Z-Transform Properties Quiz

Questions and Answers

What is the primary purpose of the Z-transform?

To convert a discrete-time signal into a time-frequency representation

To analyze continuous-time signals

To perform arithmetic operations on signals

To convert a discrete-time signal into a complex frequency domain representation (correct)

The Final Value Theorem can be applied regardless of the pole placement of the function in the Z-domain.

False

What is the result of right shifting a signal $x[n]$ by $k$?

z^{-k} X(z)

The convolution in the Z-domain is represented as $x[n] * h[n] \leftrightarrow _____$

X(z) H(z) Signup and view all the answers

Which of the following represents the Initial Value Theorem?

$x = ext{lim}_{z o ext{infinity}} X(z)$ Signup and view all the answers

Match the properties of the Z-transform with their descriptions:

Linearity = Combining Z-transforms of individual signals Time Shifting = Multiplying by $z^{-k}$ for right shifts Conjugate Symmetry = Relationship between real signals in time and Z-domains Parseval's Theorem = Relating energy of signal in time and Z-domain Signup and view all the answers

The Z-transform of a real signal exhibits complex conjugate symmetry.

True Signup and view all the answers

What must be identified to ensure the Z-transform converges?

Region of Convergence (ROC) Signup and view all the answers

Study Notes

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.

Description

Test your understanding of the properties of the Z-transform, which is essential for analyzing discrete-time signals. This quiz covers concepts such as linearity, time shifting, time scaling, and important theorems associated with Z-transforms. Perfect for students in digital signal processing or control systems.