Direct method compute the roots of pz direct

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Unformatted text preview: . i.e. H0(z) is a minimum phase filter. Example: 6 = 3 . 3 0 P(z) H0(z) H0(z-1) (Minimum phase) (Maximum phase) 11 11 Practical Algorithms: 1. Direct Method: compute the roots of P(z) Direct numerically. numerically. 2. Cepstral Method: First factor out the zeros which lie on the unit First circle circle P(z) = [(1 + z-1)(1 + z)]p Q(z) Now we need to factor Q(z) into R(z) R(z -1) such that i. R(z) has all its zeros inside the unit circle. ii. R(z) is causal. 12 12 Then use logarithms to change multiplication into Then addition: addition: Q(z) = Q(z) ln Q(z) = Q(z) 123 123 ^ Q(z) R(z) ln R(z) ln R(z) 123 ^ R(z) • + R(z-1) ln R(z-1) ln 123 ^ R(z-1) Take inverse z transforms: ^ q[n] = Complex cepstrum Complex cepstrum of q[n] ^ r[n] + ^ -n] r[ 13 13 Example: 5 5X X X X R(z) ^ R(z) = ln R(z) R(z) ln R(z) has all its zeros and all its poles inside the unit R(z) ^ circle, so R(z) has all its singularities inside the unit circle. (ln0 = - ∞ , ln = ∞ .) circle. ln0 ln .) 14 14 All singularities inside the unit circle leads to a causal All sequence, e.g. sequence, 1 X(z) = X(z) 1 - zk z-1 Pole at z = zk Pole 1 X(ω) = 1 - zk ?-iω If |zk| < 1, we can write If ∞ X(ω) = ∑ (zk)n e-iωn n=0 678 678 678 ⇒x[n] is causal ^ So ^ is the causal part of q[n]: r[n] ½^ q[0] ; n = 0 ^ ^ q[n] ; n>0 r[n]...
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