# FIR frequency response using the FFT

The frequency response of a finite impulse response filter can be efficiently computed as a Fast Fourier Transform. Without loss of generality, we consider a causal FIR filter, with impulse response $$h[n]$$ non-zero in $$[0, N-1]$$. With $$\mathbf h^N = (h, \ldots, h[N-1])$$ the vector holding the non-zero coefficients of the impulse response, the DFT $$\mathbf H^N$$ of $$\mathbf h^N$$ yields samples of the frequency response of the filter: $\mathbf H^N_k = H\left(\frac{k}{N}\right) \mbox{ pour } 0 \leq k < N.$

The sampling of the frequency response is too coarse for a faithful plot of the frequency response. Zero-padding can be used. It consists of computing the DFT of $$\mathbf h^L = (h, \ldots, h[N-1], 0, \ldots, 0)$$ of length $$L > N$$. We get a finer sampling of the frequency response: $\mathbf H^L_k = H\left(\frac{k}{L}\right) \mbox{ pour } 0 \leq k < L.$

We can then center the plot around the origin to highlight the symmetry of the frequency response of a filter with real coefficients: $$H(-\nu) = H(\nu)^\star$$.