In the design of digital filters, we often choose a filter with a specified magnitude ch...

In the design of digital filters, we often choose a filter with a specified magnitude characteristic that has the shortest duration. That is, the impulse response, which is the inverse Fourier transform of the complex frequency spectrum, should be as narrow as possible. Assuming that h[n] is real, we wish to show that if the phase θ(ω) associated with the frequency response is zero, the duration of the impulse response is minimal. Let the frequency response be expressed as