Window Method for FIR Filter Design

The window method for digital filter design is fast, convenient, and robust, but generally suboptimal. It is easily understood in terms of the convolution theorem for Fourier transforms, making it instructive to study after the Fourier theorems and windows for spectrum analysis. It can be effectively combined with the frequency sampling method, as we will see in §4.6 below.

The window method consists of simply ``windowing'' a theoretically ideal filter impulse response $h(n)$ by some suitably chosen window function $w(n)$ , yielding

$$h_w(n) = w(n)\cdot h(n), \quad n\in{\cal Z}.$$ (5.8)

For example, as derived in Eq.(4.3), the impulse response of the ideal lowpass filter is the well known sinc function

$$h(n) \eqsp B\cdot (B n) \isdefs B\frac{\sin(\pi B n)}{\pi B n}, \quad n\in\mathbb{Z},$$ (5.9)

where $B=2f_c$ is the total normalized bandwidth of the lowpass filter in Hz (counting both negative and positive frequencies), and $f_c$ denotes the cut-off frequency in Hz. As noted earlier, we cannot implement this filter in practice because it is noncausal and infinitely long.

Since $h(n) = B\,$sinc$(BnT)$ decays away from time 0 as $1/n$ , we would expect to be able to truncate it to the interval $[-N,N]$ , for some sufficiently large $N$ , and obtain a pretty good FIR filter which approximates the ideal filter. This would be an example of using the window method with the rectangular window. We saw in §4.3 that such a choice is optimal in the least-squares sense, but it designs relatively poor audio filters. Choosing other windows corresponds to tapering the ideal impulse response to zero instead of truncating it. Tapering better preserves the shape of the desired frequency response, as we will see. By choosing the window carefully, we can manage various trade-offs so as to maximize the filter-design quality in a given application.

Window functions are always time limited. This means there is always a finite integer $N_w$ such that $w(n)=0$ for all $\vert n\vert>N_w$ . The final windowed impulse response $h_w(n) = w(n)\cdot h(n)$ is thus always time-limited, as needed for practical implementation. The window method always designs a finite-impulse-response (FIR) digital filter (as opposed to an infinite-impulse-response (IIR) digital filter).

By the dual of the convolution theorem, pointwise multiplication in the time domain corresponds to convolution in the frequency domain. Thus, the designed filter $h_w$ has a frequency response given by

$$H_w(f) = \left(W\ast H\right)(f), \quad f\in\left[\,-1/2,\,1/2\,\right)$$ (5.10)

where $H=\hbox{\sc DTFT}(h)$ is the ideal frequency response and $W=\hbox{\sc DTFT}(w)$ is the window transform. For the ideal lowpass filter, $H$ is a rectangular window in the frequency domain. The frequency response $H_w(f)$ is thus obtained by convolving the rectangular window with the window transform $W(f)$ . This implies several points which can be immediately seen in terms of this convolution operation:

• The pass-band gain is primarily the area under the main lobe of the window transform, provided the main lobe ``fits'' inside the pass-band (i.e., the total lowpass bandwidth $B\isdeftext 2f_c$ is greater than or equal to the main-lobe width of $W$ ).

• The stop-band gain is given by an integral over a portion of the side lobes of the window transform. Since side-lobes oscillate about zero, a finite integral over them is normally much smaller than the side-lobes themselves, due to adjacent side-lobe cancellation under the integral.

• The best stop-band performance occurs when the cut-off frequency is set so that the stop-band side-lobe integral traverses a whole number of side lobes.

• The transition bandwidth is equal to the bandwidth of the main lobe of the window transform, again provided that the main lobe ``fits'' inside the pass-band.

• For very small lowpass bandwidths $B$ , $H(\omega)$ approaches an impulse in the frequency domain. Since the impulse is the identity operator under convolution, the resulting lowpass filter $H_w(\omega)$ approaches the window transform $W(\omega)$ for small $B$ . In particular, the stop-band gain approaches the window side-lobe level, and the transition width approaches half the main-lobe width. Thus, for good results, the lowpass cut-off frequency should be set no lower than half the window's main-lobe width.