ee3010_lecture3 al-dhaifallah_term332 1 3. introduction to signal and systems dr. mujahed...

Al-Dhaifallah_Term332 1EE3010_Lecture3

3. Introduction to Signal and Systems

Dr. Mujahed Al-Dhaifallah

EE3010: Signals and Systems Analysis

Term 332

Al-Dhaifallah_Term332 2

Lecture Objectives

General properties of signalsEnergy and power for continuous &

discrete-time signalsSignal transformationsSpecific signal types

EE3010_Lecture3


Reading List/Resources

EssentialAV Oppenheim, AS Willsky: Signals and

Systems, 2nd Edition, Prentice Hall, 1997

Sections 1.1-1.4

RecommendedS. Haykin and V. Veen, Signals and

Systems, 2005.

Sections 1.4-1.9

EE3010_Lecture3


Last Lecture Material

What is a Signal? Examples of signal. How is a Signal Represented? Properties of a System What is a System? Examples of systems. How is a System Represented? Properties of a System How Are Signal & Systems Related (i), (ii), (iii)

EE3010_Lecture3


Signals and Systems

Signals are variables that carry information.

Systems process input signals to produce output signals.

Today: Signals, next lecture: Systems.

EE3010_Lecture3


Examples of signals

Electrical signals --- voltages and currents in a circuit

Acoustic signals --- audio or speech signals (analog or digital)

Video signals --- intensity variations in an image (e.g. a CAT scan)

Biological signals --- sequence of bases in a gene

EE3010_Lecture3


Signal ClassificationType of Independent Variable

Time is often the independent variable. Example: the electrical activity of the heart recorded with chest electrodes –– the electrocardiogram (ECG or EKG).

EE3010_Lecture3


The variables can also be spatial

Eg. Cervical MRI

In this example, the signal is the intensity as a function of the spatial variables x and y.

EE3010_Lecture3


Independent Variable Dimensionality

An independent variable can be 1-D (t in the ECG) or 2-D (x, y in an image).

EE3010_Lecture3


Continuous-time (CT) Signals

Most of the signals in the physical world are CT signals, since the time scale is infinitesimally fine, so are the spatial scales. E.g. voltage & current, pressure, temperature, velocity, etc.

EE3010_Lecture3


Discrete-time (DT) Signals

x[n], n — integer, time varies discretely

Examples of DT signals in nature:— DNA base sequence— Population of the nth generation of

certain species

EE3010_Lecture3


Transformations of the independent Variable

A central concept in signal analysis is the transformation of one signal into another signal. Of particular interest are simple transformations that involve a transformation of the time axis only.

A linear time shift signal transformation is given by:

Time reversal

EE3010_Lecture3

( ) ( )y t x t b

( ) ( )y t x t


Transformations of the independent Variable

Time scaling

represents a signal stretching if 0<|a|<1, compression if |a|>1

EE3010_Lecture3


Periodic Signals

An important class of signals is the class of periodic signals. A periodic signal is a continuous time signal x(t), that has the property

where T>0, for all t. Examples:

cos(t+2p) = cos(t)sin(t+2p) = sin(t)

Are both periodic with period 2pThe fundamental period is the smallest t>0 for which

EE3010_Lecture3

)()( Ttxtx

2p

)()( Ttxtx


Odd and Even Signals

An even signal is identical to its time reversed signal, i.e. it can be reflected in the origin and is equal to the original or x[n] = x[−n]

Examples:x(t) = cos(t)x(t) = c

EE3010_Lecture3

)()( txtx



An odd signal is identical to its negated, time reversed signal, i.e. it is equal to the negative reflected signal

or x[n] = − x[−n]

Examples:x(t) = sin(t)x(t) = t

This is important because any signal can be expressed as the sum of an odd signal and an even signal.

EE3010_Lecture3

)()( txtx



EE3010_Lecture3


Exponential and Sinusoidal Signals

Exponential and sinusoidal signals are characteristic of real-world signals and also from a basis (a building block) for other signals.

A generic complex exponential signal is of the form:

where C and a are, in general, complex numbers. Lets investigate some special cases of this signal

Real exponential signals

EE3010_Lecture3

atCetx )(

0

0

C

aExponential growth 0

0

C

a

Exponential decay


Right- and Left-Sided Signals

EE3010_Lecture3


Bounded and Unbounded Signals

EE3010_Lecture3


Periodic Complex Exponential & Sinusoidal Signals

EE3010_Lecture3

Consider when a is purely imaginary:

By Euler’s relationship, this can be expressed as:

This is a periodic signals because:

when T=2p/w0

A closely related signal is the sinusoidal signal:

We can always use:

tjCetx 0)(

tjte tj00 sincos0

tj

Ttj

etjt

TtjTte0

0

00

00)(

sincos

)(sin)(cos

ttx 0cos)( 00 2 f

)(

0

)(0

0

0

sin

cos

tj

tj

eAtA

eAtA

T0 = 2p/w0

= p

cos(1)

T0 is the fundamental time periodw0 is the fundamental frequency


Exponential & Sinusoidal Signal Properties

Periodic signals, in particular complex periodic and sinusoidal signals, have infinite total energy but finite average power.

Consider energy over one period:

Therefore:

Average power:

EE3010_Lecture3

00

0

2

0

00

1 Tdt

dteE

T

T tjperiod

E

11

0

periodperiod ET

P


General Complex Exponential Signals

So far, considered the real and periodic complex exponential Now consider when C can be complex. Let us express C is polar

form and a in rectangular form:

So

Using Euler’s relation

EE3010_Lecture3

0

jra

eCC j

tjrttjrjat eeCeeCCe )()( 00

))sin(())cos(( 00)( 0 teCjteCeeCCe rtrttjrjat


“Electrical” Signal Energy & Power It is often useful to characterise signals by measures

such as energy and power For example, the instantaneous power of a resistor is:

and the total energy expanded over the interval [t1, t2] is:

and the average energy is:

How are these concepts defined for any continuous or discrete time signal?

EE3010_Lecture3

)(1

)()()( 2 tvR

titvtp

2

1

2

1

)(1

)( 2t

t

t

tdttv

Rdttp

2

1

2

1

)(11

)(1 2

1212

t

t

t

tdttv

Rttdttp

tt


Generic Signal Energy and Power

Total energy of a continuous signal x(t) over [t1, t2] is:

where |.| denote the magnitude of the (complex) number.

Similarly for a discrete time signal x[n] over [n1, n2]:

By dividing the quantities by (t2-t1) and (n2-n1+1), respectively, gives the average power, P

Note that these are similar to the electrical analogies (voltage), but they are different, both value and dimension.

EE3010_Lecture3

2

1

2)(

t

tdttxE

2

1

2][

n

nnnxE


Energy and Power over Infinite Time

For many signals, we’re interested in examining the power and energy over an infinite time interval (-∞, ∞). These quantities are therefore defined by:

If the sums or integrals do not converge, the energy of such a signal is infinite

EE3010_Lecture3

dttxdttxET

TT

22)()(lim

n

N

NnN nxnxE22][][lim

T

TT dttxT

P2)(

2

1lim

N

NnN nxN

P2][

12

1lim


Classes of Signals

Two important (sub)classes of signals

1. Finite total energy (and therefore zero average power).

2. Finite average power (and therefore infinite total energy). x[n]=4

3. Neither average power or total energy are finite. x(t)=t

EE3010_Lecture3

1 0 1( )

0

tf t

otherwise

28/25

Discrete Unit Impulse and Step Signals The discrete unit impulse signal is defined:

Useful as a basis for analyzing other signals

The discrete unit step signal is defined:

Note that the unit impulse is the first difference (derivative) of the step signal

Similarly, the unit step is the running sum (integral) of the unit impulse.

01

00][][

n

nnnx

01

00][][

n

nnunx

]1[][][ nunun

29/25

Continuous Unit Impulse and Step Signals

The continuous unit impulse signal is defined:

Note that it is discontinuous at t=0 The arrow is used to denote area, rather than

actual value Again, useful for an infinite basis

The continuous unit step signal is defined:

0

00)()(

t

tttx

tdtutx )()()(

01

00)()(

t

ttutx

ee3010_lecture3 al-dhaifallah_term332 1 3. introduction to signal and systems dr. mujahed...

Documents