The Performance of Digital Matched Filters for Multilevel Signals

The analysis and optimization of digital matched filters (DMF's) with multilevel quantizers, which are matched to multilevel amplitude and phase modulated (
AM/\phi M
) signals, are considered here. Quite general expressions for the DMF's output signal-to-noise ratio (SNR
0
) have been obtained for the case in which the interference has a symmetric, differentiable probability density distribution. For such interference, SNR
0
has been shown to be separable into two factors, which respectively represent the effects of the quantization and correlation operations of the DMF. The two factors can be optimized independently of each other. In particular, the optimal quantizer strategy for Gaussian interference is proven to be (within the limits of practical approximation and also asymptotically) one with equal spacings of the quantizer levels and thresholds. All the SNR
0
calculations have been carried out for a complex output
X
rather than the commonly used real-valued, physical output
Z
. The validity of the relationship
SNR_{0}X = 2 SNR_{0}Z
has been proven for all types of interference considered in this work. Finally, the optimal
SNR_{0}X
for Gaussian interference has been proven to be bounded from above by
SNR_{0}X
for the analog matched filter and from below by that of DMF with hard limiter as a quantizer.