Properties of Convolution (Part 3)

Main Ideas and Concepts

The lecture covers properties of the convolution operation, focusing on:

• Property 7: Time Delay
• Property 8: Time Scaling

The goal is to show how the convolution output changes when the input signals are delayed or time-scaled, and then apply each property to example problems.

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Property 7: Time Delay

Setup

Start with two signals (X_1(t)) and (X_2(t)). Their convolution is:

[ Y(t) = X_1(t) * X_2(t) ]

Now apply delays:

• Delay (X_1(t)) by (T_1): (X_1(t - T_1))
• Delay (X_2(t)) by (T_2): (X_2(t - T_2))

Core Lesson (Resulting Rule)

Convolving the delayed signals delays the output by the sum of the delays:

[ X_1(t - T_1) * X_2(t - T_2) = Y\bigl(t - (T_1 + T_2)\bigr) ]

Conclusion: the resultant delay is (T_1 + T_2).

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Example 1 (as presented)

Given a convolution structure involving delayed unit-step/ramp-related signals (subtitles are slightly garbled), the intended form is:

• First signal: a unit step delayed by 1
• Second signal: a unit step delayed by 2, multiplied by (t) / ramp-related expression

Using the property:

• (T_1 = 1)
• (T_2 = 2)

So the output delay is:

[ T_1 + T_2 = 3 ]

The subtitles’ conclusion is that the output is the original result evaluated at a shifted time, i.e. using (t-3) (via (t-1+2=3)).

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Example 2 (assigned as homework)

Given:

• Convolution of ((t+2)) with (u(t-3))

Instruction:

• Solve using Property 6 and Property 7.

Output request:

• “Find out the resultant signal”

The subtitles indicate the answer will be posted later (e.g., in a comment section).

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Property 8: Time Scaling

Setup

Start with two signals (X_1(t)) and (X_2(t)), with convolution:

[ Y(t) = X_1(t) * X_2(t) ]

Apply time scaling by the same factor (a) to both signals:

• Scale (X_1(t)) by (a)
• Scale (X_2(t)) by (a)

Core Lesson (Resulting Rule)

The time-scaled convolution output becomes:

[ X_1(at) * X_2(at) = \frac{1}{|a|}\,Y!\left(\frac{t}{a}\right) ]

Constraint: (a \neq 0).

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Example for Property 8 (as presented)

Given

You know:

[ X_1(t) * X_2(t) = Y(t) ]

But the problem provides a scaled convolution result:

[ \frac{1}{3}Y(3t) ]

Interpretation

Match this to the time-scaling form:

[ \frac{1}{|a|}Y!\left(\frac{t}{a}\right) ]

From (\frac{1}{3}Y(3t)), the scaling factor is:

• (a = 3)

Determine the two scaled input signals

Both inputs are scaled by (a=3):

• Subtitles state: (X_1(t) = t^2 + t)

Scale it:

[ X_1(3t) = (3t)^2 + 3t = 9t^2 + 3t ]

• Subtitles state: (X_2(t) = t)

Scale it:

[ X_2(3t) = 3t ]

Convolution claim

The convolution of:

• (9t^2 + 3t) and (3t)

is concluded (by the subtitles) to give:

[ \frac{1}{3}Y(3t) ]

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Speakers or Sources Featured

• No specific named speaker is identified.
• The content is delivered by an unnamed lecturer/teacher.
• Music/applause appear as non-speaker audio cues.