Frequency-response masking approach for the synthesis of sharp linear phase digital filters

If each delay element of a linear phase low-pass digital filter is replaced by delay elements, an -band filter is produced. The transition-width of this -band filter is that of the prototype low-pass filter. A complementary filter can be obtained by subtracting the output of the -band filter from a suitably delayed version of the input. The complementary filter is an -band filter whose passbands and stopbands are the stopbands and passbands, respectively, of the original -band filter. If the frequency responses of the original -band filter and its complementary filter are properly masked and recombined, narrow transition-band filter can be obtained. This technique can be used to design sharp low-pass, high-pass, bandpass, and bandstop filters with arbitrary passband bandwidth.