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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
Fourier Transform

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Fourier Transform

  • 2. 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
  • 3. 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.
  • 4. 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
  • 5. 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.
  • 6. Fourier Transform-more formally • Fourier Transform: • Inverse Fourier Transform: Spatial Domain (x) Frequency Domain (u)
  • 7. 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=
  • 9. Types • Continuous Fourier Transform (CFT) • Discrete Fourier Transform (DFT) • Fast Fourier Transform (FFT)
  • 10. 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:
  • 11. Fourier Transform and Convolution • Convolution in spatial domain= Multiplication in frequency domain (and vice versa)
  • 13. 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