When Is Differential Detection Optimum for Ideal and Partially Coherent Reception/Demodulation of -PSK? arvin Simon, Dariush Divsalar Jet Propulsion Laboratory, California Institute of Technology 4800 Oa Grove Drive Pasadena, CA 909 USA {arvin.k.simon, Dariush.Divsalar}@jpl.nasa.gov Abstract We investigate the low SNR behavior of the optimum AP sequence (blocwise) detection metric for ideal and partially coherent detection of differentially encoded - PSK modulation. In particular, we show that based on an observation of N symbols in the presence of a phase ambiguity introduced by the carrier tracing loop and assumed to be constant over the observation, this optimum coherent metric approaches that of multiple symbol differential detection (SDD) asymptotically as the SNR approaches zero. ore importantly, we show that for practical low SNR applications (e.g., error-correction coded systems), the region of SNR where the optimum coherent metric becomes synonymous with SDD grows with the modulation order. Finally, we consider the SDD metric itself modified for symbol-by-symbol detection and demonstrate that for sufficiently low SNR (asymptotically as the SNR approaches zero), its behavior is equivalent to conventional N differential detection, i.e., no gain to be had be extending the observation interval beyond two symbols. Unfortunately, unlie the sequence detection case, here the SNR region of validity of the low SNR approximation does not increase with increasing modulation order. I. INTRODUCTION A number of years ago, the authors investigated [] the optimality of coherent receivers for differentially encoded - PSK. In that paper, it was demonstrated for an observation of N symbols that the so-called classical coherent receiver [], i.e., the one that maes coherent hard decisions on two successive symbol phases and then differentially decodes them to arrive at a decision on the information symbol, differs from the blocwise sequence detection scheme for >. At the same time, the true optimum In using the term -PSK we implicitly mae the conventional assumption that the phases are uniformly distributed around the circle and are transmitted independently from symbol to symbol. Later on, it was shown in Ref. 6 that the blocwise sequence detection scheme with N = coincides with that found from AP symbol detection at which point it could now be formally concluded that what was classically thought to be the optimum coherent receiver was in fact not optimal in the receiver for directly detecting the information phases, assuming ideal carrier phase coherence (perfect carrier demodulation except for the possibility of an -phase ambiguity introduced by the carrier tracing loop), was derived and illustrated. Finally, it was also shown in [] that under an approximation of the lielihood function (LF) used to derive the bloc detection sequence receiver that is suitable for high SNR, the classical coherent receiver becomes optimum. What was not investigated there and turns out to be particularly interesting is the behavior of the bloc detection sequence receiver at low SNR. In a recently published paper [3], the authors demonstrated that in the asymptotic limit of low SNR, for > the AP symbol detection scheme, which coincides with the bloc detection sequence scheme based on a two-symbol observation, becomes equivalent to that of conventional differential detection (DD) which suggests that at sufficiently low SNR, one needn t trac the carrier at all. This paper starts by extending the above notions to detection of a sequence of N- differentially encoded symbols based on observation of N symbols still maintaining the assumption of ideal (aside from the phase ambiguity) carrier phase demodulation. After specifying the optimum coherent blocwise sequence metric for an N-symbol observation, we show that asymptotically at low SNR, the receiver based on this metric implements multiple symbol differential detection (SDD) [4], which again requires no carrier tracing function. Furthermore, we show that under these same low SNR conditions, modifying the SDD sequence metric to one that maes AP symbol-by-symbol decisions, reduces to conventional differential detection based on two-symbol observation, i.e., the detection performance is not improved by extending the observation interval beyond N=. The important question that was not previously addressed is: How low in SNR must one have so that the asymptotic behavior mentioned above is valid? Furthermore, is the SNR region of validity a function of the order of the sense of minimizing symbol error probability.
modulation? In this paper, we attempt to answer these questions using a combination of analysis and simulation. In particular, we analytically perform a series (aclaurin) expansion of the LF corresponding to the optimum coherent metric based on an N-symbol observation in terms of powers of the SDD metric and demonstrate that the combination of the first / terms has an equivalence with this metric. Based on this, we conjecture that as the order of the modulation increases, the low SNR region of validity for which the SDD is equivalent to that of optimum sequence coherent detection also increases. The validity of this conjecture is later supported by simulation results obtained from computer simulation. Finally, if now one allows for a non-ideal carrier tracing loop, i.e., partially coherent detection, even then the optimum metric (an extension of the results presented in Ref. 4 to the case where the carrier demodulation reference has associated with it an -phase ambiguity as in an -PSK tracing loop) reduces to conventional DD. II. SYSTE ODEL AND DERIVATION OF THE ASYPTOTIC BEHAVIOR OF THE OPTIU RECEIVER AT LOW SNR (PERFECT CARRIER PHASE TRACKING) We model the problem at complex baseband as is customary for ideal coherent reception. In particular, the transmitted signal in the th signaling interval is given by s Pe j Pe j, T s t T s () where P denotes the transmitted power, T s denotes the symbol time, is the actual transmitted phase in this interval, and is the information phase for that same interval. The phase taes on values i / /, i,,..., whereas the phase taes on values i /, i,,..., each with probability / where m, m a positive integer. 3 The receiver observes the sequence 3 It has become traditional in the literature to represent -PSK constellations with balanced (in amplitude) I and Q components (except for = where only an I component exists). Defining the transmitted constellation phases by i / /, i,,..., taes care of both the binary and higher order -PSK cases. At the same time since the information phases are represented by the difference of two successive transmitted phases, their constellation consists of the phases i /, i,,...,. r r N, r N,..., r, r with r l s l e j c n l where n l is a zero mean complex Gaussian additive noise random variable (RV) with variance n per dimension and c is the received carrier phase. After ideal demodulation by the carrier tracing loop, the received signal vector becomes x x N, x N,..., x, with x l r x l e j c where is the phase ambiguity introduced by this loop (assumed to be constant over the N-symbol observation) and taes on the discrete set of values i /, i,,..., with probability /. By a straightforward extension of the approach taen in [], it can be shown, analogous to the result given in [, Eq. (9)], that the LF to be maximized for maing an optimum joint decision on the information phase sequence ˆ N, N 3,...,, is given by LF / i P cosh Re e j i x N n and thus, the decision rule becomes 4 Choose ˆ argmax ˆ / i N l 0 P cosh Re e j i x N n x l exp j N l 0 x l exp j Nl lm m 0 () Nl lm m 0 (3) where the notation arg max y means the value of x that x maximizes y and i i / /, i,,..., corresponds to the allowable values for the total received phase of each symbol after demodulation. We now examine the behavior of this decision rule in the limit of small SNR. Defining V P n x N N i 0 x i exp j Ni im m 0 4 Note that this result differs from that given in Ref. 6 where it is concluded that for AP sequence detection the optimum decision rule in the presence of an -phase ambiguity associated with the carrier tracing loop corresponds to that of the classical coherent receiver, i.e., pairwise symbol detection [, Fig. 4.3b]. (4)
3 then the LF of () can be expressed as / LF coshre{e j i V / } cosh V i cos(argv i ) i (5) We recall from Ref. 4 that V P x N N x i exp j N i im n i0 m0 P n r N N r i exp j Ni m0 im i0 is the decision metric 5 used for SDD and was derived from the LF for that detection scheme. What we are first interested in showing is that at low SNR, the LF of (5) approaches the same behavior. Since from (6), it is clear that V is directly (6) proportional to SNR, then for this purpose, we propose expanding (5) in a aclaurin series in powers of V. Proceeding this way, we can show that the LF can be partitioned into two parts, one composed of the first / terms that depends only on even powers of V and a second part composed of the remaining terms that depend on arg V as well as even powers of V. Specifically, we derive the following closed-form results. For > LF where with c n, n 0 n n V n LF n! n LF n n! c n n, V n / n p 0 In (9), p, p 0, p 0 n n p () n 0 n n! n V LF (7) (8) p cos(p argv ) p (9) is the Neumann coefficient. For = 5 We note that any monotonic function of V, in particular, a series containing powers of V with constant (i.e., independent of arg V ) coefficients is also an SDD decision statistic. LF n n n! n p 0 n p cos( pargv ) n p (0) V n We observe that the first (summation) term of (7) is purely a monotonic function of V and thus represents a decision metric for SDD, i.e., Choose ˆ arg max r N N i 0 r i exp j Ni im m 0 () On the other hand, LF, which represents the remainder of the aclaurin series in V, has coefficients that depend on argv and thus is not appropriate to an SDD decision rule. Furthermore, we note from (0) that for =, no portion of LF behaves lie an SDD decision metric. While it may seem surprising at first that a differential detection metric results as being optimum for decision-maing when in fact ideal phase coherent reception is postulated at the outset, the following intuitive argument is offered to support this finding. In deriving the optimum receiver for an N-symbol observation of differentially encoded -PSK received in a phase noncoherent environment, one forms the lielihood function of the received signal conditioned on the unnown carrier phase and then averages it over this phase which is assumed to be a continuous RV with a uniform distribution. This results in a metric that incorporates a modified Bessel function of the first ind with an argument equal to the magnitude of V as in (6). Then, as shown in Ref. 4, maing use of the monotonicity of the Bessel function, one arrives at the SDD with decision rule in accordance with () which is optimum for all SNR. In the case of ideal coherent demodulation of differentially encoded PSK, the lielihood function of the received signal is computed conditioned on the phase ambiguity associated with the carrier tracing function and then averaged over this phase which now is a discrete RV with a uniform distribution defined on the set of uniformly spaced values i, i,,...,. This averaging results in a sum of exponentials which because of the symmetry of the -PSK constellation around the circle can be rewritten as a sum of / hyperbolic cosine functions as seen in (). Then, as we have seen by expanding this sum of hyperbolic cosine function in a power (aclaurin) series in V, the first / terms of this series results in the same SDD metric. In effect, it is now the unnown discrete (uniformly distributed) carrier phase ambiguity, assumed to be constant over the observation interval, that lie the unnown continuous carrier phase in the noncoherent case, introduces memory into the
4 decision process and results in a sequence (as opposed to a symbol-symbol) decision metric being optimum. The question that remains at this point is: how low must the SNR be in order for the approximation of the LF by the SDD portion or equivalently the decision rule of () to be valid? From the form of (7), we have already observed that the larger the value of, the more terms in the aclaurin series can be retained (in particular, the first / terms) in the SDD portion. One possible outcome of this is that the larger the value of, the higher the value of SNR can be, that will result in the SDD metric being optimum for sequence detection. Indeed the validity of this conjecture will be born out by the simulation results presented later on in the paper. Before providing this validation, however, we first investigate the low SNR behavior of a symbol-by-symbol version of the detection metric derived from a multiple symbol observation. III. EVALUATION OF THE ULTIPLE SYBOL OBSERVATION SYBOL-BY-SYBOL DETECTION ETRIC AND ITS ASYPTOTIC BEHAVIOR AT LOW SNR We now show that at low SNR, the maximum-lielihood approach that led to the SDD decision rule of (6) when applied to the case of symbol-by-symbol detection reduces to that corresponding to conventional (two-symbol observation) differential detection as described by the particular case of this decision rule corresponding to N=. For SDD, the lielihood function (LF) is of the form [4] LF I 0 V () In view of the monotonicity of the Bessel function, the L sequence decision is given by (). If instead one was interested in L symbol detection based on an observation of N symbols, then one must for each symbol in the sequence other than the one to be detected, average the LF over the other N- symbols that are uniformly distributed around the circle. Thus, for example, to detect the information symbol N, one must compute the average LF LF I 0 V,,..., N3 (3) where now each of,,..., N 3 ranges over the set of uniformly distributed phases i i /, i,,...,. As an example, consider the case N 3 for which V P n x x e j x e j (4) To find the LF for detection of say we must compute P LF I 0 r r e j r e j n P I 0 r r e j r e j i i n (5) which cannot be evaluated analytically. Once again expanding the LF in a power series in V (suitable for low SNR considerations), then (5) becomes LF n n n! n P n i r r e j r e j i (6) We examine the behavior of the n = term which, ignoring constant multiplicative factors since they do not affect the ultimate decision on, becomes LF i r r e j r e j i Expanding the squared magnitude in (7) gives LF i r r r Re r e j i Re e j r * r * (7) n r r e j i (8) In the final result in (8), the first three magnitude squared terms do not affect the decision based on the LF and the averaged over the uniformly discrete distribution of i (represented by the summation on i from to ) of the fourth term equates to zero. Hence, we can simplify the LF of (8) to LF i i Re Re e j r r e j i Im Im e j r r e j i r * r r r r Re r e j Re Re r e j i Im Im r e j i (9) However, for fixed, the terms Re r e j i Im r e j i when averaged over i (represented by and
5 the summation on i from to ) again become equal to zero. Thus, since the first term in (9) is independent of the summation index i, then ignoring the constant multiplier of, the LF reduces to LF Re * r r e j (0) which is the conventional LF for maing a differential detection of based on only a two symbol observation. Following an analogous approach, one can show that the average LF needed for maing a decision on becomes LF Re r * r e j () which again is the conventional LF for differential detection of based on only a two-symbol observation. Thus, at least from this example, one can conclude that for sufficiently low SNR, the SDD algorithm reduces to symbol-by-symbol detection using conventional (overlapping two-symbol observation) DD and thus from an L standpoint there is no gain to be had by extending the observation interval from two to three symbols. If we now consider the n = term in (6), then analogous to (7), an N = 3 symbol observation requires evaluating the term r r e j r e j 4 i r r r Re r * r e j i i i r * r r e j i Re e j () for inclusion in the LF. Expanding the squared sum of terms in () and evaluating the resulting sums maing use of the cos i, 0, 4 i sin i 0 for all i (3) then after dropping terms that do not affect the decision on as well as a constant multiplication factor, we obtain after much manipulation 6 i r r e j r e j i 4 * r r e j Re * r r r Re r r e j, 4 (4) Thus, while the first term in (4) suggests conventional DD, unfortunately the second term does not. Hence, we conclude that for all, including the n = terms in the power series expansion of the LF does not suggest conventional DD as being optimum for symbol-by-symbol detection. Equivalently, independent of the value of, conventional DD is optimum for symbol-by-symbol detection only at low enough values of SNR such that the first two terms, (i.e., n = 0,), of the power series expansion of the LF are sufficient. The extension of the above to arbitrary N > 3 follows along the same lines but with rapidly increasing mathematical complexity. Nevertheless, we have been able to show via simulation that the same conclusions as demonstrated by the above example continue to bear truth. IV. EVALUATION OF THE PARTIALLY COHERENT ULTIPLE SYBOL DIFFERENTIAL DETECTION ETRIC AT LOW SNR We now consider the more practical scenario where the - PSK carrier tracing loop is non-ideal, i.e., in addition to the -phase ambiguity that results in the loop locing at one of uniformly distributed points around the circle, the loop also exhibits a continuous phase error, c, around its loc point with a statistical distribution typically characterized by the Tihonov probability density function (PDF) p c exp cos c I 0 (5) where characterizes the loop SNR. In Ref. 5, the optimum detection metric was derived for a partially coherent scenario such as that characterized by (5); however, the presence of the -fold phase ambiguity was not accounted for there since implicitly it was assumed that a pilot signal (unmodulated carrier) was included in the transmitted signal which then allows for carrier tracing with a phase-loced loop (PLL) that has no such phase ambiguity. If now we include the presence of the uniformly distributed -phase ambiguity in the analysis approach taen in Ref. 5, then it is straightforward to show that analogous to Eq. (9) of that reference, the LF for maing a sequence decision based on an N-symbol observation becomes (eliminating terms that have no bearing on the decision) 6 For additional terms are present in the LF; however, they do not affect the conclusion reached above.
6 N LF I 0 e j l P x i e j i n l n i 0 (6) Note that for 0, i.e., noncoherent reception, and no phase ambiguity ( l 0 for all l,,..., ), the LF of (6) differs from that in () together with (4) in that in the case of the latter, differential encoding has been assumed a priori and thus there the metric is expressed in terms of the information phases i rather than the transmitted phases i. Specifically, it was shown in Ref. 4 that because of the presence of the dc term n, the resulting metric was nonambiguous and thus differential encoding was not necessary; hence, an N-symbol observation directly results in an N- symbol sequence decision on the transmitted phases that are now also the information phases. If we now again expand the LF of (6) in a power series and maintain only the n = term, then eliminating the constant additive terms through the derivation we obtain LF l P n 4 N 4 e j l P x i e j i n n N i 0 i 0 x i e j i N Re e j l P x i e j i n (7) However, the second (summation) term in the second line of (7) which represents the average over the uniformly distributed l equates to zero and thus the LF further simplifies to (ignoring the scaling constant) LF N i 0 x i e j i l N i 0 i 0 r i e j i (8) which is independent of the partial coherence and is in fact the SDD LF prior to the necessity of invoing differential encoding to resolve the inherent phase ambiguity, i.e., a constant phase shift of each i, i 0,..., N results in the same LF. V. NUERICAL RESULTS Plotted in Figs.,, and 3 are illustrations of the symbol error probability (SEP) performance of the receiver that implements the optimum coherent sequence decision rule of (3) for values of equal to, 4, and 8 and observation intervals of N,5 symbols. The results displayed illustrate SEP versus bit energy-to-noise ratio, E b /N 0, and were obtained by computer simulation. We observe from these plots that, interestingly enough, the performance is virtually independent of N. Furthermore, while it was previously shown in Ref. that for the binary case and a twosymbol observation N the decision rule of (3) is mathematically equivalent to maing independent hard (binary) decisions on adjacent symbols (bits) followed by differential decoding, i.e., the classical coherent receiver, for any other combination of and N (including the binary case with N ), the decision rule does not assume this form. Nevertheless, it is of interest to mae a further comparison with the performance of this classical coherent receiver of differentially encoded -PSK, which for a given value of maes symbol-by-symbol decisions on overlapping pairs of symbols and then differentially decodes each pair. The performance of this receiver is well documented in the literature, e.g., [, Eq. (4.00)] and is superimposed on the plots in Figs.,, and 3 in dashed lines. It is interesting to observe from these figures that, in addition to the exact equivalence already mentioned for the case,n, for 4 and 8for the entire range of SNRs illustrated, the SEP of the classical coherent receiver is virtually indistinguishable from the SEP of the optimum coherent sequence receiver determined by simulations. 7 Thus, one is led to the conclusion that for all practical purposes, the optimum and classical coherent receivers have equivalent performance for all SNRs, modulation orders, and in the case of the former, all observation interval lengths. Previous to this, it was only demonstrated mathematically [] (but not numerically) that the large SNR asymptotic performance of the optimum coherent receiver approaches that of the classical coherent receiver for. Now on to the more interesting comparisons with the performance of the SDD. Superimposed on the plots in Figs.,, and 3 are the SEP performances corresponding to the SDD metric as described by (6) for observation intervals of, 3 and 5 symbols. We observe as suggested by the characterization of the LF in (7), the larger the value of, the larger the value of SNR can be, that will result in the SDD metric being optimum for sequence detection. Furthermore, as mathematically discussed in Ref. 3 but now numerically illustrated in Figs.,3, for the case,n the low SNR asymptotic performance of the optimum coherent metric approaches that of classical differential detection. Before concluding, we remind the reader that the numerical results presented and comparisons made thus far have assumed that, aside from the -phase ambiguity, the carrier loop was assumed to be ideal, i.e., the effective loop SNR was sufficiently large that the phase error introduced by the loop 7 Additional simulation results taen for SEPs as low as 0-4 continue to show the same lac of distinction between the performances of the two different forms of receiver.
7 could be assumed negligible. At the same time, we mae note of the fact that in a practical -PSK system, the effective SNR of the carrier tracing loop, e.g., an -phase Costas loop, is made up of the product of the loop SNR corresponding to a linear loop such as a phase-loced loop (PLL) and the nonlinear -phase loss (in the binary case often referred to as squaring loss ) due to the signal times signals S, signal time noises N, and noise time noise N N products. The distortions produced by these products in so far as their effect on the evaluation of the - phase loss become increasing dramatic as the symbol SNR decreases and the value of increases. This further enhances the importance and significance of the results presented here in that in a real situation the optimum coherent SEP performance illustrated in Figs., and 3 will be further degraded by the increased phase jitter that comes about as a result of the reduction of effective loop SNR due to the excessive -phase loss. Thus, in practice, the region of SNR for which SDD is equivalent to that of coherent detection for any N will extend beyond that illustrated in these figures which correspond to the ideal (except for the phase ambiguity) carrier tracing situation. Fig.. Symbol error probability performance of various coherent and differentially coherent detection schemes for differentially encoded -PSK; = 4. Fig.. Symbol error probability performance of various coherent and differentially coherent detection schemes for differentially encoded -PSK; =. Fig. 3. Symbol error probability performance of various coherent and differentially coherent detection schemes for differentially encoded -PSK; = 8.
8 VI. CONCLUSIONS When transmitting differentially encoded -PSK, the optimum sequence decision metric at sufficiently low SNR becomes synonymous with that corresponding to multiple symbol differential detection (SDD) where the SNR region of validity increases with the order of the modulation,. Furthermore, for sufficiently low SNR, the SDD algorithm reduces to symbol-by-symbol detection using conventional (overlapping two-symbol observation) DD and thus from an L standpoint there is no gain to be had by extending the observation interval beyond two symbols. However, here the SNR region of validity does not grow with the order of the modulation. Finally, for partially coherent detection of - PSK, at sufficiently low SNR, the optimum sequence metric reduces to that corresponding to SDD (prior to the necessity of invoing differential encoding to resolve the inherent phase ambiguity) and as such it is independent of the partial coherence. ACKNOWLEDGENT The research in this paper was performed at the Jet Propulsion Laboratory, California Institute of Technology under a contract with the National Aeronautics and Space Administration. REFERENCES [].K. Simon and D. Divsalar, On the optimality of classical coherent receivers of differentially encoded -PSK, IEEE Comm. Lett., vol., no. 3, ay 997, pp. 67 70. []. K. Simon, S. Hinedi and W. C. Lindsey, Digital Communication Techniques: Signal Design and Detection, Upper Saddle River, NJ: Prentice Hall, 995. [3] V. R. Kanchumarthy T. Choudhari and R. Viswanathan, Performance analysis of optimum receivers for differentially encoded -PSK in low SNR, IEEE Comm. Lett., vol. 9, no. 5, ay 005, pp. 400 40. [4]. K. Simon and D. Divsalar, "ultiple Symbol differential detection of PSK," IEEE Trans. Commun., vol. 38, no. 3, arch 990, pp. 300-308. [5]. K. Simon and D. Divsalar, "ultiple Symbol Partially Coherent Detection of PSK," IEEE Trans. Commun., vol. 4, no. /3/4, February/arch/April 994, Part I, pp. 430-439. [6] G. Colavolpe, Classical coherent receivers for differentially encoded -PSK are optimal, IEEE Commun. Lett., vol. 8, no. 4, April 004, pp. 3.