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📘 Parseval’s Theorem – Fourier Transform Version

Parseval’s theorem for Fourier Transform states that energy in time domain equals energy in frequency domain.

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🔹 1. Statement

If:

x(t) ↔ X(ω)

Then:

∫ |x(t)|² dt = (1/2π) ∫ |X(ω)|² dω

This is energy conservation.

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🔹 2. Physical Meaning

Signal energy can be measured in: • Time domain • Frequency domain Both give same result.

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🔹 3. Example 1 – Exponential Signal

Given:

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

Time domain energy:

E = ∫₀^∞ e^{-2at} dt = 1/(2a)

Frequency domain:

X(ω) = 1/(a + jω)

Compute:

(1/2π) ∫ |1/(a + jω)|² dω

Result:

= 1/(2a)

Both match ✔

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🔹 4. Example 2 – Rectangular Pulse

Rectangular pulse: Time energy = Area under square. Fourier Transform = sinc function. Energy computed using:

(1/2π) ∫ |sinc(ω)|² dω

Matches time domain result.

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🔹 5. Important Observations

• Higher frequency components contribute energy
• Energy spectrum = |X(ω)|²
• Used in power spectral density

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🎯 GATE Important Points

• Remember 1/2π factor
• Energy signal → Parseval applies
• Connects directly to communication systems
• Conceptual + numerical questions possible

Energy in Time = Energy in Frequency