Doubly Iterative MMSE Equalization of Binary

Continuous Phase ModulationBarn§ Ozgul*, Mutlu Kocat, and Hakan DelitDepartment of Electrical and Electronics Engineering

Bogaziqi UniversityBebek 34342 Istanbul, Turkey

Email: {*ozgulbar,tmutlu.koca,tdelic}@boun.edu.tr

Abstract- In this paper, a suboptimal, but low-complexitytechnique is introduced to achieve the turbo equalization for bi-nary continuous phase modulation (CPM). The proposed receiverconsists of a soft-information aided minimum mean squared errorequalizer (MMSE) at its front-end, and the soft-input soft-outputdemodulator and decoder blocks at its back-end. The receiver isdoubly iterative with outer iterations for equalization and inneriterations for CPM demodulation and channel decoding at eachouter iteration. Turbo processing overcomes the performancedegradation due to suboptimality, where the iterations do notresult in high computational complexity as it is in the conven-tional methods. Simulation results are fumished for performancecomparisons.

I. INTRODUCTIONTurbo equalization is first introduced in [3] and achieves re-

markable processing gains against the impacts of intersymbolinterference (ISI). Here, equalizer and decoder blocks employthe maximum a posteriori probability (MAP) algorithm [2],and exchange soft information on the transmitted symbols.In [8] and [9], lower-complexity turbo receivers are proposedby replacing the MAP equalizer with soft-input soft-output(SISO) linear or decision feedback equalizers, whose param-eters are optimized according to the minimum mean squarederror (MMSE) criterion. However, such techniques have sofar been applied to linear modulation schemes, where themodulated symbols are uncorrelated. This paper considers theSISO MMSE equalization of continuous phase modulation(CPM), which is a popular modulation scheme because of itsspectral efficiency, along with its constant envelope property[1].The conventional methods for the equalization of CPM

signals [10] employ decoding algorithms such as the Viterbialgorithm (VA) [4], or the MAP algorithm. However, thesemethods results in high computational complexity whichmakes iterative processing inapplicable. In this paper, a sub-optimal, but low-complexity receiver is proposed to achievethe turbo equalization for binary CPM. Iterative nature of thereceiver overcomes the performance degradation that resultsfrom suboptimality. The proposed receiver consists of the soft-information aided MMSE equalizer at its front-end, and SISOdemodulator and channel decoder blocks at its back-end. Thereceiver is doubly iterative and performs two types of itera-tions. The outer iterations are for equalization, where the inneriterations at each outer iteration are for the CPM demodulation

and channel decoding. The inner iterations are performed toimprove the a priori information for the MMSE equalizer.During the outer iterations, soft interference cancellation takesplace by feeding the corresponding extrinsic information fromthe CPM demodulator to the MMSE equalizer. The equaliza-tion algorithm requires symbol autocorrelations for which theresults in [5] are employed.

The organization of this paper is as follows. In Section 2,the signal model and the transmitter are described. Section 3presents the receiver structure and the proposed doubly itera-tive method, as well as the computation of the CPM symbolautocorrelations used in the equalizer algorithm. The reason toapply the extrinsic information from the CPM demodulator forthe soft interference cancellation and complexity comparisonsare also given here. Following the simulation results in Section4, conclusions can be found in Section 5.

II. SIGNAL MODELThe transmission system is displayed in Fig. 1. At the

transmitter, Ld data bits di E {0, 1} are convolutionallyencoded. The code bits are denoted as cn E {-1, + 1}, where

Encoder Intedeaver CPMModulator

Fig. 1. The transmitter and the channel.

n = 0,... ,Lc - 1, and the code rate is Ld/Lc. Then, thebits are interleaved and passed through the CPM modulator toform [1]

0 < t < LUT, (1)with

n

p(t, x) = 27rhE xkq(t - kT)k=O

= 0(t,x)+On, nT<t<(n+1)T (2)n

0(t, x) = 2rrh E Xkq(t - kT)k=n-L+1n-L

On = (7h E xk) mod 2wrk=O

(3)

(4)

0-7803-9206-X/05/$20.00 ©2005 IEEE108

y (t) = expfj .c (t. x) 1,

where T is the symbolling interval. The modulation index htakes values equal to 2m/p, where m and p are relativelyprime integers. q(t) = fo g(T)dT is the phase smoothingfunction, where g is the modulation pulse shape that is zerooutside the interval 0 < t < LT. q(t) = 0 for t < 0 andq(t) = 1/2 for t > LT. O,n E ), and

(5)

Equation (4) represents the cumulative phase, and thereare 2L- 1 non-cumulative phase states, regarding the bits{Xn-L+1,.. ,Xn -I} in (3). Thus, the total number of statesin the binary CPM trellis is Q = p2L-1. The bit xn results inthe state transition.The CPM symbols are transmitted through ISI channel, and

the channel output is sampled at the symbol rate. Assumingthat the channel is time-invariant, i.e. the sampled channelimpulse response hm := h(mT) has finite duration for 0 <m < M - 1, sampled noisy signal model at the receiver isexpressed as

M-1

rn = hmynnm + Wn, (6)1=0

where rn := r(nT), yn: y((nT). The additive complex whiteGaussian noise term wn has zero mean and variance o'2.

III. DOUBLY-ITERATIVE EQUALIZATION OF CPM

The doubly-iterative receiver consists of three serially con-catenated blocks, as shown in Fig. 2. Initially, no soft interfer-ence cancellation takes place, and the a priori information forall three blocks is regarded as zero. Then, the MMSE equalizerproduces soft information for the CPM symbols, which aretransferred to the CPM demodulator. The CPM demodulatorand the channel decoder exchange soft information for theencoded bits at each inner iteration. In case of an outeriteration, the extrinsic information from the CPM demodulatoris fed back for the soft interference cancellation and MMSEequalization. Then, the MMSE equalizer outputs the new apriori information for the CPM symbols to the demodulator,prior to proceeding with the inner iterations. It might also bepossible to compute the soft information for the CPM symbolsby considering the extrinsic information for the encoded bitsfrom the channel decoder. However, this approach does notresult in soft interference cancellation for turbo processing,as shown in Section Ill.A. That is why we use the extrinsicinformation from the CPM demodulator. The log-domain MAPalgorithm employed by the demodulator and the channeldecoder is well-known in the literature [7], [1 1]. Therefore, wefocus on the details of the SISO MMSE equalization algorithmin the rest of the paper.

Defining

rn := [rn-K rn-K+1 ... 7n+N-1 rn+N]T,Yn := [Yn-MA-K+1 ... Yn-1 Yn Yn+i ... Yn+NJ,Wn = [wn-K Wn-K+1 ... Wn+N-1 Wn+N]T

the received data vector rn can be expressed as

rn =Hy + Wn,

with

hm-, .. hoh0hM .

H =

0 .i.. 0 h

0 .... 1ho ... 0 K

tm-1 *.. ho J

as the P x (P + M - 1) channel convolution matrix (P =K + N + 1) that includes the channel coefficients.

Considering perfect channel state information and perform-ing soft interference cancellation prior to equalization, thesymbol estimate at the output of the equalizer filter at timen can be expressed as

Yn = fn (rn -HYn)where f, includes the K + N + 1 filter coefficients, and Ynis the soft interference cancellation vector, with

fn = [fn,-K fn.-K+1 fn,N-1 fn,NT,Yn = [Yn-M-K+1 ... Yn-1 0 Yn+1 ... Yn+N]

Here, Yk denotes the instantaneous mean value computed byusing the symbol probabilities from the CPM demodulator attime k.By inserting (7) into (9), we get the following expression.

(10)The error between the estimated and the actual symbol ise(n) := Yn-Yn. Then, using (10), MSE cost function canbe expressed as

JMSE = E[ene*] E Ifn (H[yn-Y] +Wn) - Y ] (11)

We assume that the CPM symbols are wide sense stationary.Moreover, they are considered to have zero mean (E[yk] = 0,see Section IHLA). Then, to find the optimum coefficients

aim= = (HCUHH ±+ 2 I)f -S,

where the P x P matrix Cn, is

Cn = R + -ynYn i

with the correlation matrix

- R(0) R(T) ...

R(T) R(0) ...

R = RR(2T) R(T)

R((P- 1)T)

R((P - 1)T)R((P - 2)T)R((P - 3)T)

R(O)and the (K +N + 1) x 1 vector s is

s = Hq

109

(7)

(8)

(9)

(12)

(13)

(14)

(15)

E) = 01 27r147 I2(p 1)7

.n= f,. (H [Yn Y-, I + Wn)

L(d)

y

Fig. 2. Receiver structure.

withT

q = [R((K + Al-1)T) ... R(T) R(O) R(T) ... R(NT)(~16)

Then, (12) yields

fn = (HCnHH + 2 I)-I1s. (17)In order to compute (14) and (16), we employ the PAM

decomposition of binary CPM in [5], where the autocorrelationfunction is expressed as

Q-1Q-1 oc

R(T) = E E z [cos(hr)]A(,jP)Cij(7-pT)i=O j=o p=-oo

- R(-T), (18)

Here

with channel gain ,u, and the complex Gaussian noise term v,with zero mean and variance a2 [8], [9]. Assuming E[yn] 0O(see Section Ill.A), these parameters can be calculated at eachtime instant as

/Un = E[ynY*] = fnHE[(H[yn - n] + Wn)Y*]- fn Hr = fn s, (23)

an = E[JYn - nYn|I]- fnI(HC,HH + an I)fn-fn'S1 - ,insHfn+ |,unI12 = fHS -fnHSP*- fns( sHfn). (24)

There are Q possible CPM symbol values at time n so thatYn E T = {Yo,... , YQ-l}. Using (23) and (24), the symbolprobabilities can be calculated as

L-1

A(i,j,p) = IP + ,(3i,k +/3j,k)k=l

- k<L-1,-p- I k<L-l,p-1

-2 [ E O5i,k + Ek>l k>l

k<L-1,L-1-p+ E p i,nk /3j,k+p1,

k>1,1-p J

where

PQAIYn = Yi) =

_ lYn-linYi i2012e an

_ IfYn-Lnyr i2

EY,eTa-n2Re(y*pn Y )_ rL

e an(25)

EYrE(19)

L-1

k =Z 2-13k,i, 0 < k < Q-1, (20)i=l1

and !k,i E {0, 1}. In addition,

Cij(T) = Cji(-T) = f Ci(t)Cj(t +T)dt, (21)

where Ci(t), for i = 0,... ,Q - 1, denotes the PAM wave-forms.

Plugging fn into (9), the symbol estimate Yn is found. Then,the extrinsic information for the CPM demodulator is extractedfrom the MMSE equalizer output. The estimate Yn of theequalizer can be viewed as the output of an AWGN channelwith input yn so that

Yn =InYn + Vn (22)

2Re(y*TnYr) I

Ale °

for i = 0,1,... , Q - 1, where Re(.) stands for the realoperation and * is for complex conjugation. Then, the extrinsicinformation in (25) is fed to the CPM demodulator.

A. Evaluating the Soft Information for the CPM SymbolsUsing the Bit Probabilities

In this section, we show the reason to use the extrinsicinformation for the CPM symbols from the CPM demodulator,instead of evaluating the necessary soft information by usingthe extrinsic information for the encoded bits from the channeldecoder.

In [5], the binary CPM signal is represented as the super-position of PAM waveforms, as follows.

Q-1 n

y(t)= Z Ak,mCk(t-mT), nT< t <(n+ 1)T,k=O m=-oo

(26)

110

with the pseudo-symbols

Akin = ejh7r [ZEl-=Z. x Im-3] (27)

where xj are the input bits. We need to compute E[Ak, ]to find E[yn], as seen in (26). Using (27) and assumingindependent bits,

m L-1

E[Ak,m] = J7 E[ejhrx] 11 E[e-jh7x,-iOk,]i (28)1=-oo i=O

It is obvious that

|E[ejh7rxi] < E[eihjxl] = 1. (29)Then, the magnitude of the complex mean E[eJhrxl] will beless than one unless P(xl = +1) = 1 or P(xl = -1) = 1.Thus, the magnitude of (28) goes to zero, as m increases.Since the MMSE equalizer is fed with zero mean values, andthe soft interference cancellation does not take place, turboprocessing is not possible unless the channel decoder resultsin bit probabilities very close to one.

Similarly, in case of stationary and binary-distributed xl, theCPM symbols can be considered to have zero mean, which isa property we use in the derivation of the MMSE equalizeralgorithm.

B. Complexity ComparisonFor the proposed receiver, the symbol estimate in (9)

requires the inversion of a P x P matrix in (17), whereP = K + N + 1 is the MMSE equalizer filter length. Then,considering the real multiplications and additions, the com-putational complexity per symbol is 0(P3) at each iteration.The trellis for CPM modulation has Q = p2L-1 states, asmentioned in Section II. Regarding the computation of -ynquantities, and the an, /n recursions for the MAP algorithm ateach trellis section, complexity introduced by the equalizer andthe CPM demodulator becomes 0(P3) +0(p2L) per iteration.

In the conventional case, a MAP decoder is employedfor combined equalization and demodulation, which is againfollowed by the channel decoder. The SISO equaliza-tion/demodulation block and the channel decoder exchangesoft information regarding the encoded bits. The state of thecombined trellis for the ISI channel and CPM modulation is

Sn = (On,Xn-MA-L+2, * * *,Xn-M-1, ...,Xn-1).Considering the signal model in Section II, the complexity tocompute an, an, and /3n per trellis section is Q(p2M±+L-1),which grows exponentially in case of longer channel re-sponses, as well as the higher modulation depth.

Complexity from the channel decoder is omitted abovesince it is the same for both receivers. For comparison, let usconsider the CPM modulation with h = 1/2 and L = 3, wherethe symbols are transmitted over a 5-tap channel, i.e. ProakisC [6]. Assuming K = 4 and N = 5, the complexity fromthe MMSE equalizer and the CPM modulator is 0(1, 000) +0(32), where the conventional method seems less complexwith 0(512). However, it is not easy to conduct computations

over such a huge trellis with memory requirements for -y anda. Moreover, due to the doubly iterative nature of the proposedreceiver, number of outer iterations with complexity 0 (1, 000)can be significantly reduced by improving the soft informationto the equalizer with sufficient number of inner iterations(see Fig. 5), where the complexity for CPM demodulationalone is only 0(32) per iteration. On the other hand, theconventional turbo receiver should employ the MAP decoderfor combined equalization and demodulation at every iteration.As another example, considering an 1 l-tap channel (suchas Proakis A) with h = 1/2 and L = 5 for CPM, thecomplexity for the conventional case at each trellis sectionbecomes 0(131,072) per iteration. Assuming K = 10 andN = 11, the proposed receiver requires 0(10, 648) complexityfor each outer iteration with 0(128) complexity for the CPMdemodulation per inner iteration.

IV. SIMULATIONSIn this section, the BER performance of the doubly iterative

receiver is presented for different number of inner and outeriterations. The performance of the proposed receiver is com-pared to the case without ISI, as well. During the simulations,3RC binary CPM is considered, where the modulation index his 1/2. Before applying the CPM modulation, bits are encodedby rate- 1/2 convolutional code with generator polynomial(7,5). ISI channels are Proakis B and the more severe Proakis

1 7~,0

102

m 104 \

.

10-6 no ISI (8 iter.): Proakis B (8 outer with 1 inner iter.)

- ProakisC (8outerwith 1 inneriter.)-7100.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5

Eb/No(dB)

Fig. 3. BER performance for the 1SI and AWGN channels.

C with channel tap weights [0.407 0.815 0.407] and [0.2270.460 0.688 0.460 0.227], respectively. The filter parametersare set as K = 3 and N = 3 for Proakis B channel, whereK = 4 and N = 5 for Proakis C. Each data packet consistsof 1,000 bits.The BER performance for the Proakis B and C channels,

and the no ISI case are available in Fig. 3. When the ISI chan-nel is not present, the receiver consists of a CPM modulatorfollowed by the channel decoder. Both the demodulator and

111

the channel decoder employs the log-domain MAP algorithm.

Eight iterations are run for the turbo CPM demodulation and

channel decoding. In case of turbo equalization, one inner

iteration is run for each of the eight outer iterations. Due to the

suboptimality of the proposed receiver, the BER for the first

outer + inner iteration is below i0-5 only after 19 dB and 22

dB, for Proakis B and C, respectively. Thus, after eight outer

iterations with one inner iteration, we observe a performance

gain of about 15 dB for both channels.

In Fig. 4, the BER performance of the doubly iterative

receiver is presented for twelve outer iterations with one inner

w

5.5 6 6.5

EbWN0(dB)

7

Fig. 4. BER performance for cases I and II.

10

ccw

co

103

5i.5 6 6.5

EbINo(dB)

7

Fig. 5. BER performance of case III compared to cases I and

iteration (say case I) and one outer iteration with twelve inner

iterations (case II), where the 1ST channel is Proakis C. The

performance gain for case I is about 0.5 dB with respect to case

However, the MMSE equalizer with complexity 0(1,000)is employed twelve times more in case I compared to case

The comparison of three outer iterations with four inner

iterations (case III) with cases I and is shown in Fig. 5. In

case HII, only two extra outer iterations are performed with

respect to case to obtain the similar performance in case

Moreover, high complexity that comes from outer iterations

is four times less in case 11 (30(1, 000) complexity for outer

iterations) with respect to case I (1 20(1,000) complexity for

outer iterations), where the total number of inner iterations is

the same for both cases. The performance gain due to SISO

equalization is not significant for moderate channels. For

instance, during the simulations for Proakis B channel, it is

observed that the BER performance for one outer iteration with

eight inner iterations is similar to the performance in Fig. 3

for eight outer iterations with one inner iteration. Thus, for

such channels, faster convergence is possible by performing

just one outer iteration.

V. CONCLUSIONS

In this paper, a low-complexity turbo receiver is introduced

for the equalization of CPM. Although the proposed method

is suboptimal, turbo processing helps to overcome the per-

formance degradation. The doubly-iterative receiver performs

an outer iteration for equalization and inner iterations for

the CPM demodulation and channel decoding at each outer

iteration. Performing a few inner iterations for each outer

iteration, a priori information for the equalizer is improved,

and faster convergence to low bit error rates is observed with

less complexity. At each outer iteration, extrinsic information

from the CPM demodulator is fed to the S150 equalizer to

achieve soft interference cancellation.

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112

-1 outer with 12 inner

--A-12 outer with 1 inner

---4th outer with 1 inner

1t outer with12 inner12th outerwith1 inner~~~~~~~

3 outer with 42inner

10

10

ir

io-4

1 0-5