GATE Electrical Engineering - Complete Master Library

 

📘 Continuous Time Convolution – Advanced Worked Problems

This section focuses on convolution involving finite duration signals. Careful limit selection is the key to solving these problems.

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🔹 Problem 1

Given:

x(t) = u(t) − u(t−2) h(t) = u(t) − u(t−3)

Both are rectangular pulses.

Step 1: Understand Signals

x(t) → Pulse from 0 to 2 h(t) → Pulse from 0 to 3

Step 2: Convolution Formula

y(t) = ∫ x(τ) h(t−τ) dτ

Step 3: Overlap Regions

Case 1: t < 0 → No overlap → y(t)=0 Case 2: 0 ≤ t < 2 Overlap length = t

y(t) = t

Case 3: 2 ≤ t < 3 Full overlap of width 2

y(t) = 2

Case 4: 3 ≤ t < 5 Overlap decreasing

y(t) = 5 − t

Case 5: t ≥ 5 → No overlap → 0

Final Answer:

Triangular shaped output.

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🔹 Problem 2

Given:

x(t) = e^{-t} u(t) h(t) = u(t) − u(t−1)

Step 1: Limits

h(t) active only from 0 to 1

Step 2: Convolution

y(t) = ∫ e^{-τ} dτ

Limits depend on overlap: Case 1: 0 ≤ t ≤ 1

y(t) = 1 − e^{-t}

Case 2: t > 1

y(t) = e^{-(t-1)} − e^{-t}

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🔹 Problem 3

Convolution of two rectangular pulses of width T:

x(t) = u(t) − u(t−T) h(t) = same

Output:

y(t) = t for 0 ≤ t ≤ T 2T − t for T ≤ t ≤ 2T

Triangular waveform.

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🔹 Key Observations

• Rectangle * Rectangle → Triangle
• Step * Step → Ramp
• Impulse * x(t) → x(t)

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🎯 GATE Important Tips

• Always draw signals first
• Find overlap region carefully
• Split solution into time intervals
• Most mistakes happen in limits

Convolution = Area of Overlap as One Signal Slides Over Another