Optimum Pre- and Postfiltering of Sampled Signals with Application to Pulse Modulation and Data Compression Systems

The following classical problem is solved and the results are applied to minimize the mean-square error ε of communication and data compression systems: "a stationary random input signal
x(t)
is prefiltered, corrupted by noise
n(t)
, sampled every
T
seconds, and finally postfiltered to yield output
\hat{x}(t)
. Let the desired output
z(t) = a(t) \bigotimes x(t)
where
a(t)
is any real function of
t
and
\bigotimes
denotes convolution. Determine the linear timeinvariant pre- and postfilters which jointly minimize
\epsilon = E\{\frac{1}{T}\int^{T}_{0}[z(t) - \hat{x}(t)]^{2}dt\}
subject to power constraint
P = E\{[k(t) \bigotimes f(t) \bigotimes x(t)]^{2}\}
.'' Operator
E{\dot}
denotes expected value,
k(t)
is a real function of
t
, and
f(t)
is the prefilter impulse response. Appropriate choice of
T
,
P,k(t)
, and
n(t)
makes the solution applicable to amplitude, angle, pulse-amplitude (PAM), pulse-code (PCM), and differential pulsecode (DPCM) modulation systems, and data compression systems. In this analysis no restrictions are placed on the input-signal spectrum, the noise spectrum, or the passbands of the filters; furthermore, the cross correlation between signal and noise is taken into consideration. Necessary and sufficient conditions are obtained for the jointly optimum pre- and postfilters. The performance obtainable using these optimum filters is compared with that obtainable using suboptimum filters. One suboptimum filtering scheme is derived which yields virtually the same performance as optimum filters and has the practical advantage that the filter transfer characteristics are independent of noise
n(t)
in many cases of interest. In applying the results to PCM, DPCM, and data compression systems, the filters, sampling rate, and quantizer are jointly optimized. The performance obtainable for various filtering schemes and various communication systems is compared with the optimum attainable as calculated from information theory.