The Z-transform is a powerful tool for analyzing discrete-time signals 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 linearity, time-shifting, 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

• Z-transform converts a discrete-time signal into a complex frequency-domain representation
  • Defined as $X(z) = \sum_{n=-\infty}^{\infty} x[n]z^{-n}$, 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
• Region of convergence $ROC$ 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

• 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 $ROC$s 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 $x_1[n] \leftrightarrow X_1(z)$ and $x_2[n] \leftrightarrow X_2(z)$, then $ax_1[n] + bx_2[n] \leftrightarrow aX_1(z) + bX_2(z)$, where $a$ and $b$ are constants
• Time-shifting property relates the Z-transform of a shifted signal to the original Z-transform
  • If $x[n] \leftrightarrow X(z)$, then $x[n-k] \leftrightarrow z^{-k}X(z)$, 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 $x[n] \leftrightarrow X(z)$, then $a^nx[n] \leftrightarrow X(\frac{z}{a})$, where $a$ is a non-zero constant
  • Scaling a signal by $a^n$ corresponds to replacing $z$ with $\frac{z}{a}$ 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 $x_1[n] \leftrightarrow X_1(z)$ and $x_2[n] \leftrightarrow X_2(z)$, then $x_1[n] * x_2[n] \leftrightarrow X_1(z)X_2(z)$, where $*$ denotes convolution
  • The $ROC$ of the convolution is the intersection of the $ROC$s of the individual signals

Z-transform Theorems

Initial Value Theorem

• Initial value theorem allows for the calculation of the initial value of a discrete-time signal from its Z-transform
  • If $x[n] \leftrightarrow X(z)$, then $x[0] = \lim_{z \to \infty} X(z)$, 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

• Final value theorem allows for the calculation of the steady-state value of a discrete-time signal from its Z-transform
  • If $x[n] \leftrightarrow X(z)$, then $\lim_{n \to \infty} x[n] = \lim_{z \to 1} (z-1)X(z)$, 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