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546 Chapter 10 ROUND-OFF EFFECTS IN DIGITAL FILTERS
and the variance is
σ 2 e R
△
=E
[(e R (n) − m eR ) 2] =
= ∆2
12
Using (6.45), we obtain
σ 2 e R
= 2−2B
12
∫ ∆/2
−∆/2
e 2 f ER (e)de =
or σ eR = 2−B
2 √ 3
( ) 1
e 2 de
−∆/2 ∆
∫ ∆/2
(10.8)
(10.9)
Since the samples of the sequence e R (n) are assumed to be independent
of each other, the variance of [e R (n)+e R (n − 1)]/2 isgiven by
[ ]
eR (n)+e R (n − 1)
var
= 1 ( ) 2
−2B
2
4 12 + 2−2B
= 2−2B
= 1 12 24 2 σ2 e R
(10.10)
or the standard deviation is σ eR / √ 2.
From the model assumptions and (10.6) or (10.9), the covariance of
the error sequence (which is an independent sequence) is given by
E[e(m)e(n)] = △ C e (m − n) = △ C e (l) = 2−2B
δ (l) (10.11)
12
where l △ = m − n is called the lag variable. Such an error sequence is also
known as a white noise sequence.
10.1.5 MATLAB IMPLEMENTATION
In MATLAB, the sample mean and standard deviation are computed
using the functions mean and std, respectively. The last argument of the
function StatModelR is a vector containing sample means and standard
deviations of unnormalized errors e(n) and [e(n) +e(n − 1)]/2. Thus,
these values can be compared with the theoretical values obtained from
the statistical model.
□ EXAMPLE 10.3 The plots in Example 10.1 also indicate the sample means and standard deviations
of the errors e(n) and [e(n) +e(n − 1)]/2. For B =2,these computed
values are shown in Figure 10.3. Since e(n) isuniformly distributed over the
interval [−2 −3 , 2 −3 ], its mean value is 0, and so is the mean of [e(n)+e(n−1)]/2.
The computed values are 3.4239 × 10 −5 and 3.4396 × 10 −5 , respectively, which
agree fairly well with the model. The standard deviation of e(n), from (10.9), is
0.072169, while that from the top plot in Figure 10.3 is 0.072073, which again
agrees closely with the model. The standard deviation of the average of the two
consecutive samples, from (10.10), is 0.051031, and from the bottom plot in
Figure 10.3 it is 0.063851, which clearly does not agree with the model. Hence
the samples of e(n) for B =2are not independent. This was confirmed by the
bottom plot in Figure 10.3.
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