Signals and Systems: Multiplication of Signals

Multiplication of Continuous-Time Signals

The product of two continuous-time signals can be obtained by multiplying their values at every instant of time. Consider two continuous time signals x₁(t) and x₂(t) as shown in the figure.

Explanation

The multiplication of the two signals can be performed by considering different time intervals as follows −

For 0 ≤ t ≤ 1

$$\mathrm{x_1(t) \:=\: 3 \:\:and\:\: x_2(t) \:=\: 2,\:\: thus}$$

$$\mathrm{x_1(t)x_2(t)\:=\: 3 \:\times\: 2 \:=\: 6}$$

For 1 ≤ t ≤ 2

$$\mathrm{x_1(t) \:=\: 2 \:\:and\:\: x_2(t) \:=\: 2 \:+\: (t \:-\: 1),\:\: hence,}$$

$$\mathrm{x_1(t)x_2(t) \:=\: 2[2 \:+\: (t \:-\: 1)] \:=\: 4 \:+\: 4(t \:-\: 1)}$$

For 2 ≤ t ≤ 3

$$\mathrm{x_1(t) \:=\: 2 \:-\: (t \:-\: 2)\:\: and\:\: x_2(t) \:= \:3,\:\: hence,}$$

$$\mathrm{x_1(t)x_2(t) \:=\: [2 \:-\: (t \:-\: 2)]3 \:=\: 6 \:-\: 3(t \:-\: 2)}$$

The multiplication of the signals (i.e., x₁(t)x₂(t)) is shown in the figure.

Multiplication of Discrete-Time Signals

The multiplication of two discrete-time signals x₁(n) and x₂(n) can be performed by multiplying the corresponding sample values.

Consider two discrete time sequences x₁(n) and x₂(n) as follows −

$$\mathrm{x_1(n) \:=\: \{-3,\: 1,\: 5,\: 1,\: 2\}}$$

$$\mathrm{x_2(n) \:=\: \{2,\: -1,\: 1,\: 3,\: -3\}}$$

Then, the product of these two signal is given by,

$$\mathrm{x_1(t)x_2(t) \:=\: \{-3 \:\times\: 2,\: 1 \:\times\: (-1),\: 5 \:\times\: 1,\: 1 \:\times\: 3,\: 2 \:\times \:(-3)\}}$$

$$\mathrm{\Rightarrow\: x_1(t)x_2(t) \:=\: \{-6,\: -1,\: 5,\: 3,\: -6\}}$$