7/29/2019 DecodDemod of Trellis Coded.pdf

1/5

Iterative Demodulation and Decoding of Trellis Coded CPM

Krishna R. Narayanan

Department of Electrical Engineering

Texas A&M University,College Station, TX 77843, USA

ABSTRACT

Interleaved trellis coded systems with continuous phasemodulation (TCCPM) are considered. It is shown that thecombination of a trellis code, interleaver and the CPM mod-ulator results in a significantly enhanced distance spectrumfor the transmitted signals. Upper bounds on the bit errorrate are derived by considering the combination of the trelliscode, interleaver and the CPM modulator as serially con-catenated code. Finally, iterative coherent and non-coherent

receivers are proposed and their performance is evaluatedthrough computer simulations.

I Introduction

Many communication systems employ non-linear poweramplifiers and, hence, require a low peak-to-average powerratio for the modulated signal. Examples of such systemsinclude satellite and mobile communications systems. Con-tinuous phase modulation (CPM) is a constant envelope mod-ulation technique and, hence, a good choice for such systems.Trellis coded modulation (TCM) is a technique that com-

bines modulation and coding in order to achieve coding gainswithout sacrificing data rate or bandwidth. When TCM iscombined with CPM, high power and bandwidth efficiencycan be achieved.

The distance spectrum and error performance of convo-lutionally encoded CPM on additive white Gaussian noise(AWGN) channels has been analyzed by many researchers.However, most of the work has not considered the use of aninterleaver between the trellis code and the CPM modula-tor for AWGN channels. Similarly, they have also ignoredthe effect of the recursive nature of CPM modulator in de-signing TCCPM schemes. The performance of TCCPM forfading channels has b een evaluated in [1],[2],[3]. A fadingchannel typically exhibits correlation in time and, when the

correlation is high, the channel exhibits prolonged deep fades.Therefore, some form of interleaving has to be used to com-bat the fading correlation. When TCM is used with PSK,PAM, or QAM, the modulated symbols are interleaved usinga block or convolutional interleaver. Since interleaving at-tempts to destroy the correlation in the fading process, per-formance bounds can be derived by assuming that the channelis uncorrelated. However, when CPM is used, the modulatedsignal cannot be interleaved because interleaving destroys thecontinuous phase property of the modulated signal. There-fore, a system was proposed in [1] where an interleaver is used

between the trellis code and the CPM modulator. At the re-ceiver, a soft-output CPM demodulator generates metrics forthe coded symbols. These metrics are then deinterleaved andused to derive the branch metrics for the Viterbi decoder thatis used to decode the trellis code. This deinterleaving makesthe channel appear uncorrelated to the trellis code. Perfor-mance bounds can then be developed by suitably modifyingthe transfer function of the trellis code and assuming that thechannel is uncorrelated [2].

All of the aforementioned approaches completely ignorethe contribution of the interleaver to the overall distance spec-trum of the transmitted signals. From the recent advancesin concatenated schemes, we know that the interleaver dras-tically influences the distance spectrum and, hence, will sig-nificantly change the system performance [4].

Our approach is to treat TCCPM as a serial concatena-tion of a trellis code and a CPM modulator, which representsa rate-1 recursive inner code. In contrast to the other pa-pers, we explicitly consider the effect of the interleaver onthe distance spectrum of the transmitted signals. The sameapproach was used in [5] to analyze the performance of in-terleaved convolutionally encoded systems with differentialphase-shift keying (DPSK). In [5], performance bounds were

derived for Rayleigh fading channels by assuming perfect in-terleaving of the modulator output. In this paper, we focuson TCCPM schemes for fading channels. As mentioned be-fore, unlike the case of PSK, for CPM the modulated symbolscannot be interleaved prior to transmission. Consequently,we cannot assume an interleaved fading channel in the caseof CPM signals. This makes a significant difference in theperformance bounds developed in this paper in comparisonto the ones in [5].

Finally, we consider iterative receivers for coherent andnon-coherent reception of TCCPM. The coherent receiver isidentical to the one in [6]. A non-coherent receiver for TC-CPM was proposed in [3] where a non-coherent front end isused to generate metrics for the coded bits and a Viterbi de-

coder is used for the trellis code. In this paper, we derivea non-coherent detector for iterative demodulating and de-coding TCCPM. Our receiver structure is similar to the onein [7] and based on the maximum-likelihood block detectionprinciple in [8].

II System Model

The TCCPM system considered in this paper is similar tothe system in [1], [2], but with a few differences. At each timeinstant, m bits of the binary data sequence a are encoded

0-7803-5538-5/99/$10.00 (c) 1999 IEEE

7/29/2019 DecodDemod of Trellis Coded.pdf

2/5

by a rate-m/n convolutional code and binary outputs of theconvolutional encoder are multiplexed into the sequence b.The binary sequence b is then interleaved in to the sequenceb. Then, p-tuples of the sequence b are mapped on to symbolsck {1, 3, . . . , (2

p 1)} and the sequence c is input tothe CPM modulator. If p = n, the symbol rate is the sameas the uncoded data rate and, hence, there is no bandwidth

expansion. We refer to such systems as trellis coded CPM.If p < n, the required bandwidth is higher for the codedsystem than the uncoded system and, we refer to such systemsas convolutionally encoded CPM. Note that we consider bitinterleaving of the outer code words in contrast to the symbolinterleaving used in [1], [2].

The low-pass equivalent CPM signal s(t, c) for nT t (n + 1)T is given by

s(t,c) =

2pEbmT

expj{(t, c) + 0} (1)

The information is contained in the phase (t, c) = n +

2n

i=nJ+1ciq(t iT), where q(t) =

t

0g(t) is a fre-

quency shaping pulse such that q(t) = 1/2 for t > J T ,n =

nJi=0

ci is the accumulated phase at the begin-ning of the nth epoch, and is the modulation index. Wewill assume that is rational. If J = 1, the CPM scheme iscalled full-response CPM, and if J > 1 it is called partial-response CPM. In this paper, we will only consider full-response CPM. A full response CPM signal can also be rep-resented by s(t, n, cn) to emphasize the finite-state nature ofCPM and, hence, suggest a trellis representation similar tothat of convolutional codes.

When the modulated signal is transmitted over a fre-quency non-selective channel, under the assumption that thechannel gain remains constant over one modulated symbol,the received signal can be expressed as

r(t) = ks(t) + n(t), kT < t (k + 1)T (2)

where k = (kT). For the AWGN channel, k = 1, k andfor the flat Rayleigh fading channel |k| is a random variablewith a Rayleigh probability density function (PDF).

III CPM modulator as recursive inner code

Consider the low pass equivalent representation ofMaryCPM schemes in Section II. It was shown in [9] that an equiv-alent way to represent the CPM signal is in terms of the physi-cal tilted phase, defined as (t) =

(t) + (M1)t

T

mod 2.

The CPM modulator can then be represented as a finitestate machine with inputs un = (cn + (M 1))/2 and state

n = [n, un] as shown belows(t,c) = f(n, t)

n+1 = g(n, un) (3)

Note that since ck {1, 2, . . . , (2n 1)}, uk

{0, 1, . . . , M 1}. It was shown in [9] that g is time-invariantand can be synthesized as a differential encoder with arith-metic operations in the ring of integers modulo B. That is,n = 2vn/B where vn is given by

vk = vk1 uk1 (4)

where denotes modulo-B addition. It was also shown in[9] that for n T < t (n + 1)T, f is a memoryless map-per that maps n on to any one of the BM

J possible CPMsignals from the CPM signal set S. From (3) and (4), wecan see that the CPM modulator is equivalent to a recursiverate-J/J+1 convolutional encoder, also called the continuousphase encoder (CPE) followed by a memoryless mapper. We

can consider the combination of the trellis code, interleaverand the recursive CPE as a SCCC. In the following, we takethis approach. Since the inner code in the SCCC is recursive,large interleaving gains similar to conventional SCCCs shouldbe possible [4].

IV Performance Analysis

The combination of the convolutional code, interleaverand the modulator can be considered as an equivalent blockcode, whose code words are transmitted over the corre-lated fading channel. Therefore, we need to derive the pair-wise error probability between the transmitted CPM signalx0 = A expj0(t) and another signal

x1 = A expj1(t) fora correlated fading channel. By extending the derivation in

[10], the pairwise error probability for flat Rayleigh fadingchannels can be shown to be [11]

P(x0 x1)

1

detIL +

EsD4No

(5)

where D is a diagonal matrix with entries Dii =tiT+TtiT

|ej0(t) ej1(t)|2dt, and IL is the L L identity

matrix. Further, = [t1, t2, . . . , tL] is a vector of L symbolpositions at which x0 differs from x1, denotes an L Lcorrelation matrix with ij = E[(ti mti )(tj mtj )],

where mtj denotes the mean of tj .Consider the trellis coded CPM system with bit interleav-

ing in Section II. Let us assume that the CPM signal x0corresponding to the input sequence a0 is transmitted. Letxi,L denote the ith CPM signal which differs from x0 in ex-actly L epochs. Ifxi,L corresponds to the input sequenceai,L, then the difference sequence e = a0 ai,L is called theerror sequence. Let W(e) denote the input weight of the er-ror sequence. The union bound on the probability of errorfor correlated flat fading channels is

Pb(e) a0

P(a0)

L=Lmin

xi,L

W(e)

NP(x0 xi,L) (6)

a0

P(a0)

L=Lmin

xi,L

D

W(e)N

P(L,,D)(7)

where P(L,,D) denotes the pairwise error probability be-tween two sequences that differ in L time instants, with and D defined earlier, and N is the block length. In general,for CPM schemes P(x0 xi,L) depends on x0 and, hence,the transmitted code word cannot be assumed to be the all-zeroes code word. In [11] approximations to evaluate (7) insuch situations are discussed. In this paper, we consider only

0-7803-5538-5/99/$10.00 (c) 1999 IEEE

7/29/2019 DecodDemod of Trellis Coded.pdf

3/5

minimum shift keying (MSK) for which P(x0 xi,L) doesnot depend on x0 and, hence, (7) can be shown to be

Pb(e)

L=Lmin

D

A(L,,D)

NP(L,,D) (8)

where A(L,

,D

) is the average number of CPM signals (or,codewords) which differs from the transmitted sequence in Lepochs, with and D defined before. Since the CPE is recur-sive, with the assumption of a uniform interleaver [4], it can

be shown that for small values of L, A(L,,D) Ndo

f/2

[4],[11]. This suggests that increasing the block length N re-sults in a drastic reduction of the number of codewords withsmall diversity L and, hence, in a large interleaving gain.

V Receiver Structures

V-A Coherent Receiver

The coherent receiver considered is identical to the one in

[4],[6]. The receiver performs iterative demodulation and de-coding of the received signal using the Bahl, Jelinek, Cocke,and Raviv (BCJR) algorithm for both demodulation and de-coding. The receiver assumes that perfect channel state infor-mation is available. After each iteration, the decoded data ischecked for errors using a cyclic redundancy check (CRC) andif the CRC passes, the iterations are stopped; otherwise, theiterative process continues up to a maximum of 8 iterations.

V-B Non-coherent Receiver

From Sect. IV and Sect. VI we can see that a very sig-nificant interleaving gain is possible for the TCCPM scheme

and, hence, TCCPM schemes can operate at very low Eb/No.At such low Eb/No, channel parameter estimation becomevery difficult. Non-coherent receivers do not require channelstate information and, hence, are attractive for use at suchlow Eb/No. Iterative non-coherent detection is a rather newtopic and there is very little work in this area. An itera-tive non-coherent receiver structure for iterative demodula-tion and decoding of DPSK signals in AWGN channels wasproposed in [7]. The overall receiver structure is identical tothe coherent receiver except that a non-coherent soft-outputdemodulator is used instead of the BCJR algorithm.

Unlike in the case of coherent reception, the BCJR algo-rithm cannot be used for optimum soft-output demodulationof the CPM signals. This is mainly due to the fact that the

optimum metric for non-coherent demodulation is not addi-tive and, hence, cannot be used in the BCJR algorithm. Ingeneral, non-coherent demodulation of CPM signals is accom-plished either by using a sub-optimum metric in conjunctionwith an optimum trellis based algorithm or by using the op-timum metric with a sub-optimum trellis based algorithm.One such approach is block detection, where a small windowof data is used to generate soft-output instead of the entiredata sequence [8]. A sub-optimal non-iterative non-coherentdemodulator using this principle was used in [3] for Rayleighfading channels. Here, we derive the optimum soft-output

non-coherent demodulator for CPM signals for Rayleigh fad-ing channels using block detection. Our approach here isbased on non-coherent block detection of CPM signals pro-posed by Simon and Divsalar in [8].

Let x(t, c) denote the transmitted CPM signal correspond-ing to the sequence c. Then, the received signal is given by

r(t) = x(t, c) |(t)|ej(t) + n(t) (9)

We assume that |(t)| and (t) remain constant over the in-terval kT N1T < t < kT + N2T. Further, let |k| and kdenote the magnitude of the channel gain and phase of thechannel gain, respectively and rk+N2kN1 the received signal overthe interval kT N1T < t < kT + N2T.

To generate the LLRs for ck, the non-coherent receiverobserves the received signal over the window (kT N1T)