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Method for representing DT signals as superpositions of complex geometric (exponential) functions.

Lecture #15:

THE BILATERAL Z-TRANSFORM

Motivation: Method for representing DT signals as superpositions of complex geometric (exponential) functions

Outline:

  • Review of last lecture
  • The bilateral Z-transform

– Definition

– Properties

  • Inventory of transform pairs
  • Conclusion

Review of last lecture

Solve linear difference equation for a causal exponential input

k = 0 K a k y [ n + k ] = l = 0 L b l x [ n + l ] for x [ n ] = Xz n u [ n ] size 12{ Sum cSub { size 8{k=0} } cSup { size 8{K} } {a rSub { size 8{k} } y \[ n+k \] ={}} Sum cSub { size 8{l=0} } cSup { size 8{L} } {b rSub { size 8{l} } x \[ n+l \]} matrix { {} # {}} ital "for" matrix { {} # {}} x \[ n \] = ital "Xz" rSup { size 8{n} } u \[ n \]} {}

Solve homogeneous equation for n>0

k = 0 K a k y h [ n + k ] = 0 by assu min g y h [ n ] = n size 12{ Sum cSub { size 8{k=0} } cSup { size 8{K} } {a rSub { size 8{k} } y rSub { size 8{h} } \[ n+k \] ={}} 0 matrix {{} # {} } ital "by" matrix {{} # {} } ital "assu""min"g matrix {{} # {} } y rSub { size 8{h} } \[ n \]=Aλ rSup { size 8{n} } } {}

Solve characteristic polynomial for λ.

k = 0 K a k λ n + k = 0 size 12{ Sum cSub { size 8{k=0} } cSup { size 8{K} } {a rSub { size 8{k} } λ rSup { size 8{n+k} } ={}} 0} {}

Solve for a particular solution for n>0

k = 0 K a k y p [ n + k ] = l = 0 L b l x [ n + l ] for x [ n ] = Xz n u [ n ] size 12{ Sum cSub { size 8{k=0} } cSup { size 8{K} } {a rSub { size 8{k} } y rSub { size 8{p} } \[ n+k \] ={}} Sum cSub { size 8{l=0} } cSup { size 8{L} } {b rSub { size 8{l} } x \[ n+l \]} matrix { {} # {}} ital "for" matrix { {} # {}} x \[ n \] = ital "Xz" rSup { size 8{n} } u \[ n \]} {}

Assuming y p [ n ] = Yz n size 12{y rSub { size 8{p} } \[ n \] =" Yz" rSup { size 8{n} } } {} and solving for Y yields

Y = H ~ ( z ) X = l = 0 L b l z l k = 0 K a k z k X size 12{Y= {H} cSup { size 8{ "~" } } \( z \) X= { { Sum cSub { size 8{l=0} } cSup { size 8{L} } {b rSub { size 8{l} } } z rSup { size 8{l} } } over { Sum cSub { size 8{k=0} } cSup { size 8{K} } {a rSub { size 8{k} } } z rSup { size 8{k} } } } X} {}

Logic for an analysis method for DT LTI systems

  • H ~ ( z ) size 12{ {H} cSup { size 8{ "~" } } \( z \) } {} characterizes system  compute H ~ ( z ) size 12{ {H} cSup { size 8{ "~" } } \( z \) } {} efficiently.
  • In steady state, response to Xz n size 12{"Xz" rSup { size 8{n} } } {} is H ~ ( z ) z n size 12{ {H} cSup { size 8{ "~" } } \( z \) z rSup { size 8{n} } } {} .
  • Represent arbitrary x[n] as superpositions of Xz n size 12{"Xz" rSup { size 8{n} } } {} on z.
  • Compute response y[n] as superpositions of H ~ ( z ) Xz n size 12{ {H} cSup { size 8{ "~" } } \( z \) "Xz" rSup { size 8{n} } } {} on z.

I. THE BILATERAL Z-TRANSFORM

1/ Definition

The bilateral Z-transform is defined by the analysis formula

X ~ ( z ) = n = x [ n ] z n size 12{ {X} cSup { size 8{ "~" } } \( z \) = Sum cSub { size 8{n= - infinity } } cSup { size 8{ infinity } } {x \[ n \] z rSup { size 8{ - n} } } } {}

X ~ ( z ) size 12{ {X} cSup { size 8{ "~" } } \( z \) } {} is defined for a region in z — called the region of convergence — for which the sum exists.

The inverse transform is defined by the synthesis formula

x [ n ] = 1 2πj C X ~ ( z ) z n 1 dz size 12{x \[ n \] = { {1} over {2πj} } Int rSub { size 8{C} } { {X} cSup { size 8{ "~" } } \( z \) } z rSup { size 8{n - 1} } ital "dz"} {}

Since z is a complex quantity, X ~ ( z ) size 12{ {X} cSup { size 8{ "~" } } \( z \) } {} is a complex function of a complex variable. Hence, the synthesis formula involves integration in the complex z domain. We shall not perform this integration in this subject. The synthesis formula will be used only to prove theorems and not to compute time functions directly.

a/ Approach

An inventory of time functions and their Z-transforms will be developed by

  • Using the Z-transform properties,
  • Determining the Z-transforms of elementary DT time functions,
  • Combining the results of the above two items.

b/ Notation

We shall use two useful notations — Z{x[n]} signifies the Z-transform of x[n]and a Z-transform pair is indicated by

x [ n ] Z X ~ ( z ) size 12{x \[ n \] { dlrarrow } cSup { size 8{Z} } {X} cSup { size 8{ "~" } } \( z \) } {}

2/ Properties

a/ Linearity

ax 1 [ n ] + bx 2 [ n ] Z a X ~ 1 ( z ) + b X ~ 2 ( z ) size 12{ ital "ax" rSub { size 8{1} } \[ n \] + ital "bx" rSub { size 8{2} } \[ n \]{ dlrarrow } cSup { size 8{Z} } a {X} cSup { size 8{ "~" } } rSub { size 8{1} } \( z \) +b {X} cSup { size 8{ "~" } } rSub { size 8{2} } \( z \) } {}

The proof follows from the definition of the Z-transform as a sum.

X ~ ( z ) = n = ( ax 1 [ n ] + bx 2 [ n ] ) z n X ~ ( z ) = a n = x 1 [ n ] z n + b n = x 2 [ n ] z n X ~ ( z ) = a X 1 ~ ( z ) + b X 2 ~ ( z ) alignl { stack { size 12{ {X} cSup { size 8{ "~" } } \( z \) = Sum cSub { size 8{n= - infinity } } cSup { size 8{ infinity } } { \( ital "ax" rSub { size 8{1} } \[ n \]+ ital "bx" rSub { size 8{2} } \[ n \] \) z rSup { size 8{ - n} } } } {} #{X} cSup { size 8{ "~" } } \( z \) =a Sum cSub { size 8{n= - infinity } } cSup { size 8{ infinity } } {x rSub { size 8{1} } \[ n \] z rSup { size 8{ - n} } } +b Sum cSub { size 8{n= - infinity } } cSup { size 8{ infinity } } {x rSub { size 8{2} } \[ n \] z rSup { size 8{ - n} } } {} #{X} cSup { size 8{ "~" } } \( z \) =a {X rSub { size 8{1} } } cSup { size 8{ "~" } } \( z \) +b {X rSub { size 8{2} } } cSup { size 8{ "~" } } \( z \) {} } } {} {}

b/ Delay by k

x [ n k ] Z z k X ~ ( z ) size 12{x \[ n - k \] { dlrarrow } cSup { size 8{Z} } z rSup { size 8{ - k} } {X} cSup { size 8{ "~" } } \( z \) } {}

This result can be seen using the synthesis formula,

x [ n k ] = 1 2πj X ~ ( z ) z n k 1 dz x [ n k ] = 1 2πj z k X ~ ( z ) z n 1 dz alignl { stack { size 12{x \[ n - k \]= { {1} over {2πj} } Int { {X} cSup { size 8{ "~" } } \( z \) z rSup { size 8{n - k - 1} } ital "dz"} } {} # x \[ n - k \]= { {1} over {2πj} } Int {z rSup { size 8{ - k} } {X} cSup { size 8{ "~" } } \( z \) z rSup { size 8{n - 1} } ital "dz"} {} } } {}

c/ Multiply by n

nx [ n ] Z z d X ~ ( x ) dz size 12{ ital "nx" \[ n \] { dlrarrow } cSup { size 8{Z} } - z { {d {X} cSup { size 8{ "~" } } \( x \) } over { ital "dz"} } } {}

This result can be seen using the analysis formula.

d X ~ ( z ) dz = n = nx [ n ] z n 1 z d X ~ ( z ) dz = n = nx [ n ] z n alignl { stack { size 12{ { {d {X} cSup { size 8{ "~" } } \( z \) } over { ital "dz"} } = Sum cSub { size 8{n= - infinity } } cSup { size 8{ infinity } } { - ital "nx" \[ n \]z rSup { size 8{ - n - 1} } } } {} # - z { {d {X} cSup { size 8{ "~" } } \( z \) } over { ital "dz"} } = Sum cSub { size 8{n= - infinity } } cSup { size 8{ infinity } } { ital "nx" \[ n \]z rSup { size 8{ - n} } } {} } } {}

Most proofs of Z-transform properties are simple. Some of the important properties are summarized here.

R, R1, and R2 are the ROCs of X ~ ( z ) size 12{ {X} cSup { size 8{ "~" } } \( z \) } {} , X 1 ~ ( z ) size 12{ {X rSub { size 8{1} } } cSup { size 8{ "~" } } \( z \) } {} , and X 2 ~ ( z ) size 12{ {X rSub { size 8{2} } } cSup { size 8{ "~" } } \( z \) } {} , respectively. * Exceptions may occur at z = 0 and z = ∞.

II. Z-TRANSFORMS OF SIMPLE TIME FUNCTIONS

1/ Unit sample function

δ [ n ] = { 1 if n = 0 0 otherwise size 12{δ \[ n \] = left lbrace matrix {1 matrix { {} # {}} ital "if" matrix { {} # {}} n=0 {} ## 0 matrix {{} # {} } ital "otherwise"{}} right none } {}

The Z-transform of the unit sample is

Z { δ [ n ] } = n = δ [ n ] z n = z 0 = 1 size 12{Z lbrace δ \[ n \] rbrace = Sum cSub { size 8{n= - infinity } } cSup { size 8{ infinity } } {δ \[ n \]z rSup { size 8{ - n} } =z rSup { size 8{0} } =1} } {}

for all values of z, i.e., the ROC is the entire z plane.

2/ Unit step function

u [ n ] = { 1 for n 0 0 for n < 0 size 12{u \[ n \] = left lbrace matrix {1 matrix { {} # {}} ital "for" matrix { {} # {}} n>= 0 {} ## 0 matrix {{} # {} } ital "for" matrix {{} # {} } n<0{} } right none } {}

The unit step and unit sample functions are simply related.

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Source:  OpenStax, Signals and systems. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10803/1.1
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