Editing Finite impulse response (section)

=== Window design method ===

In the window design method, one first designs an ideal IIR filter and then truncates the infinite impulse response by multiplying it with a finite length [[window function]]. The result is a finite impulse response filter whose frequency response is modified from that of the IIR filter. Multiplying the infinite impulse by the window function in the time domain results in the frequency response of the IIR being [[convolved]] with the Fourier transform (or DTFT) of the window function. If the window's main lobe is narrow, the composite frequency response remains close to that of the ideal IIR filter.

The ideal response is often rectangular, and the corresponding IIR is a [[sinc function]].  The result of the frequency domain convolution is that the edges of the rectangle are tapered, and ripples appear in the passband and stopband.  Working backward, one can specify the slope (or width) of the tapered region (''[[transition band]]'') and the height of the ripples, and thereby derive the frequency-domain parameters of an appropriate window function.  Continuing backward to an impulse response can be done by iterating a filter design program to find the minimum filter order.  Another method is to restrict the solution set to the parametric family of [[Kaiser window]]s, which provides closed form relationships between the time-domain and frequency domain parameters.  In general, that method will not achieve the minimum possible filter order, but it is particularly convenient for automated applications that require dynamic, on-the-fly, filter design.

The window design method is also advantageous for creating efficient [[half-band filter]]s, because the corresponding sinc function is zero at every other sample point (except the center one).  The product with the window function does not alter the zeros, so almost half of the coefficients of the final impulse response are zero.  An appropriate implementation of the FIR calculations can exploit that property to double the filter's efficiency.