Lecture 20 - Notes

March 1, 2016

The Inverse Z-Transform

If we express the Z-transform as a Fourier Transform of an equally weighted sequence,

$$\begin{aligned} x[n] = \frac{1}{2 \pi j} \oint X(z) z^{n-1} ~dz \end{aligned}$$

For rational Z-transforms we can compute the inverse Z-transforms alternate ways,

• Inspection
• Partial Fraction Expansion
• Power Series Expansion

Inspection

Use a table of Z-transform properties and pairs to convert back to the time domain.

e.g.

Determine $h[n]$ given,

$$\begin{aligned} H(z) = \frac{1 - z^{-1}}{1 + \frac{3}{4} z^{-1}} \end{aligned}$$

Consider,

$$\begin{aligned} H(z) &= \frac{1 - z^{-1}}{1 + \frac{3}{4} z^{-1}} \\\\ &= \underbrace{\frac{1}{1 + \frac{3}{4} z^{-1}}}_\text{exponential} - \underbrace{\frac{z^{-1}}{1 + \frac{3}{4} z^{-1}}}_\text{shifted exponential} \\\\ &\downarrow\\\\ h[n] &= \left( \frac{3}{4} \right)^n u[n] - \left( \frac{3}{4} \right)^{n-1} u[n-1] \end{aligned}$$

Partial Fraction Expansion

Expand two sums which are partial fractions,

$$\begin{aligned} X(z) &= \frac{\sum_{k=0}^M b_k z^{-k}}{\sum_{k=0}^N a_k z^{-k}} \\\\ &\downarrow \\\\ X(z) &= \sum_{k=1}^N \frac{A_k}{1-d_k z^-1} \end{aligned}$$

e.g.

Determine the inverse Z-transform of,

$$\begin{aligned} X(z) &= \frac{3z^2 - \frac{5}{6}z}{\left(z - \frac{1}{4}\right)\left(z - \frac{1}{3}\right)} \end{aligned}$$

where $\left| z \right| \lt \frac{1}{3}$.

We identify $M = 1$ and $N = 2$ and perform the partial fraction expansion,

$$\begin{aligned} \frac{3z - \frac{5}{6}z}{\left(z - \frac{1}{4}\right)\left(z - \frac{1}{3}\right)} &= \frac{A}{\left(z - \frac{1}{4}\right)} + \frac{B}{\left(z - \frac{1}{3}\right)} \\\\ 3z^2 - \frac{5}{6}z &= A \left(z - \frac{1}{3}\right) + B \left(z - \frac{1}{4}\right) \\\\ \end{aligned}$$

TODO

Power Series Expansion

$$\begin{aligned} X(z) &= \frac{\sum_{k=0}^M b_k z^{-k}}{\sum_{k=0}^N a_k z^{-k}} \\\\ &\downarrow \\\\ X(z) &= \sum_{n=-\infty}^{+\infty} x[n]z^{-1} \end{aligned}$$

e.g.

Determine the inverse Z-transform of,

$$\begin{aligned} X(z) &= (1 + 2z)(1 + 3z^{-1}) \end{aligned}$$

Start by expanding,

$$\begin{aligned} X(z) &= (1 + 2z)(1 + 3z^{-1}) \\\\ &= 1 - 2z + 3z^{-1} + 6 \\\\ &= 3z^{-1} + 7 - 2z \\\ &\downarrow \\\\ x[n] &= \begin{cases} 3 & n = 1 \\\\ 7 & n = 0 \\\\ 2 & n = -1 \\\\ \end{cases} \end{aligned}$$

e.g.

Determine the inverse Z-transform of,

$$\begin{aligned} X(z) &= \frac{1}{1 + \frac{1}{2}z^{-1}} \end{aligned}$$

TODO