# Reduced state adaptive SISO algorithms for serially concatenated

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Reduced state adaptive SISO algorithms for serially concatenated

Reduced State Adaptive SISO Algorithms for Serially Concatenated CPM over Frequency-Selective Fading Channels Kyuhyuk Chung, Keith M. Chugg, and Jun Heo Electrical Engineering–Systems Dept., University of Southern California, Los Angeles, CA 90089-2565 Abstract — Iterative detection (ID) has proven to be a near-optimal solution for concatenated Finite State Machines (FSMs) with interleavers over an additive white Gaussian noise (AWGN) channel. When perfect channel state information (CSI) is not available at the receiver, an adaptive ID (AID) scheme is required to deal with the unknown, and possibly time-varying parameters. The basic building block for ID or AID is the soft-input soft-output (SISO) or adaptive SISO (A-SISO) module. The complexity of SISOs and A-SISOs increases exponentially with constraint length. In this paper, Reduced State SISO (RS-SISO) algorithms have been applied for complexity reduction of the A-SISO module. We show that serially concatenated CPM (SCCPM) with AID has turbo-like performance over fading ISI channels and also RS-A-SISO systems have large iteration gains. Various design options for RS-A-SISO algorithms are evaluated and performance and complexity is compared. I. I NTRODUCTION Iterative detection (ID) or turbo decoding is a good approximation to the optimal decoder for a concatenated network of Finite State Machines (FSMs) constructed from constituent codes and intersymbol interference (ISI) channel memory. In [9], Trellis Coded CPM (TCCPM) on multipath fading ISI channels was studied without iterative detection. It is shown in [10] that by exploiting the recursive nature of CPM, TCCPM can have large interleaver gains similar to Serially Concatenated Convolutional Codes (SCCCs) [3]. Similar independent work with iterative detection was reported in [7]. None of papers above considers SCCPM [7] over fading ISI channels with iterative detection. Adaptive SISO (A-SISO) modules [4] were developed as a natural extension of Soft-Input Soft-Output (SISO) modules [2] for use when perfect CSI is not available at the receiver. In [4], various A-SISO algorithms were defined and derived, which produces a family of suboptimal practical algorithms. Since the complexity of SISO algorithms increases exponentially with constraint length, Reduced State (forward-backward) SISO (RS-SISO) algorithms were proposed independently for complexity reduction at the same time in [11] [12]. In this paper, the RS-SISO algorithm is used for complexity reduction of the A-SISO module, namely RS-A-SISO algorithms. RS-A-SISO modules are applied to SCCPM over frequency-selective fading channels when perfect CSI is not available and trade-offs between performance and complexity are considered. RS-SISO algorithms are of paramount importance in A-SISO algorithms because the complexity of A-SISO algorithms is higher than that of SISO algorithms due to channel estimation. When the input of a joint CPM-ISI trellis is binary, multiple-step RS-A-SISO algorithms [13] can be a design option for improved performance with slightly increased complexity. This paper organized as follows. Section II describes the system and channel model. In section III we explain various RSA-SISO algorithms. Section IV presents numerical results and Section V concludes the paper. II. S YSTEM D ESCRIPTION AND C HANNEL M ODEL A block diagram of SCCPM on an ISI and AWGN channel is shown in Figure 1. After the source bits b n are fed to a convolutional code (CC), the coded bits c m are bit-interleaved and continuous-phase-modulated. The modulated signals a(t) are transmitted through an ISI and AWGN channel. The carrier phase is modulated by a positive normalized frequency pulse g (t), containing no impulses and being non-zero for L symbol intervals [8]. Its corresponding phase pulse q (t) is the integral of g (t), normalized to q (LT ) = 1=2 with a symbol time T . The modulation index is assumed to be rational and irreducible, h = K=P , and then the CPM modulator can be viewed as a continuous-phase encoder (CPE) followed by a memoryless modulator (MM) [17]. For practical purposes, we can assume that W is the bandwidth occupied by the real bandpass CPM signal. Then the band occupancy of the complex baseband equivalent lowpass CPM signal is jf j W=2. A tapped-delayline (TDL) model is usually used for a time-varying frequencyselective fading channel The lowpass impulse response for the channel is f (t; ) = Nc X i=0 fi (t)Æ ( iTc ) (1) where Tc is the channel resolution which satisfies T c 1=W , fi (t) = Tc f (t; iTc) is the tap weight coefficient, and (N c +1) is the length of the channel in terms of T c . Tc is chosen such that symbol time T = Ns Tc with Ns being an integer. In complex baseband representation, the received continuous-time signal is given by r(t) bn CC encoder x(t) + n(t) cm ck Interleaver CPM modulator a(t) ISI fading channel w(t) + AWGN SO(bn) CC SISO 0-7803-7206-9/01/$17.00 © 2001 IEEE 1162 = Deinterleaver Interleaver joint CPM-ISI A-SISO Fig. 1. System model. r(t) LPF = = Z1 0 Nc X i=0 f (t; )a(t fi (t)a(t )d + n(t) iTc ) + n(t) (2) where n(t) is the AWGN. Conventionally, the received signal r(t) is passed through a matched-filter x ( t), followed by a sampler and a noise whitening filter. But since CPM is a nonlinear modulation, a bank of matched-filters has to be implemented. In [9] a low-pass filter (LPF) with two-sided bandwidth W is used, followed by fractionally-spaced sampling every T c . Then the k th sample of r(t) is rk r(kTc ) Nc X = i=0 Nc X = i=0 fi (kTc )a(kTc iTc ) + n~ (kTc ) fi;k ak i + nk (3) where n ~ (t) is filtered white noise and n k is complex zeromean Gaussian distributed with variance N 0 . The coefficient ffi;k gNi=0c is a time-varying tap vector at time k and fa k i giN=0c are fractionally-spaced signal samples during an ISI-channel length (Nc + 1). The equation (3) can be used as an equivalent Discrete-Time (DT) model for CPM on fading ISI channel [9]. The low-pass-filtered and sampled observations are fed to the inner joint CPM-ISI A-SISO module [4]. The outer CC SISO [2] and the inner A-SISO are connected by an interleaver and a deinterleaver to exchange soft information iteratively. The inner A-SISO re-estimates the ISI channel each iteration based on [4]. III. R EDUCED S TATE A DAPTIVE SISO A LGORITHMS The SISO module [2] implements an algorithm performing an update of the reliability information of the input and the output symbols of a FSM. A-Posteriori Probability (APP) forwardbackward algorithms (FBAs) and Minimum Sequence Metric (MSM) FBAs [1] produce soft information that is thresholded to obtain the Maximum A-posteriori Probability (MAP) Symbol Detection and the MAP Sequence Detection respectively. By replacing min-sum operations by sum-product operations in the MSM FBA, one directly obtain the APP FBA. In the following we consider only the MSM FBA. MSM is defined by MSM(u) = min[ a:u ja)p (a))] ln(p(z MSMK k 1 (sk ) = = ^ f(sk+1 ) = ^f(sk ) + [zk T ^ f (sk ) a(tk )]a (tk ) (7) c where a(tk ) = fak i gN i=0 are fractionally-spaced signal samples during an ISI-channel length (N c + 1), is a suitable constant, and the forward channel estimate is initialized by running the Recursive Least Squares (RLS) algorithm on a training sequence. RS-SISO algorithms also have two options, namely forward decision feedback, backward decision feedback (FDF/BDF) structures and forward decision feedback (FDF) structures [11] [12]. In the FDF/BDF structure, survivors are constructed both in the forward and in the backward direction. In the FDF structure, survivors are constructed only in the forward direction, all transition metrics calculated in the forward recursion are saved, and the backward recursion is performed without survivors. In this paper, the FDF structure is used. RS-SISO with the FDF structure is consistent with the FA structure in A-SISO. The forward and backward state metrics F(v k ) and B(vk ) are defined for the RS-SISO algorithm recursively as F(vk+1 ) = min [F(vk ) + Mk (k )] (8) B(vk ) vk :vk+1 = vk+1 :vk min [B(vk+1 ) + Mk (k )] (9) (4) where z is an observation sequence, u is any quantity derived from the system input-output sequence pair (a; x(a)), and the notation a : u means all a consistent with u. The forwardAdd Compare Select(ACS) and backward-ACS recursions in the MSM algorithm are MSMk0 (sk+1 ) where MSMkk12 (sk ) denotes soft information of a state at time k based on observation z kk12 and Mk (tk ) is the transition metric of tk . When perfect CSI is not available at the receiver, the SISO module should be able to deal with unknown, and possibly timevarying parameters. SISO modules that jointly estimate the parameters and generate soft information will be referred to as Adaptive SISO (A-SISO) modules [4]. When we construct ASISO algorithm whether Fixed-Interval (FI) or Fixed-Lag (FL), two options are available, namely forward adaptive, backward adaptive (FABA) structures and forward adaptive (FA) structures [1] [4] [5]. In the first one, adaptive processing is performed both in the forward and in the backward direction. In the second approach, adaptive processing is only performed in the forward direction and all transition metrics associated with the forward adaptive step are saved for the non-adaptive backward recursion. In this paper, FA structure is used assuming a training sequence only at the beginning of data blocks. The Least Mean Square (LMS) algorithm is used for forward channel estimation.The LMS algorithm for updating a channel tap vector ^f(sk ) in the forward recursion is given by where vk is the truncated trellis state at time k and k = is the corresponding truncated trellis transition at time k . After executing both the forward and backward recursions, the soft outputs for a k can be obtained by (vk ; vk+1 ) k 1 min [MSM0 (sk ) + Mk (tk )] (5) tk :sk+1 K 1 min [MSMk+1 (sk+1 ) + Mk (tk )] (6) tk :sk 1163 MO(ak ) = min [F(vk ) + Mk (k ) vk+1 :ak +B(vk+1 )] MI(ak ): (10) sk standard PSP sk+1 s x s x 10 ME-A-SISO Perfect CSI AWGN s x x s s : survivor o o o o 10 -1 10 -2 10 -3 x : non-survivor multiple-step PSP sk+2 x sx x x x xs o : channel estimation Bit error rate sk 0 sk+2 64 8 16 Fig. 2. Standard PSP and multiple-step PSP. 8 RS-SISO algorithms can be used in A-SISO algorithms to produce soft-outputs for iterative detection with reduced complexity when perfect CSI is not available. These combined SISO algorithms will be referred to as RS-A-SISO algorithms. In RS-A-SISO algorithms the state-survivor paths selected for updating channel tap vectors are used for calculating transition metrics. In the following subsections various RS-A-SISO algorithms are described. A. Multiple-Estimator (ME) RS-A-SISO Algorithm In the ME-RS-A-SISO each state has a estimated channel tap vector ^f(vk ) that is updated in the Per-Survivor Processing (PSP) manner [15]. The reduced number of states means reduced channel-estimation complexity as well as ACS complexity. For the calculation of the truncated trellis transition metric both the estimated channel tap vector ^f(vk ) and the statesurvivor corresponding to the transition are required. The same state-survivors are also used in updating channel tap vectors. B. Single-Estimator (SE) RS-A-SISO Algorithm In the SE-RS-A-SISO every state has the common estimated channel tap vector ^fk that is updated employing the decisionfeedback (DF) assumption introduced in [16]. For updating the estimated channel tap vector ^fk the state with minimum metric is first chosen, the state-survivor path is traced back as many steps as tentative delay d, and then ^fk is updated with the truncated trellis transition k d that has the corresponding statesurvivor path. The single estimator structure reduces estimator complexity more than the multiple estimator structure. TABLE I C OMPLEXITY C OMPARISON : S = NUMBER OF STATES , M- ARY INPUT Algorithm ME-RS-A-SISO SE-RS-A-SISO MS-RS-A-SISO Number of transitions SM SM SM2 1/2 Channel updates S 1 (SM+S)1/2 64 10 16 -4 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 E /N [dB] b 0 Fig. 3. Performance of AWGN channel, perfect CSI, and multiple-estimator ASISO (d = 0:002). (For AWGN channel, the 8-state CPM SISO is used. For perfect CSI and ME-A-SISO, 64 is full state and 16 or 8 are reduced states.) C. Multiple-Step (MS) RS-A-SISO Algorithm The A-SISO algorithm based on a trellis in a fading channel is suboptimal due to a forced-folding of a tree [6]. But it is almost impossible to use a tree structure A-SISO for detection of large size sequences so that trellis structures are used in practice. The MS-RS-A-SISO uses a tree structure during two trellis steps instead of a trellis structure [13]. The channel adaptation scheme in MS-RS-A-SISO is shown in Figure 2. The channel updates are executed on all transitions instead of survivor paths in the first step of each multiple step, which means that non-survivors in standard trellis-based PSP can be a candidate for a survivor depending on the decision of the next step. In the second step of each multiple step, the channel updates are executed only on survivor paths. Complexity increases slightly with a relatively large performance gain. Complexity comparison is described in Table I using the number of transitions and channel updates per trellis step. IV. N UMERICAL R ESULTS In this section we present simulation results for the system described in Section II. The outer convolutional code of rate 1=2 and constraint-length 4 is used. The code-bits of the convolutional encoder are bit-interleaved using an interleaver with size 1800 = 30 60. Each block of 60 interleaved code-bits with an additional training sequence is referred as a burst. Bursts are continuous-phase-modulated with an M = 2, L = 2-RC, h = K=P = 3=4 partial response CPM scheme (L > 1), which 1164 10 0 10 0 ME-RS-A-SISO MS-RS-A-SISO 10 -1 10 -2 Bit error rate Bit error rate SE-A-SISO ME-A-SISO 10 -1 10 -2 10 -3 10 -4 8 16 8 16 10 -3 64 10 -4 5 6 7 8 9 E /N [dB] b 10 11 5 12 6 7 8 9 10 11 12 E /N [dB] 0 b 0 Fig. 4. Performance of single-estimator A-SISO and multiple-estimator ASISO (d = 0:002). (64 is full state and 16 or 8 are reduced states) Fig. 5. Performance of multiple-step RS-A-SISO and multiple-estimator RSA-SISO (d = 0:002). (16 or 8 are reduced states) has the following frequency pulse function 64-state joint trellis. The joint CPM-ISI FSM is recursive because the CPE FSM is recursive. The state of recursive FSM includes the accumulation (memory) of all the previous input symbols. When the number of states of the joint CPM-ISI FSM is reduced, survivors must have the accumulated value. For every Eb =N0 we use fixed values of the step size = 0:011 and = 0:04 for d = 0:002 and d = 0:01 respectively assuming the receiver doesn’t estimate signal-to-noise ratio (SNR). The optimized tentative delays found via simulation in the singleestimator scheme are d = 5 and d = 3 for d = 0:002 and d = 0:01 respectively. In this section, each system was identified by the number of states in the joint CPM-ISI A-SISO, i.e., 64 is full-state, 16 or 8 are reduced-state. This figure also includes a perfect CSI case and an AWGN channel case for comparison. For the AWGN channel, the 8state CPM-SISO is used. Note that ME-A-SISO algorithms show steep slopes as an AWGN channel and a perfect CSI channel. In Figure 4 the comparison of the adaptive iterative detection using SE-RS-A-SISO algorithms and ME-RS-A-SISO algorithms is shown for the fifth iteration. As the number of states is reduced, the performance of the single estimator schemes becomes closer to that of the multiple estimator schemes. At low Doppler spread with d = 0:002, the performance of SE-RSA-SISO algorithms is quite close to that of ME-RS-A-SISO algorithms. It is shown in Figure 6, however, that for more rapidly varying channels with d = 0:01 the SE-A-SISO with 64 full-states reaches an error floor above a BER 10 2 . At this Doppler spread d = 0:01 no error floor was observed for the ME-A-SISO with 64 full-states while the ME-RS-A-SISO with g (t) = 1 2LT [1 0; t )]; cos( 2LT 0 t LT otherwise (11) where T is a symbol time. The CPM signals of each burst are sent over Nc + 1 = 7 tap equal power Rayleigh fading channels with maximum normalized Doppler spreads d = fd T = 0:002 and d = 0:01. The taps of the DT channel model are generated under the wide sense stationary, uncorrelated scatter (WSSUS) assumption from an independent Gaussian process with the Clarke spectrum Rf (m) = J0 (2d m) (12) where J0 () is the zero-order Bessel function of the first kind. The channel resolution T c is chosen as T=2, which means Ns = 2 samples per symbol 1 . Then the length of the channel ISI in terms of the symbol time T is L c = bNc =Ns c + 1 = 4. In Figure 3 the performance of the adaptive iterative detection using ME-A-SISO algorithms is shown for the fifth iteration over the ISI channel with d = 0:002. If we decompose the CPM into the CPE and the MM [17], then the CPE has P M L 1 = 8 states and the MM produces one of P M L = 16 signals. The ISI channel can be described separately by (P M L )Lc 1 = 16 16 16 = 4096 states, but this number of states is infeasible for SISO algorithms. If we describe the CPM and the ISI by a joint trellis, only P M L 1 M Lc 1 = 64 states are needed. A joint CPM-ISI A-SISO is used based on the 1 The normalized double-sided bandwidth of this CPM signal is 1:165 [14]. This means that 2 samples per symbol are sufficient. 1165 10 0 10 0 SE-A-SISO (64 states) ME-A-SISO 10 ME-RS-A-SISO MS-RS-A-SISO -1 10 -1 10 -2 8 10 Bit error rate Bit error rate 8 -2 16 16 10 -3 10 -4 64 8 10 12 14 16 18 20 22 24 10 12 14 16 18 20 22 24 26 [4] [5] [6] [7] [8] SCCPM with AID showed turbo-like performance over fading ISI channels. RS-A-SISO algorithms were proposed and various design options for RS-A-SISO algorithms were evaluated for SCCPM over frequency-selective fading channels. RSA-SISO systems also showed large iteration gains. As the number of states used decreases, the performance of the SE-RS-ASISO algorithms approaches that of the ME-RS-A-SISO algorithms at low Doppler spread. For more rapidly varying channels full-state ME-A-SISO algorithms did not show an error floor but full-state SE-A-SISO algorithms showed an error floor. MS-RS-A-SISO algorithms can be a design option to improve performance. [9] [10] [11] [12] [13] [14] [15] R EFERENCES K. M. Chugg, A. Anastasopoulos, and X. Chen, Iterative Detection, Kluwer Academic Press, 2001. S. Benedetto, D. Divsalar, G. Montorsi and F. Pollara, “Soft-input softoutput modules for the construction and distributed iterative decoding of code networks,” ETT, vol. 9, No. 2, Mar./Apr. 1998, pp. 155-172. S. Benedetto, G. Montorsi, D. Divsalar and F. Pollara, “Serial concate- 0 Fig. 7. Performance of multiple-step RS-A-SISO and multiple-estimator RSA-SISO (d = 0:01). (16 or 8 are reduced states) V. C ONCLUSIONS [3] -4 b 0 16 reduced-states hits an error floor below a BER 8 10 4 . The comparisons of MS-RS-A-SISO algorithms and ME-RSA-SISO algorithms are shown for the fifth iteration in Figure 5 and Figure 7 for d = 0:002 and d = 0:01 respectively. Note that the gain of MS-RS-A-SISO algorithms over ME-RS-ASISO algorithms increases as the number of states is reduced. The gains at BER 10 3 are 0.5 dB and 1 dB for a 16 reducedstate and an 8 reduced-state scheme respectively at d = 0:002 and 2 dB for a 16 reduced-state scheme at d = 0:01. [2] 10 E /N [dB] Fig. 6. Performance of single-estimator A-SISO and multiple-estimator ASISO (d = 0:01). (64 is full state and 16 or 8 are reduced states) [1] -3 8 26 E /N [dB] b 10 [16] [17] 1166 nation of interleaved codes: Performance analysis, design, and iterative decoding,” IEEE Trans. Inf. 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