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To analyze the effect of the round-off noise sources at the output of the filter, it is necessary to know how noise propagates through a filter. If the input to a linear shift-invariant filter with a unit sample response h(n) is wide-sense stationary white noise, e(n). with a mean nr, and a variance the filtered noise, f (11) = h(n) * e(n), is a wide-sense stationary process with a mean
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Fig. 8-18. Analysis of round-off noise. (a) A second-order direct form 1 filter. (b)Quantization of products in the filter. ( c )An additive noise model for the round-off noise. .
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The variance may also be evaluated using 2-transforms as follows:
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EXAMPLE 8.6.1
Consider the first-order all-pole tilter with a system function
1 H(z)=1 - cuz-'
If the input to this filter, e ( n ) , is zero mean white noise with a variance a, , variance of the output will be the
a = a,, ;
,,=-C
h(n)12 = 4
lul2''= a,:
1 - laI2
Returning to the direct form I filter, note that the model in Fig. 8-18(c) may be represented in the equivalent form shown in Fig. 8-19 where
Fig. 8-19.
An additive noise model after combining noise sources.
Thus, the quantization noise is filtered only by the poles of the filter, and the output noise satisfies the difference equation
If the noise sources are uncorrelated, as assumed by the third property above, the variance of e,(n) is the sum of the variances of the five noise sources, or
Assuming that the filter is stable, and that the poles of the filter are complex,
CHAP. 81
IMPLEMENTATION OF DISCRETE-TIME SYSTEMS
the variance of the output noise is
a2 = 5 -, .
z-' d z
Using Cauchy's residue theorem to evaluate this integral, we find that
Note that as the poles move closer to the unit circle, r + 1, the variance of the output noise increases. The noise performance of digital filters may be improved by using (2B 1)-bit adders to accumulate sums of products prior to quantization. In this case, the difference equation for the direct form I network becomes
Thus, the sums are accumulated with an accuracy of 2B 1 bits, and the sum is then quantized to B 1 bits in order to store j(n - I) and j ( n - 2) in ( B ])-bit delay registers and to generate the (B I)-bit output j(n). Because there is only one quantizer, which quantizes the sum of products. the variance of the noise source in Fig. 8-19 is reduced from 50; to 0: ,.
8.6.4 Pairing and Ordering
For a fiIter that is implemented in cascade or parallel form, there is considerable flexibility in terms of selecting which poles are to be paired with which zeros and in selecting the order in which the sections are to be cascaded for a cascade structure. Pairing and ordering may have a significant effect on the shape of the output noise power and on the total output noise variance. The rules that are generally followed for pairing and ordering are as follows:
1. The pole that is closest to the unit circle is paired with the zero that is closest to it in the z-plane, and this pairing is continued until all poles and zeros have been paired. 2. The resulting second-order sections are then ordered in a cascade realization according to the closeness of the poles to the unil circle. The ordering may be done either in terms of increasing closeness to the unit circle or in terms of decreasing closeness to the unit circle. Which ordering is used depends on the consideration of a number of factors, including the shape of the output noise and the output noise variance.
Another issue in fixed-point implementations of discrete-time systems is overflow. If each fixed-point number is taken to be a fraction that is less than 1 in magnitude, each node variable in the network should be constrained to be less than I in magnitude in order to avoid overflow. If we let h k ( n )denote the unit sample response of the system relating the input x ( n ) to the kth node variable, wk(n),
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