Page 211 - Mechanical Engineers' Handbook (Volume 2)

200 Signal Processing
h(i) sinc i 1 i N (27)

2 
N
c
c
Where x is the greatest integer less than or equal to x and is the desired cutoff frequency
c
divided by the Nyquist frequency.
Windowing
This FIR filter will have ripple in the passband and in the stop band. It is possible to suppress
these ripples and smooth the frequency response, but the trade-off will be an increased
transition width. The method for suppressing these ripples is with the application of a win-
dowing function. There is a large class of windowing functions that allow the designer to
determine how he or she wishes to trade off the transition width and how much ripple is to
be tolerated. The design of an FIR filter with windowing involves the use of Eq. (26) for
the determination of the FIR coefficients followed by the component-by-component product
of the coefficients with the windowing values, that is, h (i) h(i) win(i). Below, is a list of
common windows :
5
Rectangular:
win(i) 1 0 i N 1 (28)
Bartlett:

N 1 0 i N 1
2i
2
win(i) 2i N 1 (29)
2
N 1 2 i N 1
Hanning:
– 1 cos
win(i) 2 i
2 1 0 i N 1 (30)
N 1
Hamming:
win(i) 0.54 0.46 cos
2 i
N 1 0 i N 1 (31)
Blackman:
win(i) 0.42 0.5 cos 0.08 cos
4 i
2 i
N 1 N 1 0 i N 1 (32)
Table 2 gives a list of several common windowing functions together with their char-
acteristics. 5
Figure 10 demonstrates the effect of a Hanning window.
FIR High-Pass and Bandpass Design
The design of a high-pass filter is simply 1 H(z). In the time domain, this is

h(i) sinc i
1 i N (33)
2 
2 
N
N
c
c
c
where