Properties of Convolution (Part 4)

Main ideas / lessons

• The video is a continuation of a series on properties of convolution, concluding the remaining two properties:
  • Property #9: Area property
  • Property #10: Duration (Extension) property
• It emphasizes that convolution has predictable effects on:
  1. The total area of the resulting waveform
  2. The time support / extent (where the result is non-zero)

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Property #9 (Area property)

Statement of the property

Given two signals:

• (X_1(t)) with area (a_1)
• (X_2(t)) with area (a_2)

Their convolution produces: [ Y(t) = X_1(t) * X_2(t) ]

Then the area of the resultant signal (Y(t)) is: [ \boxed{a = a_1 \cdot a_2} ]

Example calculation shown in the video

1. Signal (X_1(t)) (rectangular-like in the explanation)

   • Width = 3
   • Height = 2
   • Area: [ a_1 = 3 \times 2 = 6 ]

2. Signal (X_2(t)) (triangular)

   • Base = 4
   • Altitude = 2
   • Area: [ a_2 = \frac{1}{2} \times 4 \times 2 = 4 ]

3. Resultant signal (Y(t)) area: [ a = a_1 \cdot a_2 = 6 \times 4 = 24 ]

Cross-verification using a prior trapezoid example

The speaker references an earlier convolution where:

• An impulse response (rectangular/step-like) is convolved with
• A rectangular input, producing a trapezoid output

They verify that the area of the trapezoid output equals the product of areas of the two original signals.

Trapezoid area verification (as described)

• Impulse response area: [ a_1 = 1 \times 1 = 1 ]

• Input signal area: [ a_2 = 2 \times 1 = 2 ]

• Expected output area (property): [ a = a_1 a_2 = 1 \times 2 = 2 ]

Then they compute the trapezoid area directly by splitting it:

• Area of square part: [ 1 \times 1 = 1 ]

• Area of triangle parts:

  • each triangle: [ \frac{1}{2} \times \text{base} \times \text{altitude} = 0.5 ]

  • two triangles total: [ 0.5 + 0.5 ]

Total area: [ 1 + 0.5 + 0.5 = 2 ]

Conclusion: The computed trapezoid area matches the area-property expectation, suggesting no mistake in the convolution result.

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Property #10 (Duration / Extension property)

Statement of the property

If:

• (X_1(t)) is non-zero only over a time interval ([T_1, T_2])
• (X_2(t)) is non-zero only over a time interval ([T_3, T_4])

Then the convolution result (Y(t) = X_1(t) * X_2(t)) will be non-zero only over: [ [T_1 + T_3,\; T_2 + T_4] ]

Example verification shown

From the example intervals:

• For the first signal (HD in the subtitles):

  • non-zero range: (0 \to 1)
  • so (T_1 = 0,\; T_2 = 1)

• For the second signal (X(t) in the subtitles):

  • non-zero range: (0 \to 2)
  • so (T_3 = 0,\; T_4 = 2)

Predicted non-zero range of convolution output:

• Start: [ T_1 + T_3 = 0 + 0 = 0 ]

• End: [ T_2 + T_4 = 1 + 2 = 3 ]

Conclusion: The video states this matches the observed/non-zero support of the convolution output, confirming the duration/extension property.

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Instructions / practice guidance included

• “Try to practice more problems”
• “If you face any difficulty you may ask in the comment section”

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Speakers / sources featured

• No specific named speaker or external source is mentioned in the subtitles.
• Only the course presenter/lecturer (unidentified) is speaking.