signal & system inverse z-transform

signal & system inverse z-transform

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  SIGNAL AND SYSTEM ELECTRONICS AND COMMUNICATION 1
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  SARDAR VALLABHBHAI PATEL INSTITUTEOF TECHNOLOGY 2ELECTRONICS AND COMMUNICATION
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  INVERSE Z-TRANSORM MADE BY: VISHALHASRAJANI 130410111033 RAJSI JADHAV 130410111035 MIHIR JAIN 130410111036 3ELECTRONICS AND COMMUNICATION
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  DEFINITION z-TRANSFORM • z-transformprovides a valuable technique for analysis and design of discrete time signals and discrete time LTI system. ELECTRONICS AND COMMUNICATION 4
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  The z-Transform Definition • Thez-transform of sequence x(n) is defined by ∑ ∞ −∞= − = n n znxzX )()(  Let z = e−jω . ( ) ( )j j n n X e x n eω ω ∞ − =−∞ = ∑ Fourier Transform 5ELECTRONICS AND COMMUNICATION
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  z-Plane Re Im z = e−jω ω ∑ ∞ −∞= − = n n znxzX)()( ( ) ( )j j n n X e x n eω ω ∞ − =−∞ = ∑ Fourier Transform is to evaluate z-transform on a unit circle. Fourier Transform is to evaluate z-transform on a unit circle. 6ELECTRONICS AND COMMUNICATION
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  Advantages of z-transform •Stability of LTI system can be determined using z-transform. • By calculating z-transform of a given signal, DFT and FT can be determined. • The solution of differential equations can be simplified using z-transform. ELECTRONICS AND COMMUNICATION 7
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  Inverse z-transform Where Cis a counterclockwise closed path encircling the origin and is entirely in the ROC. Contour C must encircle all the poles of X(z). 8ELECTRONICS AND COMMUNICATION ∫ − = c n n dzzzX j x 1 )( 2 1 π
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  The Inverse Z-Transform •There are generally three types of inverse z-transform: – Synthetic division method – Partial fraction expansion – Power series expansion 9ELECTRONICS AND COMMUNICATION
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  The Inverse zTransform Synthetic Division For rational z transforms of the form H z( ) = bM zM + bM −1zM −1 ++ b1z + b0 aN zN + aN−1zN−1 ++ a1z + a0 we can always find the inverse z transform by synthetic division. For example, H z( ) = z −1.2( ) z + 0.7( ) z + 0.4( ) z − 0.2( ) z − 0.8( ) z + 0.5( ) , z > 0.8 H z( ) = z3 − 0.1z2 −1.04z − 0.336 z3 − 0.5z2 − 0.34z + 0.08 , z > 0.8 10ELECTRONICS AND COMMUNICATION
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  Synthetic Division The Inversez Transform 11ELECTRONICS AND COMMUNICATION
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  Partial Fraction Expansion 12ELECTRONICSAND COMMUNICATION
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  Partial Fraction Expansion 13ELECTRONICSAND COMMUNICATION
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  Inverse z TransformExample 14ELECTRONICS AND COMMUNICATION
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  Inverse z TransformExample 15ELECTRONICS AND COMMUNICATION
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  Inverse Z-Transform byPower Series Expansion • The z-transform is power series • In expanded form • Z-transforms of this form can generally be inversed easily • Especially useful for finite-length series • Example ( ) [ ]∑ ∞ −∞= − = n n znxzX ( ) [ ] [ ] [ ] [ ] [ ]  ++++−+−+= −− 2112 z2xz1x0xz1xz2xzX ( ) ( )( ) 12 1112 z 2 1 1z 2 1 z z1z1z 2 1 1zzX − −−− +−−= −+      −= [ ] [ ] [ ] [ ] [ ]1n 2 1 n1n 2 1 2nnx −δ+δ−+δ−+δ= [ ]          = = =− −=− −= = 2n0 1n 2 1 0n1 1n 2 1 2n1 nx 16 ELECTRONICS AND COMMUNICATION
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  ELECTRONICS AND COMMUNICATION 17