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Fourier Transforms

Introduction​

Fourier transforms are a generalization of Fourier series that extend the concept to non-periodic functions. They are named after Joseph Fourier and play a crucial role in many areas of mathematics, physics, and engineering.

Basic Concept​

The Fourier transform decomposes a function of time (or space) into its constituent frequencies. It's a reversible operation, allowing us to move between the time (or space) domain and the frequency domain.

Continuous Fourier Transform​

For a function f(t), its Fourier transform F(ω) is given by:

F(ω) = ∫ f(t) e^(-iωt) dt (from -∞ to +∞)

Where:

  • ω represents angular frequency
  • i is the imaginary unit
  • e is Euler's number

The inverse Fourier transform is:

f(t) = (1/2π) ∫ F(ω) e^(iωt) dω (from -∞ to +∞)

Properties of Fourier Transforms​

  1. Linearity: The transform of a sum is the sum of the transforms.
  2. Scaling: Scaling in one domain causes inverse scaling in the other.
  3. Shift: A shift in one domain causes a phase change in the other.
  4. Convolution: Convolution in one domain becomes multiplication in the other.

Discrete Fourier Transform (DFT)​

For discrete, finite-length signals, we use the DFT:

X[k] = Σ x[n] e^(-i2πkn/N) (n from 0 to N-1)

Where:

  • x[n] is the input sequence
  • X[k] is the output sequence
  • N is the number of samples

Fast Fourier Transform (FFT)​

The FFT is an efficient algorithm to compute the DFT, reducing computational complexity from O(N²) to O(N log N).

Applications​

Fourier transforms are widely used in:

  1. Signal processing
  2. Image processing
  3. Data compression
  4. Solving differential equations
  5. Spectral analysis

Fourier Transform Pairs​

Some common Fourier transform pairs include:

  1. Delta function ↔ Constant function
  2. Gaussian function ↔ Gaussian function
  3. Rectangular function ↔ Sinc function

Relationship to Fourier Series​

For periodic functions, the Fourier transform produces a discrete spectrum, which is directly related to the coefficients of the Fourier series.

Generalizations​

  1. Short-time Fourier transform: For analyzing non-stationary signals.
  2. Fractional Fourier transform: Rotates signals in the time-frequency plane.
  3. Wavelet transform: Provides time-frequency localization.

Exercises​

  1. Compute the Fourier transform of a simple function, like e^(-|t|).
  2. Explore the effect of scaling and shifting on the Fourier transform of a function.
  3. Implement a simple DFT algorithm and compare its output with a built-in FFT function.