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Analysis of A/D Quantization Noise 547
Similarly, for B =6computed statistical values are shown in Figure 10.4.
The computed values of the two means are −4.1044 × 10 −6 , which agree very
well with the model. The standard deviation of e(n), from (10.9), is 0.0045105,
while that from the top plot in Figure 10.4 is 0.0045076, which again agrees
closely with the model. The standard deviation of the average of the two consecutive
samples, from (10.10), is 0.0031894, while from the bottom plot in
Figure 10.4 it is 0.00318181, which clearly agrees with the model. Hence the
samples of e(n) for B =6are independent. This was also confirmed by the
bottom plot in Figure 10.4.
□
Similar calculations can be carried out for the signal in Example 10.2.
The details are left to the reader.
10.1.6 A/D QUANTIZATION NOISE THROUGH DIGITAL FILTERS
Let a digital filter be described by the impulse response, h(n), or the frequency
response, H(e jω ). When a quantized input, Q[x(n)] = x(n)+e(n),
is applied to this system, we can determine the effects of the error sequence
e(n) onthe filter output as it propagates through the filter, assuming
infinite-precision arithmetic implementation in the filter. We are generally
interested in the mean and variance of this output-noise sequence, which
we can obtain using linear system theory concepts. Details of these results
can be found in many introductory texts on random processes,
including [27].
Referring to Figure 10.8, let the output of the filter be ŷ(n). Using
LTI properties and the statistical independence between x(n) and e(n),
the output ŷ(n) can be expressed as the sum of two components. Let y(n)
be the (true) output due to x(n) and q(n) the response due to e(n). Then
we can show that q(n) isalso a random sequence with mean
m q
△
=E[q(n)] = me
∞
∑
−∞
h(n) =m e H(e j0 ) (10.12)
where the term H(e j0 )istermed the DC gain of the filter. For truncation,
m eT = −∆/2, which gives
m qT = − ∆ 2 H(ej0 ) (10.13)
For rounding, m eR =0or
m qR =0 (10.14)
x(n) ˆ = x(n) + e(n)
h(n), H(e jω )
y(n) ˆ = y(n) + q(n)
FIGURE 10.8
Noise through digital filter
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