Fourier Transform

By- Nidhi Baranwal MCA 5th sem

How to Represent Signals?

• Option 1: Taylor series represents any function using polynomials.

• Polynomials are not the best - unstable and not very physically meaningful.

• Easier to talk about “signals” in terms of its “frequencies” and makes filtering easy

Origin• Jean Baptiste Joseph Fourier Had crazy idea :Any periodic

function can be rewritten as a weighted sum of Sines and Cosines of different frequencies-called Fourier Series

• F(t) = a0 + a1cos (ωt) + b1sin(ωt) +a2cos (2ωt) + b2sin(2ωt)+..

=

• In other words , a function can be described by a summation of waves with different amplitudes and phases.

What?• Fourier transform is the generalization of Fourier series • For every w from 0 to inf, F(w) holds the amplitude A and

phase f of the corresponding sine

f(x) F(w)Fourier Transform

F(w) f(x)Inverse Fourier Transform

Conditions

• The sufficient condition for the Fourier transform to exist is that the function g(x) is square integrable,

• g(x) may be singular or discontinuous and still have a well defined Fourier transform.

Fourier Transform-more formally

• Fourier Transform:

• Inverse Fourier Transform:

Spatial Domain (x) Frequency Domain (u)

Definitions

• F(u) is a complex function:

• Magnitude of FT (spectrum):

• Phase of FT:

• Magnitude-Phase representation:

• Power of f(x): P(u)=|F(u)|2=

2D Example

Types

• Continuous Fourier Transform (CFT)

• Discrete Fourier Transform (DFT)

• Fast Fourier Transform (FFT)

Convolution

• A mathematical operator which computes the “amount of overlap” between two functions. Can be thought of as a general moving average

• Discrete domain:

• Continuous domain:

Fourier Transform and Convolution

• Convolution in spatial domain= Multiplication in frequency domain (and vice versa)

Properties of Fourier Transform

Applications of Fourier Transform

• Physics– Solve linear PDEs (heat conduction, wave propagation)

• Antenna design– side scan sonar, GPS

• Signal processing– 1D: speech analysis, enhancement– 2D: image restoration, enhancement

THANKS