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560 Chapter 10 ROUND-OFF EFFECTS IN DIGITAL FILTERS
Amplitude
No Limit Cycles
1
0.75
0.5
0.25
0
−0.25
−0.5
−0.75
−1
0 20 40 60 80
Sample Index n
Amplitude
Granular Limit Cycles
1
0.75
0.5
0.25
0
−0.25
−0.5
−0.75
−1
0 20 40 60 80
Sample Index n
FIGURE 10.12 Comparison of limit cycles in Example 10.8
Amplitude
Overflow Limit Cycles
1
0.75
0.5
0.25
0
−0.25
−0.5
−0.75
−1
0 20 40 60 80
Sample Index n
As shown in these examples, the limit-cycle behaviors of many different
filters can be studied for different quantizer characteristics using the
MATLAB function QFix.
10.2.4 MULTIPLICATION QUANTIZATION ERROR
Amultiplier element in the filter implementation can introduce additional
quantization errors since multiplication of two B-bit fractional numbers
results in a 2B-bit fraction and must be quantized to a B-bit fraction.
Consider a multiplier in fixed-point arithmetic with B =8.The number
√1
3
is represented as 0.578125 in decimal. The square of 0.578125 rounded
to 8 bits is 0.3359375 (which should not be confused with 1/3 rounded to
8 bits, which is 0.33203125). The additional error in the squaring operation
is
0.3359375 − (0.578125) 2 =0.001708984375
This additional error is termed as the multiplication quantization error. Its
statistically equivalent model is similar to that of the A/D quantization
error model, as shown in Figure 10.13.
Statistical model Consider the B-bit quantizer block following the
multiplier element shown in Figure 10.13a. The sequence x(n) and the constant
c are quantized to B fractional bits prior to multiplication (as would
be the case in a typical implementation). The multiplied sequence {cx(n)}
is quantized to obtain y(n). We want to replace the quantizer by a simpler
linear system model shown in Figure 10.13b, in which y(n) =cx(n)+e(n),
c
c
x(n) Q Q[cx(n)] ⇒ x(n) cx(n) + e(n)
e(n)
(a) Quantizer
(b) Linear system model
FIGURE 10.13 Linear system model for multiplication quantization error
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