Fourier Transform

The Fourier Transform

Developed by Ayush Tripathi

Introduction

The Fourier Transform is a mathematical tool that decomposes a function (often a function of time, representing a signal) into its constituent frequencies. It's like separating a musical chord into its individual notes. It's a cornerstone of signal processing, image processing, and many areas of physics and engineering.

The Core Idea

Any function can be represented as a sum (or integral, for continuous functions) of sine and cosine waves of different frequencies and amplitudes. The Fourier Transform tells us what those frequencies and amplitudes are.

Continuous Fourier Transform (CFT)

For a continuous-time function \( f(t) \), the Fourier Transform, \( F(\omega) \), is defined as:

\( F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-j\omega t} dt \)

Where

• \( f(t) \) is the input function (e.g., a signal) in the time domain.
• \( F(\omega) \) is the Fourier Transform of \( f(t) \) in the frequency domain. This is, in general, a *complex* function. The magnitude \(|F(\omega)|\) represents the amplitude of the frequency component \(\omega\), and the angle (argument) of \(F(\omega)\) represents its phase.
• \( \omega \) (omega) is the angular frequency (radians per second). It's related to frequency \( f \) (in Hertz) by \( \omega = 2\pi f \).
• \( j \) is the imaginary unit (\( j = \sqrt{-1} \)).
• \( t \) is time.
• The integral goes from \( -\infty \) to \( \infty \), covering the entire time domain.

Note that \( e^{-j\omega t} \) is a complex exponential, which can be expressed in terms of sines and cosines using Euler's formula: \( e^{j\theta} = \cos(\theta) + j\sin(\theta) \). This highlights how the Fourier Transform relates the function \(f(t)\) to sines and cosines.

Inverse Fourier Transform

We are having Fourier Transform as. \(F(ω) \) to \( f(t) \) as . The Fourier Transformation \(f(t) \) by.

\( f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{j\omega t} d\omega \)

The structure similar , and constant and imaginary is positve.

Discrete Fourier Transform (DFT)

For Digital Data with Sequence of values which comes for samples taken.

The Equation Looks Similar To Discrete Equation Looks Similar to Integral But is summation Form

\( X(k) = \sum_{n=0}^{N-1} x(n) e^{-j(2\pi/N)kn} \)

Were:

• \( x(n) \) signal in time domain for input index n = \( 0 to n = N -1)
• \(X(k) \): represents for fourier coefficients that represents for kth index with k=0 ... N-1.
• N Represents the length, sample Points.
• K : is Index for all component and output domain, ranging for frequency sample to taken

Inverse Discrete Fourier Transform

From discrete to again continues. Inverse transform will recover Original.

\( x(n) = \frac{1}{N} \sum_{k=0}^{N-1} X(k) e^{j (2\pi/N)kn} \)

Properties and Applications

• Linearity: The Fourier Transform of a sum of functions is the sum of their individual Fourier Transforms. If `F[ ]` denotes the Fourier transform operator: `F[a*f(t) + b*g(t)] = a*F[f(t)] + b*F[g(t)]`
• Time Shifting: Shifting a function in time introduces a linear phase shift in the frequency domain. `F[f(t - t0)] = e^(-jωt0) * F[f(t)]`
• Frequency Shifting: Multiplying a function by a complex exponential shifts its spectrum in frequency. `F[f(t) * e^(jω0t)] = F(ω - ω0)`
• Convolution Theorem: The Fourier