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586 Chapter 10 ROUND-OFF EFFECTS IN DIGITAL FILTERS
or
( M−1
)
∑
SNR dB =6.02(B+1)+10 log 10 |h(n)| 2
0
−10 log 10
( K
∑
i=1
M−2i
∑
n=1
|ĝ i (n)| 2 )
(10.95)
10.3.3 ANALYSIS USING MATLAB
Using the casfiltr function, we can compute the output of the infiniteprecision
cascade structure. Using the scaling approach outlined above,
each second-order section can be scaled and used in the simulation of
quantized outputs. Again, all calculations can be done in vector fashion,
which improves the execution speed. These and other simulation steps are
detailed in the following example.
□ EXAMPLE 10.14 Consider the 4th-order FIR filter given in Example 10.13. Its cascade-form
realization has two sections along with a gain constant b 0, which can be obtained
using the dir2cas function:
H 1(z) =1+1.4859z −1 +2.8901z −2 , H 2(z) =1+0.5141z −1 +0.3460z −2 , and b 0 =0.1
(10.96)
Note that some of these coefficients are greater than 1, which will cause problems
with coefficient quantization when only B fractional bits are used. Hence we
need to scale each section as explained. The scaled values are
Ĥ 1(z) =0.1860 + 0.2764z −1 +0.5376z −2 , Ĥ 2(z) =0.5376 + 0.2764z −1 +0.1860z −2
(10.97)
and ˆb 0 =1.Thus wedonot need to scale the input. Now ĝ 1(n) =ĥ2(n) and
ĝ 2(n) =1in(10.94). Thus, from (10.95) the output SNR is 70.96 dB, which
compares well with the one-multiplier direct-form implementation (72.41 dB).
These calculations and error histogram plotting are illustrated in the following
MATLAB script.
% Example Parameters
B = 12;
% # of fractional bits
N = 100000;
% # of samples
xn = (2*rand(1,N)-1); % Input sequence - Uniform Distribution
h = [0.1,0.2,0.4,0.2,0.1]; % Filter parameters
M = length(h);
% Filter length
[b0,Bh,Ah] = dir2cas(h,1); % Cascade sections
h1 = Bh(1,:);
% Section-1
h2 = Bh(2,:);
% Section-2
h1 = h1/sum(h1);
% Scaled so Gain=1
h2 = h2/sum(h2);
% Scaled so Gain=1
% Local variables
bM = 7; DbM = 2^bM; % bin parameter
BB = 2^B;
% useful factor in quantization
K = round(DbM/2); % Half number of bins
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