3.2 Filter design by windowing

Fir filter design by windowing

Desired ideal filter response

Unlike the design of IIR filters, the design of FIR filters starts from the spectrum of the desired filter in theDTFT domain. Let $H_d(e^{\frac{i}{}})$ be the ideal desired frequency response. Then, the impulse response $h_d(n)$ corresponding to $H_d(e^{\frac{i}{}})$ is obtained by the inverse DTFT as

Windowing of impulse response

To obtain an FIR filter approximating the frequency response of the desired ideal filter, we need to find a filterimpulse response a causal and finite $h(n)$ that approximates $h_d(n)$ . The simplest way to obtain such $h(n)$ is to define a new system with impulse response $h(n)$ given by

Frequency response of windowed impulse response

Because the multiplication by the window function in time domain corresponds to a convolution in the DTFT domain, wecan easily visualize the spectrum of the designed FIR filter. Different window functions have different main lobewidth and peak sidelobe heights. The width of the main lobe governs the property of the frequency transition at bandedges. The height of the peak sidelobes is related to the oscillations near the transition frequencies.