Frequency domain analysis and Fourier Transform Compiled from: S. Narasimhan Njegos Nincic Other 1

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” (how fast/often signals change, etc). 2

Jean Baptiste Joseph Fourier (1768-1830) Had crazy idea (1807): Any periodic function can be rewritten as a weighted sum of Sines and Cosines of different frequencies. Don’t believe it? Neither did Lagrange, Laplace, Poisson and other big wigs Not translated into English until 1878! But it’s true! called Fourier Series Possibly the greatest tool used in Engineering 3

Time and Frequency example : g(t) = sin(2pi f t) + (1/3)sin(2pi (3f) t) 4

Time and Frequency example : g(t) = sin(2pi f t) + (1/3)sin(2pi (3f) t) = + 5

Frequency Spectra example : g(t) = sin(2pi f t) + (1/3)sin(2pi (3f) t) 6

Frequency Spectra Usually, frequency is more interesting than the phase 7

Frequency Spectra = + = 8

Frequency Spectra = + = 9

Frequency Spectra = + = 10

Frequency Spectra = + = 11

Frequency Spectra = + = 12

ringing near discontinuity Gibbs phenomenon: ringing near discontinuity Source: http://mathworld.wolfram.com/FourierSeries.html

Frequency Spectra = 14

DIRICHLET CONDITIONS Suppose that f(x) is defined and single valued except possibly at finite number of points in (-l,+l) f(x) is periodic outside (-l,+l) with period 2l f(x) and f’(x) are piecewise continuous in(- l,+l)

Then the Fourier series of f(x) converges to f(x) if x is a point of continuity b)[f(x+0)+f(x-0)]/2 if x is a point of discontinuity

METHOD OF OBTAINING FOURIER SERIES OF 1. 2. 3. 4.

(7.3) Expressing cos nx and sin nx in exponential form, we may rewrite Eq.(7.1) as (7.4) in which (7.5) and (7.6)

COMPLEX FORM OF FOURIER SERIES Recall from earlier that we can write a Fourier series in the complex form in which and From the earlier definitions of we can show that

Real and Complex Sinusoids M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 20

Fourier Transform We want to understand the frequency of our signal. So, let’s reparametrize the signal by  instead of x: f(x) F() Fourier Transform For every  from 0 to inf, F() holds the amplitude A and phase of the corresponding sine How can F hold both? Complex number trick! F() f(x) Inverse Fourier Transform 21

Transforms Transform: Can be thought of as a substitution In mathematics, a function that results when a given function is multiplied by a so-called kernel function, and the product is integrated between suitable limits. (Britannica) Can be thought of as a substitution

Transforms Example of a substitution: Original equation: x + 4x² – 8 = 0 Familiar form: ax² + bx + c = 0 Let: y = x² Solve for y x = ±√y 4

Transforms Transforms are used in mathematics to solve differential equations: Original equation: Apply Laplace Transform: Take inverse Transform: y = Lˉ¹(y)

Fourier Transform Property of transforms: Fourier Transform: They convert a function from one domain to another with no loss of information Fourier Transform: converts a function from the time (or spatial) domain to the frequency domain

Fourier Series to Fourier Transform For periodic signals, we can represent them as linear combinations of harmonically related complex exponentials To extend this to non-periodic signals, we need to consider aperiodic signals as periodic signals with infinite period. As the period becomes infinite, the corresponding frequency components form a continuum and the Fourier series sum becomes an integral (like the derivation of CT convolution) Instead of looking at the coefficients a harmonically –related Fourier series, we’ll now look at the Fourier transform which is a complex valued function in the frequency domain

Fourier Series to Fourier Transform

Effect of increasing period T a/T a/T a/T

Frequency Spectra 29

FT: Just a change of basis M * f(x) = F() * = . 30

IFT: Just a change of basis M-1 * F() = f(x) = * . 31

Fourier Transform – more formally Represent the signal as an infinite weighted sum of an infinite number of sinusoids Note: Arbitrary function Single Analytic Expression Spatial Domain (x) Frequency Domain (u) (Frequency Spectrum F(u)) Inverse Fourier Transform (IFT) 32

Fourier Transform Also, defined as: Note: Inverse Fourier Transform (IFT) 33

Review of Cauchy Principal Value Integrals x 1/x Recall for real integrals, but a finite result is obtained if the integral interpreted as because the infinite contributions from the two symmetrical shaded parts shown exactly cancel. Integrals evaluated in this way are said to be (Cauchy) principal value integrals (or “deleted” integrals) and are often written as 34

Delta function Definition Area for any  > 0 Sifting property since

Dirac delta Function This allows an arbitrary sequence x(n) or continuous-time function f(t) to be expressed as:

Example 1: Decaying Exponential Consider the (non-periodic) signal Then the Fourier transform is: a = 1

Example 2: Single Rectangular Pulse Consider the non-periodic rectangular pulse at zero The Fourier transform is: Note, the values are real T1 = 1

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl The Sinc Function M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 39

Example 3: Impulse Signal The Fourier transform of the impulse signal can be calculated as follows: Therefore, the Fourier transform of the impulse function has a constant contribution for all frequencies  X(j)

Example 4: Periodic Signals A periodic signal violates condition 1 of the Dirichlet conditions for the Fourier transform to exist However, lets consider a Fourier transform which is a single impulse of area 2 at a particular (harmonic) frequency =0. The corresponding signal can be obtained by: which is a (complex) sinusoidal signal of frequency 0. More generally, when Then the corresponding (periodic) signal is The Fourier transform of a periodic signal is a train of impulses at the harmonic frequencies with amplitude 2ak

Fourier Transform Pairs (I) Note that these are derived using angular frequency ( ) 42

Fourier Transform Pairs (I) Note that these are derived using angular frequency ( ) 43

Fourier Transform and Convolution Let Then Convolution in spatial domain Multiplication in frequency domain 44

Fourier Transform and Convolution Spatial Domain (x) Frequency Domain (u) So, we can find g(x) by Fourier transform FT IFT 45

Properties of Fourier Transform Spatial Domain (x) Frequency Domain (u) Linearity Scaling Shifting Symmetry Conjugation Convolution Differentiation Note that these are derived using frequency ( ) 46

Properties of Fourier Transform 47

Exercícios

Example use: Hearing mechanism Human ears do not hear wave-like oscilations, but constant tone Often it is easier to work in the frequency domain

Example use: Smoothing/Blurring We want a smoothed function of f(x) Let us use a Gaussian kernel Then H(u) attenuates high frequencies in F(u) (Low-pass Filter)! 57

Image Processing in the Fourier Domain 58

Low-pass Filtering Let the low frequencies pass and eliminating the high frequencies. Generates image with overall shading, but not much detail 59

High-pass Filtering Lets through the high frequencies (the detail), but eliminates the low frequencies (the overall shape). It acts like an edge enhancer. 60