1050: principles of communication system (i)

Commun. I Lecture2 - Signal and Linear System ([email protected])

2-1

1050: Principles of Communication System (I)

1050: Principles of 1050: Principles of Communication System (I)Communication System (I)

Lecture 2 Signal and Linear Lecture 2 Signal and Linear System AnalysisSystem Analysis


2-2

Introduction• Mathematical models for signals and

systems– Characteristics of signals and systems– Time-domain signal representation– Frequency-domain signal representation– Time-frequency signal analysis– Linear system – Time-invariant system


2-3

Signal Models• Two classes of signals

– Deterministic signals• Completely specified by a function of time• E.g.• Periodic and Aperiodic signals

– Random signals

• Signal representation– Waveform– Phasor– Complex signal :

• Amplitude A, phase θ, frequency ω0

– Spectra

tAtx 0cos)( ω=

∞<<∞−= ttAtx ,cos)( 0ω

tjtjjtj etxeAeAetx 000 )()(~ )( ωωθθω === +

00 2 fπω =

nindex at time tossedshown when valued)( icenx =


2-4

WaveformsTime-domain representation


2-5

PhasorsA (graphic) vector representation

Amplitude : AAngle or phase: θ

ProjectionDecomposition/addition

Information is contained in A and θ


2-6

Complex Signals• By Euler’s theorem• Time-domain representation

• Frequency-domain representation

– The parameters amplitude A and phase θ are sufficient to represent a signal, for a given or fixed f0

}Im{}Re{sincos θθθ θθ jjj ejeje ±=±=±

}Re{

)}(~Re{)cos()()(

0

0 θω

θω+=

=+=tjAe

txtAtx

)()(

*0

00

21

21

)(~21)(~

21)cos()(

θωθω

θω

+−+ +=

+=+=

tjtj AeAe

txtxtAtx

Easy mathematical analysis for signal


2-7

Spectra Frequency-domain representation

even symmetryodd symmetry

single-sided double-sided


2-8

Remarks• Two equivalent frequency representation for a real signal

– Single-sided (SS) band representation• It exists only for positive frequency component

– Double-sided (DS) band representation• It exists both the positive and negative frequency components

• For a DS frequency representation for a real signal– The amplitude spectrum has even symmetry and the magnitude

is a half of the DS amplitude spectrum.– The phase spectrum has odd symmetry about f=0.

• Every real signal can be represented by a series of sine and cosine functions (i.e. sinusoids) (complete property)

• A sum of sinusoids of differing frequencies consists of a multiplicity of lines, with one/pair line/lines for each sinusoidal component of the sum (superposition property)


2-9

Example 2.1• Skills:

– Leave the cosine form unchanged– Turn all the sine forms into cosine forms


2-10

Unit Impulse Function• A special aperiodic function• One of the singularity functions opposed to

regular functions• Definition

– Suppose that x(t) is continuous at t=0– The unit impulse function δ(t) is defined by

– Note that, if x(t) is continuous at t0, then

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

0

0

0xdttxdtttxdtttx === ∫∫∫

+

−

+

−

∞

∞−δδδ

1)(0

0=∫

+

−

dttδ

)()()()()()()( 00000

0

0

0

txdttttxdttttxdtttxt

t

t

t=−=−= ∫∫∫ +

−

+

−

∞

∞−δδδ

1)(0

00 =−∫ +

−

t

tdtttδ This is the shifting property

0 if ,0)( ≠= ttδ

00 if ,0)( tttt ≠=−δ


2-11

Properties of δ(t)• It defines a precise sample point of a continuous

function x(t) at time t=t0

•

• even function

•

•

dttttxtx )()()( 00 −= ∫∞

∞−δ

)(||

1)( ta

at δδ =

)()( tt −= δδ

⎪⎩

⎪⎨

⎧

=

<<=−∫

210

2010

0

or for undefined,otherwise,0

),()()(2

1

ttt

ttttxdttttx

t

tδ

0000 at continuous is )( that suppose ),()()()( tttxtttxtttx =−=− δδ


2-12

Realization of δ(t)• By definition, the impulse function is just a function having

unit area in an infinitesimally width.• Two examples

1)(0

0=∫

+

−

dttδ

0 if ,0)( ≠= ttδ

⎪⎩

⎪⎨⎧ ≤=Π=

otherwise,0

,21

)2

(21)( ε

εεεδε

ttt2

sin1)( ⎟⎠⎞

⎜⎝⎛=

επ

πεδε

tt

t


2-13

Unit Step Function• A special aperiodic function• One of the singularity functions opposed to

regular functions• Definition

dttdut

ttt

dtut

)()(or

;0,undefine;0,1;0,0

)()(

=

⎪⎩

⎪⎨

⎧

=><

== ∫ ∞−

δ

λλδ


2-14

Energy Signal vs. Power Signal• For an arbitrary signal x(t), we define the total

energy E as well as the average power P as

• x(t) is an energy signal if and only if

• x(t) is an power signal if and only if

dttxdttxET

TT ∫∫∞

∞−−∞→=≡ 22 |)(||)(|lim dttx

TP

T

TT ∫−∞→≡ 2|)(|

21lim

0 that implies ,0 =∞<< PE

)(E 0 ∞=∞<< P

∞=∞<< EP that implies ,0


2-15

Remarks• If x(t) is periodic of period T0, then it is meaningless to find

its energy, we only need to check its power

• Noise is often persistent and is often a power signal • Deterministic and aperiodic signals are often energy signals • A realizable LTI system can be represented by a signal and

mostly is a energy signal (BIBO property)• Power measure is useful for signal and noise analysis • 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

dttxT

PT

Tt∫+≡0

0

2

0|)(|1

page. 23 ….


2-16

Examples• 2-2

• 2-2

• 2-4

)()(1 tuAetx tα−=

)()(2 tAutx =

)cos()( 03 θω += tAtx


2-17

Recall: Vector Space

aA

cbaX CBA ++=

V3

V2

bB

cC

X

on projected orthogonal is where value theis then ,1 if

aXa A=

Any vector in V3 can be represented by a, b, c

basis vector


2-18

Basis Function for Functional Space• Consider the following correlation function

• If we define the orthogonality as

• Then we have

• We call as orthonormal basis

∑ ∫∫∞

−∞=

∞

∞−

∞

∞−

=n

mnnm dtttXdtttx )()()()( φφφ

⎩⎨⎧ =

≡−≡∫∞

∞− o.w.,0,1

)()()(mn

mndttt mn δφφ

mm Xdtttx =∫∞

∞−

)()( φ

∞−∞=mm t)}({φ


2-19

Remarks• Using Freshman calculus can show that the

function approximation expansion by orthogonal basis functions is an optimal least-square error (LSE) approximation

• Is there a good set of orthogonal basis functions?– cos(mω0t)? Check its unit and orthogonal properties– ejω0t ?


2-20

Fourier Who…?

Jean B. Joseph Fourier(1768-1830)

“An arbitrary function, continuous or with discontinuities, defined in a finite interval by an arbitrarily capricious graph can always be expressed as a sum of sinusoids”

J.B.J. FourierDecember, 21, 1807

∑∫−

=

− ==1

0

/2/2 ][21)( )(][

N

i

NktjNktj ekFtfdtetfkF πππ

Laplace, Lagrange, Legendre

15 years…

1822 published

1829 Dirichlet proved his claim


2-21

Fourier Series• Definition

– Suppose that x(t) is defined exactly in the interval (t0, t0+T0), except at a point of jump discontinuity where it converges to the arithmetic mean of left-hand and right-hand limits, that is (Dirichlet’s condition) x(t) is defined and bounded on the range (t0, t0+T0) and have only a finite number of maxima and minima and a finite number of discontinuities on that range, then the Fourier series is defined by

0002 ,)( 0 TttteXtx

n

tnfjn +<<= ∑

∞

−∞=

π

∫+ −= 00

0

02

0

)(1 Tt

t

tnfjn dtetx

TX π

where

A sum of periodic rotating phasors with harmonic freq.


2-22

More on Fourier Series• If x(t) is real, then Xn*=X-n. Writing then

– For real signals, the magnitude of the Fourier coefficients is an even function of n, and the phase is odd

• The Fourier series can be represented by

[ ]

[ ]

∑

∑

∑

∞

=

∞

=

∠+−−

∠+

∞

=

−−

∠++=

++=

++=

100

1

)2()2(0

1

)(220

)2cos(2

)(

00

00

nnn

n

Xtnfjn

Xtnfjn

n

tfnjn

tnfjn

XtnfXX

eXeXX

eXeXXtx

nn

π

ππ

ππ

,nXjnn eXX ∠=

nnn Xjn

Xjn

Xjnn eXeXeXX ∠−∠−∠

−− === − *

DC component + Cosine FS


2-23

Trigonometric Form of the FS• Since then

• DC or average component of x(t) + the fundamental harmonic of x(t) + the second harmonic x(t) + …

,)2cos(2)(1

00 ∑∞

=

∠++=n

nn XtnfXXtx π

[ ]

dttnftxT

XXXB

dttnftxT

XXXA

tnfBtnfAX

tnfXtnfXXXtx

Tt

tnnnn

Tt

tnnnn

nnn

nnn

n

∫

∫

∑

∑

+

+

∞

=

∞

=

=−=∠−=

==∠=

++=

∠−∠+=

00

0

00

0

)2sin()(2}Im{2)sin(2

,)2cos()(2}Re{2)cos(2 where

,)2sin()2cos(

)2sin()sin()2cos()cos()(

00

00

1000

001

0

π

π

ππ

ππ

DC + ACs


2-24

DC and AC Coefficients• DC coefficient

• AC coefficients

– If x(t) is even and real, that is x(t) = x(− t), the second term is zero. Hence Xn is purely real and even

– If x(t) is odd and real, that is x(t) = −x(− t), the first term is zero. Hence Xn is purely imaginary and odd

)( of average

)()(1 00

0

00

0

0)0(2

00

tx

dttxdtetxT

XTt

t

Tt

t

tfj

=

== ∫∫++ − π

dttnftxT

jdttnftxT

dttnfjtnftxT

X

T

T

T

T

Tt

tn

∫∫

∫

−−

+

−=

−=

2

2 00

2

2 00

000

0

0

0

0

00

0

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

)]2sin()2)[cos((1

ππ

ππ


2-25

Properties of the FS• Symmetry (2.4.2)

• Linearity

• Time reversal

• Time shift

• Time scaling

• Multiplication and Convolution Theorem

• Parseval’s Theorem (2.4.4)

bYaXtbytax F ±⎯→←± )()(

nF Xtx −⎯→←− )(

ntπnfjF Xettx 002

0 )( −⎯→←−

an

F Xa

atx 1)( ⎯→←

∑∞

−∞=−=∗⎯→←

llnlnn

F YXYXtytx )()(

∑∫∞

−∞=

+==

nn

Tt

to

x XdttxT

P 2201

1

)(1

Power in time domain = Power in frequency domain


2-26

Examples of FS


2-27


2-28


2-29

Remarks• Fourier found that the sinusoids are good orthonormal basis

functions to expand a periodic function• The Fourier series is derived from the good orthonormal

basis functions for a periodic function, defined over a period interval (t0, t0+T0)

• How about the aperiodic signal?– We consider the aperiodic energy signal x(t), that is x(t) is

integrable in the interval (-∞,∞)– Note that aperiodic signals are mostly finite duration– We may interpret the aperiodc fnction as a special case

of periodic function with infinite period

dfedex

TtedexT

tx

ftjfj

n

tnfjT

Tfj

T

πλπ

πλπ

λλ

λλ

22

022

2

2

0

)(

2 ,)(1lim)( 0

0

0

0

0

∫ ∫

∑ ∫∞

∞−

∞

∞−

−

∞

−∞=−

−

∞→

⎥⎦⎤

⎢⎣⎡⇒

<⎥⎦

⎤⎢⎣

⎡=


2-30

Fourier Transform• Then

∫

∫ ∫∑∞

∞−

∞

∞−

∞

∞−

−∞

−∞=>−

≡

⎟⎠⎞⎜

⎝⎛==

dfefX

dfedexeXtx

ftj

ftjfj

n

ndftjndf

π

πλππ λλ

2

222

0

)(

)(lim)(

)( of response frequency )( of TransformFourier

)()( 2

txtx

dtetxfX ftj

=≡

= ∫∞

∞−

− π


2-31

Energy Spectral Density• For periodic signal, we have power spectral

density• For aperiodic energy signal, we have the similar

energy spectral density

∫∫

∫ ∫∫ ∫

∫∫

∞

∞−

∞

∞−

∞

∞−

∞

∞−

∞

∞−

∞

∞−

∞

∞−

∞

∞−

==

=⎟⎠⎞⎜

⎝⎛=

==

dffXdffXfX

dfdtetxfXdtdfefXtx

dttxtxdttxE

ftjftj

2*

2*2*

*2

|)(|)()(

)()()()(

)()(|)(|

ππ

2|| nX

2|)(|)( fXfG ≡

Parseval’s theorem


2-32

Fourier Transform Theorems• Section 2.5.5

– Superposition (linearity) theorem– Time-delay theorem– Scale-change theorem– Duality theorem– Frequency translation (shift) theorem– Modulation theorem– Differentiation theorem– Integration theorem– Convolution theorem– Multiplication theorem


2-33

Signals & Systems, Oppenheim et al., pp. 223


2-34


2-35

Signals & Systems, Oppenheim et al., pp. 225


2-36


2-37

FT of δ(t), Example 2.12• δ(t), a singular function and is not an energy signal

(not satisfy Dirichlet’s condition)• The Fourier transform of δ(t) is obtained by

formal definition

,1)( ⎯→⎯FTtδ )(1 fFT δ⎯→⎯

,)( 020

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

1)(sinclim1lim)]([00

==⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛Π⎟

⎠⎞

⎜⎝⎛ℑ=ℑ

→→τ

ττδ

ττftt


2-38

FT of Periodic Signals• Periodic signals are not energy signals

– This implies that they don’t satisfy Dirichlet’s conditions– The Fourier transform of a periodic signal doesn’t exist

• For a periodic signal x(t), then

• By applying the convolution theorem and the sampling theorem, then

)()()()( tpmTtmTtptxm

sm

s ∗⎥⎦

⎤⎢⎣

⎡−=−= ∑∑

∞

−∞=

∞

−∞=

δ

∑∑

∑∞

−∞=

∞

−∞=

∞

−∞=

−=−=

⎭⎬⎫

⎩⎨⎧

−ℑ=

nsss

nss

ms

nffnfPfnfffPf

fPmTtfX

)()()()(

)()()(

δδ

δ

∑∑∞

−∞=

∞

−∞=

−↔−n

sssm

s nffnfPfmTtp )()()( δ


2-39

Poisson Sum Formula

∑∑

∑∑∞

−∞=

∞

−∞=

−

∞

−∞=

−∞

−∞=

−

=−ℑ=

−ℑ=−==ℑ

n

tnfjss

nsss

nsss

ms

senfPfnffnfPf

nffnfPfmTtptxfX

πδ

δ

21

11

)(})({)(

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

∑∑∞

−∞=

∞

−∞=

=−n

tnfjss

ns

senfPfnTtp π2)()(

∑∑∞

−∞=

∞

−∞=

−↔−n

ssm

s nffnfPfmTtp )()()( 0δSince

This is the Poisson sum formula

Hence

The sample values P(nfs) of P(f)=ℑ{p(t)} are the Fourier series coefficients of Ts∑p(t-mTs)


2-40

Samples of Fourier Transform

• Discrete Fourier transform (DFT) is identical to samples of Fourier transforms

• In DSP applications, we are able to store only a finite number of samples

• We are able to compute the spectrum only at specific discrete values of ω


2-41

Power/Energy Spectral Density• Recall energy signals, we define the energy spectral

density G(f) of a signal x(t) as a real and nonnegative function of frequency:

• For power signals, define the power spectral density S(f) of a signal x(t) as a real, even, nonnegative function of frequency:

∫∫∫∞

∞−

∞

∞−

∞

∞−=== dttxdffXdffGE 22 )()()(

∫∫ −∞→

∞

∞−==

T

TTdttx

TdffSP 2)(

21lim)(

Frequency-domain representation


2-42

Autocorrelation Function• Time-average autocorrelation function

– A measure of the similarity, or coherence, between a signal and a delayed version of itself.

• For energy signals:

– Note that, φ(0)=E, the signal energy• For power signals:

)}({)]()([)]([)]([

)()()()()()(lim)(1*1*11 fGfXfXfXfX

xxdxxdxxT

TT−−−−

∞

∞−−∞→

ℑ=ℑ=ℑ∗ℑ=

−∗=+=+≡ ∫∫ ττλτλλλτλλτφ

Time-domain representation

⎪⎪⎩

⎪⎪⎨

⎧

+

+=

+≡

∫

∫−∞→

signal.power periodic if,)()(1

signal;power aperiodic if,)()(21lim

)()()(

0

*

0

*

*

T

T

TT

dttxtxT

dttxtxT

txxR

τ

τ

τττ


2-43

More on Autocorrelation Function• For energy signals, we have the fact that the

autocorrelation function and energy spectral density are Fourier transform pairs and φ(0)=E, the signal energy

• For power signals, by definition, we note that

• In fact, we have the similar relation for power signals

∫∞

∞−= dffSR )()0(

∫∞

∞−

−=ℑ= τττ τπ deRRfS fj2)()}({)(This is the Wiener-Khinchine theorem


2-44

Properties of R(τ)

The correlation function and spectral density function are important tools for system analysis…


2-45

Signal and System

• x(t) and y(t) are input signal and output signal, respectively

• H is the characteristic of the system• y(t)=H[x(t)]• The most simple system is the linear time-

invariant (LTI) system

)(tx )(tyΗ


2-46

LTI System• Two properties

– Linear property or the superposition property

– Time-invariant propertyif then

• The characteristic of the LTI system can be represented by its impulse response

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

2211

22112211

tytytxHtxHtxtxHty

αααααα

+=+=+=

where α1 and α2 are any constants

)}({)( txHty =

)]([)( txHty = )()]([ 00 ttyttxH −=−

)]([)( tHth δ≡

∑∑∑===

−=−=−=N

nnn

N

nnn

N

nnn tthttHtytttx

111)()]([)( then ,)()( if αδαδα


2-47

Impulse Response

• As Δt→0, we have

λλδλ dtxtx )()()( −= ∫∞

∞−

∑ <<ΔΔΔ−Δ=n

tttnttnxtx 1 ,)()()(~ δ

∑ ΔΔ−Δ==n

ttnthtnxtxHty )()()](~[)(~

λλλ

λλδλλλδλ

dthx

dtHxdtxHtxHtyty

)()(

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

−=

−=−==→

∫∫∫

∞

∞−

∞

∞−

∞

∞−

dteththfH ftj π2)()}({)( −∞

∞−∫=ℑ= The transfer function…


2-48

Properties of LTI Systems• Duality of convolution of x(t) and h(t)

• BIBO stability– The output of any bounded input signal is bounded

• Causality– A system is called causal if it does not anticipate the

input, that is, the current output does not depend on future input

λλλλλλ dtxhtxthtythtxdthx )()()()()()()()()( −≡∗==∗≡− ∫∫∞

∞−

∞

∞−

conditionDirichlet ofelement main |)(|

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

⇒∞<⇒

∞<≤−=

∫

∫∫∞

∞−

∞

∞−

∞

∞−

λλ

λλλλλ

dh

dhtxdtxhty

0for ,0)(

)()()()()(0

<=⇒

−=−= ∫∫∞∞

∞−

tth

dtxhdtxhty λλλλλλ


2-49

Remarks• Causality property and Paley-Wiener criterion

• For a causal system, if , then

• Conversely, given any square-integrable function |H(f)| and satisfies (2.183), then there exists an h(t)which is a causal system

∞<∫∞

∞−dtth 2|)(|

∞<+∫

∞

∞−df

ffH21

|)(|ln (2.183)


2-50

I/O Relationships for Spectra

• If x(t) and, therefore, y(t) are energy signals with its energy spectral density Gx(f) and Gx(f),respectively, then

• A similar relationship holds for power signals

)()()( 2 fGfHfG xy =

)(tx )(tyΗ

)()()( 2 fSfHfS xy =


2-51

Eigenfunction of LTI Systems• For a LTI system with its impulse response h(t), consider

the input signalThen, the output is

– The output is the same input complex signal with a constant H(f0)

– Since any arbitrary periodic input can be represented by a summation of complex exponential, consequently, its output will be

tfje 02π

)(

)()()(

02

22)(2

0

000

fHe

dehedehtytfj

fjtfjtfjei

π

λππλπ λλλλ

=

== ∫∫∞

∞−

−∞

∞−

−

∑∞

−∞=

=n

tnfjn enfHXty 02

0 )()( π

The summation of complex exponential Fourier series

Eigen value


2-52

Remarks• Since almost any signal x(t) can be represented by a linear

combination of orthogonal sinusoidal basis function {ej2πft}, hence, we only need to input the signal Aej2πft to the system.

• If the system is distortionless, then we only to characterize the system h(t) (or its transfer function), and the eigen value H(f)carries all the system information responding to Aej2πft

• For distortionless system y(t)=Ax(t-t0):– the amplitude response is constant and the phase shift is linear with

frequency• In reality, there are existing transmission distortion

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

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

– Nonlinear distortion: nonlinear system


2-53

Group Delay and Phase Delay• Phase delay:

– When input a single sinusoid through LTI systems, the noise effect, if any, may cause the change of amplitude and/or the phase

– The measurement of the phase delay experienced by a single sinusoid is defined by

• Group delay:– The delay measurement of a group of two or more frequency

components undergo in passing through a linear system– Definition

• In distortionless LTI system– Since the phase shift is directly proportional to frequency, hence, the

derivation of phase with respect to frequency is constant– The group delay is constant– The distortionless system has equal group and phase delays

response phase the),()( where,)(21)( fHf

dffdfTg ∠=−= θθ

π

response phase the),()( where,2

)()( fHffffTp ∠=−≡ θπ

θ


2-54

Example 2.22


2-55

Nonlinear Distortion

• An illustrative example: y(t)=a1x(t)+ a2x2(t)• Suppose the input is

• The nonlinear distortion– Harmonic of the input frequencies– Sums and differences of harmonics of the input frequencies

tAtAtx 2211 coscos)( ωω +=

More complicated !!


2-56

Ideal Filters• Constant amplitude response and linear phase response

lowpass

highpass

bandpass Frequency-domain

Time-domain

Non-causal system !!


2-57

Realizable Filters• For lowpass filters

– Butterworth filter : simple– Chebyshev filter: smaller maximum deviation – Bessel filter : approximately linear phase

• For bandpass and highpass filters– Start from lowpass filters– Followed by suitable frequency transformation

• The details, in DSP or ADSP course…


2-58

Pulse Resolution and Bandwidth• The fact is a narrow time signal has a wide

bandwidth, and vice versa– The rule of thumb: the uncertainty principle

– Exampleconstant)bandwidth()duration( ≥×

)0(|)()(|)(|)0( : (a) From 0 XfXdttxdttxTx f ==≥= =

∞

∞−

∞

∞− ∫∫

)0()(|)(|)0(2 : (b) From xdffXdffXWX =≥= ∫∫∞

∞−

∞

∞−

211

)0()0(2 have weHence, ≥⇒≥≥ TW

TXxW

)]([)( txfX ℑ≡


2-59

Sampling• Sample period: Ts Sample frequency: fs=1/Ts

• Impulse train

• Time domain sampled signal

• Frequency domain representation

– A superposition of infinity shifted replicas of X(f)

( ) ( )∑∞

−∞=

−=n

snTtts δ

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )∑∑

∑∞

−∞=

∞

−∞=

∞

−∞=

−=−=

−==

nss

ns

ns

nTtnTxnTttx

nTttxtstxtx

δδ

δδ

( ) ( ) ( ) ( )

( ) ( ) ( )∑ ∑

∑∞

−∞=

∞

−∞=

∞

−∞=

−=−∗=

−∗=∗=

n nssss

nss

nffXfnfffXf

nffffXfSfXfX

δ

δδ

][)(


2-60

Band-limited Sampling

No aliasing

Aliasing !!fs-W ≥ W

Wfs 2≥

distortion


2-61

Recovery of Band-limited Sampling

By passing the equivalent impulse train xδ through an ideal lowpassfilter Hr with a cutoff at fc and a gain of Ts

WffW sc −≤≤

cf cf−

sTHr

f

fW-W

X(f)

xδ(t)Hr


2-62

Reconstruction Filter• Ideal reconstruction filter

• An alternative expression

WfBWeBfHfH s

ftj −≤≤Π= − ,)2

()( 020

π

)()()()(

00

20

0

ttxHftyefXHffY

s

ftjs

−=⇒=⇒ − π

∑∑∞

−∞=

∞

−∞=

−−=−=n

ssn

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


2-63

Ideal Band-limited Interpolation

x(t)

xδ(t)

y(t)

Ts

Ts

y(t)xδ(t)x(t)

B = ½ fs, H0=Ts, and t0=0

∑∞

−∞=

−=n

ss ntfnTxty )][(sinc)()(


2-64

Two Types of Sampling Distortions


2-65

Nyquist Sampling Theorem• If a signal x(t) contains no frequency components

for frequencies above f=W Hertz, then it is completely described by instantaneous sample values uniformly spaced in time with period Ts<1/2W. The band-limited signal can be exactly reconstructed from the sampled waveform by passing it through an ideal lowpass filter with bandwidth B, where W<B<fs-W with fs=1/Ts . The frequency 2W is referred to as the Nyquistfrequency.


2-66

Bandpass Sampling• Bandpass signal can be obtained by modulating a

lowpass signal.• Frequency range: fL ≤ |f | ≤fH

• Bandwidth BW=Δf=fH – fL

• One can of course sample the bandpass signal with fs ≥ 2fH to prevent aliasing (by sampling theorem)

• How about fH >>0 or fH >> Δf ?

Bandpass Spectrum

fL fH

f–fH –fL


2-67

BP-Sampling Example

BP sampled signal is just a down-shifted version of the bandpass signal


2-68

BP-Sampling: Simple Case• Consider the simple case fH= LB• L is an integer: choose Fs=2B

Odd

Even


2-69

BP Sampling: General Case• The general case: fH ≠ L (Δf), L an integer• Choose an frequency interval [f0, f2] such that

– [f0, f2] ⊇ [f1, f2]– Δf = f2 – f0 satisfies f2 = L(Δf ) , L an integer

• BP-Sample the signal at Fs=2 Δf

f0


2-70

General BP-Sampling( ) ( )

⎥⎦⎥

⎢⎣⎢

=

⎥⎦

⎥⎢⎣

⎢−

=⎟⎠⎞

⎜⎝⎛=

⎥⎦

⎥⎢⎣

⎢−

=−

≤≤=−

−=Δ−=Δ=

Bff

ffff

LfF

fffL

fffLfff

LL

fL

LffLffLf

s2

2

12

2

22

12

2

12

2102

2002

222Then .3

Or . then ,1 Since .2

1 isThat .1 then , Since 1.

Fs ≈ 2B, if f2 >> BP. 75, Theorem…


2-71

Hilbert Transform• Will be introduced in Lecture 3


2-72

4 Forms of Fourier Transform

“Sampled” frequency


2-73

Continuous-Time and Continuous-Frequency

ContinuousAperiodic

ContinuousAperiodic


2-74

Continuous-Time and Discrete-Frequency

Fourier series of periodic continuous signals

PeriodicContinuous

Discrete Aperiodic


2-75

Discrete-Time and Continuous-Frequency

Fourier transform of aperiodic discrete signals

DiscreteAperiodic Continuous

Periodic


2-76

Discrete Fourier Transform

• DFT is identical to samples of Fourier transforms• In DSP applications, we are able to store only a finite number of samples• we are able to compute the spectrum only at specific discrete values of ω


2-77

Discrete Fourier Transform• Discrete Fourier transform (DFT) pairs

knN

jknN

N

k

knNkn

N

n

knNnk

eW

NnWXN

x

NkWxX

π2

1

0

1

0

where

,1,,1,0 ,1

1,,1,0 ,

−−

−

=

−

−

=

=

−==

−==

∑

∑

K

K

• DFT/IDFT can be implemented by using the same hardware• It requires N2 complex multiplications and N(N-1) complex additions

N complex multiplicationsN-1 complex additions


2-78

Decimation in Time

N+2(N/2)2 complex multiplications vs. N2 complex multiplication

twiddle factor


2-79


2-80

Flow Graph of the DIT FFT


2-81


2-82

8-point DIT DFT


2-83

Remarks• It requires v=log2N stages• Each stage has N complex multiplications and N complex

additions• The number of complex multiplications (as well as additions)

is equal to N log2N• By symmetry property, we have (butterfly operation)

222 NN

jrN

NN

rN

NrN WeWWWW −=== −+ π

2 complex multiplications2 complex additions

1 complex multiplications2 complex additions


2-84

Example: 8-point FFT

Normal orderBit-Reversed order


2-85

DFT v.s. Radix-2 FFT• DFT is one of most important mathematical tools• DFT: N2 complex multiplications and N(N-1)

complex additions

• Recall that each butterfly operation requires one complex multiplication and two complex additions

• FFT: (N/2) log2N multiplications and N log2Ncomplex additions

1050: principles of communication system (i)

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