Introduction to Convolution Operation

Main ideas / lessons

• Convolution as a core operation for LTI systems

  • Convolution is a mathematical tool to compute the output (y(t)) of an LTI (Linear Time-Invariant) system when:
    • the input (x(t)) is known, and
    • the impulse response (h(t)) is known.
  • In this course, convolution is emphasized mainly for this purpose.

• Connection to earlier “signal operations”

  • The lecturer reviews five basic signal operations studied earlier:
    1. Shifting
       • amplitude shifting and time shifting
    2. Scaling
       • amplitude scaling and time scaling
       • reversal is treated as a special case of scaling:
         • amplitude reversal and time reversal
    3. Differentiation (graphical method)
    4. Integration (graphical method)
    5. Convolution (deferred until the basics of LTI systems and integration are ready)
  • Key intuition: convolution ultimately relates to overlap, which can be understood via integration.

• Definition of convolution (overlap interpretation)

  • Convolution expresses the amount of overlap between two functions when one function is shifted over the other.
  • The definition highlights:
    • an integral
    • measuring overlap
    • involving two functions
    • one function being shifted
    • and the overlap being integrated.

---

Methodology: steps to compute convolution (graphical procedure)

Given input (x(t)) and impulse response (h(t)), the output is: [ y(t) = (x * h)(t) = \int_{-\infty}^{\infty} x(\tau)\, h(t-\tau)\, d\tau ]

The lecturer provides five steps to carry out the computation using the convolution formula:

1. Replace (t) by a dummy variable (\tau)

   • Start from (x(t)) and rewrite as (x(\tau))
   • Similarly work with (h) in a form consistent with the formula

2. Time reversal (special case of time scaling)

   • Implement the form that leads to:
     • (h(-\tau)) (as part of obtaining (h(t-\tau)))

3. Time shifting (performed with respect to (\tau))

   • Shift the reversed signal so that it becomes:
     • (h(t-\tau))
   • Emphasis: time shifting should be applied relative to the variable (\tau)
   • Avoid confusion between the specific time instant and the variable by using the negative sign correctly.

4. Multiply the two signals

   • Multiply pointwise:
     • (x(\tau)\cdot h(t-\tau))

5. Integrate the product

   • Integrate over all (\tau):
     • (\int_{-\infty}^{\infty} x(\tau)\, h(t-\tau)\, d\tau)
   • In practice for piecewise signals, the integral limits can be reduced to only where overlap occurs.

Efficiency tip from the example

• Instead of integrating over every tiny time region, identify the time instants where waveform values change (edges).
• Perform the integration only over intervals where overlap exists and the product is nonzero.

---

Example: piecewise/rectangular signals leading to a trapezoidal output

• Given

  • Input (x(t)): square pulse that is wider
  • Impulse response (h(t)): square pulse that is narrower

• Graphical convolution outcome

  • The output (y(t)) becomes trapezoidal because:
    • convolving two unequal-width rectangular pulses yields a trapezoid.

Case analysis based on (t)

The lecturer computes (y(t)) by considering five ranges of (t) (overlap changes as the shifted (h(t-\tau)) slides across (x(\tau))):

1. Case 1: (t < 0)

   • No overlap [ y(t)=0 ]

2. Case 2: (0 < t < 1)

   • Partial overlap growing linearly [ y(t)=t ]

3. Case 3: (1 < t < 2)

   • Complete overlap over the shorter pulse region (constant product area) [ y(t)=1 ]

4. Case 4: (2 < t < 3)

   • Partial overlap decreasing linearly [ y(t)=3-t ]

5. Case 5: (t > 3)

   • No overlap [ y(t)=0 ]

So overall, the output is: [ y(t)= \begin{cases} 0, & t<0\ t, & 0<t<1\ 1, & 1<t<2\ 3-t, & 2<t<3\ 0, & t>3 \end{cases} ] (Endpoints are discussed informally via the piecewise graphing.)

---

Rewriting the output using ramp functions

The lecturer then expresses the trapezoid using ramp signals (R(t-a)) (where a ramp term turns on after (t=a)).

• Using the “turning points” at (t=0,1,2,3), and noting upward/downward slopes:
  • combine ramps with positive or negative signs depending on whether the slope increases or decreases.

Final simplified ramp-form expression (as stated): [ y(t)= t \;-\; R(t-1)\;-\; R(t-2)\;+\; R(t-3) ] (The structure corresponds to subtracting (R(t-1)) and (R(t-2)) and adding (R(t-3)).)

---

Animation interpretation (what the animation demonstrates)

• One window shows:
  • stationary (x(\tau))
• The other window shows:
  • moving (h(t-\tau))

As (h(t-\tau)) shifts:

• overlap increases → convolution integral value increases (starts at 0, rises)
• overlap reaches maximum → convolution becomes constant
• overlap then decreases → convolution decreases back to 0

---

Speakers / sources featured

• Single unnamed lecturer (presenter/instructor) — no other identified speakers or external sources mentioned.