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268 Chapter 6 IMPLEMENTATION OF DISCRETE-TIME FILTERS
1
0.75
0.5
0.25
xhat
0
−0.25
−0.5
−0.75
x
xhat
−1
−1 −0.75 −0.5 −0.25 0
x
0.25
0.5
0.75
1
FIGURE 6.25
Truncation error characteristics in the sign-magnitude format
□ EXAMPLE 6.20 Let −1 < x < 1 and B = 2. Using MATLAB, verify the truncation error
characteristics.
Solution
The resolution is ∆ = 2 −2 =0.25. Using the following MATLAB script, we can
verify the truncation error e T relations given in (6.48) and (6.49).
x = [-1+2^(-10):2^(-10):1-2^(-10)]; % Sign-Mag numbers between -1 and 1
B = 2;
% Number of bits for Truncation
xhat = fix(x*2^B)/2^B
% Truncation
plot(x,x,’g’,x,xhat,’r’,’linewidth’,1); % Plot
The resulting plots of x and ˆx are shown in Figure 6.25. Note that the plot of
ˆx has a staircase shape and that it satisfies (6.48) and (6.49).
□
One’s-complement format For x ≥ 0, we have the same characteristics
for e T as in sign-magnitude format—that is,
−2 −B ≤ e T ≤ 0, for x ≥ 0 (6.50)
For x