Fourier Transform Topics Notes

1. Introduction to Fourier Analysis

• History and motivation
• Applications in signal processing, physics, and engineering

2. Fourier Series (Prerequisite)

• Periodic functions
• Trigonometric and exponential Fourier series
• Even and odd function symmetry
• Half-range expansions

3. Continuous Fourier Transform (CFT)

• Definition and basic formula:
  F(ω) = ∫ from -∞ to +∞ of f(t) × e^-jωt dt
• Inverse Fourier Transform:
  f(t) = (1 / 2π) × ∫ from -∞ to +∞ of F(ω) × e^jωt dω
• Conditions for existence (Dirichlet conditions)

4. Fourier Transform Properties

• Linearity: FT of (a × f(t) + b × g(t)) = a × F(ω) + b × G(ω)
• Time shifting: f(t - t₀) ↔ e^-jωt₀ × F(ω)
• Frequency shifting: e^jω₀t × f(t) ↔ F(ω - ω₀)
• Scaling: f(at) ↔ (1/|a|) × F(ω/a)
• Convolution theorem: f(t) * g(t) ↔ F(ω) × G(ω)
• Differentiation in time domain: dⁿf(t)/dtⁿ ↔ (jω)ⁿ × F(ω)
• Integration in time domain: ∫ f(t) dt ↔ (1/jω) × F(ω) + π × F(0) × δ(ω)
• Parseval’s theorem: ∫ |f(t)|² dt = (1/2π) × ∫ |F(ω)|² dω

5. Common Fourier Transform Pairs

• Delta function (Dirac delta): δ(t) ↔ 1
• Rectangular pulse: rect(t/T) ↔ T × sinc(ωT/2π)
• Sine and cosine functions:
  • cos(ω₀t) ↔ π [δ(ω - ω₀) + δ(ω + ω₀)]
  • sin(ω₀t) ↔ jπ [δ(ω + ω₀) - δ(ω - ω₀)]
• Gaussian function: e^-at² ↔ √(π/a) × e^-(ω²/4a)
• Exponential decay: e^{-at} u(t) ↔ 1 / (a + jω), where Re(a) > 0}

6. Discrete-Time Fourier Transform (DTFT)

• Defined for discrete-time aperiodic signals x[n]
• Formula: X(ω) = Σ from n=-∞ to +∞ of x[n] × e^-jωn
• Properties similar to continuous FT
• Relation to z-transform: X(z) with z = e^jω

7. Discrete Fourier Transform (DFT)

• Defined for finite-length discrete signals x[n], n=0 to N-1
• Formula: X[k] = Σ from n=0 to N-1 of x[n] × e^-j2πkn/N, k = 0, 1, ..., N-1
• Inverse DFT recovers x[n] from X[k]
• Used extensively in digital signal processing

8. Fast Fourier Transform (FFT)

• Efficient algorithm to compute the DFT
• Reduces computation from O(N²) to O(N log N)
• Radix-2 algorithm most common: splits DFT into smaller parts recursively
• Bit reversal and decimation-in-time/frequency techniques

9. Applications of Fourier Transform

• Signal analysis (audio, speech, image processing)
• Filtering in time and frequency domains
• Spectral analysis and power spectrum estimation
• Modulation and demodulation in communications
• Data compression (JPEG, MP3)

10. 2D Fourier Transform

• Extension of Fourier Transform to two dimensions (x, y)
• Used for image processing and analysis
• Formula: F(u, v) = ∫∫ f(x, y) × e^{-j2π(ux + vy)} dx dy
• Filtering and frequency domain operations on images

11. Fourier Transform in Complex Analysis

• Use of contour integration and residue theorem to evaluate Fourier integrals
• Fourier inversion formula with complex analysis tools

12. Windowed Fourier Transform (Short-Time Fourier Transform - STFT)

• Analyzes signals whose frequency content changes over time
• Applies a window function to isolate signal parts in time
• Limitation: trade-off between time and frequency resolution

13. Fourier Transform vs Laplace Transform

• Region of convergence: FT integrates over jω axis; Laplace over complex s-plane
• Laplace transform useful for causal and unstable systems
• FT mainly used for steady-state or stable signals

Advanced Topics

• Wavelet Transform: better time-frequency localization for non-stationary signals
• Fourier Integral Theorem
• Generalized Fourier Transforms
• Multidimensional Fourier Transforms beyond 2D
• Spectrograms and time-frequency representations