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📘 Continuous Time Convolution – Complete Explanation

For an LTI system, output is obtained using convolution of input and impulse response. This is the most important operation in Signals & Systems.

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🔹 1. Convolution Definition

y(t) = x(t) * h(t) = ∫ x(τ) h(t − τ) dτ

Where:

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

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🔹 2. Important Idea

Convolution means: 1. Flip one signal 2. Shift it 3. Multiply 4. Integrate

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🔹 3. Step-by-Step Procedure

To compute y(t):

• Replace h(t) by h(−τ)
• Shift → h(t − τ)
• Multiply with x(τ)
• Integrate over overlapping region

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🔹 4. Example 1 – Step with Step

Given:

x(t) = u(t) h(t) = u(t)

Then:

y(t) = ∫₀ᵗ 1 dτ = t u(t)

Result: Ramp signal.

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🔹 5. Example 2 – Exponential Case

Given:

x(t) = e^-tu(t) h(t) = u(t)

Then:

y(t) = ∫₀ᵗ e^-τ dτ = 1 − e^-t

Output:

y(t) = (1 − e^-t)u(t)

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🔹 6. Important Properties

• Commutative → x*h = h*x
• Associative
• Distributive

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🔹 7. Graphical Understanding

Output depends on overlapping area of signals. As shift increases → overlap changes → output changes.

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🎯 GATE Important Points

• Unit step convolution gives ramp
• Know integral limits carefully
• Draw overlap region before integrating
• Practice piecewise signals

Convolution = Input Spread by Impulse Response