GATE Electrical Engineering - Complete Master Library

 

📘 Fourier Transform – Convolution Property

One of the most powerful results: Convolution in time domain becomes multiplication in frequency domain.

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🔹 1. Statement

If:

y(t) = x(t) * h(t)

Then:

Y(ω) = X(ω) · H(ω)

Very important result.

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🔹 2. Dual Property

Multiplication in time domain becomes convolution in frequency domain:

x(t) h(t) ↔ (1/2π) X(ω) * H(ω)

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🔹 3. Why This is Powerful?

Instead of solving difficult convolution integral: We can: 1. Take Fourier Transform 2. Multiply 3. Take Inverse Transform Much easier.

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🔹 4. Example 1

Given:

x(t) = e^{-at}u(t) h(t) = e^{-bt}u(t)

Time domain convolution is complex. Instead:

X(ω) = 1/(a + jω) H(ω) = 1/(b + jω)

Multiply:

Y(ω) = 1 / [(a + jω)(b + jω)]

Then inverse transform to get y(t). Much simpler.

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🔹 5. Example 2 – Rectangular Pulse

Rectangular pulse in time:

Rect(t) ↔ sinc(ω)

If we convolve two rectangular pulses: Time domain → Triangle Frequency domain → sinc²(ω) Very important relation.

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🔹 6. Graphical Understanding

Time Domain: Convolution spreads signal. Frequency Domain: Multiplication shapes spectrum.

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🔹 7. Important Observations

• Filtering action explained easily
• System frequency response H(ω)
• Output spectrum = Input spectrum × System response

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🎯 GATE Important Points

• Remember time ↔ frequency dual behavior
• Very common conceptual question
• Used heavily in communication systems
• Know rectangular ↔ sinc relationship

Convolution in Time = Multiplication in Frequency