Adaptive Maximum-Likelihood Receiver for Carrier-Modulated Data ...

624

IEEE TRANSACTIONS COM-22, ON COMMUNICATIONS, VOL.

tion,” I E E E Trans. Commun.Technol., vol. COM-19, pp. 992-1006, Dec. 1971. [4]A. Habibiand P. A. Wintz, “Image coding by linear transformations and block quantization,” I E E E Trans. Comnlun. Technol., vol. COM-19, pp. 50-62, Feb. 1971. [5] J. J. Y. Huang and P. M. Schultheiss, “Block quantization of correlated Gaussian random variables,” IEEE Trans. CottLmun. Syst., vol..CS-ll, pp. 289-296, Sept. 1963. [6] A. Habibl, “Comparison of nth-order DPCM encoder with linear transformationsand block quantization techniques,” I E E E Trans.Conmun. l’echnol., vol. COM-19, pp. 948-956, Dec. 1971. 171 P. A. Wintz and A. J. Kurtenbach, “Waveformerrorcontrol in PCM telemetry,” I E E E Trans. Inform. Theory, vol. IT-14, pp. 6.50-661, Sept. 1968. [8] A: Habibi, “Performance of zero-memory quantizers using rate dlstortlon criteria,” to be published. [9] B. Smith,“Instantaneous companding of quantized signals,” Bell Slyst. Tech. J., vol. 36, pp. 44-48, Jan. 19.51. [lo] P. J. Ready and P. A. Wintz, “Multispectral data compression through transform coding and block quantization,” School of Elec. Eng., Purdue Univ., Lafayette, Ind., Tech. Rep. TR-EE 72-29, May 1972. [ll] N. Ahmed, T.Natarajan,and K. It. Rao,“Discrete cosine transform,” I E E E Trans. on Comput., vol. C-23, pp. 90-93, Jan. 1974. [12] W. I
WC(t1

-

- real signal complex signal Fig. 1. General linear carrier-modulated data-transmission system.

PCY(t),t E

TABLE I Modulation Scheme

eal

(an

DSB-AM SSB, VSB-AM PM AM-PM real

* SBB: f(t) form [I].

=

I

real I complex 1 complex

n u s t be the realization of the noise signal ~ ( t. )Hence, owing to the Gaussian-noise assumption, the likelihood function becomes (apart from a constant of proportionality) [32]

I I {a741= PCw(tl{anl)l

f 0)

=

1

{

-exp

real . complexs real

-

o[4;

I

where K-I(

T)

exp ( -&J - jpc).

(5)

In (5) the effect of low-pass-filtering the transposed noise is neglected since it affects only noise components outside the signal bandwidth of interest. Our channel model does not include frequency offset and phase jitter. It is understood that the demodulating carrier phase pcaccounts for these effects.

t2)

(9)

is the inverse of K ( T )

K ( T ) * K-'(T) and = .\riw,(t)

{an})K-'(t1-

* t u ( t z {a,)) CZtl cztz

fl(t) f j X ( f l ( t ) ] ,where X is the Hilbert trans-

w(t)

[Zb(t'I

~

= 8(T).'

(10)

The correctness of (9) for the complex-signal case is proven in Appendix I. Substituting (S) into (9) and considering only terms that depend on { a,} , yields

111. STRUCTURE OF T H E MAXIMUR'ILIKELIHOODRECEIVER The objective of the receiver is to estimate {a,)from a given signal y ( t ). Let the receiver observe y ( t ) within a time interval I which is supposed to be long enough so that the precise conditions at the boundaries of I are insignificant for thetotal observation.Lct {a,) bea 'hypothetical sequence of pulse amplitudestransmitted during I . The NILR by its definition [a], [SI determines asthe bestestimate of (a,} the sequence {a,)= { & } that maximizes the likelihood function p [ y ( t ) ,t C I 1 {a,}]. In the following paragraphstheshape of the signal element h ( t ) and the exact timing of received signal elements are assumed to be known. The noise of the transmission medium is supposed to bestationary Gaussian noise with zero mean and autocorrelation function W , ( T ) . From (5) the autocorrelation function of ~ ( t is) obtained as

w ( T )= ~ [ s ( t ) t+~ ( t =

= @(-T)

2W,( T ) exp ( - j w , ~ ) .

(6)

l?or example, if w c( t ) is white Gaussian noise (WGN) with double-sided spectraldensity No, then W C ( 7 )= N o 6 ( 7 ) and W ( 7 ) = 2NoS( T ) . In view of this important case it is

appropriate to introduce

s1 =

11 I

- 3-1,'

a t 1

- iT)K-'(t1

-

ts)k(tz - k T )

dtlclt2

I

1

=

k - i.

(13)

Thequantities zn and s1 canbeinterpretedassample values taken at the outputof a complex MF with impulse response function2 gRIF(t)

=

A( - t ) * K-'(t).

(14)

The derivation presented is mathematically weak in that it assumes I

+ cc1

e k 12

-

602)

I:=0

> 602

(

1- -

c I s1 I)

So I l#O,

[holds, if (52) is true].

(54)

Comparing (52) with the definition of S*( f) in (21) reveals that at distinct frequcncics, S*(f ) may approach 1 zero level withoutthis significantly affectingthe crror d2 ( E ) 2 - min 1 S* ( j ) ] ZOI/ (&)aO2 (50) pcrformancc: of the MLIt. This was first obscrvcd by so Kobayashi [25] for NIL decoding of (2m - 1)-ary __ cori t follows that if relative-level encoded signals. A simal of this kindcan -.-. liT just as well be interpreted asa noncoded m-ary signal with min ( ~ * ( f) t)u r r ( & ) 2 so a T ~ * ( f ) (51) intentionally introduced ISI, which causes S*(f) to be-, ___.__ come zero at f = 0 and/or 1/22". For the RILR __ - (usually) _ _ --_____-_..is satisfied, no cvcnt & can have smaller distance than 6,,. the t\vo concepts are equivalent. A conventional receiver, Hence,a sufficient butnot necessary condition for the however,. can interpret such signals only as (am - 1)-ary nonexistence of multiple error events with distance smallercoded sequences and therebyloses in the limit 3 dB, unless v

Y

cv

~

1 _

c

64'\-.

,/3,

e\. ,:,, @'-" (-& &.& 'j ~

,\;,..,?

632

IRICF, TRANSACTIONS ON COMMUNICATIOI\'S,

MAY

1974

error correcting schemes are used. But even then the losJ must be minimized as a function of these parameters, with can only partly be compensated, since the hard decisions j 3 0 held constant. Differentiating (Ci7) and applying the madebythe symbol-by-symbol decision circuit of the i Robbins-Monro stochasticapproximationmethod [46] conventional receiver cause an irreversible loss of infor- 'i leads to the stochastic steepest-descent algorithm com- prising the following recursive relations: mation. VI. AUTOMATIC RECEIVER ADAPTATION So far the exact signal and timing characteristics have been assumed to be known. However, in a realistic case t'he MLR must atleast be ableto extract the carrier phase and sample timing from the signal received. Beyond that, automatic adjustment of the MF will often be desirable or necessary. In this section we present an algorithm that simultaneously adjuststhe demodulatingcarrierphase and the sample timing, approximates the RIF by a transversal filter,and estimatesIS1 presentat theapproximated &IF output.The algorithm works in decision-directed mode in much the same way as described by Kobayashi [5] and Qureshi and Newhall [29]. In theproposed fully adaptive MLR theRiIF is approximated by a transversalfilter,similar tothe familiar adaptive equalizers described by Lucky et al. [l], [38], [39] and others[SI, [40]-[45]. An analog implementation will be assumed. Assuming N 1 taps equally spaced by rP seconds with tap gains gi,0 5 i 5 N , the output signal of the transversal filter at the nth sampling instant nT T ~ where T~ denotes the sampling phase, becomes

g;(n+1)

=

Wl(n+C

=

.*(n+1)

-

(y

o

(47agn i ,

3 1( n ) + (y,(n)7nun-2, -

= 7

(n)

-

Re

05i 5N

-

15 I I

(Fnin),

+ a,(,) Im ( ~ ~ 2 % ) .

pc(n+l)= pe(n)

II L

(58) (59) ( 60)

(61)

The step-size gains ag,a8,a'., and 0 1 ~must be positive and may depend. on n. In (60) 2, denotes the time derivative of the transversal filter output at the nthsampling instant. As the algorithm adjusts the transversal filter as RIIF, the values 31 approach the values sl required by the RILSE algorithm. Equations (58) and (59) differ from the corresponding equations o f ,an adaptive decision-feedback equalizer, or whitened --RIIF, only by the fact thathere at thetransversal filter output L preceding and L trailingIS1 terms are considered. The algorithm will force IS1outsidethis interval t o zero. The true RIF characteristic may often require a large value of L. However, since the complexity , of the MLSE algorithm increases exponentially with L, for the choice of L a compromise suggests itself [as]. I n N 1, 2, = giv(nT - iTp,Pe) P giyni(T,,p,). (55) many practical cases already with values L = 1 or L = 2, i=O i=O a good approximation of the ideal RIIF characteristic will Note that according to (4) and ( 5 ) we have y ( t,pc) a be obtained. The potential advantages of MLSE can thus y(t,p, = 0) exp ( -jpc?, where cpc is the demodulating be exploited to a commensurate degree a t a still manage-

+

+

+

c

c

7 8

carrier phase. The ideal RIF characteristic can be accom- able receiver plishcd if 1/aT, exceeds or at least is equal to the band- Introducing the wnmetry condition i - 1 = 52 into (56) I width of y(t), and N r p corresponds tothe duration of the we Obtain Of (59) signal element h ( t ) . ip+l) = 3 p as(7L)(7nun+l ?,an-$, 1 5 I 5 L. For the derivation of the adjustmentalgorithm we make the usual assumption that the transmitted data sequence (62) { a ,] is known. The decision delay of the Viterbi algorithm will be taken into account later when we devise the final This modification has the desirable effect of forcing the adaptive MLR structure. Section In I11 we have seen that transversal filter to produce at its output a symmetric the M F is rigorously defined by the fact that itminimizes signal element even if L and the transversal filter paramthe noise power relative to the instantaneous peak power eters are notfully adequate to achieve therewith the ideal of the signal element. Let il, j I I _< L, be estimated values M F characteristic. Equations (60) and (61) have been reported by of the sample values of the signal element at the transKobayashi [SI. They describe thc operation of two firstversal filter output. Then order phase-locked loops. Theoretically, if by (58) the L (complex) tap gains arc adapted, the adjustment of '7. r, = zn i1a,-2 (56) 2=-L and pe appears to benot really necessary. In practice, represents the estimated noise component of 2,. In order however, these phases must be controlled in order to compensate cdrrier and sampling frequency offsets. In case of to adjust the parameters g i , il,T,, and pC,thc variance of F,, considerable offset one might even add second-order terms N N to (60) and (61). Var (3,) = C C S i E C g n i ( ~ s , ~ c ) ~ n k ( 7 8 , ~ C ) ] g k i=o k=O The structure of the proposed MLR is seen in Fig. 6. It is basically a combination of the approaches of IcobayN L ashi [5] and Qureshi and Newhall [as], except that here - 2 Re { C C SiE[gnz(~s,pC) un-21iL! i=o l=-L the transversal filter approximates a true MI! with IS1 a t the transversal filter not being predefined. The receiver L L &E[G,-ia,-k]& (57) operatesin decision-directed mode with the MLSE exi=-L k = - L hibiting a decision delay of A4 L symbol intervals (see

+

-

+

+

1

+