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barcode generator for ssrs I 0.75~ in Software
I 0.75~ Recognizing Code128 In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Generating Code 128 Code Set C In None Using Barcode drawer for Software Control to generate, create Code128 image in Software applications. both H ( z ) and G ( z )have minimum phase. However, the sum
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Find a factorization for H(z) of the form
where Hmi,(z) has minimum phase. and Hap(z)is an allpass filter.
The system function H(z) has a nonminimum phase factor, (1  2 ~  ~which may be written as the product of a ), minimum phase term and an allpass factor as follows: Therefore, H(z) may be written as the product of a minimum phase system with an allpass system as follows: A causal linear shiftinvariant system has a system function
H (z) = Find a factorization for H ( z ) of the form
+ zI)(2  3 z  I ) 1izl
where Hmi,(z) has minimum phase, and Hlp(z)is a linear phase system.
TRANSFORM ANALYSIS OF SYSTEMS
[CHAP. 5
This system is not minimum phase because the factor ( 2  3 z  ' ) corresponds to a zero outside the unit circle at z = However, we may express this factor as the product of a minimum phase term with a linear phase term as follows. First, we reflect the zero about the unit circle and replace it with a pole: Then, we multiply this term with a linear phase factor that has a zero at z =
$ and a zero at z = :: Thus, the factorization for H ( z ) is
H ( z )= 3 + z' (2z'  3)(2  32') ( I  4z')(2zl  3 ) 53 .1 Find a realvalued causal sequence with ~ ( 0> 0 and ) I X ( @ ' ) [ ~ = (1 + a2) 20 cos w
 .elW
We begin by expressing IX (ej'")12in terms of complex exponentials: I X ( P ' " ) I ~ = (I+ a 2 ) UP^^^
Replacing d" with z, and e1'" with z  ' , we have
G ( z )= X ( Z ) X ( Z  ' ) = ( I + a 2 )  az  u z  ' = ( I
 u z f l ) ( l a z ) Therefore, a realvalued causal sequence with the given magnitude with .u(O) > 1 is
.r(n) = 6 ( n ) u6(n  I) Find the minimum phase system that has a magnitude response given by
To solve this problem, we begin by expressing ( ~ ( e j ' " )in ~ l terms of complex exponentials as follows: Replacing ei" by z, and eJWby zI, this becomes
The minimum phase system is then formed by extracting the poles and zeros that are inside the unit circle: Use the initial value theorem to show that if hmi,(n) is a minimum phase sequence, and if h ( n ) is a causal sequence with the same Fourier transform magnitude, then CHAP. 51
TRANSFORM ANALYSIS OF SYSTEMS
The initial value theorem states that for a causal sequence, the initial value may be found from the ztransform as follows: h(0) =: I % H(z) Let hmin(n) a minimum phase sequence, and let h ( n ) be the nonminimum phase sequence that is formed by be reflecting a zero from inside the unit circle at z = a to its conjugate reciprocal location at z = I/a*: have the same Fourier transform magnitude. Because ( z  '  a * ) / ( l  azI) is an allpass filter. h ( n ) and hmin(n) Using the initial value theorem, we may compare the value of h(0) to h,l,i,,(0): and because la1 < I, lh(0)I < lhm,,(0)l. Because the magnitude of Ih(O)l is reduced each time that a zero of Hmin(z) is flipped outside the unit circle, Ih(0)l < Ihrnm(0)l
for any sequence h ( n ) that has a Fourier transform with the same magnitude as that of hmi.(n). Prove the minimum energy delay property for minimum phase sequences. Let hm,,(n) be a minimum phase sequence, and let a k be a zero of Hmin(z). Then Hmin(z) be written as may HmIn(z) = ( 1 akz'Fmin(z) (5.23) where G m i n ( zis another minimum phase sequence. Because Hm,,(z) is minimum phase, lakI < I . Let H ( z ) be the ) causal nonminimum phase sequence that is formed by replacing the zero at z = a k with a zero at z = ] / a ; :

