Approximate lower bound for the SNR of matched filters

For the simple binary detection problem of detecting a known signal in the presence of additive noise, the matched filter is well known to yield the highest output signal-to-noise ratio (SNR). When the detection is carried out in discrete time, selecting an optimal filter length for a specific detection problem is important. Bounds on the SNR of the matched filter can assist in this selection. Exact bounds on the SNR can be computed in terms of the eigenvalues of the noise covariance matrix, but these bounds can be difficult to compute. An approximate lower bound for the SNR has been suggested recently by Martinez and Thomas (see Ref. (2), Franklin Inst. Vol. 321, No. 5, pp. 251–260, 1986). A supplement to this bound which is more accurate for small values of filter length is discussed in this paper. Some examples which delineate a comparison between the two approximate bounds are presented.