Time Shifting Property of Z-Transform

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Z-Transform

The Z-transform is a mathematical tool which is used to convert the difference equations in the discrete time domain into the 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 Shifting Property of Z-Transform

Statement

The time-shifting property of Z-transform states that if the sequence x(n)is shifted by $\mathrm{n_0}$ in the time domain, then it results in the multiplication by $\mathrm{z^{-n_0}}$ in the z-domain. Therefore, if

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

With zero initial conditions.

Then, according to the time shifting property,

$$\mathrm{x(n \:-\: n_0) \:\overset{ZT}\longleftrightarrow\: z^{-n_0} X(z)}$$

With ROC = R, except for the possible addition and deletion of z = 0 or $\mathrm{z \:=\: \infty}$.

Proof

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

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

$$\mathrm{\therefore\:Z[x(n \:-\: n_0)] \:=\: \sum_{n=-\infty}^{\infty}\: x(n \:-\: n_0) \:z^{-n}}$$

Substituting $\mathrm{(n \:-\: n_0) \:=\: m}$ in the above summation, then we have,

$$\mathrm{Z[x(n \:-\: n_0)] \:=\: \sum_{m=-\infty}^{\infty}\: x(m)\: z^{-(m \:+\: n_0)}}$$

$$\mathrm{\Rightarrow\:Z[x(n \:-\: n_0)] \:=\: z^{-n_0}\: \sum_{m=-\infty}^{\infty}\: x(m) z^{-m} \:=\: z^{-n_0}\: X(z)}$$

$$\mathrm{\therefore\:Z[x(n \:-\: n_0)] \:=\: z^{-n_0} X(z)}$$

Also, it can be represented as:

$$\mathrm{x(n \:-\: n_0)\:\overset{ZT}\longleftrightarrow\: z^{-n_0}\: X(z)}$$

Similarly, if the signal is advanced in time, then according to the time shifting property, we get −

$$\mathrm{x(n \:+\: n_0) \:\overset{ZT}\longleftrightarrow\: z^{n_0}\: X(z)}$$

Also, if the initial conditions are not neglected, then

1. The time shift property for time delay is,

$$\mathrm{Z[x(n\:-\:n_0)] \:=\: z^{-n_0}\: X(z) \:+\: z^{-n_0}\: \sum_{p=1}^{n_0}\: x(-p)\: z^{p}}$$

2. The time shifting property for time advance is,

$$\mathrm{Z[x(n \:+\: n_0)] \:=\: z^{n_0} X(z) \:-\: z^{n_0}\: \sum_{p=0}^{n_0 \:-\: 1}\: x(p)\: z^{-p}}$$

Numerical Examples

Example 1

Using the time shifting property of Z-transform, find the Z-transform of the sequence

$$\mathrm{x(n) \:=\: u(n \:-\: 3)}$$

Solution

The given sequence is

$$\mathrm{x(n) \:=\: u(n \:-\: 3)}$$

Since the Z-transform of a 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 shifting property of Z-transform $\mathrm{\left[\text{i.e., }x(n\:-\:n_0)\:\overset{ZT}\longleftrightarrow\:z^{-n_0}X(z)\right]}$, we get,

$$\mathrm{Z[u(n \:-\: 3)] \:=\: z^{-3} Z[u(n)] \:=\: z^{-3} \left( \frac{z}{z \:-\: 1} \right)}$$

$$\mathrm{\therefore\:Z[u(n \:-\: 3)] \:=\: \frac{1}{z^2(z\:-\:1)}, \quad \text{ROC } \:\rightarrow\: |z| \:\gt\: 1}$$

Example 2

Using the time shifting property of Z-transform, find the Z-transform of the sequence

$$\mathrm{x(n)\:=\:\delta(n\:+\:5)}$$

Solution

The given sequence is

$$\mathrm{x(n) \:=\: \delta(n \:+\: 5)}$$

Since the Z-transform of the impulse sequence is given by:

$$\mathrm{Z[\delta(n)] \:=\: 1}$$

Now, using the time shifting property of Z-transform $\mathrm{\left[\text{i.e., }\:x(n\:+\:n_0)\:\overset{ZT}\longleftrightarrow\:z^{n_0}\:X(z)\right]}$, we get,

$$\mathrm{Z[\delta(n \:+\: 5)] \:=\: z^5(1) \:=\: z^5}$$