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.
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=
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)