113_1_Week05

# 113_1_Week05 - 1 EE 113: Digital Signal Processing Week 5...

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1 EE 113: Digital Signal Processing Week 5 1. Partial fractions 2. Use the z-transform to identify DE from transfer function description 3. Use the z-transform to evaluate the transfer function of LTI systems and identify zeros and poles 4. Use the z-transform to solve CCDE with initial conditions

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2 Rational Z-transforms ± G ( z ) can be any function; rational polynomials are important class: ± By convention, expressed in terms of z -1 – matches ZT definition ± (Reminiscent of LCCDE expression. ..) Gz () = Pz Dz ( ) = p 0 + p 1 z 1 + + p M 1 z ( M 1) + p M z M d 0 + d 1 z 1 + + d N 1 z ( N + d N z N
3 Factored rational ZTs ± Numerator, denominator can be factored : ± { ζ A } are roots of numerator G ( z ) = 0 { ζ A } are the zeros of G ( z ) ± { λ A } are roots of denominator G ( z ) = ∞→ { λ A } are the poles of G ( z ) Gz () = p 0 A = 1 M 1 −ζ A z 1 ( ) d 0 A = 1 N 1 − λ A z 1 ( ) = z M p 0 A = 1 M z A z N d 0 A = 1 N z − λ A

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4 Pole-zero diagram ± Can plot poles and zeros on complex z -plane: z -plane Re{ z } Im{ z } 1 × o o o o × × poles λ A (cpx conj for real g [ n ] ) zeros ζ A
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## 113_1_Week05 - 1 EE 113: Digital Signal Processing Week 5...

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