Multiplication Property of Z-Transform

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:

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

Multiplication Property of Z-Transform

Statement

The multiplication property of the Z-transform states that the multiplication of two signals in the time domain corresponds to the complex convolution in the z-domain. For this reason, the multiplication property is also called the complex convolution property of the Z-transform. Therefore, if:

$$\mathrm{x_1(n) \:\overset{ZT}\longleftrightarrow\: X_1(z) \quad \text{and} \quad x_2(n) \:\overset{ZT}\longleftrightarrow \: X_2(z)}$$

Then, according to the multiplication property:

$$\mathrm{x_1(n) \:\cdot\: x_2(n) \:\overset{ZT}\longleftrightarrow\: \frac{1}{2 \pi j}\: \oint\: X_1(p) X_2\left( \frac{z}{p} \right) p^{-1}\: dp}$$

Proof

From the definition of Z-transform, we have:

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

$$\mathrm{\therefore\:Z[x_1(n) \:\cdot\: x_2(n)] \:=\: \sum_{n=-\infty}^{\infty}\: \left[ x_1(n) \:\cdot\: x_2(n) \right] z^{-n} \quad \dotso\:(1)}$$

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

$$\mathrm{x(n) \:=\: \frac{1}{2 \pi j} \:\oint \:X(z) z^{n-1}\: dz \quad \dotso\:(2)}$$

Replacing the complex variable z by p in equation (2), we get:

$$\mathrm{x(n) \:=\: \frac{1}{2 \pi j}\: \oint\: X(p) p^{n-1}\: dp \quad \dotso\:(3)}$$

Substituting the value of $\mathrm{x_1(n)}$ using equation (3) into equation (1), we get:

$$\mathrm{Z[x_1(n) \:\cdot\: x_2(n)] \:=\: \sum_{n=-\infty}^{\infty}\: \left[ \frac{1}{2 \pi j}\: \oint \:X_1(p) p^{n-1} dp \right] x_2(n) z^{-n}}$$

$$\mathrm{\Rightarrow\:Z[x_1(n) \:\cdot\: x_2(n)] \:=\: \frac{1}{2 \pi j} \:\oint\: X_1(p) \left[ \sum_{n=-\infty}^{\infty}\: x_2(n) p^n p^{-1}z^{-n} \right] \: dp}$$

$$\mathrm{\Rightarrow\:Z[x_1(n) x_2(n)] \:=\: \frac{1}{2 \pi j} \:\oint\: X_1(p) \left[ \sum_{n=-\infty}^{\infty}\: x_2(n) (p^{-1}z)^{-n} \right] p^{-1}\: dp}$$

$$\mathrm{\therefore\:Z[x_1(n) \:\cdot\: x_2(n)] \:=\: \frac{1}{2 \pi j} \:\oint\: X_1(p) X_2\left[ \frac{z}{p}\right]p^{-1}\:dp}$$

Also, it can be written as,

$$\mathrm{x_1(n) \:\cdot\: x_2(n) \:\overset{ZT}\longleftrightarrow\: \frac{1}{2 \pi j}\: \oint\: X_1(p) X_2\left( \frac{z}{p} \right) p^{-1}\: dp}$$