GATE Electrical Engineering - Complete Master Library

 

📘 Continuous Time Fourier Transform (CTFT) – Fundamentals

Fourier Transform represents an aperiodic signal as a continuous spectrum of frequencies. It is the extension of Fourier Series for non-periodic signals.

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🔹 1. Definition

Forward Transform:

X(ω) = ∫ x(t) e^{-jωt} dt

Inverse Transform:

x(t) = (1/2π) ∫ X(ω) e^{jωt} dω

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🔹 2. Interpretation

• X(ω) gives frequency content
• Magnitude → Strength of frequency
• Phase → Phase shift of frequency

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🔹 3. Existence Condition

Fourier Transform exists if:

∫ |x(t)| dt < ∞

Signal must be absolutely integrable.

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🔹 4. Important Standard Transforms

1. Impulse

δ(t) ↔ 1

2. Unit Step (Does not strictly exist)

Special handling required.

3. Exponential

e^{-at} u(t) ↔ 1 / (a + jω)

4. Rectangular Pulse

Rect ↔ sinc function

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🔹 5. Basic Properties

Linearity

a x₁(t) + b x₂(t) ↔ aX₁(ω) + bX₂(ω)

Time Shifting

x(t − t₀) ↔ X(ω) e^{-jωt₀}

Frequency Shifting

x(t)e^{jω₀t} ↔ X(ω − ω₀)

Time Scaling

x(at) ↔ (1/|a|) X(ω/a)

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🔹 6. Worked Example

Find Fourier Transform of:

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

Solution:

X(ω) = ∫₀^∞ e^{-at} e^{-jωt} dt

Combine exponents:

= ∫₀^∞ e^{-(a + jω)t} dt

Result:

X(ω) = 1 / (a + jω)

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🎯 GATE Important Points

• Memorize standard pairs
• Time shift → exponential multiplication
• Scaling affects amplitude
• Rectangular ↔ sinc relation very important

Fourier Transform = Continuous Frequency Spectrum of Signal