Z-Transform Properties Quiz

Z-transform Properties

• Definition: The Z-transform converts a discrete-time signal into a complex frequency domain representation.

• Linearity:

  • If ( x[n] ) has Z-transform ( X(z) ) and ( y[n] ) has Z-transform ( Y(z) ), then for constants ( a ) and ( b ): [ a \cdot x[n] + b \cdot y[n] \rightarrow a \cdot X(z) + b \cdot Y(z) ]

• Time Shifting:

  • If ( x[n] ) has Z-transform ( X(z) ):
    • Right shift: ( x[n-k] \rightarrow z^{-k} X(z) ) (for ( k > 0 ))
    • Left shift: ( x[n+k] \rightarrow z^{k} X(z) ) (for ( k > 0 ))

• Time Scaling:

  • If ( x[n] ) has Z-transform ( X(z) ):
    • ( x[n] = x[an] ) results in a non-standard Z-transform, generally not applicable.

• Conjugate Symmetry:

  • If ( x[n] ) is real, then ( X(z) = X^(\frac{1}{z^}) ).

• Initial and Final Value Theorems:

  • Initial Value Theorem: [ x[0] = \lim_{z \to \infty} X(z) ]
  • Final Value Theorem: [ x[\infty] = \lim_{z \to 1} (z-1)X(z) ]
  • Note: Final value theorem only valid if poles of ( (z-1)X(z) ) are within the unit circle.

• Differentiation in Z-domain:

  • If ( x[n] ) has Z-transform ( X(z) ): [ \frac{dX(z)}{dz} \leftrightarrow n x[n] ]

• Multiplication by ( z^{-k} ):

  • If ( x[n] ) has Z-transform ( X(z) ): [ x[n-k] \leftrightarrow z^{-k} X(z) ]

• Convolution Property:

  • If ( x[n] ) has Z-transform ( X(z) ) and ( h[n] ) has Z-transform ( H(z) ): [ x[n] * h[n] \leftrightarrow X(z) H(z) ]

• Parseval's Theorem:

  • Relates the energy of the signal in time domain to the energy in the Z-domain: [ \sum |x[n]|^2 = \frac{1}{2\pi j} \oint |X(z)|^2 \frac{dz}{z} ]

• Region of Convergence (ROC):

  • The region in the z-plane where the Z-transform converges must be identified as it affects the properties and stability of the system.

Understanding these properties is crucial for analyzing and designing discrete-time control systems.

Definition and Function of Z-transform

• Converts discrete-time signals into a complex frequency domain representation.

Linearity

• Combines two signals with constants yield a combined Z-transform: [ a \cdot x[n] + b \cdot y[n] \rightarrow a \cdot X(z) + b \cdot Y(z) ]

Time Shifting

• Right shift of a signal: [ x[n-k] \rightarrow z^{-k} X(z) \quad (k > 0) ]
• Left shift of a signal: [ x[n+k] \rightarrow z^{k} X(z) \quad (k > 0) ]

Time Scaling

• Scaling ( x[n] = x[an] ) leads to a non-standard Z-transform and is generally inapplicable.

Conjugate Symmetry

• If the signal ( x[n] ) is real, then it holds: [ X(z) = X^\left(\frac{1}{z^}\right) ]

Initial and Final Value Theorems

• Initial Value Theorem: [ x = \lim_{z \to \infty} X(z) ]
• Final Value Theorem: [ x[\infty] = \lim_{z \to 1} (z-1)X(z) ]
  • Valid only if poles of ( (z-1)X(z) ) lie within the unit circle.

Differentiation in Z-domain

• The operation relates: [ \frac{dX(z)}{dz} \leftrightarrow n x[n] ]

Multiplication by ( z^{-k} )

• Expresses time-shift: [ x[n-k] \leftrightarrow z^{-k} X(z) ]

Convolution Property

• Convolution in time domain maps to multiplication in Z-domain: [ x[n] * h[n] \leftrightarrow X(z) H(z) ]

Parseval's Theorem

• Relates energy in time and Z-domain: [ \sum |x[n]|^2 = \frac{1}{2\pi j} \oint |X(z)|^2 \frac{dz}{z} ]

Region of Convergence (ROC)

• Identifies where the Z-transform converges, impacting properties and system stability.