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Quantization of Filter Coefficients 277
To determine the exact locations of the changed poles, consider the changed
denominator
ˆD (z) =1−0.875z −1 +0.75z −2 = ( 1 − 0.866e j0.331π z −1)( 1 − 0.866e −j0.331π z −1)
Thus, the changed pole locations are ˆp 1 =0.866e j0.331π =ˆp ∗ 2. Then |△p 1| =
∣ 0.9e iπ/3 − 0.866e i0.331π∣ ∣ =0.0344, which agrees with (6.75).
□
Analysis using MATLAB To investigate the effect of coefficient
quantization on filter behavior, MATLAB is an ideal vehicle. Using functions
developed in previous sections, we can obtain quantized coefficients
and then study such aspects as pole-zero movements, frequency response,
or impulse response. We will have to represent all filter coefficients using
the same number of integer and fraction bits. Hence instead of quantizing
each coefficient separately, we will develop the function, QCoeff,
for coefficient quantization. This function implements quantization using
rounding operation on sign-magnitude format. Although similar functions
can be written for truncation as well as for other formats, we will analyze
the effects using the Qcoeff function as explained previously.
function [y,L,B] = QCoeff(x,N)
% [y,L,B] = QCoeff(x,N)
% Coefficient Quantization using N=1+L+B bit Representation
% with Rounding operation
% y: quantized array (same dim as x)
% L: number of integer bits
% B: number of fractional bits
% x: a scalar, vector, or matrix
% N: total number of bits
xm = abs(x);
L = max(max(0,fix(log2(xm(:)+eps)+1))); % Integer bits
if (L > N)
errmsg = [’ *** N must be at least ’,num2str(L),’ ***’]; error(errmsg);
end
B = N-L;
% Fractional bits
y = xm./(2^L); y = round(y.*(2^N)); % Rounding to N bits
y = sign(x).*y*(2^(-B));
% L+B+1 bit representation
The Qcoeff function represents each coefficient in the x array using
N+1-bit (including the sign bit) representation. First, it determines the
number of bits L needed for integer representation for the magnitude-wise
largest coefficient, and then it assigns N-L bits to the fraction part. The
resulting number is returned in B. Thus all coefficients have the same bit
pattern L+B+1. Clearly, N ≥ L.
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