Properties of Convolution (Part 2)

Main ideas / lessons

Property 5: Derivative of the convolution output (LTI systems)

For an LTI system with:

• Input: (x(t))
• Impulse response: (h(t))
• Output: (y(t) = x(t) * h(t))

The key result is: [ \frac{d y(t)}{dt} = x(t) * \frac{d h(t)}{dt} = \frac{d x(t)}{dt} * h(t) ]

Interpretation: to differentiate the output of an LTI system, you have two equivalent options:

1. Differentiate the impulse response and convolve with the input.
2. Differentiate the input and convolve with the impulse response.

Example used to demonstrate Property 5

Given:

• Output: (y(t) = r(t) * u(t))
  • (r(t)) = unit ramp signal
  • (u(t)) = unit step signal Goal: compute (\frac{d y(t)}{dt})

Method 1 (differentiate impulse response) Start: [ \frac{d y(t)}{dt} = r(t) * \frac{d u(t)}{dt} ] Use the identity:

• (\frac{d}{dt}u(t) = \delta(t))

So: [ \frac{d y(t)}{dt} = r(t) * \delta(t) = r(t) ] Result: unit ramp.

Method 2 (differentiate input) Start: [ \frac{d y(t)}{dt} = \frac{d}{dt}r(t) * u(t) ] Use the identity:

• (\frac{d}{dt}r(t) = u(t))

So: [ \frac{d y(t)}{dt} = u(t) * u(t) = r(t) ] Result matches Method 1: unit ramp.

---

Property 6: Integration of the convolution output

Conceptual idea:

• Differentiating a signal and then integrating the derivative returns the original signal.

For convolution:

• Setup:
  • Input: (x(t))
  • Impulse response: (h(t))
  • In this property, (h(t)=u(t))

Property statement: [ y(t) = x(t) * u(t) = \int_{-\infty}^{t} x(\tau)\, d\tau ]

This property is supported via Property 5:

• Differentiate (y(t)) (using Property 5) to simplify to a convolution with (\delta(t)),
• Then integrate back to recover (y(t)).

Example used to demonstrate Property 6

Given:

• (y(t) = r(t) * u(t))

Using the property: [ y(t) = \int_{-\infty}^{t} r(\tau)\, d\tau ]

Since the integral of a unit ramp gives a parabolic (second-order) function: [ y(t) = p(t) = \frac{t^2}{2}\, u(t) ]

Alternative confirmation using Property 6 form

The narration also presents an equivalence by:

• Identifying the appropriate relationship inside the convolution between (x(t)) and (u(t))/ramp,
• Leading again to the same parabolic result.

---

Methodologies / instruction-style steps (as presented)

Using Property 5 to compute (\frac{d y(t)}{dt})

1. Express the output as convolution: (y(t) = x(t) * h(t)).
2. Choose one of two equivalent routes:
   • Route A: Compute (x(t) * \frac{d h(t)}{dt})
   • Route B: Compute (\frac{d x(t)}{dt} * h(t))
3. Use standard signal derivative identities (as used in the example):
   • (\frac{d}{dt}u(t) = \delta(t))
   • (\frac{d}{dt}r(t) = u(t))
4. Use convolution with delta:
   • (f(t) * \delta(t) = f(t))

Using Property 6 to compute (y(t)=x(t)*u(t))

1. Verify the impulse response is the unit step: (h(t)=u(t)).
2. Replace convolution with the integration form:
   • (y(t) = \int_{-\infty}^{t} x(\tau)\, d\tau)
3. If needed, use known integrals of standard signals:
   • Integrating (r(t)) (unit ramp) yields a parabolic signal (\frac{t^2}{2}u(t))

---

Speakers / sources featured

• No specific named speaker/source is mentioned in the subtitles. The content appears to be delivered by an unidentified instructor/narrator.
• No other sources or external speakers are explicitly credited.