barcode generator for ssrs TRANSFORM ANALYSIS OF SYSTEMS in Software

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TRANSFORM ANALYSIS OF SYSTEMS
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[CHAP. 5
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Similarly, for a type I11 or IV linear phase filter, h(n) = -h(N - n), which implies that
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In both cases, if H ( z ) is equal to zero at z = zo, H ( z )must also be zero at z = 1 / z O . Therefore, the zeros of H(z) occur in reciprocal pairs. In addition, with h(n) being real-valued, complex zeros occur in conjugate reciprocal pairs. Thus, the constraints on the zeros of a linear phase filter are as follows. First. H(z) may have one or more zeros at z = f1. Second, H (z) may have complex-conjugate zeros on the unit circle at z = e*jq or reciprocal zeros on the real axis at z = a and z = l / a . Finally, H ( z ) may have groups of four zeros in conjugate reciprocal pairs at z = rke*jo"nd z = l e * ~ " These constraints are illustrated in Fig. 5-4. rr
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Constraints on the zeros of the system function of an FIR system with generalized linear phase and a real unit sample response. Types I11 and IV filters must have a zero at z = 1 , whereas types I1 and 111 filters must have a zero a t z = -1.
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The cases of z = 1 and z = - 1 deserve special attention. Evaluating the system function at z = -1 for a type I1 filter, we have
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Because N is odd, this implies that
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which will be true only if H(-1) = 0. Therefore, a type I1 linear phase filter must have a zero at z = -1. Similarly, evaluating H ( z ) at z = - 1 for a type 111 filter, we have
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which, because N is even, requires that there be a zero at z = -1. Because the system function evaluated at z = - 1 is equal to the frequency response at w = n, H (ejW)~,,, = 0 Types I1 and 111 filters
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(5.13)
For types I11 and IV filters, evaluating the system function at z = I, we find
which will be true only if H ( z ) is zero at z = 1. Therefore, types 111and IV linear phase filters must have a zero at z = 1, which implies that H (eio)lo,o = 0 Types 111 and IV filters
(5.14)
CHAP. 51
TRANSFORM ANALYSIS OF SYSTEMS
5.4 ALLPASS FILTERS
An allpass filter has a frequency response with a constant magnitude,
This unit magnitude constraint constrains the poles and zeros of a rational system function to occur in conjugate reciprocal pairs:
Thus, if H ( z ) has a pole at z = ak, H ( z ) must have a zero at the conjugate reciprocal location z = l/a,*. If h ( n ) is real-valued, the complex roots in Eq. (5.15)occur in conjugate pairs, and if these conjugate pairs are combined to form second-order factors, the system function may be written as
where the coefficients b k , ck, and dk are real. If an allpass filter H ( z ) is stable and causal, the poles of H ( z ) lie inside the unit circle, lak(< 1. Figure 5-5 shows a typical pole-zero plot for an allpass filter. Allpass filters are useful for group delay equalization to compensate for phase nonlinearities.
Fig. 5-5. Illustration of the conjugate reciprocal symmetry constraint that is placed on the poles and zeros of an allpass system.
A stable allpass filter has a group delay that is nonnegative for all w . This follows from the fact that, for a first-order allpass factor of the form
- a* !
H(z)=
where a! = r e j e , the group delay is
T(O)
1 - uz-'
1 -r2
11 - re~ee-~w12
Therefore, with 0 5 r < 1, it follows that s ( w ) > 0. Because a general allpass filter has a group delay that is a sum of terms of this form, the group delay of a rational, stable, and causal allpass filter is nonnegative. A filter may be cascaded with an allpass filter without changing the magnitude of the frequency response. If the pole of the allpass filter cancels a zero, the zero is replaced with one at the conjugate reciprocal location.
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