3.0 fourier series representation of periodic signals 3.1 exponential/sinusoidal signals as building...

3.0 Fourier Series Representation of Periodic Signals

3.1 Exponential/Sinusoidal Signals as Building Blocks for Many Signals

Time/Frequency Domain Basis SetsTime Domain

Frequency Domain

𝑥 (𝑡)

𝑡

𝑡

𝑡

𝑛

𝑛𝑛

𝑥 [𝑛 ]

{𝑒 𝑗𝜔𝑛 ,𝜔∈ [0 ,2𝜋 ]}

Signal Analysis (P.32 of 1.0)

Response of A Linear Time-invariant System to An Exponential Signal

Initial Observation

– if the input has a single frequency component, the output will be exactly the same single frequency component, except scaled by a constant

time-invariant

scaling property

tyety

tyetxe

tytxe

tytxe

j

jj

tj

tj

0

00

0

0

tjeyty 00 0 t

Input/Output Relationship𝑦 (𝑡)𝑥 (𝑡)

Time Domain

Frequency Domain

𝛿(𝑡) h (𝑡)0 0𝑡 𝑡

matrix vectors

eigen valueeigen vector


More Complete Analysis– continuous-time

stsst

sstts

st

esHdehe

dehedeh

dtxhty

jrsetx

, 0


More Complete Analysis– continuous-time

sH

etx

dehsH

st

s

Transfer FunctionFrequency Response

: eigenfunction of any linear time-invariant system

: eigenvalue associated with the eigenfunction est


More Complete Analysis– discrete-time

k

k

nk

k

n

k

kn

k

jn

zkhzH

zzHzkhz

zkhknxkhny

cezznx

, 0

Transfer FunctionFrequency Response

eigenfunction, eigenvalue

System Characterization

Superposition Property– continuous-time

k k

tskk

tsk

kk esHatyeatx

k

nkkk

k

nkk zzHanyzanx

– discrete-time

– each frequency component never split to other frequency components, no convolution involved

– desirable to decompose signals in terms of such eigenfunctions

3.2 Fourier Series Representation of Continuous-time Periodic Signals

Fourier Series Representation

Harmonically related complex exponentials

,Ttxtx

T

et otjk

k 2 ,..... 2, 1, 0,k ,0

k

Ttk period with

all with period T

T: fundamental period

Harmonically Related Exponentials for Periodic Signals

[n]

[n](N)

(N)integer multiples of ω0

‧Discrete in frequency domain

Fourier Series RepresentationFourier Series

k

kkk

tjkk taeatx 0

ta jj : j-th harmonic components

real

kk aa

kkkk

kk

k

kjkkkk

jCBatkCtkBa

eAatkAatx

,sincos2

,cos2

1000

100

tx

Real Signals

For orthogonal basis:

(unique representation)

Fourier Series RepresentationDetermination of ak

, 1

, 1

0

0

dttxT

a

dtetxT

a

T

T

tjnn

Fourier series coefficients

dc component

nk

nkTdte

dteeadtetx

T

tnkj

Tk

tjntjkk

tjn

T

0,

,

0

000

Not unit vectororthogonal

(分析 )

Fourier Series RepresentationVector Space Interpretation

– vector space

Ttxtx period with periodic is :could be a vector space

some special signals (not concerned here) may need to be excluded

dttxtxtxtxT

2121


– orthonormal basis

......,2 ,1 ,0 ,1

,

,0

21

kttT

jiT

jitt

kk

ji

is a set of orthonormal basis expanding a vector space of periodic signals with period T


– Fourier Series

dtetxT

tT

txT

ttxT

a

tatxT

tatx

tjn

T

n

nn

kkk

kkk

0 1

11

1

1

21

21

'21

'21

Fourier Series RepresentationCompleteness Issue

– Question: Can all signals with period T be represented this way?

Almost all signals concerned here can, with exceptions very often not important

Fourier Series RepresentationConvergence Issue

– consider a finite series

22

,0

tedtteE

txtxteeatx

NT NN

NN

N

Nk

tjkkN

It can be shown

NkdtetxT

aE tjk

TkN ,...1,0 , 1 if min 0

ak obtained above is exactly the value needed even for a finite series

Truncated Dimensions


– It can be shown

dttxT

2 if

Fourier Series RepresentationGibbs Phenomenon

– the partial sum in the vicinity of the discontinuity exhibit ripples whose amplitude does not seem to decrease with increasing N

See Fig. 3.9, p.201 of text

Discontinuities in Signals

All basis signals are continuous, so converge to average values


– x(t) has no discontinuities

Fourier series converges to x(t) at every t

x(t) has finite number of discontinuities in each period

Fourier series converges to x(t) at every t except at the discontinuity points, at which the series converges to the average value for both sides


– Dirichlet’s condition for Fourier series expansion

(1) absolutely integrable,

(2) finite number of maxima & minima in a period

(3) finite number of discontinuities in a period

dttxT

3.3 Properties of Fourier Series

kFS atx

Linearity

kk

FS

kFS

kFS

BbAatBytAx

btyatx

,

Time Shift

ktjkFS aettx 00

0

phase shift linear in frequency with amplitude unchanged

Linearity

Time Shift

Time Reversal

tkjtke

atxtjk

kFS

00 sincos0

the effect of sign change for x(t) and ak are identical

Time Scaling

::

tx positive real number

k

tjkkeatx 0

periodic with period T/α and fundamental frequency αω0

ak unchanged, but x(αt) and each harmonic component are different

Time Reversal

unique representation for orthogonal basis

Time Scaling

Multiplication

kkj

jkjkFS

kFS

kFS

babadtytx

btyatx

,

Conjugation

real if , txaa

atx

kk

kFS

Multiplication

(⋯ ,𝑎0𝑏3+𝑎1𝑏2+𝑎2𝑏1+𝑎3𝑏0 ,⋯)⏟𝑒❑

𝑗3 𝜔0❑𝑡

𝑑3=∑𝑗

𝑎 𝑗𝑏3 − 𝑗

Conjugation

unique representation

Differentiation

kFS ajk

dttdx

0

Parseval’s Relation

k

kTadttx

T221

2201

kT

tjkk adtea

T

average power in the k-th harmonic component in a period T

total average power in a period T

but

Differentiation

3.4 Fourier Series Representation of Discrete-time Periodic Signals

Fourier Series Representation

Harmonically related signal sets

NNnxnx period with periodic ,

..... 2, 1, 0,k ,2

n

Njk

k en

all with period N

N 2 , 0

,nn krNk only N distinct signals in the set

else 0,

..2 ,0 ,2

N,....N, kNeNn

nN

jk

Harmonically Related Exponentials for Periodic Signals

[n]

[n](N)

(N)integer multiples of ω0

‧Discrete in frequency domain

(P. 11 of 3.0)

Continuous/Discrete Sinusoidals(P.36 of 1.0)

Exponential/Sinusoidal SignalsHarmonically related discrete-time signal sets

all with common period N

This is different from continuous case. Only N distinct signals in this set.

}2,........1,0,k , ][{)

2(

n

Njk

k en

][][ nn kNk

(P.42 of 1.0)

Orthogonal Basis

‒ N different values in

N-dimensional vector space

‒

𝑎𝑘= �⃗� ⋅𝑣𝑘

�⃗�=∑𝑘

𝑎𝑘𝑣𝑘 (合成 )

(分析 )

Nn

nN

jk

Nn

njkk

Nk

nN

jk

kNk

njkk

enxN

enxN

a

eaeanx

2

2

1 1 0

0

,krNk aa repeat with period N

Note: both x[n] and ak are discrete, and periodic with period N, therefore summed over a period of N

Fourier Series RepresentationFourier Series


Nnxnx period with periodic is ,

is a vector space

else 0,

,

2121

rNjiNnn

kxkxnxnx

ji

Nk


Nknn

N kk ,1 21

a set of orthonormal bases

njk

Nn

kk

Nkkk

enxN

nN

nxN

a

nanx

0 1

11 21

21

Fourier Series RepresentationNo Convergence Issue, No Gibbs

Phenomenon, No Discontinuity

– x[n] has only N parameters, represented by N coefficients

sum of N terms gives the exact value

– N odd – N even

2

1 if ,

2

NMnxnx

eanx

M

M

Mk

nN

jk

kM

2

if ,

1

2

NMnxnx

eanx

M

M

Mk

nN

jk

kM

See Fig. 3.18, P.220 of text

PropertiesPrimarily Parallel with those for continuous-time

Multiplication

kFS anx

Njjkjk

FS

kFS

kFS

badnynx

bnyanx ,

periodic convolution

Time Shift

Periodic Convolution𝑎 𝑗

𝑏𝑘− 𝑗

𝑗

𝑗

𝑑𝑘= ∑𝑗=⟨𝑁 ⟩

𝑎 𝑗𝑏𝑘− 𝑗

PropertiesFirst Difference

Parseval’s Relation

kN

jkFS aenxnx

2

11

22

1

Nk

kNk

anxN

average power in a period

average power in a period for each harmonic component

3.5 Application ExampleSystem Characterization

y[n], y(t)h[n], h(t)

x[n], x(t)δ[n], δ(t)

n

n

zn , e st H(z)zn, H(s)est, z=e jω, s=j e jωn , e jωt H(e jω)e jωn, H(j)e jωt

Superposition Property– Continuous-time

– Discrete-time

0

000

jkHaa

ejkHatyeatx

kk

tjk

k kk

tjkk

Njk

kk

nN

jk

Nk

Njk

kNk

nN

jk

k

eHaa

eeHanyeanx

2

222

– H(j), H(ejω) frequency response, or transfer function

Filteringmodifying the amplitude/ phase of the different frequency components in a signal, including eliminating some frequency components entirely

– frequency shaping, frequency selective

Example 1

j

j

aeeH

nxnayny

111


Example 2

Filtering


else , 0,

,1/1

1

1

MnNMNnh

knxMN

nyM

Nk

Examples

• Example 3.5, p.193 of text

Examples


dte T

a

TT

dt T

a

T

-T

T

-T

tjkk

0

1

1

1

21 10

0,

sin 10 kk

Tk

Examples


(a)

(b)

(c)

Examples


0,

sin ,0,

2

1

,

,

,,,

1010

0

111010

kk

Tkck

TT

c

Ta

cjkbtgdtdtq

aeaebTtxTtxtq

ctgbtqatx

k

k

kk

kTjk

kTjk

k

kkk


Examples

Nj

j

nN

jN

jnN

jN

j

jnj

n

nj

nN

jnN

j

n

ere

Nnr

eeHeeHny

eeeH

eeN

nnx

nunh

2

)2(2)2(2

0

)2()2(

1

1 where

2cos

21

21

11

21

212cos

1 ,

)(

)()(

)(

y[n], y(t)h[n], h(t)

x[n], x(t)δ[n], δ(t)

x[n] y[n]h[n]

• .

• .

• .

dtteE

txtxtetatxbatx

dttt

bait

b

a NN

NNi

N

NiiN

ij

b

a ji

i

2

ˆ , ˆ , ,over signal afor

,over functions lorthonorma ofset a ,...2,1,0,

Problem 3.66, p.275 of text

...2,1,0 , 0 , 0

j

when minshown becan It

ic

E

b

E

cba

dtttxaE

i

N

i

N

iii

b

a iiN

dtttxA

a

Adttt

b

a ii

ij

b

a ji

1

normalizednot functions basisFor


• 2-dimensional signals

212112

212121

220

110

)(21

21221121

220

2 1

110

220110

1 2

220110

),(11

) ,(1

2,2

,

, all ,,,

dtedtettxTT

dtdteettxTT

a

TT

eattx

ttTtTtxttx

tjn

T T

tjm

tjntjm

T Tmn

tntmj

n mmn


• 2-dimensional signals

3.0 fourier series representation of periodic signals 3.1 exponential/sinusoidal signals as building...

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