The Z-transform is a powerful tool for analyzing discrete-time signals and systems. It converts time-domain signals into complex frequency-domain representations, simplifying the analysis of linear time-invariant systems. Understanding its properties and techniques is crucial for working with sampled-data systems.

This section covers the Z-transform definition, common signal transforms, and key properties like linearity and time-shifting. It also explores inverse Z-transform techniques, including partial fraction expansion and power series methods, essential for converting frequency-domain solutions back to the time domain.

Z-Transform Definition

Definition and Region of Convergence

• The z-transform converts a discrete-time signal into a complex frequency-domain representation
• Defined as X(z) = Σ x[n] z^(-n), where the sum is taken from n = -∞ to +∞
• The region of convergence (ROC) is the set of complex numbers (z) for which the z-transform summation converges
  • Determines the uniqueness and stability of the z-transform
  • Depends on the location of the poles of the z-transform and the boundedness of the signal

ROC for Different Signal Types

• For a right-sided signal (x[n] = 0 for n < 0), the ROC is the region outside the outermost pole
• For a left-sided signal (x[n] = 0 for n > 0), the ROC is the region inside the innermost pole
• For a two-sided signal, the ROC is the region between the outermost poles on both sides
• The ROC must include the unit circle (|z| = 1) for a stable and causal system

Z-Transform of Signals

Common Discrete-Time Signals

• Unit impulse signal δ[n]: X(z) = 1 for all z
• Unit step signal u[n]: X(z) = z / (z - 1) for |z| > 1
• Exponential signal a^n u[n]: X(z) = z / (z - a) for |z| > |a|
  • Example: 2^n u[n] has a z-transform X(z) = z / (z - 2) for |z| > 2
• Sinusoidal signal cos(ω0 * n) * u[n]: X(z) = (z * cos(ω0)) / (z^2 - 2z * cos(ω0) + 1) for |z| > 1
• Sinusoidal signal sin(ω0 * n) * u[n]: X(z) = (z * sin(ω0)) / (z^2 - 2z * cos(ω0) + 1) for |z| > 1

Calculating Z-Transforms

• Use the definition of the z-transform to calculate the z-transform of a given discrete-time signal
• Identify the ROC based on the signal type and the location of the poles
• Example: For x[n] = (1/2)^n u[n], X(z) = z / (z - 1/2) for |z| > 1/2

Z-Transform Properties

Linearity and Time-Shifting

• Linearity: The z-transform of a linear combination of signals equals the linear combination of their individual z-transforms
  • Example: If X1(z) and X2(z) are the z-transforms of x1[n] and x2[n], then a * x1[n] + b * x2[n] has a z-transform a * X1(z) + b * X2(z)
• Time-shifting: If x[n] has a z-transform X(z), then x[n-k] has a z-transform z^(-k) X(z)
  • Example: If X(z) is the z-transform of x[n], then x[n-3] has a z-transform z^(-3) X(z)

Scaling, Differentiation, and Convolution

• Scaling in the z-domain: If x[n] has a z-transform X(z), then a^n x[n] has a z-transform X(z/a)
• Differentiation in the z-domain: If x[n] has a z-transform X(z), then n * x[n] has a z-transform -z * (dX(z)/dz)
• Convolution in the time-domain: If x[n] and h[n] have z-transforms X(z) and H(z), then their convolution y[n] = x[n] * h[n] has a z-transform Y(z) = X(z) * H(z)
  • Example: If X(z) = z / (z - 1) and H(z) = z / (z - 2), then Y(z) = X(z) H(z) = z^2 / ((z - 1)(z - 2))

Initial and Final Value Theorems

• Initial value theorem: The initial value of a signal x[n] can be found using lim(z→∞) X(z), provided that the limit exists
• Final value theorem: The final value of a signal x[n] can be found using lim(z→1) (z-1) * X(z), provided that the limit exists and the poles of (z-1) * X(z) are inside the unit circle
• These theorems are useful for determining the steady-state behavior of a system or signal

Inverse Z-Transform Techniques

Partial Fraction Expansion

• Partial fraction expansion decomposes a rational z-transform into a sum of simpler terms
  • Each term can be individually inverse transformed using z-transform tables or properties
• Factor the denominator of the z-transform and express it as a sum of terms with simple poles or poles of higher multiplicity
• For distinct poles, the partial fraction expansion takes the form X(z) = Σ (Ai / (z - pi)), where Ai is the residue at pole pi
  • The residue at a simple pole pi can be calculated using Ai = lim(z→pi) ((z - pi) X(z))
• For repeated poles, the partial fraction expansion takes the form X(z) = Σ (Aik / (z - pi)^k), where Aik is the coefficient of the kth term in the expansion of the pole pi

Other Inverse Z-Transform Techniques

• Power series expansion: Expand the z-transform into a power series and match the coefficients with the time-domain signal
• Long division: Divide the numerator by the denominator to obtain a power series expansion of the z-transform
• Contour integration: Use complex integration techniques to evaluate the inverse z-transform integral
  • Requires knowledge of complex analysis and residue theory
• These techniques can be used when partial fraction expansion is not feasible or when dealing with non-rational z-transforms