GATE Electrical Engineering - Complete Master Library

 

📘 Fourier Series – Exponential Form & Frequency Spectrum

Exponential form of Fourier Series represents a periodic signal as a sum of complex exponentials. This form simplifies many calculations.

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🔹 1. Exponential Form

x(t) = Σ Cₙ e^{jnω₀t}

Where:

ω₀ = 2π / T

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🔹 2. Coefficient Formula

Cₙ = (1/T) ∫ x(t) e^{-jnω₀t} dt

Integration over one period.

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🔹 3. Relation Between Trigonometric and Exponential Form

Cₙ = (aₙ - j bₙ)/2

C_{-n} = (aₙ + j bₙ)/2

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🔹 4. Frequency Spectrum

Each term corresponds to frequency:

ω = nω₀

Spectrum consists of impulses at harmonic frequencies.

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🔹 5. Example 1 – DC Signal

Given:

x(t) = A

Then:

C₀ = A All other Cₙ = 0

Only DC component.

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🔹 6. Example 2 – Cosine Signal

Given:

x(t) = cos(ω₀ t)

Using Euler’s identity:

cos(ω₀ t) = (e^{jω₀t} + e^{-jω₀t}) / 2

Thus:

C₁ = 1/2 C_{-1} = 1/2 All others = 0

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🔹 7. Symmetry Properties

If x(t) is real:

C_{-n} = Cₙ*

If x(t) is even:

Cₙ is real

If x(t) is odd:

Cₙ is imaginary

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🔹 8. Important Observations

• Magnitude spectrum = |Cₙ|
• Phase spectrum = ∠Cₙ
• Fourier Series shows harmonic structure clearly

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🎯 GATE Important Points

• Exponential form simplifies integration
• Know relation between aₙ, bₙ and Cₙ
• Spectrum interpretation often asked
• Real signal → conjugate symmetry

Exponential Form = Compact Representation of Harmonics