Digital Signal Processing. Outline. Hani Mehrpouyan, The Z-Transform. The Inverse Z-Transform

Transcription

1 Digital Signal Processing Hani Mehrpouyan, Department of Electrical and Computer Engineering, California State University, Bakersfield Lecture (Digital Signal Processing) Feb 9 th, 206 The following references have been used in this presentation: [] Sanjit Mitra, Digital Signal Processing A Computer Based Approach. [2] John Proakis and Dimitris K Manolakis Digital Signal Processing. [3] Notes by Prof. Dan Elis at Columbia University. Hani Mehrpouyan (hani.mehr@ieee.org) Boise State c 205 The Z-Transform Outline The Inverse Z-Transform Hani Mehrpouyan (hani.mehr@ieee.org) Boise State c 205 2

2 ROC Intersections Note: Two-sided exponential g[ n] = n < n < = n µ [ n] + n µ [ n ] ROC z > No overlap in ROCs ROC z < α No overlap in ROCs > ZT does not exist (does not converge for any z) Hani Mehrpouyan (hani.mehr@ieee.org) Boise State c 205 Im Re n ZT of LCCDEs LCCDEs have solutions of form: y c [n] = i n i µ [ n] +... Hence ZT (same Y c ( z) = i i z + s) n in correspon Each term λ i in g[n] corresponds to a pole λ i of G(z)... and vice versa. rsa LCCDE sol ns are right-sided ROCs are z > λ i outside circles Hani Mehrpouyan (hani.mehr@ieee.org) Boise State c 205 4

3 Z-Plane and DTFT Z-planesurface and DTFT between and unit cylinder Slice Z-plane and DTFT j j Electric ω) is G(e ω), the surface anddtft unit cylinder Circuits ( z = zslice = e between Slice( z between and cylinder j ),unit = z =surface ej ) is G(e the DTFT ( z = z = ej ) is G(ej ), the DTFT z = ej z = ej G(ej ) Chalmers j Communications Systems Group, G(e ), Chalmers University of Technology, 0 Sweden c / rad/samp / rad/samp August 20, 205 Dan Ellis Boise State c 205 Hani Mehrpouyan (hani.mehr@ieee.org) Dan Ellis Common Z Tranforms Some common Z transforms g[n] G(z) ROC Electric Circuits [n]!!!!! z µ[n] z > z nµ[n] z > Chalmers z Communications Systems r cos z Group, Signals and 0Systems, ( )! ( ) 2 2 Sweden r sin (c 0 ) z! 2r cos! 200! ( 0 ) z +r 2 z 2 rncos( 0n)µ[n]!!! z > r Chalmers University 2r cos 0 z of+rtechnology, z rnsin( 0n)µ[n]! August 20, 205 poles at z = re±j 0 Dan Ellis Hani Mehrpouyan (hani.mehr@ieee.org) Boise State c 205 sum of rnej n + rne-j n 0 0 z > r conjugate pole pair 22 6

4 Z-Transform Properties g[n] G(z) w/roc Rg Conjugation g * [n] G * (z * ) Rg Time reversal g[-n]!! G(/z)! /Rg Time shift g[n-n 0 ]! z -n 0G(z) Rg (0/?) Exp. scaling ng[n]! G(z/ ) Rg Diff. wrt z ng[n]! z! dg(z) Rg (0/?) dz Hani Mehrpouyan (hani.mehr@ieee.org) Boise State c Z-Transform Properties Convolution g[n] h[n] g[n]! G(z) ROC G(z)H(z) Modulation g[n]h[n]! G 2j! ( v)h! z v Parseval: C g[ n]h * [ n] = n= 2j ( ) v dv at least Rg Rh at least RgRh G( v)h * ( v )v C dv Hani Mehrpouyan (hani.mehr@ieee.org) Boise State c 205 8

5 Example Hani Mehrpouyan Boise State c Example Hani Mehrpouyan (hani.mehr@ieee.org) Boise State c 205 0

6 Inverse Z-Transform Forward z transform was defined as: orward z transform was defin G(z) = Z{ g[ n] } = g[n]z n n= 3 approaches to inverting G(z) to g[n]: Generalization of inverse DTFT. Power series in z (long division). Manipulate into recognizable pieces (partial fractions). Hani Mehrpouyan (hani.mehr@ieee.org) Boise State c 205 IZT #: Generalize IDTFT Hani Mehrpouyan (hani.mehr@ieee.org) Boise State c 205 2

7 IZT #2: Long division Since if we could express G(z) as a simple power series G(z) = a + bz - + cz then can just read off g[n] = {a, b, c,...} Typically G(z) is right-sided (causal) and a rational polynomial d (causal) G( z) = P(z) D(z) Can expand as power series through long division of polynomials Hani Mehrpouyan (hani.mehr@ieee.org) Boise State c IZT #2: Long division Procedure: Express numerator, denominator in descending powers of z (for a causal fn) Find constant to cancel highest term first term in result Subtract & repeat lower terms in result Just like long division for base-0 numbers. Hani Mehrpouyan (hani.mehr@ieee.org) Boise State c 205 4

8 IZT #2: Long division Example Hani Mehrpouyan Boise State c IZT#3: Partial Fractions Basic idea: Rearrange G(z) as sum of terms recognized as simple ZTs especially or sin/cos forms i.e. given products rearrange to sums z n µ n [ ] P(z) ( z )( z ) A z + B z + Hani Mehrpouyan (hani.mehr@ieee.org) Boise State c 205 6

9 IZT#3 Partial Fractions Note that: A z + B z + C z = ( )( z ) + B( z )( z ) + C( z )( z ) ( z )( z )( z ) A z order 3 polynomial Can do the reverse i.e. go from P(z) to N ( z ) = if order of P(z) is less than D(z) order 2 polynomial u + vz - + wz -2 N = z else cancel w/ long div. Hani Mehrpouyan (hani.mehr@ieee.org) Boise State c Partial Fractions Note that: Note that A z + B z + C z = A z Can do the reverse i.e. Can do the reverse i.e. the reverse i.e. P(z) to N = ( z ) If order of p(z) is less than D(z) Else cancel w/ long division. ( )( z ) + B( z )( z ) + C( z )( z ) ( z )( z )( z ) order 3 polynomial order order 22 polynomial u u + + vz vz wz wz -2-2 Hani Mehrpouyan (hani.mehr@ieee.org) Boise State c N = z else cance

10 Procedure Procedure: Partial Fractions F(z) = no repeated poles! P(z) N ( z ) = N = z = ( )F z order N- where = z ( ) z= i.e. evaluate F(z) at at the the pole pole m but multiplied by the pole term dominates = residue of pole f [ n] = ( ) n µ [ n] Hani Mehrpouyan (hani.mehr@ieee.org) Boise State c N = (cancels term in denominator) Example Hani Mehrpouyan (hani.mehr@ieee.org) Boise State c 205