Finite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 219

frequency response, such as that of a lowpass, highpass, bandpass, or bandstop filter. The following
sections will introduce design methods to calculate the FIR filter coefficients.

7.2 FOURIER TRANSFORM DESIGN
We begin with an ideal lowpass filter with a normalized cutoff frequency U c , whose magnitude
frequency response in terms of the normalized digital frequency U is plotted in Figure 7.1 and is
characterized by

( 1 ; 0 jUjU c
0 ; U c jUjp

Hðe jU Þ¼

(7.4)

Since the frequency response is periodic with a period of U ¼ 2 p radians, as we discussed in Chapter 6,
we can extend the frequency response of the ideal filter Hðe jU Þ , as shown in Figure 7.2 .
The periodic frequency response can be approximated using a complex Fourier series expansion
(see Appendix B) in terms of the normalized digital frequency U , that is,

N

Hðe jU Þ¼ c n e ju 0 nU (7.5)

n¼ N

and the Fourier coefficients are given by

Z p

1
2 p

Hðe jU Þe ju 0 nU dU c n ¼ for N < n < N (7.6)

p

FIGURE 7.1

Frequency response of an ideal lowpass filter.

FIGURE 7.2
Periodicity of the ideal lowpass frequency response.