Signal & Linear System Analysis

Principles of Communications I (Fall, 2002) Signal & Linear System Analysis

NCTU EE 1

Signal & Linear System Analysis

Signal Model and Classifications

Deterministic vs. Random

Deterministic signals: completely specified function of time. Predictable, no uncertainty

e.g. tAtx 0cos)( ω= , ∞<<−∞ t ; where A and 0ω are constants

Random signals (stochastic signals): take on random values at any given time instant and characterized by pdf (probability density function)

“Not completely predictable”, “with uncertainty”

e.g. nindex at time tossedshown when valued)( icenx =

Model: A (large, maybe ∞ ) set of waveforms each associated with a prob-ability measure

e.g. pdf characterizing the out-of-band radio noise


NCTU EE 2

Periodic vs. Aperiodic

Periodic signal: A signal )(tx is periodic iff ∃ a constant 0T , such that )()( 0 txTtx =+ , t∀

The smallest such 0T is called “fundamental period” or “simply period”. Aperiodic signal: cannot find a finite 0T such that )()( 0 txTtx =+ , t∀

Phasor signal & Spectra

A special periodic function tjjtj eAeAetx 00 )()(~ ωθθω ⋅== +

phasor rotating)(~ ≡tx ; phasor≡θjAe ; numbers real , ≡θA

Why this complex signal? 1. Key part of modulation theory 2. Construction signal for almost any signal 3. Easy mathematical analysis for signal 4. Phase carries time delay information

More on Phasor Signal:

1. Information is contained in A and θ (given a fixed ))(or 00 ωf .

2. The related real sinusoidal function:

)}(~Re{)cos()( 0 txtAtx =+= θω or )(*~21)(~

21)cos()( 0 txtxtAtx +=+= θω

3. In vector form graphically:


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4. Frequency-domain representation

Single-sided (SS) amp. & phasor vs. double-sided (DS):

Line spectra:

E.g. find SS and DS spectra of )sin()( 0 θω += tAtx

Ans: put into cosine form first

Singularity Functions: opposed to regular functions

Unit impulse function )(tδ :

1. Defined by

)0()()0()()()()(0

0

0

0xdttxdtttxdtttx === ∫∫∫

+

−

+

−

∞

∞−δδδ ; 1)(

0

0=∫

+

−

dttδ

2. It defines a precise sample point of x(t) at time t (or 0t if )( 0tt −δ )

dttttxtx )()()( 00 −= ∫∞

∞−δ

3. Basic function for linearly constructing a time signal

ττδτ dtxtx )()()( −= ∫∞

∞−

4. Properties: (Z & T, pp. 25-26 (don’t bother properties 5 and 6))

)(||

1)( ta

at δδ = ; )()( tt −= δδ : even function


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5. What is )(tδ precisely? some of intuitive ways of realizing it:

E.g. 1 ⎪⎩

⎪⎨⎧ <= →

elsewhere)(or otherwise,0

||,21lim)( 0

εεδ ε

tt

E.g. 2 2

0sin1lim)( ⎟

⎠⎞

⎜⎝⎛=

→ επ

πεδ

ε

tt

t

Unit step function u(t):

dttdutdtu

t )()( ;)()( == ∫ ∞−δλλδ

Signal types classified by energy & power This classification will be needed for the later analysis of deterministic and random signals

Energy: dttxET

TT ∫−→∞≡ 2|)(|lim

Power: dttxT

PT

TT ∫−→∞≡ 2|)(|

21lim

Energy Signals: iff 0)(P 0 =∞<< E

Power Signals: iff )(E 0 ∞=∞<< P


NCTU EE 5

Example-1 )()(1 tuAetx tα−=

Example-2 )()(2 tAutx =

Example-3 )cos()( 03 θω += tAtx

Note: 1. If x(t) is periodic, then it is meaningless to find its energy, we only need

to check its power

dttxT

PT

Tt∫+≡0

0

2

0|)(|1

2. Noise is often persistent and is often a power signal 3. Deterministic and aperiodic signals are often energy signals 4. A realizable LTI system can be represented by a signal and mostly is a

energy signal 5. Power measure is useful for signal and noise analysis 6. The energy and power classifications of signals are mutually exclusive

(cannot be both at the same time). But a signal can be neither energy nor power signal

Signal Space & Orthogonal Basis

Applying the Sophomore’s Linear Algebra

Basis vectors (for vector space): (essential in DSP & communication theory)

N-dimensional basis vectors: Nbbb ,,, 21 L

Degree of freedom and independence:

E.g.: In geometry, any 2-D vector ⎥⎦

⎤⎢⎣

⎡=

qp

x can be decomposed into com-

ponents along two orthogonal basis vectors, (or expanded by these two vectors)

2211 bxbxx +=

Meaning of “linear” in linear algebra:

222111 )()( byxbyxyx +++=+


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Basis functions (for function space): (indispensable for general signal analysis) A general function can also be expanded by a set of basis functions (in an approximation sense)

∑∞

−∞=

≈n

nn tXtx )()( φ

or more feasibly

∑=

≈N

nnn tXtx

1)()( φ

Define the matching (or correlation) operation as

∑ ∫∫∞

−∞=

∞

∞−

∞

∞−

≈n

mnnm dtttXdtttx )()()()( φφφ

If we define orthogonality as

⎩⎨⎧ =

≡−≡∫∞

∞− ..,0,1

)()()(womn

mndttt mn δφφ

then

mm Xdtttx =∫∞

∞−

)()( φ

E.g. )cos()( 0tmtm ωφ = for periodic even x(t); set of )(tmφ orthogonal?

E.g. Taylor’s expansion for cosine function, basis functions? orthogonal? Remarks:

1. Using Freshmen calculus can show that function approximation ex-pansion by orthogonal basis functions is an optimal LSE approxima-tion

2. Is there a very good set of orthogonal basis functions? 3. Concept and relationship of spectrum, bandwidth and infinite continu-

ous basis functions


NCTU EE 7

Fourier Series & Fourier Transform

Fourier Series: ∑∞

−∞=

=n

tnfjneXtx 02)( π

∫+ −= 00

0

02

0

)(1 Tt

t

tnfjn dtetx

TX π

Sinusoidal Representation

[ ]∑∞

=

−−++=

1

)(220

00)(ˆn

tfnjn

tnfjn eXeXXtx ππ

If x(t) is real,

nXjnn eXX ∠= , nXj

nn eXX −∠−− = nXj

n eX ∠−=

][)(ˆ )2()2(

10

00 XntnfjXntnfj

nn eeXXtx ∠+−∠+

∞

=

++= ∑ ππ

)2cos(2 01

0 nn

n XtnfXX ∠++= ∑∞

=

π Cosine FS

Note: Index starts from 1 (not ∞ )

Trigonometric FS:

[ ])2sin()sin()2cos()cos()(ˆ 001

0 tnfXtnfXXXtx nnn

n ππ ∠−∠+= ∑∞

=

∑∑∞

=

∞

=

++=1

001

0 )2sin()2cos(n

nn

n tnfbtnfaX ππ

where )cos(2 nnn XXa ∠= , )sin(2 nnn XXb ∠=

Or, ∫+

=00

0

)2cos()(20

0

Tt

tn dttnftxT

a π

∫+

= 00

0

)2sin()(20

0

Tt

tn dttnftxT

b π

∑ ∑∞

=

∞

=

++=1 1

000 )2sin()2cos(

2)(ˆ

n nnn tnfbtnfaatx ππ


NCTU EE 8

Properties of Fourier series

∫+ −= 00

0

02)(1 Tt

t

tnfj

nn dtetxX π

λ

“DC” coefficient: ∫+ −= 00

0

0)0(2

00 )(1 Tt

t

tfj dtetxT

X π

∫+

= 00

0

)(1

0

Tt

tdttx

T = average value of x(t)

“AC” coefficients: ∫+

−=00

0

)]2sin()2)[cos((100

0

Tt

tn dttnfjtnftxT

X ππ

∫ ∫+ +

−= 00

0

00

0

)2sin()(1)2cos()(10

00

0

Tt

t

Tt

tdttnftx

Tjdttnftx

Tππ

If x(t) is real, then, Xnjnnnn eXXjXX ∠=+= ]Im[]Re[

where ∫+

= 00

0

)2cos()(1]Re[ 00

Tt

tn dttnftxT

X π

dttnftxT

XTt

tn ∫+

= 00

0

)2sin()(1]Im[ 00

π

Hence,

]Re[]Re[ nn XX =− ]Im[]Im[ nn XX −=− *nn XX =− nn XX =− (even function) and

nn XX −∠=∠ − (odd function)

Linearity x(t) ↔ ak

y(t) ↔ bk

Ax(t)+By(t) ↔ Aak + Bbk

Time Reversal

x(t) ↔ ak

x(-t) ↔ a-k

Time Shifting

ktfjk2

0 ae )tx(t 00π−↔−


NCTU EE 9

Time Scaling x(αt) ↔ ak But the fundamental frequency changes

Multiplication x(t) ↔ ak

y(t) ↔ bk

x(t)y(t) ↔

Conjugation and Conjugate Symmetry

x(t) ↔ ak

x*(t) ↔ a*-k

If x(t) is real ⇒ a-k = ak*

Parseval’s Theorem

Power in time domain = power in frequency domain

dttxT

P Ttt

ox

201

1)(1

∫+=

∑∑∞

−∞=

∞

−∞==⎥⎦

⎤⎢⎣⎡=

nn

nno XXT

TPx 22

0

1

∑∞

−∞=−

llklba


NCTU EE 10

Example: half-rectified sinewave

Example:


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Fourier Series Good orthogonal basis functions for a periodic function:

1. Intuitively, basis functions should be also periodic

2. Intuitively, periods of the basis functions should be equal to the period or integer fractions of the target signal

3. Fourier found that sinusoidal functions are good and smooth functions to expand a periodic function

Synthesis & analysis: (reconstruction & projection) Given periodic )(tx with period

00

1f

T = , 00 2 fπω = , it can be synthe-

sized as

∑∞

−∞=

≈n

tjnneXtx 0)( ω

nX : Spectra coefficient, spectra amplitude response

To synthesize it we must first analyze it and find out nX . By orthogonality

∫+ −= 00

0

0)(1

0

Tt

t

tjnn dtetx

TX ω

Fourier Transform

Good orthogonal basis functions for a aperiodic function:

1. Already know sinusoidal func-tions are good choice

2. Sinusoidal components should not be in a “fundamental & har-monic” relationship

3. Aperiodic signals are mostly fi-nite duration

4. Consider the aperiodic function as a special case of periodic func-tion with infinite period

Synthesis & analysis: (reconstruction & projection) Given aperiodic )(tx with period

∞→=df

T 10 , dfd πωω 20 == , we can

synthesize it as

∫

∫∑∞

∞−

∞

∞−

∞

−∞=>−

=

=≈

dfefX

deXeXtx

ftj

tj

n

tjndnd

π

ωω

ωωω

π

2

0

)(

)(21lim)(

By orthogonality

)( )(

)()( 2

txofresponsefrequencytxofTransformFourier

dtetxfX ftj

≡≡

= ∫∞

∞−

− π

Hence,

FT Inverse)()( 2 ≡= ∫∞

∞−

dfefXtx ftj π


NCTU EE 12

Frequency components: 1. Decompose a periodic signal into

countable frequency components 2. Has a fundamental freq. and many

other harmonics

|| of phase:amplitude|:|

|| 00

nn

n

tjnXjn

tjnn

XXX

eeXeX n

∠

= ∠ ωω

3. Discrete line spectra

Power Spectral Density: 2|| nX

and (by Parseval’s equality)

∑∫∞

−∞=

+==

nn

Tt

tXdttx

TP 22

0

|||)(|1 00

0

In real basis functions:

∑

∑

∑

∞

=

∞

=

∞

=

−−

+

+=

∠++=

∠+=+

10

100

100

0

sin

cos

)cos(||2)(

)cos(||2

00

nn

nn

nnn

nn

tjnn

tjnn

tnB

tnAX

XtnXXtx

XtnXeXeX

ω

ω

ω

ω

ωω

note that *nn XX =− for real x(t)

Frequency components: 1. Decompose an aperiodic signal into

uncountable frequency components 2. No fundamental freq. and contain all

possible freq.

∞<<∞−= ∠

feefXefX ftjfXjftj ππ 2)(2 |)(|)(

3. Continuous spectral density

Energy Spectral Density: 2|)(|)( fXfG ≡

and

∫∫∞

∞−

∞

∞−== dffXdttxE 22 |)(||)(|

In real basis functions: As your exercise example!


NCTU EE 13

What x(t) has Fourier series? 1. Expansion by orthogonal basis func-

tions can be shown is equivalent to finding nX using the LSE (or MSE) cost function:

}])({[})]({[ 22 0∑−=

−=N

Nn

tjnnN eXtxEteE ω

2. Would 0})]({[ 2 →teE N as ∞→N ?3. This requires square integrable condi-

tion (for the power signal):

∫ ∞<0

2|)(|T

dttx

and not necessarily 0|)(| →teN

4. Dirichlet’s conditions: (a) finite no. of finite discontinuities (b) finite no. of finite max & min. (c) absolute integrable

∫ ∞<0

|)(|T

dttx

Dirichlet’s condition implies conver-gence almost everywhere, except at some discontinuities.

What x(t) has Fourier Transform? 1. Expansion by orthogonal basis func-

tions can be shown is equivalent to finding )( fX using the LSE (or MSE) cost function:

}])()({[})]({[ 222 dfefXtxEteET

T

ftjT ∫−−= π

2. Would 0})]({[ 2 →teE T as ∞→T ?3. This requires square integrable condi-

tion (for the energy signal):

∫∞

∞−∞<dttx 2|)(|

4. Dirichlet’s conditions: (a) finite no. of finite discontinuities (b) finite no. of finite max & min. (c) absolute integrable

∫∞

∞−∞<dttx |)(|

Dirichlet’s condition implies conver-gence almost everywhere, except at some discontinuities.

Some General Symmetry Definitions

(a) Symmetric (even): real )( ),()( txtxtx −=

(b) Conjugate –symmetric: )(*)( txtx −=

(c) Anti-symmetric (odd): real )( ),()( txtxtx −−=

(d) Conjugate –anti-symmetric: )(*)( txtx −−=


NCTU EE 14

Symmetry Properties of x(t) and Its Fourier Function

For real periodic x(t), *nn XX =− , or nnnn XXXX −∠=∠= −− |,||| .

For real aperiodic x(t), )()( * fXfX −= or |)(||)(| fXfX −= , )()( fXfX −−∠=∠

Example:

Fourier Transform of Singular Functions

)(tδ is not an energy signal (hence doesn’t satisfy Dirichlet condition).

However, its FT can be obtained by formal definition.

,1)( ⎯→⎯FTtδ ),(1 fFT δ⎯→⎯

,)( 020

fjFT AettA πδ −⎯→⎯− ),( 00 ffAAe FTtfj −⎯→⎯ δπ

Example: ∑∞

−∞=

−n

nTt )( 0δ

Fourier Transform of Periodic Signals—Periodic signals are not energy signals (don’t satisfy Dirichlet’s conditions)

Given a periodic )(tx , ∑∞

−∞=

=n

tjnneXtx 0)( ω

⇒ ∑∞

−∞=

−=n

n nffXfX )()( 0δ

Example-1 tf02cos π

Example-2 ∑∞

−∞=

−n

nTt )( 0δ


NCTU EE 15

Properties of Fourier Transform (p.223 of O.W.Y)


NCTU EE 16

Fourier Transform Pairs (p.225 of O.W.Y)


NCTU EE 17

Relationship Between FT of an Aperiodic Signals & Its Periodic Extension

Let FT of an aperiodic pulse signal p(t) be )()}({ fPtp =ℑ ,

We can generate a periodic signal )(tx by duplicating )(tp at every sT in-

terval , then

∑∑∞

−∞=

∞

−∞=

−=−=n

sn

s nTtptpnTttx )()(*])([)( δ

From convolution theorem,

∑∑

∑∞

−∞=

∞

−∞=

∞

−∞=

−=×−=

×−ℑ=

nsss

nss

ns

nffnfPffPnfff

fPnTtfX

)()()()(

)(]})({[)(

δδ

δ


NCTU EE 18

Poisson sum formula: By taking inverse FT of above eq.

∑∑

∑∑∞

−∞=

∞

−∞=

−

∞

−∞=

−∞

−∞=

−

=−ℑ=

−ℑ=−==ℑ

n

tnfjss

nsss

nsss

ns

senfPfnffnfPf

nffnfPfnTtptxfX

πδ

δ

21

11

)(})({)(

})()({)()()}({

∑∑∞

−∞=

∞

−∞=

=−n

tnfjss

ns

senfPfnTtp π2)()( --> Poisson sum formula

Power Spectral Density & Correlation Energy Signal

∫∫ −∞→

∞

∞−

−−−−

+=+=−∗=

ℑ∗ℑ=ℑ=ℑ≡T

TTdxxdxxxx

fXfXfXfXfG

λτλλλτλλττ

τφ

)()(lim)()()()(

)](*[)]([)](*)([)}({)( 1111

Time-average autocorrelation function

φ(0) = E signal energy

Time average autocorrelation function of power signals

⎪⎪⎩

⎪⎪⎨

⎧

+

+=

+≡

∫∫−∞→

signalpower periodic ifsignalpower aperiodic if

,)()(1

,)()(21lim

)()()(

0

*

0

*

*

T

T

TT

dttxtxT

dttxtxT

txxR

τ

τ

τττ

∫∞

∞−= dffSR )()0(

)}({)( τRfS ℑ= Power spectral density

For periodic power signal

∑∞

−∞=

−=ℑ=n

n nffXRfS )(||)}({)( 02 δτ


NCTU EE 19

Interpretation

1. )(τφ and )(τR measure the similarity between the signal at time t

and τ+t

2. )( fG and )( fS represents the signal energy or power per unit fre-

quency at freq. f.

Properties of )(τR :

1. ττ ∀≥== ,)()()0( 2 RtxpowerR , )0()}(max{ RR =τ

2. )(τR is even for real signal: )()()()( * τττ RtxtxR =−≡−

3. If x(t) does not contain a periodic component

2

||)()(lim txR =

→∞τ

τ

4. If x(t) is periodic with period 0T , then )(τR is periodic in τ with

the same period

5. fRfS ∀≥ℑ= ,0)}({)( τ

Crosscorrelation of two power signals )(),( tytx

∫−→∞−=

+=−≡

T

TT

xy

dttytxT

tytxtytxR

)()(21lim

)()()()()(

*

**

τ

τττ

Crosscorrelation of two energy signals )(),( tytx

∫∞

∞−−≡ dttytxxy )()()( * ττφ

Remarks: )()( * ττ −= yxxy RR , )()( * τφτφ −= yxxy


NCTU EE 20

Signals & Linear Systems

1. )}({)( txHty = ,

2. H: an operator representing some mechanism and operation on x(t) and/or y(t) to produce y(t); generally it can be approximated by a differential equation.

3. More specifically, the linear constant-coefficient differential eq. suits for most applications and makes life a lot easier!

Linear & Time-Invariant (LTI) System

Linear systems:

Satisfies superposition principle

)()()}({)}({)}()({)( 2122112211 tytytxHtxHtxtxHty +=+=+= αααα

Time-invariant:

Delayed input produces a delayed output (due to the non-delayed input)

)}({)( txHty = --> )()}({ 00 ttyttxH −=−

Complete characterization of LTI systems:

The unit impulse function is key to the characterization.

)}({)( tHth δ≡

λλδλ dtxtx )()()( −= ∫∞

∞−

λλδλλλδλ dtHxdtxHtxHty )}({)(})()({)}({)( −=−== ∫∫∞

∞−

∞

∞−

)(tx )(tyΗ


NCTU EE 21

If TI,

)()()()}({

)(*)()()()}({)(

000 ttydtthxttxH

thtxdthxtxHty

−=−−=−

≡−==

∫

∫

∞

∞−

∞

∞−

λλλ

λλλ


NCTU EE 22

Convolution form holds iff LTI

1. Duality of signal x(t) & system h(t)

λλλλλλ dtxhdthxty )()()()()( −=−= ∫∫∞

∞−

∞

∞−

Convolution theorem:

)()(})()({)()}({ fXfHdtxhfYty =−ℑ=ℑ ∫∞

∞−λλλ

Key application: generally )()( fXfH is easier than )(*)( thtx ….

BIBO stability:

A system is BIBO if output is bounded, given any bounded input

conditionDirichlet ofelement main |)(|

|)(||})(max{||})()(max{||})(max{|

⇒∞<⇒

∞<=−=

∫∫∫

∞

∞−

∞

∞−

∞

∞−

λλ

λλλλλ

dh

dhtxdtxhty

Causality:

1. A system is causal if: current output does not depend on future input; or current input does not contribute to the output in the past A truth in reality.

2. All the systems in nature world are causal. Causality generally is not a problem in most circumstance. However, it matters frequently when we want to counteract the effect of hostile communication channel, such that we need an ideal equalizer, which is non-causal.

0for ,0)(

)()()()()(0

<=⇒

−=−= ∫∫∞∞

∞−

tth

dtxhdtxhty λλλλλλ


NCTU EE 23

Paley-Wiener Condition:

If (1) ∞<∫∞

∞−dffH 2)(

(2) ∞<= tth ,0)(

⇒ ∞<+∫

∞

∞−df

fH(f)

21ln

Remarks: |H(f)| cannot grow too fast. |H(f)| cannot be exactly zero over a finite band of frequency

If (1) ∞<∫∞

∞−dffH 2)(

(2) ∞<+∫

∞

∞−df

fH(f)

21ln

⇒ ) ,0)(( causal is that H(f)such H(f) ∞<=∠∃ tth

Eigenfunctions of LTI System

If )()}({ tgtg α=Η , where α is a constant w.r.t time parameter, then

⎩⎨⎧

function eigen :)(eigenvalue :

tgα

numbercomplex arbitrary an : ,)( sAetx st=

λλα

αλλλλ

λ

λλ

dAeh

txAedehdAehty

s

ststs

−∞

∞−

−∞

∞−

−∞

∞−

∫∫∫

=

===

)(

)(])([)()( )(

Let tfjs iπ2= , then )(])([)( 22 txAedehty tfjfj ii αλλ πλπ == −∞

∞−∫


NCTU EE 24

Correlation functions related by LTI systems:

1. ∫∞

∞−−=∗= λλτλτττ dRhRhR xxyx )()()()()(

2. )()()()( * ττττ xy RhhR ∗∗−=

3. )()()( fSfHfS xyx ⋅=

4. )(|)(|)( 2 fSfHfS xy ⋅=

pf) (1) & (2) from definitions. (3) & (4) from )()}({ fHh −=−ℑ τ and

)()}({ ** fHh −=ℑ τ

System Transmission Distortion & System Frequency Response (a) Since almost any input x(t) can be represented by a linear combination of

orthogonal sinusoidal basis functions ftje π2 , we only need to input

ftjAe π2 to the system to characterize the system’s properties, and the

eigenvalue

)()( 2 fHdteth ftj == −∞

∞−∫πα

carries all the system information responding to ftjAe π2 .

(b) In communication theory, transmission distortion is of primary concern in high-quality transmission of data. Hence, the system representing transmission channel is the key investigation target.

Three types of distortion of a transmission channel:

1. Amplitude distortion: linear system but the amplitude response is not constant.

2. Phase (delay) distortion: linear system but the phase shift is not a linear function of frequency.

3. Nonlinear distortion: nonlinear system


NCTU EE 25

Group delay:

)()( ,)(21)( fHf

dffdfTg ∠=−= θθ

π

For a linear phase system, 00 2)( ftHf πθ −∠=

0)( tfTg = , a constant

If Tg(f) is not a constant, sinusoidal inputs of different frequencies have different delays.

Cf. Phase delay: fffTp π

θ2

)()( −=

Ideal general filters:


NCTU EE 26

Realizable filters approximating the ideal filters:

1. Butterworth filter: simple 2. Chebyshev filter: smaller maximum deviation 3. Bessel filter: approximately linear phase in passband


NCTU EE 27

Time-Bandwidth Product—uncertainty principle It can be argued that a narrow time signal has a wide (frequency) bandwidth, and vice versa:

constant)()( ≥× bandwidthduration

1. Definition of T: equal areas

0|)()(|)(|)0( =

∞

∞−

∞

∞−=≥= ∫∫ ffXdttxdttxTx

2. Definition of W:

)0()(|)(|)0(2 xdffXdffXWX =≥= ∫∫∞

∞−

∞

∞−

3. Combine 1 and 2:

211

)0()0(2 ≥⇒≥≥ TW

TXxW


NCTU EE 28

Sampling Theory Ideal sampling

Ideal sampling signal: impulse train (an analog signal)

( ) ( )∑∞

−∞=−=

nnTtts δ , T: sampling period

Analog (continuous-time) signal: )(tx

Sampled (continuous-time) signal: )(txδ

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )∑∑

∑∞

−∞=

∞

−∞=

∞

−∞=

−=−=

−==

nss

ns

ns

nTtnTxnTttx

nTttxtstxtx

δδ

δδ

( ) ( ) ( ) ( )

( ) ( ) ( )∑ ∑

∑∞

−∞=

∞

−∞=

∞

−∞=

−=−∗=

−∗=∗=

k kssss

kss

kffXfkfffXf

kffffXfSfXfX

δ

δδ

][)(

Two Cases:

(1) no aliasing: Wf s 2> , and

(2) aliasing: Wf s 2< , where W is the highest nonzero frequency

component of )( fX .

After sampling, the replicas of )( fX overlap in frequency domain.

That is, the higher frequency components of )( fX overlap with the

lower frequency components of ( )sffX − .


NCTU EE 29

Nyquist Sampling Theorem:

Let x(t) be a bandlimited signal with 0)( =fX for Wf ≥|| . (i.e., no com-

ponents at frequencies greater than W ) Then x(t) is uniquely determined

by its samples K,2,1,0),(][ ±±== nnTxnx s , if WT

fs

s 21≥= .

-- Undersampling: Wf s 2<

-- Overdampling: Wf s 2>

Reconstruction:

Ideal reconstruction filter:

WfBWeBfHfH s

ftj −≤≤Π= − ,)2

()( 020

π

)()()()(

00

20

0

ttxHftyefXHffY

s

ftjs

−=⇒=⇒ − π

Alternative expression

∑∑∞

−∞=

∞

−∞=

−−=−=n

ssn

ss nTttBnTxBHnTthnTxty )](2[sinc)(2)()()( 00

Two types of reconstruction errors


NCTU EE 30

∑∑−

=

−

=

==1

0

1

0

)()(1)(1)(N

k

nkN

N

k

nkN WkXkH

NWkY

Nny

DFT & FFT

DFT

1,,1,0 ,

1,,1,0 ,1

1

0

2

1

0

2

−==

−==

∑

∑−

=

−

−

=

NkexX

NneXN

x

N

n

Nnkj

nk

N

k

Nnkj

kn

L

L

π

π

Fast convolution by FFT

∑∑−

=

−

=

−

=

=

===

=1

0

1

0

/22 )()(|)()(

|)()(N

n

nkN

N

n

Nnkj

Nk

j

ezj

WnhenheHkH

zHeH j

ππω

ω

ωω

Signal & Linear System Analysis

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