Finite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 282

b 0

x ()

+ +

y(n)

b 1

− 1

+

+

xn ( −1 )

− 1

b 2

xn ( − 2 )

−1

xn ( − 3 )

− 1

x n ( − 4 )

FIGURE 7.41

Linear phase FIR filter realization.

7.8 COEFFICIENT ACCURACY EFFECTS ON FINITE IMPULSE RESPONSE
FILTERS
In practical applications, the filter coefficients achieved through high-level software such as MATLAB
must be quantized using finite word length. This may have two effects. First, the locations of zeros are
changed; second, due to the location change of zeros, the filter frequency response will change
correspondingly. In practice, there are two types of digital signal (DS) processors: fixed-point
processors and floating-point processors . The fixed-point DS processor uses integer arithmetic, and the
floating-point processor employs floating-point arithmetic. Such effects of filter coefficient quanti-
zation will be covered in Chapter 9.
In this section, we will study the effects of FIR filter coefficient quantization in general, since
during practical filter realization, obtaining filter coefficients with infinite precision is impossible.
Filter coefficients are usually truncated or rounded off for the application. Assume that the FIR filter
transfer function with infinite precision is given by

K
n¼ 0 b n z n ¼ b 0 þ b 1 z 1

þ / þ b 2 M z K HðzÞ¼ (7.40)

b n

where each filter coefficient has infinite precision. Now let the quantized FIR filter transfer

function be

K
n¼ 0 b n z n ¼ b 0 þ b 1 z 1

þ / þ b K z K

H q ðzÞ¼ (7.41)

where each filter coefficient b n is quantized (rounded off) using the specified number of bits. Then the
error of the magnitude frequency response can be bounded as
Hðe jU ÞH q ðe jU Þ ¼ K

n¼ 0

b n b n Þe jU

(7.42)

b n b n ðK þ 1 Þ$ 2 B

< K
n¼ 0