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barcode lib ssrs A white noise sequence e ( n ) with variance 0; is input to a filter with a system function in Software
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= 720,' Consider the following cascade of two firstorder allpole filters: (a) Find the variance of the roundoff noise at the output of the cascade for an 8bit processor with rounding.
(h) Repeat for the case in which the order of the cascade is reversed.
(a) A model for the roundoff noise is shown in the following figure: where the variance of each noise source is equal to
The system function of the filter is
H(z) = and the unit sample response is
Note that because e l ( n ) is filtered by h ( n ) , and e2(n) is only filtered by the second filter in the cascade, which has a unit sample response hdn) = (:)nw
the output noise, f (n), is
f ( n )= Therefore, the variance of f (n) is
* h ( n )+ e d n ) * h d n ) CHAP. 81
IMPLEMENTATION OF DISCRETETIME SYSTEMS
With
we have Next, we have
Therefore, the variance of the roundoff noise at the output of the filter is
which, for an 8bit processor (B = 7), is
228 = 2.8953  = 0.2413 . 2l4 = I.4726 . 12
(b) If the order of the cascade is reversed. we have the following network: The variance of the roundoff noise due to e l ( n ) is the same as in part (a),but because the unit sample response of the second system in the cascade is now h; (n) ( i ) n u ( n ) = the variance of the noise due to e z ( n ) is
Thus, the variance of the roundoff noise at the output of the filter is
a = 1.82860: : + 1.33330: = 3.16190; which, for an 8bit processor is
With this structure, the roundoff noise is slightly larger.
Consider a linear shiftinvariant system with a system function
H (z) = 10 . 4 ~ ~ ' ( 1  0.62')(I  0.8~1) Suppose that this system is implemented on a 16bit fixedpoint processor and that the sums of products a are accumulated prior to quantization. Let : be the variance of the roundoff noise. IMPLEMENTATION OF DISCRETETIME SYSTEMS
[CHAP. 8
(a) If the system is implemented in direct form 11, find the variance of the roundoff noise at the output of the filter. (b) Repeat part (a) if the system is implemented in parallel form.
( a ) The direct form I1 implementation of this system is shown in the figure below along with the two roundoff noise sources. Because the sum x(n) + 1.4w(n  1)  0.48w(n  2 ) Similarly, because the sum
may be accumulated prior to quantization, the variance of the noise e l ( n )is . : a
may be accumulated prior to quantization, the variance of the noise e2(n)is also me2. With c l ( n )being filtered by the system and with ez(n)being noise that is simply added to the output, the quantization noise at the output of the filter is f ( n ) = h(n) * el ( n ) eAn) which has a variance equal to
To find the unit sample response of the filter, we expand H ( z ) in a partial fraction expansion as follows: H ( z )= I  0.4~I  O.6zr1)(I  0.82I) 1 1  0.62I
1  0.8~I
Therefore, h(n) = (0.6)"u(n) + 2(0.8)"u(n)  4(0.48)" lh(n)12 = [(0.6)" + 2(0.8)"]2u(n) [(0.6)'" = +4(0.8)~"]u(n) Evaluating the sum of the squares of h(n),we have
Thus, the variance of the output noise is
= 6u: (b) Using the partial fraction expansion for H ( z ) given in part ( a ) ,the parallel form implementation of this filter is shown in the following figure: CHAP. 81
IMPLEMENTATION OF DISCRETETIME SYSTEMS
As indicated in the figure, there are two noise sources. The first, cl (n), is filtered with a firstorder allpole filter that has a unit sample response hl(n) = (0.8)"1~(~) and the second, ez(n), is filtered with a firstorder allpole filter that has a unit sample response Because the output noise is + f ( n ) = cJl(n)* I I I ( ~ )r 2 ( n )* hz(n) the variance of f (n) is
A linear shiftinvariant system with a system function of the form
is to be implemented as a cascade of N secondorder sections. where each section is realized in either 1 direct form I o r 1 or in their transposed forms. How many different cascaded realizations are possible. Let us assume that each factor in H ( z ) is unique, so that there arc N different secondorder polynomials in the numerator and the same number of polynomials in the denominator. In this case. there are N! different pairings of factors in the numerator with factors in the denominator. In addition, for each of these pairings, there are N ! different orderings of these sections. Therefore. there are ( N !)' different pairings and orderings. With four different structures for each section (direct form I, direct form 11, transposed direct form I, and transposed direct form 11), there are a total of 4N(N!)2 different realizations. For a tenthorder system (N = 5). this corresponds to 14,745,600 different structures. This is why general pairing and ordering rules are important. Let H ( z ) be a pthorder allpass filter with a gain of 1 that is implemented in direct form I1 using a 1 bits. processor with B

