Z-Transform
A power-series [see page 6, representation] of a discrete-time sequence.
It decomposes a time-domain signal into a series of inputs each multiplied be decreasing increments of powers of \(z\) (begins with \(z^0\), \(z^{-1}\), etc.). We write the Z transform as \[ X(z) = \sum_{n = - \infty}^{\infty}{x[n] \times {z}^{-k}} \]
For example for the sequence:
\begin{align*}
x &=& x[0], x[1], x[2], x[3] \
X(z) &=& x[0]\times {z}^{0} + x[1]\times {z}^{-1} + x[2]\times {z}^{-2} + x[3]\times {z}^{-3}
\end{align*}
We can be write \(Z\\) to indicate we transform the contents of \(\{\}\).
Note: By convention negative powers of \( z \) are used for positive time indices, that is to \( {z}^{-1} \) is the previous sample, \( {z}^{-2} \) is the sample before that etc.
We can express a convolution of a linear function in time domain as a product of the Z-transform. Which can be re-arranged to the z-plane transfer function.
Properties
The z-transform is:
| Property | Demonstration |
|---|---|
| Linear | \(Z\{ax_1[n] + bx_1[n]\} = aZ\{x_1[n]\} + bZ\{x_1[n]\}\) |
| Time-shifting | \(Z\{x[n-n_0]\} = z^{-n_0} . Z\{x[n]\}\) |
| Convolution | \(Z\{x[n] * y[n]\} = Z\{x[n]\} . Z{y[n]}\}\) |
Relation to Fourier Transform
Observe that when:
| Z Relation | Equivalent to multiplying input sequence by |
|---|---|
| \( z > 1 \) | a rising exponential |
| \( 0 < z < 1 \) | a falling exponential |
| complex number lying on unit circle | a real cosine and imaginary sinusoid |
Observe that that final case is directly equivalent to the fourier-transform.