ELEN E4810: Digital Signal Processing Topic 4: The Z Transform. 1. The Z Transform. 2. Inverse Z Transform

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1 ELEN E480: Digital Signal Processing Topic 4: The Z Transform. The Z Transform 2. Inverse Z Transform

2 . The Z Transform Powerful tool for analyzing & designing DT systems Generalization of the DTFT: G(z) = Z g n [ ] z is complex... z = e j! DTFT z = r e j! n= { } = g[n]z n n g[ n]r n e jn Z Transform DTFT of r -n g[n] 2

3 Region of Convergence (ROC) Critical question: Does summation G(z) = n= converge (to a finite value)? In general, depends on the value of z Region of Convergence: Portion of complex z-plane for which a particular G(z) will converge ROC z > λ x[n]z n 3 Im{z} λ Re{z} z-plane

4 ROC Example e.g. x[n] = n µ[n] X(z) = n z n = n=0 z ß converges only for z - < i.e. ROC is z > (previous slide) < (e.g. 0.8) - finite energy sequence > (e.g..2) - divergent sequence, infinite energy, DTFT does not exist but still has ZT when z >.2 (in ROC) 4-2 n closed form when z - < Im{z} λ M Re{z}

5 About ROCs ROCs always defined in terms of z circular regions on z-plane (inside circles/outside circles/rings) If ROC includes unit circle ( z = ), g[n] has a DTFT (finite energy sequence) Unit circle lies in ROC DTFT OK Im{z} Re{z} z-plane 5

6 Another ROC example Anticausal (left-sided) sequence: x[ n] = n µ [ n ] n X z n ( )z n ( ) = n µ [ n ] = n z n = m z m = z n= m= z = z ROC: λ > z Same ZT as n µ[n], different sequence? 6

7 -4 ROC is necessary! A closed-form expression for ZT must specify the ROC: x[n] = n µ[n] X(z) = z Im n ROC z > x[n] = - n λ µ[-n-] X(z) = z n ROC z < A single G(z) expression can match several sequences with different ROCs -4 z-plane 7 Re DTFTs?

8 G z Rational Z-transforms G(z) expression can be any function; rational polynomials are important class: ( ) = P ( z ) D( z) = p 0 + p z + + p M z (M ) + p M z M d 0 + d z + + d N z (N) + d N z N By convention, expressed in terms of z - matches ZT definition (Reminiscent of LCCDE expression...) 8

9 Factored rational ZTs Numerator, denominator can be factored: G( z) = p 0 M ( = z ) M p 0 M = z d 0 N = z z N N d 0 = z ( ) = z ( ) ( ) { } are roots of numerator G(z) = 0 { } are the zeros of G(z) { } are roots of denominator G(z) = { } are the poles of G(z) 9

10 Pole-zero diagram Can plot poles and zeros on complex z-plane: o o o Im{z} o poles (cpx conj for real g[n]) Re{z} zeros z-plane (Value of) expression determined by roots 0

11 Z-plane surface G(z): cpx function of a cpx variable Can calculate value over entire z-plane ROC not shown!! M

12 ROCs and sidedness Two sequences have: G(z) = z ROC z > g[n] = nµ[n] RIGHT-SIDED n λ 4 ( < ) ROC z < g[n] = - nµ[-n-] LEFT-SIDED n z-plane Each ZT pole region in ROC outside or inside for R/L sided term in g[n] Dan Ellis Overall ROC is intersection of each term s

13 ZT is Linear G(z) = Z{ g[ n] } = g[n]z n n Z Transform y[n] = Æg[n] + Øh[n] Y(z) = ß(Æg[n]+Øh[n])z -n Linear = ßÆg[n]z -n + ßØh[n]z -n = ÆG(z)+ØH(z) Thus, if then y[n] = n µ[n]+ 2 2n µ[n] Y (z) = z z ROC: z > λ, λ 2 3

14 - nµ[-n-] - 2 nµ[-n-] ROC intersections G(z) = z + Consider 2 z with <, 2 >... Two possible sequences for term... Similarly for 2... n n nµ[n] no ROC specified 4 possible g[n] seq s and ROCs... 4 or or 2 nµ[n] n n

15 ROC intersections: Case G(z) = z + 2 z ROC: z > and z > 2 Im g[n] = n µ[n] + 2n µ[n] both right-sided: n Im Re λ λ 2 5

16 ROC intersections: Case 2 G(z) = z + 2 z g[n] = - n µ[-n-] - 2n µ[-n-] both left-sided: n ROC: z < and z < 2 Im Im Re λ λ 2 6

17 G(z) = ROC intersections: Case 3 z + 2 z ROC: z > and z < 2 Im Im Re λ λ 2 g[n] = n µ[n] - 2n µ[-n-] two-sided: n 7

18 G(z) = ROC intersections: Case 4 z + 2 z ROC: z < and z > 2? g[n] = - n µ[-n-] + 2n µ[n] two-sided: n Im Re λ λ 2 no ROC... 8

19 ROC intersections Note: Two-sided exponential g[ n] = n = n µ n [ ] + n µ n [ ] ROC z > Æ < n < ROC z < Æ No overlap in ROCs ZT does not exist (does not converge for any z) Im α Re n 9

20 ZT of LCCDEs LCCDEs have solutions of form: y c [n] = i i n µ n [ ] +... Hence ZT Y ( c z) = + i z Each term n i in g[n] corresponds to a pole i of G(z)... and vice versa LCCDE sol ns are right-sided ROCs are z > i 20 i outside circles (same s)

21 Z-plane and DTFT Slice between surface and unit cylinder ( z = z = e j!) is G(e j!), the DTFT z = e j! G(e j! ) 0 º! / rad/samp 2

22 Some common Z transforms g[n] G(z) ROC ±[n] z µ[n] z > Æ n µ[n] r n cos(! 0 n)µ[n] r n sin(! 0 n)µ[n] z z r cos( 0 )z 2r cos( 0 )z +r 2 z 2 r sin( 0 )z 2r cos( 0 )z +r 2 z 2 poles at z = re ±j! 0 z > Æ z > r z > r 22 sum of r n e j! 0n + r n e -j! 0n conjugate pole pair

23 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 Æ n g[n] G(z/Æ) Æ Rg Diff. wrt z ng[n] Rg (0/?) z dg(z) dz 23

24 Z Transform properties Convolution g[n] h[n] Modulation g[n]h[n] Parseval: g[n] G(z) ROC 2j C g[ n]h * [ n] = n= 2j G(z)H(z) G v ( )H z v C ( )v dv at least Rg Rh at least RgRh G( v)h * ( v )v dv 24

25 ZT Example x n [ ] = r n cos 0 n ( )µ n [ ] µ n 2 [ ] re j 0 Hence, ( ) n + ( re j ) n 0 X( z) = V ( z) + V * z * ; can express as = v n [ ] + v * [ n] v[n] = / 2 µ[n]æ n ; Æ = re j! 0 V(z) = /(2(- re j! 0 z - )) ROC: z > r ( ) ( ) = 2 re j 0 + z re j 0 z r cos( = 0 )z 2r cos( 0 )z +r 2 z 2 25

26 Another ZT example X z ( ) = ( z > Æ ) Y z ( ) = ROC z > Æ y n [ ] = n + ( ) n µ n [ ] = x[n]+ nx[n] where x[n] = Æ n µ[n] z z + z dx z ( ) dz = z d dz = z 26 z ( z ) 2 z ( z ) 2 = ( z ) 2 repeated root - IZT

27 2. Inverse Z Transform (IZT) Forward z transform was defined as: G(z) = Z g n [ ] n= { } = g[n]z 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) the useful one 27

28 IZT #: Generalize IDTFT If so z = re j then G(z) = G(re j ) = g[n]r n e j n = DTFT g[n]r n z = re j! d! = dz/jz g[n]r n = 2 = 2 j Counterclockwise closed contour at z = r within ROC Any closed contour around origin will do Cauchy: g[n] = ß[residues of G(z)z n-] 28 C G re j e j n d G (z) zn r n dz IDTFT Im Re

29 IZT #2: Long division G(z) = g[n]z n n= 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,...} Since Typically G(z) is right-sided (causal) and a rational polynomial G z ( ) = P(z) D(z) Can expand as power series through long division of polynomials 29

30 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 30

31 IZT #2: Long division e.g. H z ( ) = + 2z + 0.4z 0.2z 2 Result +.6z 0.52z z z 0.2z 2 ) + 2z + 0.4z 0.2z 2.6z + 0.2z 2.6z z z z z

32 IZT#3: Partial Fractions Basic idea: Rearrange G(z) as sum of terms recognized as simple ZTs especially n µ [ n] z or sin/cos forms i.e. given products rearrange to sums P(z) ( z )( z ) A z + B z + 32

33 Partial Fractions Note that: A z + B z + C z = A( z )( z ) + B( z )( z ) + C( z )( z ) ( z )( z )( z ) order 3 polynomial Can do the reverse i.e. go from P(z) N ( z ) = to if order of P(z) is less than D(z) order 2 polynomial u + vz - + wz -2 N = z else cancel w/ long div. 33

34 Partial Fractions Procedure: F(z) = no repeated poles! P(z) N ( z ) = N = z = order N- f n N where = ( z )F( z) z= i.e. evaluate F(z) at the pole but multiplied by the pole term dominates = residue of pole [ ] = ( ) n µ [ n] 34 = (cancels term in denominator)

35 Partial Fractions Example Given H z ( ) = factor: + 2z = + 0.6z + 2z + 0.4z 0.2z 2 ( )( 0.2z ) = (again) + 0.6z z where: = ( + 0.6z )H( z) = z=0.6 2 = + 2z + 0.6z z=0.2 = z 0.2z z=0.6 =.75 35

36 Partial Fractions Example Hence H( z) = z z If we know ROC z > Æ i.e. h[n] causal: h n [ ] =.75 ( ) 0.6 ( ) n µ n [ ] ( ) 0.2 ( ) n µ n [ ] =.75{ } +2.75{ } = { } same as long division! 36

Topic 4: The Z Transform

ELEN E480: Digital Signal Processing Topic 4: The Z Transform. The Z Transform 2. Inverse Z Transform . The Z Transform Powerful tool for analyzing & designing DT systems Generalization of the DTFT: G(z)