Time Reversal 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 Reversal Property of Z-Transform

Statement

The time reversal property of Z-transform states that the reversal or reflection of the sequence in the time domain corresponds to the inversion in the z-domain. Therefore, if:

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

Then,

$$\mathrm{x(-n) \:\overset{ZT}\longleftrightarrow\: X\left(\frac{1}{z}\right) \:=\: X(z^{-1}); \quad \text{ROC } \:=\: \frac{1}{R}}$$

Proof

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

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

Now, by reversing the sequence in the time domain, we get:

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

Substitute -n = m in the above summation, we get,

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

$$\mathrm{\Rightarrow\:Z[x(-n)] = \sum_{m=-\infty}^{\infty}\: x(m) (z^{-1})^{m} \:=\: X(z^{-1})}$$

$$\mathrm{\therefore\:Z[x(-n)]\:=\:X(Z^{-1})\:=\:Z\left(\frac{1}{z}\right)}$$

Or it can also be represented as,

$$\mathrm{x(-n) \:\overset{ZT}\longleftrightarrow\: X\left(\frac{1}{z}\right) \:=\: X(z^{-1}); \quad \text{ROC} \:=\: \frac{1}{R}}$$

Numerical Example

Using the time reversal property of Z-transform, find the Z-transform of the following sequence:

$$\mathrm{x(n) \:=\: u(-n)}$$

Solution

The given sequence is:

$$\mathrm{x(n) \:=\: u(-n)}$$

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

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

Now, by using the time reversal property of the Z-transform, $\mathrm{\left[\text{i.e., }\: x(-n) \:\overset{ZT}\longleftrightarrow\: X\left(\frac{1}{z}\right) \right]}$, we get:

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

$$\mathrm{\Rightarrow\:Z[u(-n)] \:=\: \frac{1/z}{(1/z)\:-\:1} \:=\: \frac{-1}{z-1}}$$

$$\mathrm{\therefore\:u(-n) \:\overset{ZT}\longleftrightarrow\: \frac{-1}{z\:-\:1}; \quad \text{ROC }\: \rightarrow \:|z| \:\lt\: 1}$$