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Analysis of A/D Quantization Noise 539
x(n) QUANTIZER Q[x (n)] ⇒ x(n) x(n) + e(n)
FIGURE 10.1
Statistical model of a quantizer
e(n)
output-error characteristics. In this technique, we assume that the quantized
value Q[x] isasum of the exact value x and the quantization error e,
which is assumed to be a random variable. When x(n) isapplied as an
input sequence to the quantizer, the error e(n) isassumed to be a random
sequence. We then develop a statistical model for this random sequence
to analyze its effects through a digital filter.
For the purpose of analysis, we assume that the quantizer employs
fixed-point two’s-complement number format representation. Using the
results given previously, we can extend this analysis to other formats as
well.
10.1.1 STATISTICAL MODEL
We model the quantizer block on the input as a signal-plus-noise
operation—that is, from (6.46)
Q[x(n)] = x(n)+e(n) (10.1)
where e(n) isarandom sequence that describes the quantization error sequence
and is termed the quantization noise. This is shown in Figure 10.1.
Model assumptions For the model in (10.1) to be mathematically
convenient and hence practically useful, we have to assume reasonable
statistical properties for the sequences involved. That these assumptions
are practically reasonable can be ascertained using simple MATLAB examples,
as we shall see. We assume that the error sequence, e(n) has the
following characteristics: 1
1. The sequence e(n) isasample sequence from a stationary random
process {e(n)}.
2. This random process {e(n)} is uncorrelated with sequence x(n).
3. The process {e(n)} is an independent process (i.e., the samples are
independent of each other).
4. The probability density function (pdf), f E (e), of sample e(n) for each
n is uniformly distributed over the interval of width ∆ = 2 −B , which
is the quantizer resolution.
1 We assume that the reader is familiar with the topic of random variables and processes
and the terminology associated with it.
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