The is a powerful tool for analyzing and systems. It converts time-domain signals into complex frequency-domain representations, making it easier to study system behavior and solve difference equations.
Z-transforms have several key properties, including , , and convolution. These properties simplify the analysis of complex systems by allowing engineers to break down problems into smaller, more manageable parts and manipulate signals in the frequency domain.
Z-transform and ROC
Definition and Calculation
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Z-transform converts a discrete-time signal into a complex frequency-domain representation
Defined as X(z)=∑n=−∞∞x[n]z−n, where x[n] is the discrete-time signal and z is a complex variable
Allows for the analysis of discrete-time systems in the frequency domain, similar to the Laplace transform for continuous-time systems
ROC is the set of complex numbers z for which the Z-transform converges
Determines the stability and causality of the system
For a causal system, the ROC includes the exterior of a circle centered at the origin, while for an anti-causal system, the ROC includes the interior of a circle
A system is stable if the ROC includes the unit circle ∣z∣=1
Inverse Z-transform
recovers the original discrete-time signal from its Z-transform representation
Can be calculated using partial fraction expansion or contour integration
Partial fraction expansion decomposes X(z) into a sum of simpler terms, each corresponding to a pole in the ROC
Contour integration involves evaluating a complex integral along a closed contour within the ROC
The inverse Z-transform is unique only when the ROC is specified along with X(z)
Different ROCs can lead to different time-domain signals with the same Z-transform
Properties of Z-transform
Linearity and Time-Shifting
Linearity property states that the Z-transform of a linear combination of signals is equal to the linear combination of their individual Z-transforms
If x1[n]↔X1(z) and x2[n]↔X2(z), then ax1[n]+bx2[n]↔aX1(z)+bX2(z), where a and b are constants
Time-shifting property relates the Z-transform of a shifted signal to the original Z-transform
If x[n]↔X(z), then x[n−k]↔z−kX(z), where k is an integer shift
Shifting a signal to the right by k samples corresponds to multiplying its Z-transform by z−k
Scaling and Convolution
Scaling property relates the Z-transform of a scaled signal to the original Z-transform
If x[n]↔X(z), then anx[n]↔X(az), where a is a non-zero constant
Scaling a signal by an corresponds to replacing z with az in its Z-transform
Convolution property states that the convolution of two discrete-time signals in the time domain corresponds to the multiplication of their Z-transforms in the frequency domain
If x1[n]↔X1(z) and x2[n]↔X2(z), then x1[n]∗x2[n]↔X1(z)X2(z), where ∗ denotes convolution
The ROC of the convolution is the intersection of the ROCs of the individual signals
Z-transform Theorems
Initial Value Theorem
allows for the calculation of the initial value of a discrete-time signal from its Z-transform
If x[n]↔X(z), then x[0]=limz→∞X(z), provided the limit exists
Useful for determining the initial conditions of a system or the response to an impulse input
The theorem requires that the ROC of X(z) includes infinity, which is typically the case for causal systems
Final Value Theorem
allows for the calculation of the steady-state value of a discrete-time signal from its Z-transform
If x[n]↔X(z), then limn→∞x[n]=limz→1(z−1)X(z), provided the limits exist and the poles of (z−1)X(z) are inside the unit circle
Useful for determining the steady-state response of a system to a step input or the convergence of an iterative process
The theorem requires that the ROC of X(z) includes the unit circle, which is typically the case for stable systems