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Questions and Answers
What is the primary purpose of the Z-transform?
What is the primary purpose of the Z-transform?
The Final Value Theorem can be applied regardless of the pole placement of the function in the Z-domain.
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 $k$?
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 $x[n] * h[n] \leftrightarrow _____$
The convolution in the Z-domain is represented as $x[n] * h[n] \leftrightarrow _____$
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Which of the following represents the Initial Value Theorem?
Which of the following represents the Initial Value Theorem?
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Match the properties of the Z-transform with their descriptions:
Match the properties of the Z-transform with their descriptions:
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The Z-transform of a real signal exhibits complex conjugate symmetry.
The Z-transform of a real signal exhibits complex conjugate symmetry.
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What must be identified to ensure the Z-transform converges?
What must be identified to ensure the Z-transform converges?
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Study Notes
Z-transform Properties
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Definition: The Z-transform converts a discrete-time signal into a complex frequency domain representation.
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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) ]
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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 ))
- If ( x[n] ) has Z-transform ( X(z) ):
-
Time Scaling:
- If ( x[n] ) has Z-transform ( X(z) ):
- ( x[n] = x[an] ) results in a non-standard Z-transform, generally not applicable.
- If ( x[n] ) has Z-transform ( X(z) ):
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Conjugate Symmetry:
- If ( x[n] ) is real, then ( X(z) = X^(\frac{1}{z^}) ).
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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.
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Differentiation in Z-domain:
- If ( x[n] ) has Z-transform ( X(z) ): [ \frac{dX(z)}{dz} \leftrightarrow n x[n] ]
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Multiplication by ( z^{-k} ):
- If ( x[n] ) has Z-transform ( X(z) ): [ x[n-k] \leftrightarrow z^{-k} X(z) ]
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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) ]
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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} ]
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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.
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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.
