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 ?₁(?) and ?₂(?) as shown in the figure.

Explanation

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

• For ? ≤ ? ≤ ?: ?₁(?) = 3 and ?₂(?) = 2, thus

  ?₁(?)?₂(?) = 3 × 2 = 6

• For 1≤ ? ≤ ?: ?₁(?) = 2 and ?₂(?) = 2 + (? − 1), hence,

  ?₁(?)?₂(?) = 2[2 + (? − 1)] = 4 + 4(? − 1)

• For 2≤ ? ≤ ?: ?₁(?) = 2 − (? − 2) and ?₂(?) = 3, hence,

  ?₁(?)?₂(?) = [2 − (? − 2)]3 = 6 − 3(? − 2)

The multiplication of the signals (i.e., ?₁(?)?₂(?)) is shown in the figure.

Multiplication of Discrete-Time Signals

The multiplication of two discrete-time signals ?₁(?) and ?₂(?) can be performed by multiplying the corresponding sample values.

Consider two discrete time sequences ?₁(?) and ?₂(?) as follows −

?₁(?) = {−3, 1, 5, 1, 2}

?₂(?) = {2, −1, 1, 3, −3}

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

?₁(?)?₂(?) = {−3 × 2, 1 × (−1), 5 × 1, 1 × 3, 2 × (−3)}

? ?₁(?)?₂(?) = {−6, −1, 5, 3, −6}