Fast Fourier Transform and MATLAB Implementation

(1)

Fast

Fourier

Transform

and

MATLAB

Implementation

by

Wanjun

Huang

for

Dr.

Duncan

L. MacFarlane

(2)

Signals

In the fields of communications, signal processing, and in electrical engineering more generally, a signal is any time‐varying or spatial‐varying quantity

This variable(quantity) changes in time

• Speech or audio signal: A sound amplitude that varies in time

• Temperature readings at different hours of a day

• Stock price changes over days

• Etc

• Etc.

Signals can be classified by continues‐time signal and discrete‐time signal:

• A discrete signal or discrete‐time signal is a time series, perhaps a signal that

h b l d f ti ti i l

has been sampled from a continuous‐time signal

• A digital signal is a discrete‐time signal that takes on only a discrete set of values

1

Continuous Time Signal

1

Discrete Time Signal

-0.5 0 0.5 f( t) -0.5 0 0.5 f[n ] 0 10 20 30 40 -1 Time (sec) 0 10 20 30 40 -1 n

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Periodic

Signal

periodic signal and non‐periodic signal:

1 Periodic Signal 1 Non-Periodic Signal 0 10 20 30 40 -1 0 f( t) Time (sec) 0 10 20 30 40 -1 0 f[n ] n n

• Period T: The minimum interval on which a signal repeats

• Fundamental frequency:Fundamental frequency: ff₀₀=11//TT

• Harmonic frequencies: kf₀

• Any periodic signal can be approximated

byy a sum of manyy sinusoids at harmonic frequenciesq of the signal(g (kff₀₀)) with appropriate amplitude and phase

• Instead of using sinusoid signals, mathematically, we can use the complex exponential functions with both positive and negative harmonic frequencies

) cos( ) sin( ) exp( jt  t  j t Euler formula:

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Time

‐

Frequency

Analysis

• A signal has one or more frequencies in it, and can be viewed from

two different standpoints: Time domain and Frequency domain

Time Domian (Banded Wren Song)

0 1 A m p lit ude

Time Domian (Banded Wren Song)

1 2 Po w e r Frequency Domain 0 2 4 6 8 x 104 -1 Sample Number A 0 200 400 600 800 1000 1200 0 Frequency (Hz)

Time‐domain figure: how a signal changes over time

Why frequency domain analysis?

g g g

Frequency‐domain figure: how much of the signal lies within each given frequency band over a range of frequencies

Why frequency domain analysis?

• To decompose a complex signal into simpler parts to facilitate analysis • Differential and difference equations and convolution operations in the

time domain become algebraic operations in the frequency domain • Fast Algorithm (FFT)

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Fourier Transform

Fourier

Transform

We can go between the time domain and the frequency domain

by using a tool called Fourier transform

• A Fourier transform converts a signal in the time domain to the

frequency domain(spectrum)

A i F i f h f d i

y g f

• An inverse Fourier transform converts the frequency domain

components back into the original time domain signal

Continuous‐Time Fourier Transform:

    d e j F t f   j t    ( ) 2 1 ) (       dt e t f j F (  ) ( ) jt

Continuous Time Fourier Transform:

Discrete‐Time Fourier Transform(DTFT):

Discrete Time Fourier Transform(DTFT):

     _  2 ) ( 2 1 ] [n X e e d x j j n       n n j j e n x e X (  ) [ ] 

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Fourier Representation For Four Types of Signals

Fourier

Representation

For

Four

Types

of

Signals

The signal with different time‐domain characteristics has different

frequency‐domain characteristics

1 Continues‐time periodic signal ‐‐‐> discrete non‐periodic

spectrum

q y

p

2 Continues‐time non‐periodic signal ‐‐‐> continues non‐periodic

spectrum

3 Discrete non‐periodic signal ‐‐‐> continues periodic spectrum

3 Discrete non periodic signal > continues periodic spectrum

4 Discrete periodic signal ‐‐‐> discrete periodic spectrum

The last transformation between time‐domain and frequencyq y is most

useful

The reason that discrete is associated with both time‐domain and frequency

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Periodic Sequence

Periodic

Sequence

) ( ~ ) ( ~ _n _x _n _kN x  

, where

k

is integer

A periodic sequence with period

N

is defined as:

Periodic

,

g

kn N j kn N e W  2  

For example:

) ( ) (k N n k n N kn W W W  

(it is called

Twiddle Factor

)

Properties:

Periodic

( ) k(n N ) N n N k N kn N W W W    

Symmetric

* ( ) ( ) ) ( _Nkn _NN k n _Nk N n kn N W W W W      

Properties:

Orthogonal

N_ W  _ N n_other rN k kn N 0 1 0

For

a

periodic

sequence

with

period

N

,

only

N

samples

are

independent.

So

that

N

sample

in

one

period

is

enough

to

represent the whole sequence



x

(

n

)

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Discrete Fourier Series(DFS)

Discrete

Fourier

Series(DFS)

Periodic

signals

may

be

expanded

into

a

series

of

sine

and

cosine functions

     _   1 0 1 0 ) ( ~ 1 ) ( ~ ) ( ~ ) ( ~ N kn N N n kn N W k X N n x W n x k X

cosine

functions

)) ( ~ ( ) ( ~ )) ( ~ ( ) ( ~ k X IDFS n x n x DFS k X   0 n N

is still a periodic sequence with period

N

in frequency

domain

) ( ~ k X

The Fourier series for the discrete‐time periodic wave shown below:

1

Sequence x (in time domain)

0.2 Fourier Coeffients 0 0.5 A m p lit u d e 0 4 -0.2 0 X 0 10 20 30 40 0 time 0 10 20 30 40 -0.4

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Finite Length Sequence

Finite

Length

Sequence

 x(n) 0  n  N  1

Real lift signal is generally a

fi i

l

h

If we periodic extend it by the period N then ~x(n)   x(n  rN )

    0 ) ( ) (n x n x others N n 1 0  

finite length sequence

If we periodic extend it by the period N, then  

  r rN n x n x( ) ( ) ) (n x ) ( ~_x _n  

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Relationship

Between

Finite

Length

Sequence

d P i di S

and

Periodic

Sequence

A periodic sequence is the periodic extension of a finite length

A finite length sequence is the principal value interval of the periodic

sequence

       m N n x rN n x n x ( ) ( ) (( )) ~

A finite length sequence is the principal value interval of the periodic

sequence ) ( ) ( ~ ) (n x n R n x  _N     0 1 ) (n R_N others N n 1 0    Where 0 others So that: ) ( )] ( ~ [ ) ( ) ( ~ ) (n x n R n IDFS X k R n x  _N  _N ) ( )] ( ~ [ ) ( ) ( ~ ) ( ) ( )] ( [ ) ( ) ( ) ( n R n x DFS k R k X k X _N _N N N  

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Discrete Fourier Transform(DFT)

Discrete

Fourier

Transform(DFT)

• Using

the

Fourier

series

representation

we

have

Discrete

Fourier Transform(DFT) for finite length signal

Fourier

Transform(DFT)

for

finite

length

signal

• DFT

can

convert

time

‐

domain

discrete

signal

into

frequency

‐

domain

discrete

spectrum

Assume

that

we

have

a

signal

.

Then

the

DFT

of

the

signal

is

a

sequence

for

1 0 ]} [ {x n _nN_ ] [k X k  0, , N  1      1 0 / 2 ] [ ] [ N n N jnk e n x k X 

The

Inverse

Discrete

Fourier

Transform(IDFT):

      1 0 / 2 . 1 , , 2 , 0 , ] [ 1 ] [ N k N jnk N n e k X N n x  _

Note that because MATLAB cannot use a zero or negative

Note

that

because

MATLAB

cannot

use

a

zero

or

negative

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DFT Example

DFT

Example

The

DFT

is

widely

used

in

the

fields

of

spectral

analysis,

acoustics medical imaging and telecommunications

acoustics,

medical

imaging,

and

telecommunications.

4 5 6

e

Time domain signal

For example: 0 1 2 3 Am p lit u d e ) 3 , 2 , 1 , 0 ( , 4 ], 6 1 4 2 [ ] [n   N  n  x nk nk j j n x e n x k X [ ] [ ] 3 [ ]( ) 0 2 3 0        0 0.5 1 1.5 2 2.5 3 -1 Time n n0 0 11 6 ) 1 ( 4 2 ] 0 [       X j j j X [1]  2  (4 )  1  6  3  2 10 12

Frequency domain signal

9 6 ) 1 ( ) 4 ( 2 ] 2 [         X j j j X [3]  2  (4 )  1  6  3  2 4 6 8 10 |X [k ]| 0 0.5 1 1.5 2 2.5 3 0 2

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Fast Fourier Transform(FFT)

Fast

Fourier

Transform(FFT)

• The Fast Fourier Transform does not refer to a new or different

type of Fourier transform. It refers to a very efficient algorithm foryp y g

computing the DFT

• The time taken to evaluate a DFT on a computer depends

principally on the number of multiplications involved. DFT needs principally on the number of multiplications involved. DFT needs

N2 _{multiplications. FFT only needs} _N_log

2(N)

• The central insight which leads to this algorithm is the

realization that a discrete Fourier transform of a sequence of N

points can be written in terms of two discrete Fourier transforms

of length N/2

• Thus ifThus if NN is a power of two it is possible to recursively applyis a power of two, it is possible to recursively apply this decomposition until we are left with discrete Fourier transforms of single points

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Fast Fourier Transform(cont.)

Fast

Fourier

Transform(cont.)

Re‐writing     1 0 / 2 ] [ ] [ N n N jnk e n x k X  as    1 0 ] [ ] [ N n nk N W n x k X

It is easy to realize that the same values of W nk are calculated many times as the It is easy to realize that the same values of are calculated many times as the

computation proceeds

Using the symmetric property of the twiddle factor, we can save lots of computations

N W 1 1 1 N N N ) 1 2 ( ) 2 ( ) ( ) ( ] [ ] [ 1 2 ) 1 2 ( 1 2 2 1 0 1 0 1 0 W r x W r x W n x W n x W n x k X N r k N N kr N N n odd n kn N N n even n kn N N n nk N             _        ) ( ) ( ) ( ) ( ) 1 2 ( ) 2 ( 1 2 0 2 2 1 2 0 1 2 0 0 k X W k X W r x W W r x W r x W r x k N r kr N k N N r kr N r N r N               ) ( ) ( ₂ 1 k W X k X  _Nk 

Thus the N‐point DFT can be obtained from two N/2‐point transforms, one on even

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Introduction for MATLAB

Introduction

for

MATLAB

MATLAB is a numerical computing environment developed by

MathWorks. MATLAB allows matrix manipulations, plotting ofp , p g

functions and data, and implementation of algorithms

Getting help Getting help

You can get help by typing the commands help or lookfor at

the >> prompt, e.g.

>> help fft >> help fft

Arithmetic operators

Symbol Operation Example

+ Addition 3 1 + 9 + Addition 3.1 + 9 ‐ Subtraction 6.2 – 5 * Multiplication 2 * 3 / Division 5 / 2 / Division 5 / 2 ^ Power 3^2

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Data Representations in MATLAB

Data

Representations

in

MATLAB

Variables: Variables are defined as the assignment operator “=” . The syntax of variable assignment is

i bl l ( i )

variable name = a value (or an expression)

For example,

>> x = 5 x =

5

>> y = [3*7, pi/3]; % pi is in MATLAB

Vectors/Matrices MATLAB can create and manip late arra s of 1 ( ectors) 2 

Vectors/Matrices: MATLAB can create and manipulate arrays of 1 (vectors), 2 (matrices), or more dimensions

row vectors: a = [1, 2, 3, 4] is a 1X4 matrix

column vectors: b = [5; 6; 7; 8; 9] is a 5X1 matrix, e.g.

>> A = [1 2 3; 7 8 9; 4 5 6]

A = 1 2 3

7 8 9 4 5 6 4 5 6

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Mathematical Functions in MATLAB

Mathematical

Functions

in

MATLAB

MATLAB offers many predefined mathematical functions for

technical computing, e.g.p g, g

cos(x) Cosine abs(x) Absolute value sin(x) Sine angle(x) Phase angle

exp(x) Exponential conj(x) Complex conjugate

Colon operator (:)

exp(x) Exponential conj(x) Complex conjugate sqrt(x) Square root log(x) Natural logarithm

Colon operator (:)

Suppose we want to enter a vector x consisting of points (0,0.1,0.2,0.3,…,5). We can use the command

>> x = 0:0.1:5;;

Most of the work you will do in MATLAB will be stored in files called

scripts, or m‐files, containing sequences of MATLAB commands to be executed over and over again

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Basic plotting in MATLAB

Basic

plotting

in

MATLAB

MATLAB has an excellent set of graphic tools. Plotting a given data set or the results of computation is possible with very few commands

The MATLAB command to plot a graph is plot(x,y), e.g. >> x = 0:pi/100:2*pi; 0.8 1 Sine function p / p ; >> y = sin(x); >> plot(x,y) 0.2 0.4 0.6 x

MATLAB enables you to add axis Labels and titles, e.g.

\ i -0.4 -0.2 0 Si n e o f x >> xlabel('x=0:2\pi'); >> ylabel('Sine of x'); >> tile('Sine function') -1₀ ₁ ₂ ₃ ₄ ₅ ₆ ₇ -0.8 -0.6 x=0:2 x 0:2

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Example 1: Sine Wave

Example

1:

Sine

Wave

0.5 1

Sine Wave Signal

Fs = 150; % Sampling frequency

t = 0:1/Fs:1; % Time vector of 1 second

f = 5; % Create a sine wave of f Hz.

i (2* i*t*f) -0.5 0 Am p lit u d e x = sin(2*pi*t*f); nfft = 1024; % Length of FFT

% Take fft, padding with zeros so that length(X) is equal to nfft

X = fft(x,nfft);

% FFT is symmetric, throw away second half

0 0.2 0.4 0.6 0.8 1

-1

Time (s)

80

Power Spectrum of a Sine Wave

X = X(1:nfft/2);

% Take the magnitude of fft of x

mx = abs(X); % Frequency vector f = (0:nfft/2-1)*Fs/nfft; 40 60 80 P ow e r

% Generate the plot, title and labels.

figure(1); plot(t,x);

title('Sine Wave Signal'); xlabel('Time (s)'); l b l('A lit d ') 0 10 20 30 40 50 60 70 80 0 20 Frequency (Hz) P ylabel('Amplitude'); figure(2); plot(f,mx);

title('Power Spectrum of a Sine Wave'); xlabel('Frequency (Hz)');

ylabel('Power');

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Example 2: Cosine Wave

Example

2:

Cosine

Wave

x = cos(2*pi*t*f);

0.5 1

Cosine Wave Signal

x = cos(2*pi*t*f);

nfft = 1024; % Length of FFT

X = fft(x,nfft);

-0.5 0 Am p lit u d e y , y X = X(1:nfft/2);

mx = abs(X); % Frequency vector f = (0:nfft/2-1)*Fs/nfft; 0 0.2 0.4 0.6 0.8 1 -1 Time (s) 80

Power Spectrum of a Cosine Wave

title('Sine Wave Signal'); xlabel('Time (s)'); ylabel('Amplitude'); 40 60 80 P ow e r ylabel( Amplitude ); figure(2); plot(f,mx);

title('Power Spectrum of a Sine Wave'); xlabel('Frequency (Hz)'); ylabel('Power'); 0 10 20 30 40 50 60 70 80 0 20 Frequency (Hz) P y Frequency (Hz)

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Example 3: Cosine Wave with Phase Shift

Example

3:

Cosine

Wave

with

Phase

Shift

pha = 1/3*pi; % phase shift

0.5 1

Cosine Wave Signal with Phase Shift

pha = 1/3*pi; % phase shift

x = cos(2*pi*t*f + pha); nfft = 1024; % Length of FFT

% Take fft, padding with zeros so that length(X) is equal to nfft X = fft(x,nfft); -0.5 0 Am p lit u d e ( , )

X = X(1:nfft/2);

mx = abs(X); % Frequency vector / / 0 0.2 0.4 0.6 0.8 1 -1 Time (s) 80

Power Spectrum of a Cosine Wave Signal with Phase Shift

f = (0:nfft/2-1)*Fs/nfft;

title('Sine Wave Signal'); xlabel('Time (s)'); 40 60 80 P ow e r xlabel( Time (s) ); ylabel('Amplitude'); figure(2); plot(f,mx);

title('Power Spectrum of a Sine Wave'); xlabel('Frequency (Hz)'); 0 10 20 30 40 50 60 70 80 0 20 Frequency (Hz) P q y ylabel('Power'); Frequency (Hz)

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Example 4: Square Wave

Example

4:

Square

Wave

0.5 1

Square Wave Signal _{Fs = 150;}_{% Sampling frequency}

x = square(2*pi*t*f); -0.5 0 Am p lit u d e x = square(2*pi*t*f); nfft = 1024; % Length of FFT

X = fft(x,nfft);

0 0.2 0.4 0.6 0.8 1

-1

Time (s)

100

Power Spectrum of a Square Wave

y , y

X = X(1:nfft/2);

mx = abs(X); % Frequency vector f = (0:nfft/2-1)*Fs/nfft; 40 60 80 100 Po w e r

title('Square Wave Signal'); xlabel('Time (s)'); ylabel('Amplitude'); 0 20 40 60 80 0 20 40 Frequency (Hz) ylabel( Amplitude ); figure(2); plot(f,mx);

title('Power Spectrum of a Square Wave'); xlabel('Frequency (Hz)');

ylabel('Power'); y

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Example 5: Square Pulse

Example

5:

Square

Pulse

0.8 1

Square Pulse Signal _{Fs = 150;}_{% Sampling frequency}

t = -0.5:1/Fs:0.5; % Time vector of 1 second

w = .2; % width of rectangle

x = rectpuls(t w); % Generate Square Pulse

0.2 0.4 0.6 Am p lit u d

e x = rectpuls(t, w); % Generate Square Pulse

nfft = 512; % Length of FFT

X = fft(x,nfft);

-0.50 0 0.5

Time (s)

30

Power Spectrum of a Square Pulse

y , y

X = X(1:nfft/2);

mx = abs(X); % Frequency vector f = (0:nfft/2-1)*Fs/nfft; 15 20 25 Po w e r

title('Square Pulse Signal'); xlabel('Time (s)'); ylabel('Amplitude'); 0 20 40 60 80 0 5 10 Frequency (Hz) ylabel( Amplitude ); figure(2); plot(f,mx);

title('Power Spectrum of a Square Pulse'); xlabel('Frequency (Hz)');

ylabel('Power'); y

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Example 6: Gaussian Pulse

Example

6:

Gaussian

Pulse

3 4

Gaussian Pulse Signal

Fs = 60; % Sampling frequency t = -.5:1/Fs:.5; x = 1/(sqrt(2*pi*0.01))*(exp(-t.^2/(2*0.01))); nfft = 1024; % Length of FFT 1 2 A m pl it ud e nfft = 1024; % Length of FFT

X = fft(x,nfft);

X = X(1:nfft/2);

-0.50 0 0.5

Time (s)

( )

mx = abs(X);

% This is an evenly spaced frequency vector

40 50 60

Power Spectrum of a Gaussian Pulse

r

title('Gaussian Pulse Signal'); xlabel('Time (s)'); ylabel('Amplitude'); figure(2); 0 10 20 30 Po w e r figure(2); plot(f,mx);

title('Power Spectrum of a Gaussian Pulse'); xlabel('Frequency (Hz)');

ylabel('Power');

0 5 10 15 20 25 30

0

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Example 7: Exponential Decay

Example

7:

Exponential

Decay

1.5 2

Exponential Decay Signal

x = 2*exp(-5*t); nfft 1024 % Length of FFT 0.5 1 Am p lit u d e nfft = 1024; % Length of FFT

X = fft(x,nfft);

% FFT is symmetric, throw away second

half 0 0.2 0.4 0.6 0.8 1 0 Time (s) P S t f E ti l D Si l half X = X(1:nfft/2);

mx = abs(X);

40 50 60 70

Power Spectrum of Exponential Decay Signal

er

title('Exponential Decay Signal'); xlabel('Time (s)'); 0 20 40 60 80 0 10 20 30 Po w e _xlabel(_{'Time (s)'}_); ylabel('Amplitude'); figure(2); plot(f,mx);

title('Power Spectrum of Exponential

Decay Signal'); 0 20 40 60 80 Frequency (Hz) y g ); xlabel('Frequency (Hz)'); ylabel('Power');

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Example 8: Chirp Signal

Example

8:

Chirp

Signal

0.5 1

Chirp Signal _{Fs = 200;}_{% Sampling frequency}

x = chirp(t,0,1,Fs/6); nfft = 1024; % Length of FFT -0.5 0 Am p lit u d e ; g

X = fft(x,nfft);

X = X(1:nfft/2);

0 0.2 0.4 0.6 0.8 1

-1

Time (s)

Power Spectrum of Chirp Signal

mx = abs(X);

% Generate the plot title and labels

15 20 25 p p g w e r

title('Chirp Signal'); xlabel('Time (s)'); ylabel('Amplitude'); 0 20 40 60 80 100 0 5 10 Po w y p figure(2); plot(f,mx);

title('Power Spectrum of Chirp Signal'); xlabel('Frequency (Hz)');

ylabel('Power');