GATE Electrical Engineering - Complete Master Library

 

📘 Fourier Series – Trigonometric Form (Fundamentals)

Fourier Series represents a periodic signal as a sum of sine and cosine waves. This is the foundation of frequency domain analysis.

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🔹 1. Condition for Fourier Series

Signal must be:

• Periodic
• Finite number of discontinuities
• Absolutely integrable over one period

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🔹 2. Trigonometric Form

For period T:

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

Where:

ω₀ = 2π / T

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🔹 3. Coefficient Formulas

a₀ = (2/T) ∫ x(t) dt

aₙ = (2/T) ∫ x(t) cos(nω₀t) dt

bₙ = (2/T) ∫ x(t) sin(nω₀t) dt

Integration over one period.

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🔹 4. Even and Odd Signal Simplification

If x(t) is even:

bₙ = 0

If x(t) is odd:

aₙ = 0

Very important shortcut in GATE.

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🔹 5. Example 1 – Simple Constant Signal

Given:

x(t) = 1

Then:

a₀ = 2 aₙ = 0 bₙ = 0

Only DC component.

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🔹 6. Example 2 – Square Wave

Let x(t) be symmetric square wave: Because it is odd:

aₙ = 0

Only sine terms present. Final result:

x(t) = (4/π) [ sin(ω₀t) + (1/3)sin(3ω₀t) + (1/5)sin(5ω₀t) + ... ]

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🔹 7. Key Observations

• Fourier Series gives harmonic content
• Higher harmonics → sharper transitions
• DC term gives average value

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🎯 GATE Important Points

• Even/odd simplification saves time
• Know coefficient formulas clearly
• Square wave frequently asked
• DC value equals average of signal

Fourier Series = Time Signal Expressed in Frequency Components