Signals and Systems: Multiplication of Signals

Electronics & ElectricalElectronDigital Electronics

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: 𝑥1(𝑡) = 3 and 𝑥2(𝑡) = 2, thus

    𝑥1(𝑡)𝑥2(𝑡) = 3 × 2 = 6

  • For 1≤ 𝒕 ≤ 𝟐: 𝑥1(𝑡) = 2 and 𝑥2(𝑡) = 2 + (𝑡 − 1), hence,

    𝑥1(𝑡)𝑥2(𝑡) = 2[2 + (𝑡 − 1)] = 4 + 4(𝑡 − 1)

  • For 2≤ 𝒕 ≤ 𝟑: 𝑥1(𝑡) = 2 − (𝑡 − 2) and 𝑥2(𝑡) = 3, hence,

    𝑥1(𝑡)𝑥2(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 −

𝑥1(𝑛) = {−3, 1, 5, 1, 2}

𝑥2(𝑛) = {2, −1, 1, 3, −3}

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

𝑥1(𝑡)𝑥2(𝑡) = {−3 × 2, 1 × (−1), 5 × 1, 1 × 3, 2 × (−3)}

⟹ 𝑥1(𝑡)𝑥2(𝑡) = {−6, −1, 5, 3, −6}

raja
Updated on 12-Nov-2021 11:55:04

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