Optimal demodulation of PAM signals

Kalman filtering theory is applied to yield an optimal causal demodulator for pulse-amplitude-modulated (PAM) signals in the presence of white Gaussian noise. The discrete-time data (or sampled continuous-time data) are assumed to he either a stationary or nonstationary Gaussian stochastic process, io general nonwhite. Optimal demodulation with delay is also achieved by application of Kalman filtering theory. The resulting demodulators (fixed-lag smoothers) are readily constructed and their performance represents in many cases a significant improvement over that for the optimal demodulator without delay. The fixed-lag smoothing results are in contrast to those for amplitude-modulated signals (AM) where only approximate fixed-lag smoothing is possible, and this with considerable design effort. The performance of the optimal PAM demodulator is shown to be equivalent to that of an optimal discrete filter for the discrete data. 1. INTRODIJcTION BROADLY speaking, the problem to be considered in this paper is the optimal demodulation of pulseamplitude-modulated (PAM) signals in the presence of white Gaussian channel noise. The term “optimal” is used in the minimum-least-squares-error sense. The conventional frequency-domain approaches to this problem [1], [2] yield the spectrum of the receiver (demodulator) for the unrealizable case when an infinite delay in demodulation is assumed. The frequency spectrum of the optimal receiver is not given explicitly, and complicated algorithms are presented for its calculation. Once this is obtained, standard techniques can then be employed to yield a realizable (causal) demodulator where a fixed time lag (the lag may be zero) exists between the signal production and its estimation. Of course the greater this lag, the more nearly optimal will be the system. The problem considered in this paper can now be less broadly stated as the optimal fixed-lag demodulation of PAM signals. The approach to this problem to be adopted throughout the paper is the state-variable approach. In particular, both continuous-time and discrete-time Kalman-filtering results are applied to a state-space PAM signal model to yield the various results. For those familiar with state-space methods and Kalman-filtering results and their application to communications system designs [3]–[5], the application of these methods is perfectly straightforward, although perhaps not immediately obvious. One important advantage of using the state-variable approach in filtering problems is that time-varying signal models and receivers present no special difficulties. These time-varying systems cannot be handled without the inMaouscript received September 15, 1971; revised April 24, 1972. This work was supported by the Australian Research Grants Committee. The authors are with the Department of Electrical Engineering, University of Newcastle, New South Wales, Australia, production of some simplifying assumptions using frequencydomain approaches since these are restricted to time-invariant systems. As for the PAM filtering problem under consideration, the signal model used in our paper is a timevarying one and we believe corresponds better to the practical situation than does the model of [3], [4]. As a consequence of the time-varying model, the optimal receiver is time varying, but we hasten to point out that the optimal time-invariant receiver can be determined if required. A direct comparison of the results of this paper with the frequency-domain results is difficult, as the problems solved are posed somewhat differently. Since the solutions are optimum or near optimum in either case, comparison is perhaps best attempted by considering the calculations involved, the resulting optimum system structures, and the claims of optimality made for each approach. Before proceeding with the development of the ideas of the paper, we present a precise problem statement. Consider the case of discrete data y(t~), k = 1,2,. ... which is a sample function of a discrete-time Gaussian stochastic process, which can be modeled as follows: X(tk) = r#(rk,tk~)x(tk ,) + Z/(fk ~) (1) y(rk) = hex (2) where x(t~) is a state n-vector and u(t~) is a white Guassian noise discrete process of zero mean and covariance Q(t~) d(t~ – ?,). The initial state vector is a zero-mean Gaussian random vector independent of U(”) and having a covariance PO. It is now assumed that unknown noisy measurements z([) are made of a carrier signal c(r) modulated by the data y(r~) as follows z(t) = C(t)y(tk) + u(t), tk k. Of course the discrete data .p(tk) could be in Fact samples taken from a continuous-time process. For the case when the continuous-time process can be modeled using a linear dynamical finite-dimensional system driven by white Gaussian noise, it is not difficult to determine the parameters of ( 1) and (2) from those of the continuous model [6]. MOORE AND HETRAKUL: OPTIMAL l) EMOL)ULATION OF PAM SIGNALS In the next section both continuous-time and discretetime linear optimal filtering results are reviewed in order to introduce notation and for reference purposes. In the following section these results are applied to yield optimal PAM demodulators without delay. In Section IV, the case of optimal demodulation with delay is considered, while in Section V further extensions are briefly discussed. 11. REVIEW OF LINEAR OPTIMAL FILTERING RESULTS A. Continuous-Time Case Consider the linear dynamical system described by the differential equation i(t)= F(t)x(t) + G(t)u(t) (5) z(t) = H’(t)x(t) + v(t) (6) where x(t) is the state n-vector, z(t) the measurement m-vector, and inputs u(t) and v(t)are independent white Gaussian noise vectors of zero mean and covariances Q(t) 6(t– T)and R(t) ~(t – T), respectively. The matrices F, G, and H are of appropriate dimension. The initial state vector x(to) = XO is a zero-mean Gaussian random vector independent of U(. ) and U(.) and having a covariance PO. The optimal (minimum error variance) filter for the continuous system (5), (6) consists of the following differential equations for the conditional mean 2(t/t)and error covariance matrix P(t) &t;/t)= F(t).i?(t/t) + K(t).qt), (7) with f(O/0) given