Finite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 262

7.5 FREQUENCY SAMPLING DESIGN METHOD
In addition to methods of Fourier transform design and Fourier transform with windowing discussed
in the previous section, frequency sampling is another alternative. The key feature of frequency
sampling is that the filter coefficients can be calculated based on the specified magnitudes of the
desired filter frequency response uniformly in the frequency domain. Hence, it has design flexibility.
To begin development, we let hðnÞ , for n ¼ 0 ; 1 ; / ; N 1, be the causal impulse response (FIR
filter coefficients) that approximates the FIR filter, and we let HðkÞ , for k ¼ 0 ; 1 ; / ; N 1, represent
the corresponding discrete Fourier transform (DFT) coefficients. We obtain HðkÞ by sampling the
desired frequency filter response HðkÞ¼Hðe jU k Þ at equally spaced instants in frequency domain, as
shown in Figure 7.29 .
Then, according to the definition of the inverse DFT (IDFT), we can calculate the FIR coefficients:

N 1
k ¼ 0 HðkÞW k N ;

1
N

hðnÞ¼ for n ¼ 0 ; 1 ; / ; N 1 (7.27)

where

¼ cos 2 p
N

j sin 2 p
N

W N ¼ e j

2 p
N

We assume that the FIR filter has linear phase and the number of taps is N ¼ 2 M þ 1. Equation (7.27)
can be significantly simplified as

FIGURE 7.29
Desired filter frequency response and sampled frequency response.