Mismatched Filters for Sidelobe Suppression
Matched filters obtain best SNR performance in a white Gaussian noise environment but often at expense of large temporal sidelobes. In radar applications, large temporal sidelobes can be interpreted as false targets and in communications systems, they limit our ability to distinguish various multipath components when probing the channel response function. Ideally we would like a waveform design that minimizes temporal sidelobe levels and at the same time we would like an easy implementation. BPSK waveforms combined with mismatched filter receivers specifically designed to lower temporal sidelobe levels often fit the bill.
In the figure below, a length 5 Barker code [1,1,1,-1,1] is transmitted using a BPSK signal format. In the top figure, a matched filter is used for reception and the chart shows matched filter response as a function of time. In addition to the large desired response but we also see four large temporal sidelobes at a 20 log (1/5) = -14 dB level. The lower chart shows corresponding mismatched filter responses where the FIR filter tap values are [0,-1,2,-2,0,5,-8,5,8,5,0,-2,-1,0,0] with integer chip spacing. Now, peak sidelobe levels are at -30 dB, an improvement of 16 dB. Regarding SNR loss; we loose only 0.6 dB at the correlation peak. This is because my mismatched filter design program tries to maximize output SNR while at the same time minimize peak sidelobes; two opposing constraints. Ease of implementation was also a consideration in that filter taps were limited to 3-bit precision.