GATE Electrical Engineering - Complete Master Library

 

📘 Parseval’s Theorem – Fourier Series Power Relation

Parseval’s theorem states that total average power of a periodic signal in time domain equals sum of squares of its Fourier coefficients.

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🔹 1. Statement (Trigonometric Form)

If:

x(t) = a₀/2 + Σ [ aₙ cos(nω₀t) + bₙ sin(nω₀t) ]

Then average power:

P = (a₀² / 4) + (1/2) Σ (aₙ² + bₙ²)

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🔹 2. Statement (Exponential Form)

If:

x(t) = Σ Cₙ e^{jnω₀t}

Then:

P = Σ |Cₙ|²

Very compact and easy to use.

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🔹 3. Physical Meaning

Power in time domain = Power distributed across harmonics. Each harmonic contributes:

|Cₙ|²

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🔹 4. Example 1 – Simple Cosine

Given:

x(t) = A cos(ω₀t)

Fourier coefficients:

C₁ = A/2 C_{-1} = A/2

Using Parseval:

P = |A/2|² + |A/2|² = A²/4 + A²/4 = A²/2

Correct result.

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🔹 5. Example 2 – Square Wave

For square wave:

Cₙ = 4 / (nπ) for odd n

Using Parseval:

P = Σ |Cₙ|²

This helps derive infinite series results.

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🔹 6. Important Observations

• Higher harmonics contribute less power
• DC component contributes constant power
• Useful in spectrum analysis

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🎯 GATE Important Points

• Remember power formula carefully
• Exponential form easier for calculations
• Often asked as conceptual question
• Connects Fourier coefficients with signal energy

Parseval = Time Power Equals Frequency Power