8.3 Inverse Z-transform

The inverse Z-transform is a crucial tool in digital signal processing. It allows us to convert Z-domain representations back into discrete-time signals, enabling analysis and manipulation of digital systems.

This topic covers methods for finding the inverse Z-transform, including partial fraction expansion, power series, and residue methods. These techniques help break down complex Z-domain expressions into simpler terms for easier conversion to time-domain signals.

Inverse Z-Transform

Concept of inverse Z-transform

• Converts Z-domain representation $X(z)$ back to discrete-time signal $x[n]$
  • Reverse process of Z-transform which converts discrete-time signal to Z-domain representation
• Essential for analyzing and manipulating signals in Z-domain and converting results back to discrete-time domain
  • Allows design and analysis of discrete-time systems (digital filters, control systems)

Partial fraction expansion for Z-transforms

• Technique used to decompose rational Z-transform expressions into sum of simpler terms
  • Rational Z-transform expression is ratio of two polynomials in $z$: $X(z) = \frac{P(z)}{Q(z)}$
• Process involves following steps:
  1. Factorize denominator polynomial $Q(z)$ into product of linear and irreducible quadratic factors
  2. Determine form of partial fraction expansion based on factors of $Q(z)$
     • Distinct linear factors: $\frac{A_1}{z-p_1} + \frac{A_2}{z-p_2} + \cdots + \frac{A_n}{z-p_n}$
     • Repeated linear factors: $\frac{A_1}{(z-p_1)^1} + \frac{A_2}{(z-p_1)^2} + \cdots + \frac{A_m}{(z-p_1)^m}$
     • Irreducible quadratic factors: $\frac{B_1z+C_1}{(z-\alpha_1)(z-\beta_1)} + \cdots + \frac{B_kz+C_k}{(z-\alpha_k)(z-\beta_k)}$
  3. Solve for unknown coefficients ($A_i$, $B_i$, $C_i$) by equating original expression with partial fraction expansion and comparing coefficients or evaluating at specific points
• Simplifies process of finding inverse Z-transform by breaking down expression into easier to handle terms

Methods for Finding the Inverse Z-Transform

Power series method for inverse Z-transform

• Used to find inverse Z-transform of simple Z-domain expressions (single term, geometric series)
• For Z-domain expression of form $X(z) = az^{-k}$, inverse Z-transform is:
  • $x[n] = a\delta[n-k]$, where $\delta[n]$ is unit impulse function
• For Z-domain expression representing geometric series, such as $X(z) = \frac{a}{1-bz^{-1}}$, inverse Z-transform is:
  • $x[n] = ab^nu[n]$, where $u[n]$ is unit step function
• Involves expanding Z-domain expression into infinite series and identifying corresponding discrete-time signal based on series coefficients

Residue method for inverse Z-transform

• Powerful technique for finding inverse Z-transform of rational Z-domain expressions
• After applying partial fraction expansion to decompose rational Z-transform expression into simpler terms, residue method can be used to find inverse Z-transform of each term
• For simple pole at $z = p_i$, residue is:
  • $\text{Res}[X(z)]_{z=p_i} = \lim_{z \to p_i} (z-p_i)X(z)$
• Inverse Z-transform of simple pole term $\frac{A_i}{z-p_i}$ is:
  • $x_i[n] = A_ip_i^nu[n]$
• For pole of order $m$ at $z = p_i$, residue is:
  • $\text{Res}[X(z)]_{z=p_i} = \frac{1}{(m-1)!} \lim_{z \to p_i} \frac{d^{m-1}}{dz^{m-1}} [(z-p_i)^mX(z)]$
• Inverse Z-transform of repeated pole term $\frac{A_j}{(z-p_i)^j}$ is:
  • $x_{ij}[n] = A_j\frac{n^{j-1}}{(j-1)!}p_i^nu[n]$
• Total inverse Z-transform is sum of inverse Z-transforms of all individual terms obtained from partial fraction expansion