FIR filter design - Continuous and Discrete Time Signals and Systems - page 671

Table 15.1. Comparison of the properties of the commonly used windows

Kaiser window ( b )

Width of
main lobe

Peak side lobe
amplitude ( a )

Max. stop/pass-band
error 20log 10 ( δ )

Window (dB) β transition width

Rectangular 4 π / N − 13 . 3 − 21 0 1.81 π /( N − 1)

Bartlett 8 π /( N − 1) − 26 . 5 − 25 1.33 2.37 π /( N − 1)

Hanning 8 π /( N − 1) − 31 . 4 − 44 3.86 5.01 π /( N − 1)

Hamming 8 π /( N − 1) − 42 . 6 − 53 4.86 6.27 π /( N − 1)

Blackman 12 π /( N − 1) − 58 . 0 − 74 7.04 9.19 π /( N − 1)

a The peak side lobe magnitude in column 3 is relative to the magnitude of the main lobe.
b The last two columns for the Kaiser window are explained in Section 15.1.5.

The second and third columns of Table 15.1 compare these two parameters for
the commonly used windows as a function of the length N of the window. The
fourth column of Table 15.1 quantifies the maximum difference between the
magnitude spectra within the pass and stop bands of the ideal lowpass filter and
the causal FIR filter obtained from the windowing method. In other words, it
provides an upper bound on the values of the ripples in the pass and stop bands
of the causal FIR filter. For example, the maximum pass- and stop-band error of
-21dB for the rectangular window implies that the pass- and stop-band ripples
are confined to -21dB in the FIR filter obtained with the rectangular window.
In filter design, we prefer to minimize the transition band and reduce the
strength of the ripples. These are conflicting requirements, as we see next.
To minimize the transition band in the FIR filter, the main lobe width of the
windows should be as small as possible. To reduce the pass-band and stop-
band ripples in the FIR filter, the area enclosed by the side lobes (in other
words, the relative strength of the side lobes) of the windows should be small.
Table 15.1 illustrates that these two requirements are contradictory. The rect-
angular window has the smallest width main lobe, but the relative strength of
its highest side lobe with respect to the main lobe is the largest. As a result,
for the rectangular window, the transition bandwidth is small, but the ripple
magnitude is large. On the contrary, the relative strength of the side lobe for the
Blackman window is the smallest, but the width of its main lobe is the largest.
In other words, for the Blackman window, the transition bandwidth is large, but
the ripple magnitude is small.
In the following example, we illustrate the effect of the rectangular and
Hamming windows on the frequency characteristics of an ideal lowpass filter
truncated with these windows.

Example 15.1
Calculate the impulse response of an ideal DT lowpass filter with radian cut-
off frequency Ω c

= 1. From the ideal filter, design two 21-tap FIR filters with

= 1 using the rectangular and Hamming windows.

Ω c