Window Parameters in FIR Filter Design

The previous article in this series discussed that a tapered window, such as a Bartlett, can give better PSL than a rectangular window which has abrupt variation in the time domain.

In this article, first, we will review other popular windows. Then, we will clarify the design procedure by calculating the cutoff of the ideal filter, window type, and window length from given filter specs, namely, and now we need to find the required ideal filter response, window type, and window length to design an FIR filter. The relation between these parameters is the subject of this article.

Other Popular Window Functions

Fortunately, the Bartlett and the rectangular windows are not the only options in FIR filter design and many other windows have been developed.

Table I shows some of the most popular windows along with their important properties. In Table I, Bartlett, Hann, and Hamming have equal approximate main lobe width, but we can observe the general trade-off between the PSL and the main lobe width. The rectangular window has the smallest main lobe width and the largest PSL, whereas the Blackman has the widest main lobe and the smallest PSL.

The Fourier transform of three windows, Bartlett, Hann, and Hamming with

M=21M=21

, are plotted in Figure (1). The mentioned trade-off is observed in these three windows, too. As the PSL reduces, the main lobe width increases.

 

In addition to PSL and approximate main lobe width, Table I gives, for each window, the peak approximation error, which is the deviation from the ideal response (denoted by

δδ

) expressed in dB. This is an important parameter which allows us to choose an appropriate window based on the requirements of an application. Peak approximation error determines how much deviation from the ideal response we expect for each of the window types. This is illustrated in Figure (2).

As will be discussed in the following section, the deviations from the ideal response in the pass-band and stop-band are approximately equal when using the window method to design FIR filters, i.e.,

δ1=δ2=δδ1=δ2=δ

. Therefore, we can select the suitable window based on how much ripple is allowed in the pass-band or how much attenuation is needed in the stop-band.