Steepest Descent Multimodulus Algorithm for Blind Signal Retrieval in QAM Systems
aa r X i v : . [ ee ss . SP ] A ug Steepest Descent Multimodulus Algorithmfor Blind Signal Retrieval in QAM Systems
Shafayat Abrar a a Associate ProfessorSchool of Science and EngineeringHabib University, Gulistan-e-JauharBlock 18, Karachi 75290, PakistanEmail: [email protected]
Abstract
We present steepest descent (SD) implementation of multimodulus algorithm (
MMA2-2 ) for blind signal retrieval in digital com-munication systems. In comparison to stochastic approximate (gradient descent) realization, the proposed SD implementation of
MMA2-2 equalizer mitigates inter-symbol interference with relatively smooth convergence and superior steady-state performance.
Keywords:
Blind equalization; multimodulus algorithm (
MMA2-2 ); steepest descent; adaptive filter; channel equalization
1. Introduction
The multimodulus algorithm (
MMA2-2 ) [1, 2] is given as w n + = w n + µ ( R m − y R , n ) y R , n x n − j µ ( R m − y I , n ) y I , n x n , (1)where, j = √− R m is a positive statistical constant, x n is chan-nel observation vector, w n is equalizer vector, and y n = w Hn x n = y R , n + j y I , n is equalizer output. The update (1) is probably themost popular and widely studied multimodulus algorithm capa-ble of equalizing multi-path transmission channel blindly andrecovering carrier phase jointly in quadrature amplitude mod-ulation based wireless, wired and optical communication sys-tems. The update, however, is stochastic approximate in nature,works on symbol-by-symbol basis, and is relatively slower inconvergence when compared to its batch counterparts. More-over, even in successfully converged state, the error functionin update expression is non-zero except for instances when ∣ y ⋅ , n ∣ = √ R ; these fluctuations (as quantified in [3]) cause de-lay in switching to decision-directed mode and lead to decisionerrors causing loss of information.In order to exploit full potential of MMA2-2 , there is a newpractice in literature to realize it in batch mode. In this context,Han et al. discussed a number of methods including steepest de-scent implementation for constant modulus algorithms (
CMA )and relaxed convex optimization for
MMA2-2 in [4] and [5],respectively. In [6], Shah et al. discussed batch
MMA2-2 by ex-ploiting iterative blind source separation framework and cameup with Givens and hyperbolic rotations based batch
MMA2-2 . Also in [7], authors transformed
MMA2-2 cost into an an-alytical problem and solved that for both batch and adaptiveprocessing using subspace tracking methods. The most rigor-ous treatment appeared in [8], where a batch
MMA2-2 is ob-tained which included an analytical transformation to a set ofcoupled canonical polyadic decompositions by using subspace methods. Recently, Han and Ding [9] suggested a steepest de-scent batch implementation of a class of CM algorithms wherethe update process did not require equalizer outputs (no feed-back) and rather relied directly on statistics obtained from thereceived signal. Motivated by that approach, in this correspon-dence, we present a steepest descent implementation of MMA2-2 by estimating required batch statistics iteratively while main-taining simplicity of its adaptive structure. To the best of ourknowledge, a steepest descent implementation of
MMA2-2 hasnot been realized in literature.
2. Feedforward Steepest Descent Algorithms
In order to realize a steepest descent implementation of (1),we need to estimate expected value of its error function. w n + = w n + µ E [( R m − y R , n ) y R , n x n − j ( R m − y I , n ) y I , n x n ] (2)We evaluate this expectation in forward driving manner as ad-vocated in [9]. According to which, we replace y n with w Hn x n ,and evaluate statistical average of matrix quantities involving x n conditioned on w n . Exploiting the facts that y R , n = ( w Hn x n + x Hn w n ) (3a) y I , n = ( w Hn x n − x Hn w n ) (3b)and after some manipulations, we obtainE [( R m − y R , n ) y R , n x n − j ( R m − y I , n ) y I , n x n ]= E [( R m x Hn w n − ( x Hn w n ) w Hn x n − ( w Hn x n ) ) x n ]= R m E [ x n x Hn ] w n − E [ x n x Hn w n w Hn x n x Hn w n ] − E [( w Hn x n ) x n ] , (4) Uploaded to sharing knowledge with community. August 06, 2017 e can show that E [ x n x Hn w n w Hn x n x Hn w n ] = E [ x n x Hn w n w Hn x n x Hn ] w n = E [ mat [ vec [ x n x Hn w n w Hn x n x Hn ]]] w n = E [ mat [(( x n x Hn ) T ⊗ ( x n x Hn )) vec [ W n ]]] w n = mat [ E [( x n x Hn ) T ⊗ ( x n x Hn )] vec [ w n w Hn ]] w n , (5)The matrix operation, mat[ ⋅ ], as used in (5), however, is not anorthodox procedure, and is not supported necessarily by tradi-tional digital signal processors. To resolve this, alternatively,we may obtain a more elegant expression as follows:E [( x Hn w n ) w Hn x n x n ] = E [ x Hn w n x Hn w n w Hn x n x n ]= E [ x n x Hn w n x Hn w n x Tn w ∗ n ]= E [ x n vec [ x n vec [ x n x Hn ] T ] H ] vec [ w n vec [ w n w Hn ] T ] (6)Further, one may obtain:E [( w Hn x n ) x n ] = E [ vec [( w Hn x n ) x n ]]= E [ vec [ x n w Hn x n w Hn x n w Hn x n ]]= E [( x Tn ⊗ x n )( w ∗ n ⊗ w Hn )( x Tn ⊗ x n ) vec [ w Hn ]]= E [( x Tn ⊗ x n )( w ∗ n ⊗ w Hn )( x Tn ⊗ x n )] w ∗ n (7)However, computing a statistics of x n involving w n is inadmis-sible. One of the feasible solutions is to evaluate:E [( w Hn x n ) x n ] = E [ x n ( w Hn x n w Hn x n w Hn x n )]= E [ x n vec [ x n vec [ x n x Tn ] T ] T ] vec [ w n vec [ w n w Tn ] T ] ∗ (8)Next, we can estimate required statistics either by taking en-semble average over a batch of data or iteratively updating theestimate at each time index. At index n , an iterative estimateof expectation E [ f n ] , where f n is some matrix with randomvariable’s entities, may be obtained as S n = ( − λ ) S n − + λ f n ,0 < λ <
1. Next, using S In , S IIn , and S IIIn to denote iterativeestimates of E [ X n ] = E [ x n x Hn ] , E [ x n vec [ x n vec [ x n x Hn ] T ] H ] ,and E [ x n vec [ x n vec [ x n x Tn ] T ] T ] , respectively, we obtain feed-forward steepest descent MMA2-2 ( SD-MMA2-2 ) as given by: In (5), ⊗ denotes Kronecker product where each element of ( A ⊗ B ) ∈ C mp × nq is the product of an element of A ∈ C m × n and an element of B ∈ C p × q ;the element in the [ p ( i − ) + r ] th row and [ q ( j − ) + s ] th column of A ⊗ B is the rs th element a ij b rs of a ij B [10]; vec [ A ] is vector-valued function whichassigns a (column-vector) value to A such that the i j th element of A is the [( j − ) m + i ] th element of vec [ A ] [10], and the mat [ a ] is a reverse operationwhich converts an N × a back to an N × N square matrix form [9]. SD-MMA2-2 w n + = w n + µ R m S In w n − µ S IIn vec [ w n vec [ w n w Hn ] T ] − µ S IIIn vec [ w n vec [ w n w Tn ] T ] ∗ , S In = ( − λ ) S In − + λ x n x Hn , S IIn = ( − λ ) S IIn − + λ x n vec [ x n vec [ x n x Hn ] T ] H , S IIIn = ( − λ ) S IIIn − + λ x n vec [ x n vec [ x n x Tn ] T ] T . (9a)(9b)(9c)(9d)Considering a fixed channel, assume that the (steady-state)estimates of statistics S In , S IIn and S IIIn are available, say fromthe received large batch of data. Now, solving ∂ J / ∂ w ∗ = and exploiting these available statistics, we obtain the followingo ffl ine fixed-point steepest descent algorithm: w ←Ð [ S I ] − R m ( S II vec [ w vec [ ww H ] T ] + S III vec [ w vec [ ww T ] T ] ∗ ) (10)where S I , S II and S III are o ffl ine estimates of S In , S IIn and S IIIn ,respectively. However, note that the iteration (10) is found to bediverging which is a common problem in fixed-point procedurewhen matrix inverse is involved; see [11, eq. (21) and detailstherein]. To improve this situation, we add a step-size in (10),obtaining a stabilized (o ffl ine) fixed-point algorithm: FP-MMA2-2 w ←Ð w + µ ( R m S I w − S II vec [ w vec [ ww H ] T ] − S III vec [ w vec [ ww T ] T ] ∗ ) (11)where µ is step-size which may be made adaptive with iterationcount. It is observed that a more certain convergence may beensured if a µ much smaller than unity is selected (say, 0.1 or0.01 for 4- or 16- QAM , respectively, with N =
3. Simulation Results
We examine performance of proposed algorithm for the mit-igation of interference caused by two Baud-spaced channelsfor 16-
QAM signaling. The first channel, channel-1, is avoice-band telephone channel h n = [ − . − . , . + . , − . − . , . + . , − . + . , . − . , − . + . ] taken from [15]. The second chan-nel, channel-2, has a relatively large eigen-spread, and is2iven as h n = [ − . − . , . − . , . − . , . + . , . + . , − . − . − . − . , . + . , . + . ] . Thesignal-to-noise-ratio is 30 dB. The equalizer length is 15, ini-tialized with a unit spike at center tap, and all algorithms usestep-size of 10 − .The ISI measure in dB at n th time index is ISI n =
10 log [ N runs N runs ∑ k = ∑ i ∣ t n , k ( i )∣ − max {∣ t n , k ∣ } max {∣ t n , k ∣ } ] (12)where t n , k is the overall channel-equalizer impulse responsevector at index n in the k th run of simulation. t n , k ( i ) repre-sents the i th entity of t n , k , and max {∣ t n , k ∣ } represents the largestsquared amplitude in t n , k .For fixed channels, we choose λ = / n ( n is time index)so that the required statistics are estimated over all receiveddata. Fig. 1(a) demonstrates convergence behaviors of MMA2-2 and
SD-MMA2-2 , averaged over 400 and 50 independent runs( N runs ), respectively. We notice that the ISI mitigation achievedby
SD-MMA2-2 is far better in steady-state when allowed toconverge at the same rate as that of
MMA2-2 . In Fig. 1(b),single trajectory of
ISI convergence of each
MMA2-2 and
SD-MMA2-2 is shown. We can note that the
SD-MMA2-2 exhibitsfar smoother and more stable convergence than
MMA2-2 (forfixed channel scenario), and this is the reason why we usedfewer independent runs for the ensemble averaging of
ISI tra-jectories in
SD-MMA2-2 than
MMA2-2 .
4. Conclusions
A steepest descent implementation of
MMA2-2 for blind sig-nal recovery has been proposed and demonstrated to mitigate
ISI . The proposed equalizer has been found to yield bettersteady-state performance than stochastic approximate gradientdescent
MMA2-2 . Thus, the proposed approach seems to bequite a promising substitute for traditional counterpart on fixedchannels. Future work includes: (a) application to time-varyingchannels, (b) evaluation of optimal step-sizes, and (c) applica-tion to
MIMO systems.
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