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The Process of Quantization and Error Characterizations 271
1
0.75
x
xhat
0.5
0.25
xhat
0
−0.25
−0.5
−0.75
FIGURE 6.27
−1
−1
−0.75 −0.5
−0.25
0
x
0.25
0.5
0.75
Truncation error characteristics in the two’s-complement format
1
Rounding operation In this operation, the real number x is rounded
to the nearest representable level, which we will refer to as Q R [x]. In
MATLAB, to obtain a B-bit rounding approximation, we have to first
scale the number x up by 2 B , then use the round function on the scaled
number, and finally scale the result down by 2 −B . Thus the MAT-
LAB statement xhat = round(x*2^B)/2^B; implements the desired
operation.
Since the quantization step or resolution is ∆ = 2 −B , the magnitude
of the maximum error is
|e R | max
= ∆ 2 = 1 2 2−B (6.58)
Hence for all three formats, the quantizer error due to rounding, denoted
by e R , satisfies
− 1 2 2−B ≤ e R ≤ 1 2 2−B (6.59)
□ EXAMPLE 6.23 Demonstrate the rounding operations and the corresponding error characteristics
on the signal of Examples 6.20–6.22 using the three formats.
Solution
Since the rounding operation assigns values that can be larger than the unquantized
values, which can create problems for the two’s- and one’s-complement
format, we will restrict the signal over the interval [−1, 1 − 2 −B−1 ]. The following
MATLAB script shows the two’s-complement format rounding, but other
scripts are similar (readers are encouraged to verify).
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