W2D.2

A Space-Time MSK Code and its DetectionMichael L. B. Riediger and Paul K. M. Ho

School of Engineering Science, Simon Fraser UniversityBurnaby, British Columbia, Canada

Abstract-Motivated by the desirable attributes of bothAlamouti-type space-time (ST) codes and constant envelopemodulations, we introduce in this paper an ST minimum-shiftkeying (MSK) transmission scheme and the companion receiver.Taking advantage of the block-wise differential encoding on

which the MSK transmission is based, the receiver performs non-

coherent sequence detection on a down-sampled subset of thereceived statistics. As a result of the sequence detection, the fast-fading error floor associated with symbol-by-symbol differentialdetection is essentially eliminated. Furthermore, using theoriginal received sequence and performing iterative channelestimation and sequence detection, the decisions are refined andfurther performance gains are observed. In fast fading, resultsshow that the proposed receiver performance is within 3 dB ofthe coherent detection lower bound, at a bit-error-rate of 10-3 Itattains not only space diversity, but also time-diversity from theimplicit coding in MSK. Further results also demonstrate that thereceiver experiences only a small performance degradation whenthe fading samples are not constant within each ST interval.

Keywords-space-time, continuous-phase modulation (CPM),minimum-shift-keying (MSK), non-coherent detection

I. INTRODUCTION

ontinuous-phase modulations (CPM) are attractivetransmission schemes for wireless communications due to

their constant envelope properties. Since the signal amplitudeis constant, CPM exhibits good tolerance to non-lineardistortions and hence, less expensive transmission amplifierscan be used. At present, the vast majority of investigations on

CPM are limited to systems with a single transmitter and one

or more receivers. With the recent interests in space-time (ST)coding, where more than one transmitter is employed toprovide diversity in fading channels, it is natural to investigateimplementing CPM in a ST setting [1]-[5]. However, thecontinuous phase (CP) constraint introduces a memory effectin the modulation and makes ST code design a non-trivial task.It is almost certainly for this reason that the author in [3]employed simple delay diversity for a "ST minimum-shiftkeying (MSK) code". Another example illustrating thedifficulty of ST-CPM code design is the ST block MSK codeproposed in [5], which violates the CP requirements.Nonetheless, the research in [1]-[5] have clearly demonstratedthe potential performance advantage CPMs have over linearmodulations when employed in an ST-setting.

This work was supported by the Natural Sciences and EngineeringResearch Council (NSERC).

0-7803-9282-5/05/$20.00 ©2005 IEEE 208

In this paper, we investigate a two-transmit antenna, CP ST-MSK encoding scheme, which when sampled at the symbolrate 1/ T, becomes an Alamouti-type [6] differential ST ;z / 2 -shifted BPSK code. This means that for a Rayleigh flat fadingchannel, a diversity order of two is guaranteed when a non-coherent ST multiple-symbol diferential detector (MSDD) forPSK modulation is employed to detect the T-spaced receivedsamples [7]. Furthermore, by performing iterative datadetection and channel estimation, a coherent MSK receiverbased on maximum likelihood sequence estimation (MLSE)can actually increase the diversity order to beyond two,especially at a high fading rate. This is due to the phase codinginherent in MSK, which at a high fade rate, provides a time-diversity effect [8].The encoder, transmitter, channel model and received signal

structures are described in Section 2. Sections 3 and 4 presentthe components of the proposed receiver and offer simulationresults, respectively. Conclusions are drawn in Section 5.

II. SYSTEM AND SIGNAL MODEL

First we set our notation. {xk }: ,x 1,x,xo,}l,diag{xl,x2, , X, }: a block diagonal matrix with X being thek-th sub-matrix along its main diagonal, IM: MxM identitymatrix, JM: MxM all-ones matrix, 1M: lxM all-onesvector, /Z{3: phase of a complex element, E{.}: expectation,exp{x}: ex, 0: Kronecker product, (.)T: transpose, (.)*:conjugate, (.)1: conjugate transpose, 1-1: magnitude of acomplex vector and JO (.): zeroth-order Bessel function of thefirst kind.

A. Encoding and the Transmitted SignalsWe consider the transmission of two MSK signals via ST

coding and two transmit antennas. The bit period of each MSKsignal is T, and the resultant ST-MSK code has a symbolperiod of TST = 2T. The ST data symbols are denoted by {dm }and they are chosen randomly from the setU = IUI ,U2,U3,U43, where

1 0 ll U2 = J o U3 =- ° U4 =°J 1Clearly each dm represents two bits of information. The datasymbols {dm } are used to generate transmitted symbols {cm }in a recursive manner, identical to that used in ST differentialencoding of linear modulations,

Cm = dmcm-. (2)Assuming that the transmitted symbols are initialized with anelement from the set V = IV1, V2, V3, V4} , where

ICICS 2005

VI= l]V2=_ 1]V3 [- 1] [4 1

it is easy to show that cm E V for all m. Note that the columnsof cm correspond to the two transmit antennas and the rowscorrespond to samples of the MSK signals at t = (2m +1)Tand t = (2m + 2)T. Hence, sampled at integer multiples of T,an ST-MSK signal can be described as an Alamouti-type ST; / 2 -shifted BPSK code [7].With this description, we define the phase-state symbols as

Lo1 ((2m+2I)T) 02 ((2m+2I)T)j (4)om =/Cm} [1 ((2m+2)T) 02 ((2m+2)T)]where 0, (t) is the phase of the signal transmitted from the i-thtransmit antenna at time instant t. From the phase-state symboldefinition in (4), it is clear that 0, (mT) E { / 2, 0, ,/ 2, )z fori = 1,2 and all time indices m. Furthermore, consideringunwrapped phases, (3) and (4) imply thatj0,(JT)-0,((J-1)T) =)r12 for any integer J, which bydefinition, is the phase change over one interval in an MSKsignal. At this point, it becomes clear that the signalstransmitted from the two antennas are MSK, if we impose theconditions

0, (t)= 0 ((J-1I)T)+(t(J IT) (0 (JT)- 0, ((J-1I)T)) (5)

for (J -1)T < t < JT, and that the transmitted signals maintainan amplitude of unity; i.e. the signal transmitted from the i-thantenna can be written as exp{j0, (t)} .The ST-MSK proposed here is unique in the context of

previous research, in that: 1) it is generated using simple STblock differential encoding, 2) it maintains the CP constraintrequired for bandwidth efficiency, and 3) it promises thepotential of achieving full-diversity (and more), as thetransmitted signals are Alamouti-type at the T-spacedintervals. Adopting this signaling format, we will proceed withthe channel model and received signal structure.

B. Channel Model and Received SignalThe signals are transmitted over the two wireless links to

the receiver. Each link introduces fading and the receiver'sfront-end introduces additive white Gaussian noise (AWGN).The composite received signal is then

r(t) = g, (t) exp{jOi (t)} + g2 (t) exp 1j02 (t)} + n(t), (6)where gi (t) is the complex Gaussian fading on the i-th linkwhose average power is og2 and n(t) is complex AWGN withpower spectral density N, The average signal-to-noise ratio(SNR) is then defined as

2ag2T (7)

Without loss of generality, we will let cg2 = 1.The received signal is passed though an ideal anti-aliasing

filter with bandwidth fs / 2 Hz and is sampled at a rate of fsHz. We assume that the sampling frequency is an integer

multiple of the bit-rate, i.e. f, = I/ T, where the integer I islarge enough to prevent signal distortion [8]. The sampledreceived statistics {r, I can then be written as

rk =g1, exp{j,k I +g2 kexp{j02 }+ nk, (8)where gj, , 0,', and nk are samples of gi(t), 0, (t) and n(t),respectively, at t = kT I. The samples acquired by thereceiver during the m-th ST interval, defined to be over thetime mTT < t< (m+l)TST, are then rm(21),n, n=1,...,2I. Weassume that the fading gains are constant over each ST intervalbut changes from ST symbol to ST symbol; i.e.

gim(2)+n C,_ 1<n<2I. (9)Furthermore, we assume a discrete block-wise autocorrelationmodel:

(10)A =IE{G mGiM+ I = JO (2T fDTsT),where fD is the Doppler frequency. Due to the bandwidth ofthe front-end filter, the complex Gaussian samples {n } havevariance INo and are mutually independent.

For coherent detection (CD), i.e. given the channel stateinformation (CSI) {gi } and {,g2 }, the optimum receiverminimizes the metric

MCD =, gl, exp{jil,k }+g2,k exp{j02,k }- (1 1)

over all possible hypotheses {Cm, , with corresponding phasesamples {,}I. This solution can be easily found via theViterbi algorithm (VA). In this paper however, we areprimarily concerned with non-coherent detection, in which thereceiver does not have {g1 } and {g2 }3 at its disposal. We doconsider a low-complexity, decision-aided, iterative channelestimator-Viterbi detector that takes advantage of thedifferential encoding (2) and CP properties (5) of thetransmitted signal.

III. RECEIVER

Fig. 1 presents an overview of the receiver proposed in thisinvestigation. It has two main functions: 1) acquisition of aninitial sequence estimate using a down-sampled subset of thereceived signal, and 2) performing iterative channel estimationand Viterbi sequence decoding using all of the receivedstatistics.

A. Initialization. DF-MSDDLet rDS = ( Ir2, r,IrOrIr21, ) denote the sequence of

received samples obtained by down sampling {r, by a factorof I. Clearly, the sampling period in rDs is T, the MSK bitperiod. Based on rDS and knowledge of the differentialencoding in (2), the optimum receiver is an ST-MSDD with adecision window length equal to that ofthe received signal [9].However, due to the exponential complexity associated with it,this receiver is not feasible to implement for an observationwindow larger than 3 or 4 ST symbols.

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r 1A" |O Vitrisequence|I1 II

Down MMSE channel IViterbping -2 ODF-MSDD decodingp' I IoI

MMEchne

estimation

Fig. 1. Block diagram overview of the receiver structure, performing iterativechannel estimation and sequence detection.

Recalling that T-spaced samples of an ST-MSK code isequivalent to a differential ST T/2 -shifted BPSK code, we

have the option to transform the down-sampled statistics rDS

into an equivalent receive diversity format, as established in[7]. Introducing the new variables

(12)Pi'm 1(2m+l)J + r(2m+2)1 ffmsm + ei,m

andP2m =(2m+1) '72m 2)1 f2msm +C2me (13)

where Sm is the row sum of cm, f m and f2m are equivalentfading gains, and elm and e2im are equivalent noises obtainedafter the transformations. From (3), it is clear that sm is fromthe QPSK constellation S = {±1 ± j}. Furthermore, it can beshown that, a) sm = bmsm l, where bm is the row sum of dm, b)the new fading gains {f,m and {f2m } are independent andidentically distributed (iid) and their autocorrelation is doublethat in (10), and c) {eim, and {e2im} are iid white Gaussiannoise with a variance of 2IN,. In short, (12) and (13)represent an equivalent second order receive diversity systemwith differential QPSK modulation.

Analogous to the ST situation, the optimum receiver in thecase of receive diversity is still an MSDD [10]; for an analysisand discussion of MSDD in the context of single channelcommunications, we refer the reader to [11] [12]. In the receivediversity format however, there are low-complexity detectorsavailable which were not in the previous ST format.Specifically, we are now able to implement MSDD using themethod of sphere-decoding (SD). We consider animplementation almost identical to that introduced in [13],with a straightforward metric modification to allow for receivediversity. Although this decoder requires much lesscomputations than that of a brute-force search, only anobservation window N of 10 or 12 is practical [13].

Despite the optimality of the MSDD, its performance willbe limited by the relatively large noise power that arises fromover-sampling the received signal. To recover the loss in SNR,an iterative receiver based on decision aided channelestimation and Viterbi coherent sequence detection is desired.Since the Viterbi detector only delivers optimal performanceby operating on a long sequence, we have to impose acontinuity requirement between successive (implicit) channelestimates provided by the MSDD. Failure to do so will resultin poor convergence in subsequent iterations. For this reason,we propose to further extend the SD in [13] to allow decisionfeedback (DF) [14]. We assume a DF of length equal to that ofthe MSDD.

Without loss in generality, let us consider the detection ofthe equivalent QPSK symbols

S = diag{sO, s,...,s, .3 (14)Let S = diag{js , S-N+ , } denote the decision on theprevious block, S = diag{sO, sl, ..., sN-I } a hypothesis on S.With the sequence of equivalent statistics over the 2N QPSKsymbols denoted as

P L[Pi -N,Pi-N ,Pi' 2,Pi,N-2 ] , 1= 1, 2, (15)

the DF-MSDD minimizes the metric

M(S,S)DF-MSDD =P1 (D(S,S)) P +P2 (D(S,s)) P2 (16)over all possible S, where

F(S, S) = diag{S, S} (A2 + 12. ) diag{S, Sf (17)is the conditional covariance matrix of Pi given S and S. Weuse AM to denote an MxM matrix with the (a, b) -thelement equal to 'a-bI defined in (10), for a,b=1,2,...,M.The DF-MSDD is performed consecutively on overlappingsegments of data to acquire a stream of QPSK estimates. Thefinal accumulated decision on the QPSK symbols is thenmapped back to the ST ;Tz/2 -shifted BPSK format accordingto the one-to-one mapping rule in [7], to arrive at {c2) . Theestimate of x obtained in the q-th iteration is denoted by x(q)q = 0 signifies the initialization.With the initial sequence decision {&2° }, the receiver

proceeds to obtain {Gim } and {G2m }, estimates of the actualfading gains {Gi,m } and {G2,m } in (9). Without loss ofgenerality, we focus on the estimation of GC, and G20 . Underthe assumption that {Jc&)} is correct, the receiver first obtainsraw estimates of Gm = [GIm G2m ]T by multiplying the receivedvector rm = [r(2m1)1Ir(2m 2)1 ]T by I cT(O). This yields the vectorGm = Icf°rm [Gm ,G2m]TM Further, from the raw estimatesG ,G {X+,...,G, the receiver forms the multiplexedsequences yiO =[G, ,-,Gi IG+l,G, ]Tn i=1,2, where 2a+1is the span of the estimation filter. Refined estimates of GCOand G20 are obtained by filtering Yi, and Y20 with theWeiner filter

H =(DGy.D-Y (18)yielding

=

G,,o = Hyi , i = 1, 2, (19)where

(DGy = Of }2a+1 (20)A is the autocorrelation function defined in (10) and

(Dy = I E{yj,oyJ0}o = A2ai +I IN0I20c+i (21)Refined channel estimates at other ST intervals can beobtained in a similar fashion by introducing a time shift m tothe parameters in the above respective equations. We wouldlike to remind the reader that at this stage of the receiver, onlythe down-sampled subset of received statistics, i.e. rDS, isbeing used. Successive iterations of the receiver, however, usethe entire received sequence in (8) for both data detection and

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channel estimation. The reason for increasing the utilization ofthe received samples is to offset the relatively large noisepower caused by over-sampling of the received signal. In thefollowing discussion, we will use g(4) to denote the estimateof the fading gain sample gi,k obtained in the q-th iteration,with the understanding that =, 1 < n < 2I .

B. Successive IterationsIn the successive iterations, the receiver exploits the CP

properties of MSK and performs MLSE using the ViterbiAlgorithm. The decoding metric is

M~=j~ q)glk expt{1jl0 k}92)k exp{j02 }-rk (22)

which has the same structure as (11), apart from the fadinggains being replaced by their estimates from the previousiteration. Denoting the T-spaced ST-MSK sequence thatminimizes (22) as {cm } and letting {Q(k) } be thecorresponding phase sequences, the fading gains are then re-estimated using a Weiner filter spanning 2,8+1 ST intervals(or equivalently, K = 2I(2, + 1) samples) [ 1 5] [16]. Assumingwe wish to estimate gl 0 and g2,, the receiver first constructs

Ui= diag exp Oi-2),, exp (ii-2,1+2 ) ... exp Oi,(2,+2)1i=1,2, (23)

which are two diagonal matrices containing the decisionsmade by the VA for the interval (-2,6T, (2,6 + 2)T]. Given uland u2, the covariance matrix of

010

10

10

0-310

-410_

10 15 20 25 30SNR(dB)

Fig. 2. Receiver performance; f,T, = 0.03

I ---- ---- -H- -- -- --T-- ---- -<-- -- -- -----_

010

conv. ST diff. det.E DF-MSDD (q=0)

second iteration (q=2)- -e5- - coh. det.

10

10

10-

z = [r82#81, r-2,112II r(28 2)1 ] ,

the received pattern in the same interval, is2

-Dzz= Eui (A2+1l® J21 )uiT + INoI (2/+1)21 .i=1

(24)10-4

(25)

The presence of the Kronecker product operator in (25)reflects the assumption that the fading gains are constant overone ST block. The correlation between gi, and z, on theother hand, is

&iFz (k28+1g 121 )Ui g (26)where 2g has the same structure as the vector X2a1 in (20).Finally, the fading gain estimates are

1=) (eFJi,,q')Z, (27)where (DiJZ(D-' represents a Weiner filter. Being dependant onthe most recent VA decision, (27) suggests that the estimationfilter must be either generated online or stored in memory.Thus implying a storage requirement exponential in ,6 or anonline computation with complexity in the order of0([(2, + 1)2I]3). In practice however, we can avoid thisapparent complexity by using an iterative interferencecancellation technique to perform the channel estimation.

IV. RESULTS

We present in this section the simulated bit-error rate(BER) of the proposed receiver, the ideal coherent detector

10 15 20SNR(dB)

010

-110

25 30

Fig. 3. Receiver performance; fTT = 0.06.

~~~~~-- I------_-__- - - -_- conv. ST diff. det.

E DF-MSDD (q=0)second iteration (q=2)

-e3 coh. det.

10 ------

10- Ee ' -

0-4-------l---- ~-- - - - -X - - -- ---- -

__- -------- -- 4\- --- - -10

10 15 20 25 30SNR(dB)

Fig. 4. Receiver performance with sample-to-sample variation in the channelfading;fTD 0.03.

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(denoted by "coh. det.") and the conventional ST differentialdetector (denoted by "conv. ST diff. det."), which only usesthe T-spaced samples. In general, the span of the initializationand iterative channel estimation filters, used in (19) and (27)respectively, should reflect the coherence in the channelfading; i.e. be a function of the normalized fading rate fDT7S.For our simulations, we chose a = = 6 for fDTT = 0.03,and a =g = 3 for fDTT = 0.06. Lastly, we set f = 4 IT anda DF-MSDD with N = 12 was employed. Note that an over-sampling factor of 4 will allow the MSK signal to pass throughthe anti-aliasing factor almost undistorted.

Figs. 2 and 3 show the receiver performance for theinitialization and the second iteration, for fDTT = 0.03 and0.06, respectively. Clearly, the two iterations capitalize on thecontinuous phase characteristic of the transmission;performance improvements of approximately 3.5 dB and 3 dBare made compared to the initialization performances at a BERof 10-3, for fDTsT = 0.03 and 0.06, respectively. Lookingspecifically at fDTsT = 0.03 and the same BER, we can seethat the receiver performs within 3 dB of the CD lower boundand 7 dB better than conventional ST-differential detection.Furthermore, for the range of SNR simulated, the receiversuppresses the error-floor associated with conventional ST-differential detection. Although the performance differencebetween the proposed receiver and that of CD widens forfDTST = 0.06, it is clear that the error floor is similarly avoidedand the proposed scheme follows the performance slope of theCD curve. Furthermore, it is observed that a diversity ordergreater than two is achieved. This additional diversity gain isdue to the implicit coding in MSK and the implicitinterleaving effect that comes with fast fading.We also investigate the behavior of the proposed system in

the situation where, contrary to the assumptions made in thereceiver design, the channel fading is not necessarily constantover each ST interval. Restating, we let the channel fadingsamples now exhibit the sample-to-sample autocorrelation

R = 2E{gi gi,k+ } = Jo (2ff JDTST 1(2I)). (28)Fig. 4 shows the receiver performance for this case withfDTST = 0.03. Due to the channel fading changing over eachST-interval, the orthogonality of the Alamouti-type format isdisrupted and there is a slight degradation in the CD lowerbound when compared to that given in Fig. 2. Although thereis also a slight relative performance loss in the proposedreceiver when compared to that in Fig. 2, the receiver stillachieves a performance within 3.5 dB of the CD lower bound,without any increase in receiver complexity. Hence, weconclude that the proposed receiver structure is robust tosample-by-sample variations in the channel fading.

V. CONCLUSIONS

In conclusion, we have presented an MSK transmissiondiversity scheme motivated by the Alamouti-type codestructure. The implicit block differential encoding allows forlow-complexity non-coherent sequence detection to take place

using a down-sampled subset of the received statistics. Usingiterative channel estimation and sequence detection, thereceiver avoids the error floor evident on fast-fading channels,and attains not only the intended space-diversity, but alsoadditional time-diversity from the MSK modulation.For fDT7S = 0.03, the proposed receiver achieves performancewithin 3 dB of the coherent detection lower bound.

Furthernore, even though the receiver was designed underthe assumption of the static fading over each ST interval,results show that the receiver is robust to sample-to-samplechanges in the channel fading.

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