Course Outline (Tentative) Fundamental Concepts of Signals and Systems Signals Systems Linear Time-Invariant (LTI) Systems Convolution integral and sum

Course Outline Course Outline (Tentative)(Tentative)

Fundamental Concepts of Signals and Systems Signals Systems

Linear Time-Invariant (LTI) Systems Convolution integral and sum Properties of LTI Systems …

Fourier Series Response to complex exponentials Harmonically related complex exponentials …

Fourier Integral Fourier Transform & Properties … Modulation (An application example)

Discrete-Time Frequency Domain Methods DT Fourier Series DT Fourier Transform Sampling Theorem

Z Transform Laplace Transform

Chapter IChapter ISignals and Systems

What is Signal?What is Signal?

Signal is the variation of a physical phenomenon / quantity with respect to one or more independent variable

A signal is a function.

Example 1: Voltage on a capacitor as a function of time.

RC circuit

isV I C

R

+

-cV

What is Signal?What is Signal?

Example 2 : Closing value of the stock exchange index as a function of days

M T W FT S S

Index

Fig. Stock exchange

Example 3:Image as a function of x-y coordinates (e.g. 256 X 256 pixel image)

Continuous-Time vs. Discrete Continuous-Time vs. Discrete TimeTime

Signals are classified as continuous-time (CT) signals and discrete-time (DT) signals based on the continuity of the independent variable!

In CT signals, the independent variable is continuous (See Example 1 (Time))

In DT signals, the independent variable is discrete (See Ex 2 (Days), Example 3 (x-y coordinates, also a 2-D signal))

DT signal is defined only for specified time instants! also referred as DT sequence!

Continuous-Time vs. Discrete Continuous-Time vs. Discrete TimeTime

The postfix (-time) is accepted as a convention, although some independent variables are not time

To distinguish CT and DT signals, t is used to denote CT independent variable in (.), and n is used to denote DT independent variable in [.]

Discrete x[n], n is integer Continuous x(t), t is real

Signals can be represented in mathematical form: x(t) = et, x[n] = n/2

y(t) =

Discrete signals can also be represented as sequences:

{y[n]} = {…,1,0,1,0,1,0,1,0,1,0,…}

5

502 ,t

,t

t

Continuous-Time vs. Discrete-Continuous-Time vs. Discrete-TimeTime

Graphically,

It is meaningless to say 3/2th sample of a DT signal because it is not defined.

The signal values may well also be complex numbers (e.g. Phasor of the capacitor voltage in Example 1 when the input is sinusoidal and R is time varying)

t

)(tx

0 4-3 1 3

][nx

0

]2[x ]1[x]2[x ...

2 ...-1-2...

]0[x

]3[x

]1[x

...

n

(Fig.1.7 Oppenheim)

Signal Energy and Signal Energy and PowerPower

In many applications, signals are directly related to physical quantities capturing power and energy in physical systems

Total energy of a CT signal over is

The time average of total energy is and referred to as average power of over

Similarly, total energy of a DT signal over is

Average power of over is

)(tx 21 ttt 2

1

2)(

t

t

dttx

)(tx 21 ttt

2

1

2

2 1

1( )

( )

t

t

x t dtt t

][nx

21 nnn

2

1

2][

n

n

nx

][nx

2

1

2

2 1

1[ ]

( 1)

n

n

x nn n

21 nnn

Signal Energy and Signal Energy and PowerPower

For infinite time intervals: Energy: accumulation of absolute of the signal

Signals with are of finite energy

In order to define the power over infinite intervals we need to take limit of the average:

dttxdttxET

TT

22)()(lim

2 2[ ] [ ]lim

N

T n N n

E x n x n

Total energy in CT signal

Total energy in DT signal

E

T

Edttx

TP

T

T

TT 2

lim)(2

1lim

2

12lim][

12

1lim

2

N

Enx

NP

N

N

NnN

Note: Signals with

have

E

0P

Signal Energy and PowerSignal Energy and Power

Energy signal iff 0<E<, and so P=0 e.g:

Power signal iff 0<P<, and so E= e.g:

Neither energy nor power, when both E and P are infinite

e.g:

.

0,

0,0)(

te

ttx t

...}1,1,1,1,1,1...]}[{ nx

tetx )(

Transformation of Transformation of IIndependent ndependent VVariableariable

Sometimes we need to change the independent variable axis for theoretical analysis or for just practical purposes (both in CT and DT signals)

Time shift

Time reversal

Time scaling

][][ 0nnxnx

)()( txtx

)2/()( txtx

(reverse playing of magnetic tape)

(slow playing, fast playing)

Examples of Examples of TransformationsTransformations

If n0 > 0 x[n-n0] is the delayed version of x[n]

(Each point in x[n] occurs later in x[n-n0])

x[n]

x[n-n0]

. . . . . .

. . . . . . . . . . .

n

nn0

Time shift


t0 < 0 x(t-t0) is an advanced version of x(t)

x(t)

x(t-t0)

t

tt0

Time shift


Reflection about t=0

x(t)

x(-t)

t

t

Time reversal


x(t)

t

compressed!

stretched!

x(2t)

t

x(t/2)

t

Time scaling


Given the signal x(t):

Let us find x(t+1):

Let us find x(-t+1):

t

x(t)

1

1 20

(Time reversal of x(t+1))

t

1

10-1

(It is a time shift to the left)

x(t+1)

t

x(-t+1)

10-1

1

It is possible to transform the independent variable with a general nonlinear function h(t) ( we can find x(h(t)) )

However, we are interested in 1st order polynomial transforms of t, i.e., x(t+)


For the general case, i.e., x(t+),

1. first apply the shift (),

2. and then perform time scaling (or reversal) based on . x(t+1)

t10-1

1

x((3/2)t+1)

t2/3

0-2/3

1

Example: Find x(3t/2+1)

Periodic SignalsPeriodic Signals

A periodic signal satisfies:

Example: A CT periodic signal

If x(t) is periodic with T then

Thus, x(t) is also periodic with 2T, 3T, 4T, ...

The fundamental period T0 of x(t) is the smallest value of T for which

holds

0, )()( TtTtxtx

)(tx

0 TTT2 T2

ZmmTtxtx for )()(

0, )()( TtTtxtx

Periodic SignalsPeriodic Signals

A non-periodic signal is called aperiodic. For DT we must have

Here the smallest N can be 1,

The smallest positive value N0 of N is the fundamental period

0, ][][ NnnxNnx

Period must be integer!

a constant signal

Even and Odd SignalsEven and Odd Signals

If even signal (symmetric wrt y-axis)

If odd signal (symmetric wrt origin)

Decomposition of signals to even and odd parts:

][][or )()( nxnxtxtx

][][or )()( nxnxtxtx

odd

t

x(t)

t

evenx(t)

)()(2

1)( txtxtxEV

)()(2

1)( txtxtxOD

)()()( txODtxEVtx

Exponential and Sinusoidal Exponential and Sinusoidal SignalsSignals

Occur frequently and serve as building blocks to construct many other signals

CT Complex Exponential:

where a and C are in general complex.

Depending on the values of these parameters, the complex exponential can exhibit several different characteristics

atCetx )(

x(t) x(t)

C Ctt

a < 0a > 0 Real Exponential

(C and a are real)


Periodic Complex Exponential (C real, a purely imaginary)

Is this function periodic?

The fundamental period is

Thus, the signals ejω0t and e-jω0t have the same fundamental period

tjetx 0)(

for periodicity = 1

00

2

T

tjetx 0)( Znω

πnT

0

2)(0 Ttje Tjtj ee 00 .


t0

sincos

sincos

je

jej

j

Using the Euler’s relations:

and by substituting , we can express:

tjjtjjtjtj

tjjtjjtjtj

tj

eeeej

Aee

j

AtA

eeeeA

eeA

tA

tjte

0000

0000

0

22)sin(

22)cos(

sincos

)()(0

)()(0

00

Sinusoidals in terms of complex exponentials


Alternatively, )(

0

)(0

0

0

Im)sin(

Re)cos(

tj

tj

eAtA

eAtA

cosA

)(tx0

0

2

T

t

)cos()( 0 tAtx

CT sinusoidal signal

)cos( 0 tA


Complex periodic exponential and sinusoidal signals are of infinite total energy but finite average power

As the upper limit of integrand is increased as

However, always Thus,

0

02

TT

T

tjperiod dteE

periodETTTT ,...3,2 00

1periodP

T

T

tj

Tdte

TP 1

2

1lim

20

Finite average power!

1)(

1

0

periodperiod ETTT

P

0

1TT

T

dt 00 )( TTTT

Harmonically Related Harmonically Related Complex ExponentialsComplex Exponentials

Set of periodic exponentials with fundamental frequencies that are multiplies of a single positive frequency

kth harmonic xk(t) is still periodic with T0 as well Harmonic (from music): tones resulting from variations in

acoustic pressures that are integer multiples of a fundamental frequency

Used to build very rich class of periodic signals

0

,...2,1,0for )( 0 ketx tjkk

0frequency lfundamenta with periodic is )(0

constant a is )(0

ωk txk

txk

k

k

00

0

0

2 where,

2 period lfundamenta and

Tk

T

k

General General CComplex omplex Exponential Exponential SSignalsignals

Here, C and a are general complex numbers

Say,

(Real and imaginary parts) Growing and damping sinusoids for r>0 and r<0

atj CetxjraeCC )( and 0

)(tx

0, rt

)cos()( 0 tCetx rt

0, rt

envelope

)( 0)(Then tjrtat eeCCetx )sin()cos( 00 teCjteC rtrt

DT DT CComplex omplex Exponential and Exponential and SSinusoidal inusoidal SSignalsignals

)0for n alternatioSign (

1,01,10,1

real are and C :signals lexponentia Real

of instead use tocustomary and convenient more isIt

numberscomplex generalin are and C where

for

eα

Cnxn

n

1 10 01 1

DT Sinusoidal DT Sinusoidal SSignalsignals

)(2

)cos(

case CT thesimilar to have then weeConsider

00

0

0

j

njjnjj

n

eeeeA

nA

Infinite energy, finite average power with 1.

General complex exp signals

)sin()cos(

then

,CC

as formpolar in are and C If

00

0

nCjnCC

ee

nnn

jj

)1 decaying and ,1 (growing ssinusoidal

are expcomplex general DT of partsimaginary and Real

1

1

Periodicity Periodicity PProperties of DT roperties of DT SSignalsignals

So the signal with freq 0 is the same as the signal with 0+2 This is very different from CT complex exponentials

CT exponentials has distinct frequency values 0+2k, k Z

njωe 0 :expcomplex DT heConsider t

1

22 00 find sLet'

πnjnjωπ)nj(ω eee

It is sufficient to consider an interval from 0 to 0+2 to completelycharacterize the DT complex exponential!

Result:


00 -or 20 kesusually ta One

zero. ton oscillatio of rate the,2 until more as

,n oscillatio of rate the, to0 from as exp DTFor

lyindefinite n oscillatio of rate the as exp CTFor

0

0

0

0

00

around is expcomplex DT arying)(rapidly v freqHigh

2 and 0 around is expcomplex DT varying)(slow freq low Hence,

What about the periodicity of DT complex exponentials?


(*) :conditiony Periodicit 000

itymust be un

NjωnjωN)(njω eee

!!otherwise! periodicnot number, rational a is 2

when periodic is exp DT So

integers. bemust and that (**) and (*) from conditions thehave We

0

N

m

Nm

*)*(* 2

ly equivalentOr

2 havemust weinteger somefor s,other wordIn

(**) .2 of multipleinteger an is if holds This

0

0

0

N

m

mNm

N


mN02

then isfrequency lfundamenta The

outfactor common theTake

**)*(* 2

then is period lfundamenta The0

mN

!signals.)! sinusoidal DTfor validalso ist developmen same (The

*)*(*in as 2

express toneed weexp.complex a of freq. fund. thefind toTherefore 0

Periodicity Periodicity PProperties of DT roperties of DT SSignalsignalsExamplesExamples

2 nEx: x[n] cos( ) 12

4 nEx: x[n] cos( ) 12

00 0

2 12 n x[n] cos( ) cos( n) no factors in common,12 12 2 1212

so by using (****) , N 1 12 periodic with fund period 12.1

00 0

4 2 ( 1)4 n x[n] cos( ) cos( n) ,12 12 2 12 ( 6)

12then using (****) , N 1 6 is periodic with fundamental period 6.

2

n

N


OBSERVATION: With no common factors between N and m, N in (***) is the

fundamental period of the signal Hence, if we take common factors out

Comparison of Periodicity of CT and DT Signals:

Consider x(t) and x[n]

6 6

1

20 N

x(t) is periodic with T=12, x[n] is periodic with N=12.

122cos 12

2cos )πn(x[n])πt(x(t)


if and x(t) is periodic with 31/4.

In DT there can be no fractional periods, for x[n] we have

then N=31.

If and

x(t) is periodic with 12, but x[n] is not periodic, because there is no way to express it as in (***)

.

31t8cos)( tx 31

8cos][ nnx

31

4

20

6cos)( ttx 6cos][ nnx

12

1

20

Harmonically Related Harmonically Related Complex ExponentialsComplex Exponentials (Discrete Time)(Discrete Time)

Set of periodic exponentials with a common period N

Signals at frequencies multiples of (from 0N=2m)

In CT, all of the HRCE, are distinct

Different in DT case!

,...2,1,0for ][2

kenn

Njk

k

N

2

,...2,1,0for 0 ke tjk

Harmonically Related Harmonically Related Complex ExponentialsComplex Exponentials (Discrete Time)(Discrete Time)

Let’s look at (k+N)th harmonic:

Only N distinct periodic exponentials in k[n] !!

That is,

nN

Nkj

Nk en

2

)(

][

nN

Nj

N

nN

jnN

jenenenn

2

)1(

1

4

2

2

10 ][,,][,][,1][

][][,][][ 110 nnnn NN

][.1

22

nee knj

nN

jk

Unit Impulse and Unit Step Unit Impulse and Unit Step FunctionsFunctions

Basic signals used to construct and represent other signals

DT unit impulse:

DT unit step:

Relation between DT unit impulse and unit step (?):

0,1

0,0][

n

nn

0,1

0,0][

n

nnu

]1[][][ nunun

n

u[n]

- - -- - -

δ[n]

- - -- - -n

(DT unit impulse is the first difference of the DT step)

Unit impulse (unit sample)

Unit Impulse and Unit Step Unit Impulse and Unit Step FunctionsFunctions

0

k]-[n][k

nu

][][][][ 000 nnnxnnnx

][]0[][][ nxnnx

(DT step is therunning sum of DT unit sample)

More generally for a unit impulse [n-n0] at n0 :

[n-k]

- - -- - -

nk

0

Interval of summation

n>0

[n-k]

- - -- - -

n k0

Interval of summation

n<0

Sampling property

Unit Impulse and Unit Step Unit Impulse and Unit Step Functions (Continuous-Functions (Continuous-Time)Time)

CT unit step:

CT impulse:

dt

tdut

)()(

dtut

)()(

0,1

0,0)(

t

ttu

δ(t)

t0

1

t

u(t)

CT unit step is the running integralof the unit impulse

CT unit impulse is the 1st derivative of the unit sample

Continuous-Time ImpulseContinuous-Time Impulse

CT impulse is the 1st derivative of unit step

There is discontinuity at t=0, therefore we define as

dt

tdut

)()(

)(tu

t0

1

)(tu

)(lim)(0

tutu

)(lim)()(

)(0

ttdt

tdut

t0

1/

)(t

t

0

1

)(t

Continuous-Time ImpulseContinuous-Time Impulse

REMARKS: Signal of a unit area Derivative of unit step function

Sampling property

The integral of product of (t) and (t) equals (0) for any (t) continuous at the origin and if the interval of integration includes the origin, i.e.,

)()()()( 000 tttxtttx

21 0)0()()(2

1

ttfordt

t

CT and DT SystemsCT and DT SystemsWhat is a system?What is a system?

A system: any process that results in the transformation of signals

A system has an input-output relationship Discrete-Time System: x[n] y[n] : y[n] =

H[x[n]]

Continuous -Time System: x(t) y(t) : y(t) = H(x(t))

DT System

x[n] y[n]

CT System

x(t) y(t)

CT and DT SystemsCT and DT SystemsExamplesExamples

In CT, differential equations are examples of systems

Zero state response of the capacitor voltage in a series RC circuit

In DT, we have difference equations Consider a bank account with %1 monthly interest

rate added on:

RC circuit

isV I C

R

+

-cV )(

1)(

1)(tv

RCtv

RCdt

tdvsc

c

vc(t): output, vs(t): input

][]1[01.1][ nxnyny y[n]: output: account balance at the end of each monthx[n]: input: net deposit (deposits-withdrawals)

Interconnection of Interconnection of SystemsSystems

Series (or cascade) Connection: y(t) = H2( H1( x(t) ) )

e.g. radio receiver followed by an amplifier

Parallel Connection: y(t) = H2( x(t) ) + H1( x(t) )

e.g. phone line connecting parallel phone microphones

System 1H1

x(t) System 2H2

y(t)

System 1H1

x(t) y(t)

System 1H2

+

Interconnection of SystemsInterconnection of Systems

Previous interconnections were “feedforward systems” The systems has no idea what the output is

Feedback Connection: y(t) = H2( y(t) ) + H1( x(t) )

In feedback connection, the system has the knowledge of output

e.g. cruise control Possible to have combinations of connections..

System 1H1

x(t) y(t)

System 2H2

+

System PropertiesSystem PropertiesMemory vs. Memoryless SystemsMemory vs. Memoryless Systems

Memoryless Systems: System output y(t) depends only on the input at time t, i.e. y(t) is a function of x(t).

e.g. y(t)=2x(t) Memory Systems: System output y(t) depends on

input at past or future of the current time t, i.e. y(t) is a function of x() where - < <.

Examples: A resistor: y(t) = R x(t)

A capacitor:

A one unit delayer: y[n] = x[n-1]

An accumulator:

t

dxC

ty )(1

)(

n

k

kxny ][][

System PropertiesSystem PropertiesInvertibilityInvertibility

A system is invertible if distinct inputs result in distinct outputs.

If a system is invertible, then there exists an inverse system which converts output of the original system to the original input.

Examples:

)(4

1)( tytw

)(4)( txty

n

k

kxny ][][

Systemx(t) Inverse

Systemw(t)=x(t)y(t)

t

dttxty )()(

]1[][][ nynynwdt

tdytw

)()(

)()( 4 txty Not invertible

System PropertiesSystem PropertiesCausalityCausality

A system is causal if the output at any time depends only on values of the input at the present time and in the past

Examples: Capacitor voltage in series RC circuit (casual)

Systems of practical importance are usually casual However, with pre-recorded data available we do not constrain

ourselves to causal systems (or if independent variable is not time, any example??)

)4(2)( txty

M

Mk

knxM

ny ][12

1][

)1cos()4(2)( ttxty

Non-causal

][][ nxny Non-causal (For n<0, system requires future inputs)

Causal

Averaging system in a block of data

System PropertiesSystem PropertiesStabilityStability

A system is stable if small inputs lead to responses that do not diverge

More formally, a system is stable if it results in a bounded output for any bounded input, i.e. bounded-input/bounded-output (BIBO).

If |x(t)| < k1, then |y(t)| < k2.

Example:

Averaging system:

Interest system:

M

Mk

knxM

ny ][12

1][

t

dttxty0

)()( ][100][ nxny stable

x(t)

y(t)

Ball at the base of valley

x(t)y(t)

Ball at the top of hill

][]1[01.1][ nxnyny

stable

unstable (say x[n]=[n], y[n] grows without bound

System PropertiesSystem PropertiesTime-InvarianceTime-Invariance

A system is time-invariant if the behavior and characteristics of the system are fixed over time

More formally: A system is time-invariant if a delay (or a time-shift) in the input signal causes the same amount of delay (or time-shift) in the output signal, i.e.:

x(t) = x1(t-t0) y(t) = y1(t-t0)

x[n] = x1[n-n0] y[n] = y1[n-n0]

Examples:

][][][][ 012012 nnnxnynnxnx

][][ nnxny )2()( txty )(sin)( txty

][][][ 111 nnxnynx

][][)(][ 201001 nynnxnnnny

Not TIV

Not TIV(explicit operationon time)

TIV

When showing a system is not TIV, try tofind counter examples…

System PropertiesSystem PropertiesLinearityLinearity

A system is linear if it possesses superposition property, i.e., weighted sum of inputs lead to weighted sum of responses of the system to those inputs

In other words, a system is linear if it satisfies the properties:

It is additivity: x(t) = x1(t) + x2(t) y(t) = y1(t) + y2(t) And it is homogeneity (or scaling): x(t) = a x1(t) y(t) = a y1(t),

for a any complex constant. The two properties can be combined into a single

property: Superposition:

x(t) = a x1(t) + b x2(t) y(t) = a y1(t) + b y2(t)

x[n] = a x1[n] + b x2[n] y[n] = a y1[n] + b y2[n]

How do you check linearity of a given system?

System PropertiesSystem PropertiesLinearityLinearity

Examples:

]1[2][ nxny

)()( 2 txty ]}[Re{][ nxny 3][2][ nxny

nonlinear

][][][][][ 11 nrnynjsnrnx

][][][

])[][(][][

2

12

njrnsnx

jafornjsnrjnaxnx

][][][][ 122 naynsnynx

nonlinear933.2][][

732.2][][

3][,2][

22

11

21

nynx

nynx

nxnx

][1697][][

1335.2][][][

532][][

2121

2121

21

nynyny

nynxnx

nxnx

nonlinear

linear

Superposition in LTI Superposition in LTI SystemsSystems

For an LTI system: given response y(t) of the system to an input signal x(t) it is possible to figure out response of the system to any

signal x1(t) that can be obtained by “scaling” or “time-shifting” the input signal x(t), i.e.:

x1(t) = a0 x(t-t0) + a1 x(t-t1) + a2 x(t-t2) + …

y1(t) = a0 y(t-t0) + a1 y(t-t1) + a2 y(t-t2) + …

Very useful property since it becomes possible to solve a wider range of problems.

This property will be basis for many other techniques that we will cover throughout the rest of the course.

Superposition in LTI Superposition in LTI SystemsSystems

Exercise: Given response y(t) of an LTI system to the input signal x(t) below, find response of that system to the input signals x1(t) and x2(t) shown below.

2x(t)

1t

1

y(t)

-1 1t

2x1(t)

1 t2

x2(t)

1

t-1

3

4

1/2-1/2

Documents

Course Outline (Tentative) Fundamental Concepts of Signals and Systems Signals Systems Linear Time-Invariant (LTI) Systems Convolution integral and sum