Time Expansion Property of Z-Transform

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Z-Transform

The Z-transform is a mathematical tool used to convert the difference equations in the discrete time domain into algebraic equations in the z-domain. Mathematically, if x(n) is a discrete-time function, then its Z-transform is defined as:

$$\mathrm{Z[x(n)] \:=\: X(z) \:=\: \sum_{n=-\infty}^{\infty} \:x(n) z^{-n}}$$

Time Expansion Property of Z-Transform

Statement

The Time Expansion Property of the Z-transform states that if:

$$\mathrm{x(n) \:\overset{ZT}\longleftrightarrow\: X(z); \quad \text{ROC} \:\rightarrow\: R}$$

Then:

$$\mathrm{x_m(n) \:\overset{ZT}\longleftrightarrow\: X(z^m); \quad \text{ROC} \:\rightarrow\: R^{1/m}}$$

Where:

$$\mathrm{x_m(n) \:=\: \begin{cases} x\left(\frac{n}{m}\right),\: & \text{ when } \: n \: \text{ is an integral multiple of }\: m\\\\ \:0; & \text{ otherwise} \end{cases}}$$

Also, $\mathrm{x_m(n)}$ has $\mathrm{(m \:-\: 1)}$ zeros inserted between successive values of the original signal.

Proof

From the definition of the Z-transform, we have:

$$\mathrm{Z[x(n)] \:=\: X(z) \:=\: \sum_{n=-\infty}^{\infty}\: x(n)\: z^{-n}}$$

$$\mathrm{\therefore\:Z[x_m(n)] \:=\: \sum_{n=-\infty}^{\infty}\: x_m(n) z^{-n} \:=\: \sum_{n=-\infty}^{\infty}\: x\left(\frac{n}{m}\right) z^{-n}}$$

Substitute (n/m) = k in the summation, we get:

$$\mathrm{Z[x_m(n)] \:=\: \sum_{k=-\infty}^{\infty}\: x(k) z^{-mk}}$$

$$\mathrm{\Rightarrow\:Z[x_m(n)] \:=\: \sum_{k=-\infty}^{\infty}\: x(k) (z^m)^{-k} \:=\: X(z^m)}$$

$$\mathrm{\therefore\:x_m(n) \:\overset{ZT}\longleftrightarrow\: X(z^m); \quad \text{ROC } \:\rightarrow\: R^{1/m}}$$

Numerical Example

Using the time expansion property of the Z-transform, find the Z-transform of the signal $\mathrm{x(n) \:=\: u\left(\frac{n}{2}\right)}$.

Solution

The given signal is:

$$\mathrm{x(n) \:=\: u\left(\frac{n}{2}\right)}$$

Since, the Z-transform of the unit step sequence is given by,

$$\mathrm{Z[u(n)] \:=\: \frac{z}{z \:-\: 1}; \quad \text{ROC } \:\rightarrow\: |z| \:\gt\: 1}$$

Therefore, Using the time expansion property of the Z-transform $\mathrm{\left[x_m(n) \:\overset{ZT}\longleftrightarrow\: X(z^m)\right]}$, we get:

$$\mathrm{Z\left[u\left(\frac{n}{2}\right)\right] \:=\: \left[\frac{z}{z\:-\:1} \right]_{z=z^2} \:=\: \frac{z^2}{z^2 \:-\: 1}}$$

$$\mathrm{u\left(\frac{n}{2}\right) \:\overset{ZT}\longleftrightarrow\: \frac{z^2}{(z^2 \:-\: 1)}, \quad \text{ROC } \:\rightarrow\: |z| \:\gt\: (1)^{1/2}}$$