Solution:

The efficiency of non-matched filters: In practice, the matched filter cannot always be obtained exactly. It is appropriate, therefore, to examine the efficiency of non-matched filters compared with the ideal matched filter.

The measure of efficiency is taken as the peak signal-to-noise ratio from the non-matched filter divided by the peak signal-to-noise ratio (2E/No) from the matched filter.

Figure. plots the efficiency for a single-tuned (RLC) resonant filter and a the rectangular-shaped filter of half-power bandwidth Bτ when the input is a rectangular pulse of width τ.

The maximum efficiency of the single-tuned filter occurs for Bτ ≈ 0.4. The corresponding loss in signal-to-noise ratio is 0.88 dB as compared with a matched filter.

which maximizes the signal-to-noise ratio (SNR) for various combinations of filters and pulse shapes. It can be seen that the loss in SNR incurred by the use of these non-matched filters is small.