GATE Electrical Engineering - Complete Master Library

 

📘 Fourier Transform – Duality Property

Duality property shows symmetry between time and frequency domains. If a signal has a Fourier Transform, then its transform pair can be swapped with modification.

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🔹 1. Statement of Duality

If:

x(t) ↔ X(ω)

Then:

X(t) ↔ 2π x(−ω)

Time and frequency interchange roles.

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🔹 2. Intuition

Fourier Transform treats time and frequency symmetrically. Duality shows that structure in one domain appears in the other.

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🔹 3. Example 1 – Rectangular & sinc

We know:

Rect(t) ↔ sinc(ω)

By duality:

sinc(t) ↔ 2π Rect(−ω)

Very powerful shortcut.

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🔹 4. Example 2 – Impulse Pair

We know:

δ(t) ↔ 1

By duality:

1 ↔ 2π δ(ω)

Important transform pair.

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🔹 5. Practical Use

Instead of calculating transform: 1. Use known pair 2. Apply duality 3. Adjust scaling Saves time in exam.

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🔹 6. Important Observations

• Duality swaps time and frequency
• Sign reversal appears
• 2π factor must be remembered

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🎯 GATE Important Points

• Often asked as conceptual MCQ
• Rect ↔ sinc dual pair very common
• Remember 2π scaling carefully
• Helps derive new transforms quickly

Duality = Time and Frequency Mirror Each Other