Review on Fourier …

Slides edited from:

• Prof. Brian L. Evans and Mr. Dogu Arifler Dept. of Electrical and Computer Engineering The University of Texas at Austin course:

EE 313 Linear Systems and Signals Fall 2003

Fourier Series

Spectrogram Demo (DSP First)• Sound clips

– Sinusoid with frequency of 660 Hz (no harmonics)

– Square wave with fundamental frequency of 660 Hz

– Sawtooth wave with fundamental frequency of 660 Hz

– Beat frequencies at 660 Hz +/- 12 Hz

• Spectrogram representation– Time on the horizontal axis

– Frequency on the vertical axis

Frequency Content Matters• FM radio

– Single carrier at radio station frequency (e.g. 94.7 MHz)

– Bandwidth of 165 kHz: left audio channel, left – right audio channels, pilot tone, and 1200 baud modem

– Station spacing of 200 kHz

• Modulator/Demodulator (Modem)

ChannelTransmitter

Receiver

Receiver

Transmitter

Home ServiceProvider

upstream

downstream

Residential Application Downstream rate (kb/s)

Upstream rate (kb/s)

Willing to pay Demand Potential

Database Access 384 9 High Medium On-line directory; yellow pages 384 9 Low High Video Phone 1,500 1,500 High Medium Home Shopping 1,500 64 Low Medium Video Games 1,500 1,500 Medium Medium Internet 3,000 384 High Medium Broadcast Video 6,000 0 Low High High definition TV 24,000 0 High Medium

Business Application Downstream rate (kb/s)

Upstream rate (kb/s)

Willing to pay Demand Potential

On-line directory; yellow pages 384 9 Medium High Financial news 1,500 9 Medium Low Video phone 1,500 1,500 High Low Internet 3,000 384 High High Video conference 3,000 3,000 High Low Remote office 6,000 1,500 High Medium LAN interconnection 10,000 10,000 Medium Medium Supercomputing, CAD 45,000 45,000 High Low

Demands for Broadband Access

Cou

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y of

Mil

os M

ilos

evic

(Sc

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DSL Broadband Access Standards

Cou

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Sha

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McC

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n (C

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xDSL Meaning Data Rate Mode Applications ISDN Integrated Services

Digital Network 144 kbps Symmetric Internet Access, Voice,

Pair Gain (2 channels) T1 T-Carrier One

(requires two pairs) 1.544 Mbps Symmetric Business, Internet

Service HDSL High-Speed Digital

Subscriber Line (requires two pairs)

1.544 Mbps Symmetric

Pair Gain (12 channels), Internet Access, T1/E1 replacement

SHDSL Single Line HDSL 1.544 Mbps Symmetric Same as HDSL except pair gain is 24 channels

Splitterless ADSL

Splitterless Asymmetric DSL (G.Lite)

Up to 1.5 Mbps Up to 512 kbps

Downstream Upstream

Internet Access, Video Phone

Full-Rate ADSL

Asymmetric DSL (G.DMT)

Up to 10 Mbps Up to 1 Mbps

Downstream Upstream

Internet Access, Video Conferencing, Remote LAN Access

VDSL Very High-Speed Digital Subscriber Line (proposed)

Up to 22 Mbps Up to 3 Mbps Up to 6 Mbps

Downstream Upstream

Symmetric

Internet Access, Video-on-demand, ATM, Fiber to the Hood

channel frequency response

a subchannel

frequency

ma

gn

itude

a carrier

Multicarrier Modulation• Discrete Multitone (DMT) modulation

ADSL (ANSI 1.413) and proposed for VDSL

• Orthogonal Freq. Division Multiplexing (OFDM)Digital audio/video broadcasting (ETSI DAB-T/DVB-T)

Cou

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Harmonically related carriers

Periodic Signals• f(t) is periodic if, for some positive constant T0

For all values of t, f(t) = f(t + T0)

Smallest value of T0 is the period of f(t).

sin(2fot) = sin(2f0t + 2) = sin(2f0t + 4): period 2.

• A periodic signal f(t)Unchanged when time-shifted by one periodTwo-sided: extent is t (-, )May be generated by periodically extending one periodArea under f(t) over any interval of duration equal to the

period is the same; e.g., integrating from 0 to T0 would give the same value as integrating from –T0/2 to T0 /2

Sinusoids• f0(t) = C0 cos(2 f0 t + )

• fn(t) = Cn cos(2 n f0 t + n)

• The frequency, n f0, is the nth harmonic of f0

• Fundamental frequency in Hertz is f0

• Fundamental frequency in rad/s is = 2 f0

Cn cos(n 0 t + n) = Cn cos(n) cos(n 0 t) - Cn sin(n) sin(n 0 t) = an cos(n 0 t) + bn sin(n 0 t)

Fourier Series• General representation

of a periodic signal

• Fourier seriescoefficients

• Compact Fourierseries

1

000 sincosn

nn tnbtnaatf

n

nn

nnn

nnn

a

b

bacac

tncctf

1

2200

100

tan

and, , where

cos

0

0

0

0 00

0 00

00

0

sin2

cos2

1

T

n

T

n

T

dttntfT

b

dttntfT

a

dttfT

a

Existence of the Fourier Series• Existence

• Convergence for all t

• Finite number of maxima and minima in one period of f(t)

0

0

Tdttf

ttf

Example #1

. as amplitudein decrease and

161

8 504.0 2sin

2

161

2 504.0 2cos

2

504.0121

2sin2cos

20

2

20

2

2

0

20

10

nba

n

ndtnteb

ndtntea

edtea

ntbntaatf

nn

t

n

t

n

t

nnn

• Fundamental periodT0 =

• Fundamental frequencyf0 = 1/T0 = 1/ Hz

0 = 2/T0 = 2 rad/s

0

1e-t/2

f(t)

12

2sin42cos161

21504.0

n

ntnntn

tf

Example #2

• Fundamental periodT0 =

• Fundamental frequencyf0 = 1/T0 = 1/ Hz

0 = 2/T0 = rad/s

0

A

f(t)

-A

,15,11,7,38

,13,9,5,18

even is 0

22

22

nn

A

nn

An

bn

2/3

2/1

2/1

2/1

0

10

) sin( ) 22(2

) sin( 22

symmetric) odd isit (because 0

plot) theof inspection(by 0

sin) cos(

dttnπtAA

dttnπtAb

a

a

tnπbtnπaatf

n

n

nnn

Example #3

• Fundamental periodT0 =

• Fundamental frequencyf0 = 1/T0 = 1/ Hz

0 = 2/T0 = 1 rad/s

,15,11,7,3

,15,11,7,3 allfor 0

odd

2even 0

2

10

n

n

nn

nC

C

n

n

1

f(t)

Fourier Analysis

Periodic Signals• For all t, x(t + T) = x(t)

x(t) is a period signal

• Periodic signals havea Fourier seriesrepresentation

• Cn computes the projection (components) of x(t) having a frequency that is a multiple of the fundamental frequency 1/T.

2

2

2

2

1

T

T

tT

mj

n

m

tT

mj

n

dtetxT

C

eCtx

Fourier Integral

Processing SignalSystemsion Communicat

2

1

2

2

deXtxdfefGtg

dtetxXdtetgfG

tjtfj

tjtfj

dttg

• Conditions for the Fourier transform of g(t) to exist (Dirichlet conditions):x(t) is single-valued with finite maxima and minima in

any finite time interval

x(t) is piecewise continuous; i.e., it has a finite number of discontinuities in any finite time interval

x(t) is absolutely integrable

Laplace Transform• Generalized frequency variable s = + j

• Laplace transform consists of an algebraic expression and a region of convergence (ROC)

• For the substitution s = j or s = j 2 f to be valid, the ROC must contain the imaginary axis

dsesFtf

dtetfsF

ts

ts

2

1

Fourier Transform• What system properties does it possess?

Memoryless CausalLinear Time-invariant

• What does it tell you about a signal?• Answer: Measures frequency content• What doesn’t it tell you about a signal?• Answer: When those frequencies occurred in time

Useful Functions• Unit gate function (a.k.a. unit pulse function)

• What does rect(x / a) look like?• Unit triangle function

2

11

2

1

2

12

10

rect

x

x

x

x

2

121

2

10

xx

xx

0

1

1/2-1/2x

rect(x)

0

1

1/2-1/2x

(x)

Useful Functions• Sinc function

– Even function

– Zero crossings at

– Amplitude decreases proportionally to 1/x

it? handle toHow 0. togoingboth arer denominatoandnumerator 0, As

sinc(0)? compute toHow

sinsinc

x

x

xx

0

1

x

sinc(x)

... ,3 ,2 , x

Fourier Transform Pairs

0

1

/2-/2t

f(t)

0

F()

2sinc

2

2sin

2sin2

1

rect

2/2/

2/

2/

jj

tjtj

eej

dtedtet

F

F

Fourier Transform Pairs

1

impulse, theofproperty sampling theFrom

0

tjtj edtettF

0

1

t

f(t) = 1

F() = 2 ()

(2)F

(2) means that the area under the spike is (2)

0

Fourier Transform Pairs

000

0

00

001

cos2

1cos Since

or 2

12

1

2

1

00

00

0

t

eet

ee

edeF

tjtj

tjtj

tjtj

0

F()

000t

f(t)

F()()

Fourier Transform Pairs

ja

j

jaja

tueFtueFtF

tuetuet

tt

a

a

atat

a

atat

a

22lim

11lim

limsgn

lim01

01sgn

220

0

0

0

1

t

sgn(t)

-1

Fourier Transform Properties

Fourier vs. Laplace Transform Pairsf(t) F(s) Region of Convergence F()

e-at u(t) 1 s + a

Re{s} > -Re{a} 1 j + a

e-a|t| 2a a2 – s2

-Re{a} < Re{s} < Re{a} 2a 2 + a2

(t) 1 complex plane 1

1 2(s) complex plane 2()

u(t) 1 s

Re{s} > 0 () + 1/(j

cos(0t) [( + 0) + ( – 0)]

sin(0t) j[( + 0) - ( – 0)]

eat u(t) 1 s - a

Re{s} > Re{a}

Assuming that Re{a} > 0

dtetfF tj

deFtf tj

2

1

Ftf ftF 2

0

1

/2-/2t

f(t)

0

F()

Duality• Forward/inverse transforms are similar

• Example: rect(t/) sinc( / 2)– Apply duality sinc(t /2) 2 rect(-/)

– rect(·) is even sinc(t /2) 2 rect(/)

Scaling• Same as Laplace

transform scaling property|a| > 1: compress time axis, expand frequency axis

|a| < 1: expand time axis, compress frequency axis

• Effective extent in the time domain is inversely proportional to extent in the frequency domain (a.k.a bandwidth).f(t) is wider spectrum is narrower

f(t) is narrower spectrum is wider

aF

aatf

Ftf

1

Time-shifting Property• Shift in time

– Does not change magnitude of the Fourier transform

– Does shift the phase of the Fourier transform by -t0 (so t0 is the slope of the linear phase)

Fettf tj 0 0

Frequency-shifting Property

000

000

000

000

0

0

sin

2

1 sin

2

1

2

1 cos

2

1 cos

2

20

0

Fj

Fj

tft

FjFjtft

FFtft

FFtft

Ftfe

Ftfetj

tj

00

000

00

0

2

1

2

1

So,

thatRecall2

1

:domainfrequency thein nconvolutio

is domain time thein tionMultiplica

cos

FFY

ttxdtxttttx

txdtxttx

FY

ttfty

Modulation

Modulation• Example: y(t) = f(t) cos(0 t)

f(t) is an ideal lowpass signal

Assume 1 << 0

• Demodulation is modulation followed by lowpass filtering

• Similar derivation for modulation with sin(0 t)

0

1

-

F()

0

Y()

- - - +

F

- +

F

Time Differentiation Property• Conditions

f(t) 0, when |t| f(t) is differentiable

• Derivation of property:Given f(t) F()

tdfe

dtedt

tdfB

tfdt

dFB

tj

tj

)(Let

Fjdt

tdf

Fjdt

tdf

Fjdtetfj

edtftfeB

ejdu

tdfdveu

duvv - udvu

n

n

n

tj

tj

t

tj

tj

tj

so

),( and Let

thatrecall rule, chain theFrom

Time Integration Property

j

FFdxxf

j

FF

jF

tutf

dxxtuxfdxxf

dxxf

0

Therefore,

0

1

:nconvolutio timeofproperty theFrom

? Find

t

-

-

t

-

t

-

Summary• Definition of Fourier Transform

• Two ways to find Fourier Transform– Use definitions

– Use properties

dtetfF tj