5.13 Filtering in the frequency domain

Because we are interested in actual computations rather than analytic calculations, we must consider the detailsof the discrete Fourier transform. To compute the length- $N$ DFT, we assume that the signal has a duration less than or equal to $N$ . Because frequency responses have an explicit frequency-domain specification in terms of filter coefficients, we don't have a direct handle on whichsignal has a Fourier transform equaling a given frequency response. Finding this signal is quite easy. First of all, notethat the discrete-time Fourier transform of a unit sample equals one for all frequencies. Since the input and output oflinear, shift-invariant systems are related to each other by $Y(e^{i\times 2\pi f})=H(e^{i\times 2\pi f})X(e^{i\times 2\pi f})$ , a unit-sample input, which has $X(e^{i\times 2\pi f})=1$ , results in the output's Fourier transform equaling the system's transfer function .

In the time-domain, the output for a unit-sample input is known as the system's unit-sample response , and is denoted by $h(n)$ . Combining the frequency-domain and time-domain interpretations of alinear, shift-invariant system's unit-sample response, we have that $h(n)$ and the transfer function are Fourier transform pairs in terms of the discrete-time Fourier transform .

Returning to the issue of how to use the DFT to perform filtering, we can analytically specify the frequency response,and derive the corresponding length- $N$ DFT by sampling the frequency response.