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540 Chapter 10 ROUND-OFF EFFECTS IN DIGITAL FILTERS
These assumptions are reasonable in practice if the sequence x(n) issufficiently
random to traverse many quantization steps in going from time
n to n +1.
10.1.2 ANALYSIS USING MATLAB
To investigate the statistical properties of the error samples, we will have
to generate a large number of these samples and plot their distribution
using a histogram (or a probability bar graph). Furthermore, we have to
design the sequence x(n) sothat its samples do not repeat; otherwise, the
error samples will also repeat, which will result in an inaccurate analysis.
This can be guaranteed either by choosing a well-defined aperiodic
sequence or a random sequence.
We will quantize x(n) using B-bit rounding operation. A similar implementation
can be developed for the truncation operation. Since all
three error characteristics are exactly the same under the rounding operation,
we will choose the sign-magnitude format for ease in implementation.
After quantization, the resulting error samples e(n) are uniformly
distributed over the [− ∆ 2 , ∆ 2 ]interval. Let e 1(n) bethe normalized error
given by
e 1 (n) △ = e(n)
∆ = e(n)2B ⇒ e 1 (n) ∈ [−1/2, 1/2] (10.2)
Then e 1 (n)isuniform over the interval [− 1 2 , + 1 2
], as shown in Figure 10.2a.
Thus the histogram interval will be uniform across all B-bit values, which
will make its computation and plotting easier. This interval will be divided
into 128 bins for the purpose of plotting.
To determine the sample independence we consider the histogram of
the sequence
e 2 (n) = △ e 1(n)+e 1 (n − 1)
(10.3)
2
which is the average of two consecutive normalized error samples. If
e 1 (n) is uniformly distributed between [−1/2, 1/2], then, for sample
independence, e 2 (n) must have a triangle-shaped distribution between
[−1/2, 1/2], as shown in Figure 10.2b. We will again generate a 128-
bin histogram for e 2 (n). These steps are implemented in the following
MATLAB function.
f 1 (n)
1 2
f 2 (n)
e 1 (n)
−12 12 −12 12
e 2 (n)
FIGURE 10.2
Probability distributions of the normalized errors e 1(n) and e 2(n)
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