The concepts associated with frequency-selective filtering are often used to separate tw...

The concepts associated with frequency-selective filtering are often used to separate two signals that have been added together. If the spectra of the two signals do not overlap, ideal frequency-selective filters are desirable. However, if the spectra overlap, it is often preferable to design the filter to have a gradual transition between passband and stopband. In this problem, we explore one approach for determining the frequency response of a filter to be used for separating signals with overlapping spectra. Let x(t) denote a composite continuous-time signal consisting of two signals

Let us define a measure of the error between y(t) and s(t) at each frequency ω as