111

Lecture 2Signals and Systems (II)

Principles of Communications

Fall 2008

NCTU EE Tzu-Hsien Sang

2

Outlines

• Signal Models & Classifications

• Signal Space & Orthogonal Basis

• Fourier Series &Transform

• Power Spectral Density & Correlation

• Signals & Linear Systems

• Sampling Theory

• DFT & FFT

2

More on LTI Systems• A system is BIBO if output is bounded, given

any bounded input.

• 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.

3

conditionDirichlet ofelement main

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

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

dhdhtx

dtxhty

0for ,0)(

)()()()()(0

tth

dtxhdtxhty

• Paley-Wiener Condition:

• Remarks: (1) |H(f)| cannot grow too fast.

(2) |H(f)| cannot be exactly zero over a finite band of frequency.

• 2nd ver.:

4

.1

ln

,for 0)( and ,)( If

2

2

dff

H(f)

tthdffH

,1

ln and )( If

2

2

df

f

H(f)dffH

).for 0)(( causal is )(such that tthfHH

Eigenfunctions of LTI Systems

• Another way of taking complicated things part.• Instead of trying to find a set of orthogonal basis

functions, let’s look for signals that will not be changed “fundamentally” when passing them through an LTI system.

• Why?• Consider the key words: analysis/synthesis.• Note: Eigen-analysis is not necessarily consistent

with orthogonal basis analysis.5

• If , where is a constant, then is the eigenvalue for the eigenfunction g(t).

• Let

6

)()}({ tgtgΗ

numbercomplex arbitrary an : ,)( sAetx st

.)( where,)(

])([)()( )(

dAehtx

AedehdAehty

s

ststs

).(])([)(then

,2Let

22 txAedehty

tfjs

tfjfj

i

ii

7

• (Cross)correlation functions related by LTI systems:

• Note: In proving them, we use:

)(|)(|)( .4

)()()( .3

)()()()( .2

)()()()()( .1

2

*

fSfHfS

fSfHfS

RhhR

dRhRhR

xy

xyx

xy

xxyx

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

Filters!!!

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• Since almost any input x(t) can be represented by a linear combination of orthogonal sinusoidal basis functions , we only need to input to the system to characterize the system’s properties, and the eigenvalue

carries all the system information responding to . (Frequency response!!!)

• In communications, signal distortion is of primary concern in high-quality transmission of data. Hence, the transmission channel is the key investigation target.

ftje 2

ftjAe 2

)()( 2 fHdteth ftj

ftjAe 2

9

• Three major types of distortion caused by 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. (Q: What good is linear phase?)

3. Nonlinear distortion: nonlinear system

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• Example: Group Delay

).()( ,)(

2

1)( :Definition fHf

df

fdfTg

delays.different

have componentsfrequency different constant a

not is )(phaselinear not is system theIf

constant. a is which ,)(

2)( system,linear aFor

0

00

fT

tfT

ftHf

g

g

11

.componentsfrequency output theand

input ebetween th relations phase relative theshowsIt

.2

)()( :delay Phase

f

ffTp

12

• Example: Ideal general filters

13

• Realizable filters approximating ideal filters

14

The Uncertainty Principle• It can be argued that a narrow time signal has

a wide (frequency) bandwidth, and vice versa:constant)()( bandwidthduration

2

11

)0(

)0(2

)0()(|)(|)0(2 and

,|)()(|)(|)0(

proof), anot is (thisargument area-equal By the

0

TWTX

xW

xdffXdffXWX

fXdttxdttxTx f

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Sampling Theory

• You’ve probably heard of “signal processing.” But how to process a signal?

• For instance, the rectifier – max{x(t), 0}.• But, how to do Fourier transform of an arbitrary

signal x(t)?• Computers seem a good idea. But computers can only

work on numbers.• We need to “transform” the signal first into numbers.• Q: Tell discrete signals from digital signals.

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• Ideal sampling signal: impulse train (an analog signal) , T: the sampling period

• Analog (continuous-time) signal:

• Sampled (continuous-time) signal:

n

nTtts

)(tx)(tx

nss

ns

ns

nTtnTxnTttx

nTttxtstxtx

k kssss

kss

kffXfkfffXf

kffffXfSfXfX

][)(

Hopefully the math becomes easier in ideal case. The concept actually is harder.

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• Aliasing: If The replicas of X(f) overlap in frequency domain. That is, the higher frequency components of overlap with the lower frequency components of X(f-fs).

2 .sf W

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– Nyquist Sampling Theorem:

• Let x(t) be a bandlimited signal with X(f) = 0 for . (i.e., no components at frequencies greater than W.) Then x(t) is uniquely determined by its samples if .

Wf ||

,2,1,0),(][ nnTxnx s

WT

fs

s 21

Undersampling: 2sf W

Oversampling: 2sf W

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• In other words, oversampling preserves all the information that x(t) contains. It is possible to reconstruct x(t) purely by its samples.

• Ideal reconstruction filter (interpretation in frequency domain:

• In time domain:

WfBWeB

fHfH s

ftj ,)2

()( 020

)()(

)()(

00

20

0

ttxHfty

efXHffY

s

ftjs

0 0( ) ( ) ( ) 2 ( )sinc[2 ( )]s s s sn n

y t x nT h t nT BH x nT B t t nT

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• Two types of reconstruction errors

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DFT & FFT

• You can view DFT as a totally new definition for a totally different set of signals. Or you can try to connect it to the Fourier Transform.

1,,1,0 ,

1,,1,0 ,1

1

0

2

1

0

2

NkexX

NneXN

x

N

n

N

nkj

nk

N

k

N

nkj

kn

1

0

1

0

/22 )()(|)()(

|)()(

N

n

nkN

N

n

Nnkj

N

kj

ezj

WnhenheHkH

zHeH j

22

• Fast Fourier Transform (FFT) is not a new transform, it is simply a fast way to compute DFT. So, don’t use FFT to denote the object that you want to compute; only use it to denote the tool that you use to compute it. (Gauss knew the method already!)

• Application example: Fast convolution via FFT:

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1

0

1

0

)()(1

)(1

)(N

k

nkN

N

k

nkN WkXkH

NWkY

Nny