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272 Chapter 6 IMPLEMENTATION OF DISCRETE-TIME FILTERS
(b) Sign-Magnitude Format
(b) Ones-Complement Format
(b) Two-Complement Format
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FIGURE 6.28
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Rounding error characteristics in the fixed-point representation
B = 2;
% Select bits for Rounding
x = [-1:2^(-10):1-2^(-B-1)]; % Sign-Magnitude numbers between -1 and 1
y = sm2tc(x);
% Sign-Mag to Two’s Complement
yq = round(y*2^B)/2^B;
% Rounding
xq = tc2sm(yq);
% Two’-Complement to Sign-Mag
The resulting plots for the sign-magnitude, ones-, and two’s-complement
formats are shown in Figure 6.28. These plots do satisfy (6.59).
□
Comparing the error characteristics of the truncation and rounding
operations given in Figures 6.25 through 6.28, it is clear that the rounding
operation is a superior one for the quantization error. This is because the
error is symmetric with respect to zero (or equal positive and negative
distribution) and because the error is the same across all three formats.
Hence we will mostly consider the rounding operation for the floatingpoint
arithmetic as well as for further analysis.
6.7.2 FLOATING-POINT ARITHMETIC
In this arithmetic, the quantizer affects only the mantissa M. However,
the number x is represented by M × 2 E where E is the exponent. Hence
the quantizer errors are multiplicative and depend on the magnitude of
x. Therefore, the more appropriate measure of error is the relative error
rather than the absolute error, (Q[x] − x). Let us define the relative error,
ε, as
ε = △ Q[x] − x
(6.60)
x
Then the quantized value Q[x] can be written as
Q[x] =x + εx = x (1 + ε) (6.61)
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