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FastFourierTransformand

MATLABImplementation

by

WanjunHuang

forfor

Dr.DuncanL.MacFarlane

1

Signals

In the fields of communications, signal processing, and in electrical engineering more generally, a signal is any timeͲvarying or spatialͲvarying 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 SignalscanbeclassifiedbycontinuesͲtimesignalanddiscreteͲtime signal: hbldfti tiil h as b eensamp l e d f romacon ti nuousͲ ti mes i gna l 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 2

PeriodicSignal

periodicsignalandnonͲperiodicsignal: 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] nn

•PeriodT:Theminimumintervalonwhich

asignalrepeats

Fundamentalfrequency:

f 0 1 T

Fundamental

frequency: f 0 1 T •Harmonicfrequencies:kf 0 •Anyperiodicsignalcanbeapproximated b y asumofman y sinusoidsatharmonicfre q uenciesofthesi g nal k f 0 with yyqg( f 0 appropriateamplitudeandphase )cos()sin()exp( t j t t j

Eulerformula:

3

TimeͲFrequencyAnalysis

Time Domian (Banded Wren Song)

01

Amplitude

Time

Domian

(Banded Wren Song) 12 Power

Frequency Domain

0 2 4 6 8 x 10 4 -1

Sample Number

A 0 200
400
600
800
1000
1200
0

Frequency (Hz)

TimeͲdomain fi

g ure: how a si g nal chan g es over time

Whyfrequencydomainanalysis?

g g g FrequencyͲdomain figure: how much of the signal lies within each given frequency band over a range of frequenciesWhy frequency domain analysis? 4

FourierTransformFourier

Transform

We can go between the time domain and the frequency domain b y usin g a tool calledFourier trans f orm frequencydomain(spectrum)

AiFifhfdi

y g f A n i nverse F our i ertrans f ormconvertst h e f requency d oma i n

Continuous

TimeFourierTransform:

dejFtf tj )(21)( dtetfjF tj

Continuous

Time

Fourier

Transform:

Discrete

r

TimeFourierTransform(DTFT):

Discrete

Time

Fourier

Transform(DTFT):

2 )(21][deeXnx njj nnjj enxeX 5

Representation

For Four Types of

Signals

fre q uenc y

Ͳdomaincharacteristics

s p ectrum qy p spectrum3

Discrete

non periodicsignal rrr >continuesperiodic spectrum 3

Discrete

non periodic signal continues periodic spectrum q uenc y ismost qy domainisbecausecom p uterscanonl y takefinitediscretetimesi g nals py g 6

PeriodicSequencePeriodic

Sequence

kNnxnx wherekis inte g erA periodic sequence with periodNis defined as:

Periodic

g knNjknN eW 2

For example:

N n k n N k kn W W W (it is calledTwiddle Factor)

Properties

Periodic

N n k N n N k N kn N W W W

Symmetric

nNk NnkN Nkn Nkn N WWWW

Properties

Orthogonal

otherrNnNW N kknN 0 1 0 x(n) represent the whole sequence 7

DiscreteFourierSeries(DFS)Discrete

Fourier

Series(DFS)

1 0 1 0 )(~1)(~)(~)( NknNN nknN WkX N nxWnxkX cosine functions kXIDFSnxnxDFSkX 0 n N is still a periodic sequence with periodNin frequency domain kX The Fourier series for the discreteͲtime periodic wave shown below: 1

Sequence x (in time domain)

0.2

Fourier Coeffients

0quotesdbs_dbs17.pdfusesText_23