe had access to total of rv 700 measured power delayprofile (PDP) realizations, and we obtained additional channelimpulse responses by employing random phase rotations tothe measured PDP tap coefficients.

The obtained realizations of the channel impulse responseare used in computer simulations of the M2M communicationsystem. Equally likely QPSK symbols are generated, groupedinto blocks of size Nand precoded. The blocks of symbols aretransmitted through a channel with the impulse response beingone of the experimentally obtained channel impulse responses.A response is normalized to 1 such that the channel gainis 1 and hence the evaluated BER performance correspondsto the received SNR. Since the considered block sizes arerelatively short, we assume the channel is time-invariant duringthe transmission of a single block.

The BER versus received SNR performance of the receiversbased on the MMSE, PDA, SU-PDA, G-PDA with a singleGaussian, G-PDA with a Gaussian mixture and G-PDA withpartial update (i.e. weights in the mixture are not updated) anda5 = 1, is shown in Fig. 6 for block size N = 16 and in Fig. 7for block size N = 32.

Two types of precoders are considered in this study: theDFT precoder and identity precoder. Since the channel isassumed invariant during the transmission of a block, pre­coding does not help in achieving the diversity. In addition,it destroys the diversity offered by the multi-tap (frequencyselective) channel. On the other hand, if precoding is skipped(formally, identity precoder is used), the PDA algorithm andits modifications extract the diversity offered by the channel.This can be seen from the performance plots.

The SU-PDA and G-PDA behave worse than the originalPDA at relatively small SNR. However, the proposed algo­rithms start outperforming the PDA at around 12 dB andthe performance gain increases as SNR increases. Thus, at

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Fig. 6. Block size N = 16, QPSK modulation, measured channel, teniterations. Remark: all algorithms yield the same performance with DFTprecoder. All other plots correspond to identity precoder.

Fig. 7. Block size N = 32, QPSK modulation, measured channel, teniterations. Remark: all algorithms yield the same performance with DFTprecoder. All other plots correspond to identity precoder.

SNR of 16dB and above, the SU-PDA and G-PDA attainat least one order of magnitude lower BER compared to theoriginal PDA. More importantly, this happens at SNR's whichcorrespond to BER's of the interest for M2M modem. Amongthe PDA modifications, the G-PDA with Gaussian mixtureapproximation behaves slightly better than the SU-PDA.

VI. CONCLUSIONS

PDA algorithm has shown promising performance in sym­bol detection and interference cancellation in a variety ofcommunication techniques. One such application is symboldetection in a SISO system with block transmission anddetection. This technique is envisioned to provide fast andhighly reliable transmission of short messages over wirelesschannels. These requirements constitute design goal for theM2M communication system.

This paper proposes new algorithms based on PDA whichoutperform the conventional PDA algorithm for short blocks« 100 symbols) and at SNR's corresponding to BER'srequired for M2M communications. The proposed algorithmsdiffer from the original PDA in the way they treat the symbolbeing estimated. As such, the PDA-SU models such a symbolas a sum of a discrete deterministic variable and zero meannoise which captures the uncertainty in our knowledge aboutthat symbol. On the other hand, the G-PDA models thissymbol as a random variable distributed either as a singleGaussian or Gaussian mixture.

The proposed algorithms have been tested via computer sim­ulations using both simulated and real channel. The simulatedchannel is multi-tap with fast fading, while for real channelwe use the channel impulse responses that were experimentallymeasured. The tests on both channel types have shown that theproposed algorithms outperform the conventional PDA.

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