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Fourier SeriesFourier Transform

Filtering

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• Let’s consider a set of orthogonal functions, βi(x)• And a real-valued function f(x)• Then the function f(x) can be represented in terms of βi(x)

• A generalized Fourier Series of f(x)

• αi are called the Fourier constants of f(x)• βi are called a set of “basis” functions• This can be viewed as signal expansions or signal

decomposition

baxxi ,)},({ Îb

i

xxxf

i

iii

"¹

++== å¥

=

,0 where

...)()()( 22111

a

bababa

3

• In real life, however, we cannot represent a function f(x) exactly with the bases

• So we approximate the function with the bases in the mean-squared error sense.

• Find αi such that is minimized => approximation principle

i

xxxxf

i

nn

n

iii

"¹

+++== å=

,0 where

)(...)()()( 22111

a

babababa)

2)()( xfxf

)-

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Signal Decomposition and Synthesis

Lecture No. 6

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Fourier Representation of Signals• Fourier basis

– Orthogonal basis

frequency lfundamenta,

],[,...}cos,sin,...,cos,sin,{

T

Rttktktt

pw

wwww

2

0

1

0

0000

=

Î

k-th harmonics

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Basic Idea of Signal Representation

Sum of Fourier basis represents any signal

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Fourier Series Approximation

of a Square Wave

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Approximating Sawtooth Wave

• First five successive partial Fourier series.

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Historical Note• Fourier series and integral is known as a most beautiful (?) development.• It is indispensable in problem solving in math, science, and engineering.• Maxwell once called it a great mathematical poem.• Some key words of Fourier: a tailor’s son, math genius, French Revolution,

Napoleon, Institute of Egypt, Rosetta Stone, …• Fourier claimed that an arbitrary function defined in a finite interval by an

arbitrarily capricious graph can always be expressed as a sum of sinusoids (i.e., Fourier series).

• But Fourier could not publish his results as a paper.• Why? Mostly opposition by Lagrange. Even Fourier unable to prove his

claim because the tools required for operations with infinite series were unavailable at that time.

• Fifteen years later, after several attempts and disappointments, Fourier finally published his results as a text (now it is a classic).

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Fourier Representations for Four Classes of Signals

• Describe complicated signals as a function of frequency

• The weight associated with a sinusoid of a given frequency represents the contribution of that sinusoid to the overall signal

• Fourier representations for four classes of signals– Continuous-time, periodic → Fourier Series (F.S.)– Continuous-time, non-periodic → Fourier Transform (F.T.)– Discrete-time (D.T.), periodic → D.T.F.S– Discrete-time, non-periodic → D.T.F.T

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Continuous-Time Periodic Signals: The Fourier Series

• Trigonometric Fourier Series: A signal can be constructed as a linear combination of harmonically related sinusoid signals

• CT periodic signals can be represented by the FS

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Trigonometric FS• Trigonometric FS for real-valued periodic signals

)(sin][)cos(][]0[)( 01

0 tkkAtkkBBtxk

wwå¥

=

++=

Fourier Coefficients or FS Coefficients (a.k.a a frequency domain representation of x(t))

dttktxT

kA

dttktxT

kB

dttxT

B

T

T

T

)sin()(2][

)cos()(2][

)(1]0[

00

00

0

w

w

ò

ò

ò

=

=

=

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Guitar String Harmonics and Chords

• http://www.youtube.com/watch?v=q9D6ceOWa3g

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Fourier Spectrum• Compact FS indicates a periodic signal f(x) can be expressed

as a sum of sinusoids of frequencies 0 (i.e., dc), , 2, …, ,… whose amplitude are , , , …, , … and whose phases are 0, , , …, , …

• Amplitude spectrum: vs. • Phase spectrum: vs. • These frequency spectra provide an alternative description of

f(x), the frequency domain description of f(t). • A signal has a dual identity: the time domain identify and its

frequency domain identity.• The two identities complement each other and provide a

better understanding of a signal.

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Complex Fourier Series Expansion

tkjtke

jetjk

j

00 sincos

sincos0 ww

qqw

q

+=

+=

Tke tjk pww 2,...,2,1,0},{ 00 ==

åå¥

-¥=

¥

-¥=+==

kk

tjk tkjtkkXekXtx )sin](cos[][)( 000 www

Complex sinusoidsBases

Def.

ò -= T tjk dtetxT

kX 00)(1][ w

Continuous-time periodic signals are represented by the FS

The FS coefficients of the signal x(t)

FS Pair

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Magnitude and Phase

c=a+jbMagnitude |c|=sqrt(a2+b2)Phase Arg{c}=arctan(b/a)

c=a+jbIn a polar form c=|c|ejarg{c}

For X[k]|X[k]|=magnitude of spectrum X[k]Arg{X[k]}=phase of X[k] or phase spectrum of X[k]

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Why the FS?• In some problems, it is advantageous to represent the

signal in the time domain as x(t), but in others the FS coefficients X[k] offers a more convenient description

• The FS coefficients are known as a frequency-domain representation of x(t) because each FS coefficient is associated with a complex sinusoid of a different frequency

• The variable k determines the frequency of the complex sinusoid associated with X[k].

• The FS representation is used in engineering to analyze the effect of systems on periodic signals.

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Discrete-Time Periodic Signals:The Discrete-Time Fourier Series

• A periodic signal x[n] with fundamental period of N and fundamental frequency N

p20 =W

å-

=

W=1

0

0][][N

k

njkekXnx

where å-

=

W-=1

0

0][1][N

k

njkenxN

kX DTFS coefficients of the signal x[n]

No infinity, Why?

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Continuous-Time Nonperiodic Signals: The Fourier Transform

- The FT is used to represent a CT nonperiodic signal as a superposition of complex sinusoids- The signal is represented as a weighted integral of complex sinusoids where the variable of integration is the sinusoid’s frequency- An integral over the entire frequency interval

dtetxjX

dejXtx

tj

tj

ò

ò¥¥-

-

¥¥-

=

=

w

w

w

wwp

)()(

)(21)(FT Pair

FT

Inverse FT

- X(jω) describes the signal x(t) as a function of frequency ω- Therefore called as the frequency-domain representation of x(t)

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Back to Convolution Properties

)()()(][*][][)()()()(*)()(WWW =Û=

=Û=jjj eXeHeYnxnhny

jXjHjYtxthty www

Filtering

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Frequency response of ideal continuous- (left panel) and discrete-time (right panel) filters. (a) Low-pass characteristic. (b) High-pass

characteristic. (c) Band-pass characteristic.

Ideal Filters (1)

Note the difference between and

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Ideal Filters (2)• Low-pass filter: filter out high-frequency components of the input.

Passes lower frequency components• High-pass filter: filter out low frequency parts. Passes high frequency

parts• Band-pass filter: passes signals within a certain frequency band

• Pass band=the band of frequencies that are passed by the system• Stop band=the range of frequencies that are attenuated by the system• Decibel (dB)=a common unit for the magnitude response of a filter.

dB=20log|H(jω)|• -3dB corresponds to 1/sqrt(2) of a magnitude response• -3dB corresponds to frequencies of which the filter only passes half of

the input power.

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Why Elementary Signals & Systems?

Lego Blocks

Systems Systems