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Analysis of A/D Quantization Noise 545
Since practical signals processed using a DSP chip are typically random
in nature (or can be modeled as such), we conclude from these two
examples that the statistical model, with its stated assumptions, is a very
good model.
10.1.3 STATISTICS OF A/D QUANTIZATION NOISE
We now develop a second-order statistical description of the error sequence
e(n) for both the truncation and rounding operations.
10.1.4 TRUNCATION
From (6.57), the pdf f ET (e) ofe T (n) isuniform over [−∆, 0], as shown in
Figure 10.7a. Then the average of e T (n) isgiven by
m eT
△
=E[eT (n)] = −∆/2 (10.4)
and the variance is
∫ 0
σe 2 △
T
=E
[(e T (n) − m eT ) 2] =
∫ ∆/2
( ) 1
= e 2 de = ∆2
−∆/2 ∆ 12
−∆
(e − ∆/2) 2 f ET (e)de
(10.5)
Using ∆ = 2 −B ,weobtain
σ 2 e T
= 2−2B
12
or σ eT = 2−B
2 √ 3
(10.6)
Rounding From (6.59), the pdf f ER (e) of e R (n) is uniform over
[−∆/2, ∆/2], as shown in Figure 10.7b. Then the average of e R (n) is
given by
△
m eR =[EeR ]=0 (10.7)
f T (e)
1∆
1∆
f R (e)
e
−∆ −∆2
0 −∆2 0
∆2
(a)
(b)
FIGURE 10.7 Probability density functions: (a) truncation and (b) rounding
e
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