Properties of Convolution (Part 1)

Main ideas / lessons

• The video explains key properties of convolution as a way to compute results faster—especially useful for exam-style multiple choice questions.
• It states there are 10 properties total, and this lecture (Part 1) covers 4 properties.
• Each property is explained:
  • mathematically (how signals behave under convolution),
  • and conceptually in terms of LTI systems (input/impulse response roles, and how systems combine in series (cascade) or parallel).

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Properties covered (4)

1) Commutative property (convolution order doesn’t matter)

For two signals (X_1(t)) and (X_2(t)): [ X_1(t) * X_2(t) = X_2(t) * X_1(t) ]

Interpretation (process-wise): During convolution, the “movement/overlap” can be described starting with either signal first, and the final result is the same.

System interpretation: For an LTI system with input (x(t)) and impulse response (h(t)), [ y(t) = x(t) * h(t) ] By commutativity, [ y(t) = h(t) * x(t) ] So, the input and impulse response can be swapped while producing the same output (with the appropriate interpretation).

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2) Associative property (grouping doesn’t matter)

For three signals (X_1(t)), (X_2(t)), (X_3(t)): [ (X_1 * X_2) * X_3 = X_1 * (X_2 * X_3) ]

System interpretation (series/cascade): If two systems are connected in cascade (series), with impulse responses (h_1(t)) and (h_2(t)), then the equivalent single system has: [ h_{\text{eq}}(t) = h_1(t) * h_2(t) ] With input (x(t)), the output matches applying both systems in cascade.

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3) Distributive property (convolution distributes over addition)

Convolving (X_1(t)) with ([X_2(t) + X_3(t)]): [ X_1 * (X_2 + X_3) = (X_1 * X_2) + (X_1 * X_3) ]

System interpretation (parallel): Two LTI systems in parallel share the same input (x(t)):

• System 1: impulse response (h_1(t)), output (y_1(t) = x(t) * h_1(t))
• System 2: impulse response (h_2(t)), output (y_2(t) = x(t) * h_2(t))

Combined output: [ y(t) = y_1(t) + y_2(t) = x(t) * h_1(t) + x(t) * h_2(t) ] Equivalent single system impulse response: [ h_{\text{eq}}(t) = h_1(t) + h_2(t) ]

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4) Delta function property (convolution with an impulse “shifts”)

Core rule: convolving with a shifted delta function performs a time shift.

If: [ x(t) * \delta(t - t_1) = x(t - t_1) ]

Special cases:

• If (t_1 = 0): [ x(t) * \delta(t) = x(t) ]

• If the impulse has weight (a): [ x(t) * \big(a\,\delta(t - t_1)\big) = a\,x(t - t_1) ] [ x(t) * (a\,\delta(t)) = a\,x(t) ]

Example applications shown

• Example 1

  • Given a ramp signal (x(t)=r(t)), compute: [ r(t) * \delta(t - 2) ]

  • Result: [ r(t - 2) ]

• Example 2

  • Compute: [ [u(t+3)] * \delta(t - 1) ]

  • Using the shift rule (replace (t) with (t-1) inside the signal): [ u(t - 1 + 3) = u(t + 2) ]

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Speakers / sources featured

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