barcode lib ssrs IMPLEMENTATION OF DISCRETE-TIME SYSTEMS in Software

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IMPLEMENTATION OF DISCRETE-TIME SYSTEMS
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[CHAP. 8
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The general strategy for pairing poles with zeros is to first find the two poles that are closest to the unit circle, in this case those at z = 0.9e*jnI6, and pair these with the two zeros that are closest to these poles, which are those on the unit circle at z = e*j"I6. Thus, the first pole-zero pairing yields the second-order section
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Of the remaining poles, we next find the pair that is closest to the unit circle, which are either those at z = 0.8ei'R/4 or those at z = 0 . 8 e f j 3 x / 4Let us arbitrarily select the first of these, Paired with these poles would then be the zeros . at z = j, which gives us the second-order section
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Finally, for the last section we have
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The cascade is then done in the reverse order, with the first second-order section being H3(z),followed by H&), and then H l ( z ) .
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Supplementary Problems
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Structures for FIR Systems
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Find the unit sample response for the following network:
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What is the frequency response of the following network _-I z-I
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What is the minimum number of multiplications and additions and delays required to implement a linear phase filter with h(n) = 0 for n < 0 and n r 63 How many multiplies, adds, and delays are required to implement the filter
otherwise
using ( a ) a direct form structure and (b) a frequency sampling structure
CHAP. 81
IMPLEMENTATION OF DISCRETE-TIME SYSTEMS
Draw a frequency sampling structure for the FIR high-pass filter of length N = 32 with
H(k) =
k = l S , l 6 , 17
else
Structures for IIR Systems
Find the system function for the following network, where a z - ' is a unit delay combined with a multiplication by a :
.x(n) ~(n)
Find the unit sample response of the following network:
Find the constant A so that the second-order all-pole filter
H(z) =
+ a , z - ' + a 2 zr2
has unit gain at w = 0, and find a structure that only requires two multiplications.
What is the system function corresponding to the following filter structure
Find the transposed direct form 11 realization of the system described by the difference equation
Lattice Filters
8.52 What is the frequency response of the FIR filter that has reflection coefficients I-lo = I
r 2 = . . . = r 9 = 0 and
1MPLEMENTATION OF DISCRETE-TIME SYSTEMS Draw a lattice filter implementation for the allpass filter H (z) = 0.25 - 0.52-' z-2 1 - 0.5~-I 0 . 2 5 r 2
[CHAP. 8
If the system function of a causal filter is
H(z) = is this filter stable
+ I. lz-I + 0 . 9 r 2+ 1
+0 . 5 ~ - ~
If H(z) is an FIR filter with reflection coefficients T I , r2. . . , r,,, what is the system function of the filterG(i) with . reflection coefficients rI. r2. rl, . . . - 0 , r,, 0. 0, Suppose that the last reflection coefficient of an FIR lattice filter. not necessarily minimum phase, has unit magnitude Ir,,l = I. What general statements can be made about the system function H(z) Find the system function of the following lattice filter:
r(n)
Finite Word-Length Effects
(a) Whal fraction does the binary number x = 1.01 101 represent in one's complemenr notation' (b) What about two's complemenl notation White noise with a variance
< is input to a linear shifi-invariant filter with a system function
Find the variance of the noise at the output of the filter. Consider the following cascade of two tirst-order filters, where la1 > Ibl:
Without explicitly evaluating the variance, determine whether the variance of the round-off noise at the output of this filter will be larger or smaller if the order of the cascade is reversed. Consider a linear shift-invariant system with a system function
a Assume that each product is rounded to 16 bits before any additions are performed, and let : be the variance of the round-off noise.
CHAP 81
IMPLEMENTATION OF DISCRETE-TIME SYSTEMS
(a) If this system is implemented in direct form 11, compute the variance of the round-off noise at the output of the : a filter, , due to all noise sources.
(h) Repeat part (a) if the system is implemented in direct form I
Suppose that an FIR filter with a system function
is implemented on a 16-bit fixed-point processor. If sums of products are accumulated prior to rounding, find the variance of the round-off noise at the output of the filter.
The second-order system
H (z) =
1 1 - 1.2728~-'+ O.XIZ--~
is implemented in direct form I1 using 16-bit fixed point arithmetic. Assuming that all sums of products are accumulated prior to rounding, find the quantization noise power at the filter output.
To minimize the effects of round-off noise, what is the best pairing of poles and zeros into second-order sections for the system (1 0 . 9 ~ - ~ ) ( 12.4cos(0.75n)z-' 1.44~-~) H(z) = (1 - 1.4 cos(O.251r)z-' 0.49r2)(1 - 1.8 cos(O.9lr)z' 0.8 1 z r Z )
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