Lecture # 1 - NPRUhome.npru.ac.th/sopapun/Lecture1.pdfSignals Forexample: Electrical signals voltages and currents in a circuit Acoustic signals audio or speech 6552111 Signals and

Signals and SystemsSignals and Systems

6552111 Signals and Systems6552111 Signals and Systems

Sopapun Suwansawang

Lecture #1

1

Lecture #1

Elementary Signals and Systems

Week#1-2

SignalsSignals

Signals are functions of independent variables that

carry information.

FUNCTIONS OF TIME AS SIGNALS


Sopapun Suwansawang 2

Figure : Domain, co-domain, and range of a real function of continuous time.

)(tfv =

SignalsSignals

For example:

Electrical signals voltages and currents in a circuit

Acoustic signals audio or speech


Acoustic signals audio or speech signals (analog or digital)

Video signals intensity variations in an image

Biological signals sequence of bases in a gene

Sopapun Suwansawang 3

There are two types of signals:

Continuous-time signals (CT) are functions of a continuous variable (time).

Discrete-time signals (DT) are functions of


SignalsSignals

Discrete-time signals (DT) are functions ofa discrete variable; that is, they aredefined only for integer values of theindependent variable (time steps).

4Sopapun Suwansawang

CT and DT Signals CT and DT Signals


CT DT


Signal such as : )(tx ),...(),...,(),( 10 ntxtxtx

or in a shorter form as :

,...,...,,

],...[],...,1[],0[

10 nxxx

nxxx

or

where we understand that


)(][ nn txnxx ==

and 's are called samples and the time intervalbetween them is called the sampling interval.When

nx



between them is called the sampling interval.Whenthe sampling intervals are equal (uniform sampling),then

n

)()(][ snTtn nTxtxnxxs

=== =

where the constant is the sampling intervalsT


<

≥=

0,0

0,8.0)(

t

t

tx

t

<

≥=

0,0

0,8.0][

n

n

nx

n



)(tx

t

0

1

0

1

][nx

n

1 2 3 4 5

A discrete-time signal x[n] can be defined in two

ways:

1. We can specify a rule for calculating the nthvalue of the sequence. (see Example 1)



value of the sequence. (see Example 1)

2. We can also explicitly list the values of thesequence. (see Example 2)



DT Signals DT Signals

≥

==

0][ 2

1n

xnx

n

n

Example 1


<

00 n

...,81

,41

,21

,1 =nx



Example 1: Continue


DT SignalsDT Signals


Example 2




The sequence can be written as

Example 2 : continue

,...0,0,2,0,1,0,1,2,2,1,0,0..., =nx


2,0,1,0,1,2,2,1 =nx

We use the arrow to denote the n = 0 term.We shall use theconvention that if no arrow is indicated, then the first termcorresponds to n = 0 and all the values of the sequence arezero for n < 0.

Example 3 Given the continuous-time signal

specified by



≤≤−−

=otherwise

tttx

0

111)(

Determine the resultant discrete-time sequence

obtained by uniform sampling of x(t) with a

sampling interval of 0.25 s


otherwise0

Solve :

Ts=0.25 s,

Ts=1 s,



=

↑,...0,25.0,5.0,75.0,1,75.0,5.0,25.0,0...,][nx

= ,...0,1,0...,][nxTs=1 s,


=

↑,...0,1,0...,][nx

Analog signals


Analog and Digital Signals Analog and Digital Signals

If a continuous-time signal x(t) can take on any

value in the continuous interval (-∞∞∞∞ , +∞∞∞∞), thenthe continuous-time signal x(t) is called an analog

Digital signals

A signal x[n] can take on only a finite number of

distinct values, then we call this signal a digital

signal.


the continuous-time signal x(t) is called an analogsignal.

CT and DT


Digital Signals Digital Signals


CT

Binary signal Multi-level signal


Digital Signals Digital Signals


Intuitively, a signal is periodic when it repeats

itself.

A continuous-time signal x(t) is periodic if there exists a positive real T for which


Periodic SignalsPeriodic Signals

for all t and any integer m.The fundamental period

T0 of x(t) is the smallest positive value of T


)()( mTtxtx +=

00

2ωπ

=T

Fundamental frequency



00

1T

f = Hz

Fundamental angular frequency


000

22

Tf

ππω == rad/sec

A discrete-time signal x[n] is periodic if there exists a positive integer N for which



][][ mNnxnx +=

for all n and any integer m.The fundamental period N0 of

x[n] is the smallest positive integer N


00

2Ω

=π

N

Any sequence which is not periodic is called a non-periodic (or aperiodic) sequence.


NonperiodicNonperiodic SignalsSignals




CT


DT

Example 3 Find the fundamental frequency

in figure below.




HzT

f4

11

00 ==.sec40 =T

(sec.)

Exercise Determine whether or not each of the

following signals is periodic. If a signal is periodic,

determine its fundamental period.



ttx )4

cos()(.1π

+=


nnnx

nnx

enx

tttx

tttx

nj

4sin

3cos][.6

41

cos][.5

][.4

2sincos)(.3

4sin

3cos)(.2

4

)4/(

ππ

ππ

π

+=

=

=

+=

+=

Solve EX.1



14

cos)4

cos()( 00 =→

+=+= ω

πω

πtttx

ππ 22


ππ

ωπ

21

22

00 ===T

x(t) is periodic with fundamental period T0 = 2π.

Solve EX.2



)()(4

sin3

cos)( 21 txtxtttx +=+=ππ

( ) .62

cos3/cos)( 111 ==→==π

ωπ Ttttxwhere


( )

.84/

2sin)4/sin()(

.63/

cos3/cos)(

222

111

==→==

==→==

ππ

ωπ

πωπ

Ttttx

Ttttxwhere

numberrationalaisT

T

43

86

2

1 ==

x(t) is periodic with fundamental period T0 = 4T1=3T2=24.

Note : Least Common Multiplier of (6,8) is 24


Even and Odd Signals:Even and Odd Signals:

A signal x(t) or x[n] is referred to as an even signal if

][][

)()(

nxnx

txtx

−=

−=


A signal x(t) or x[n] is referred to as an odd signal if

][][

)()(

nxnx

txtx

−=−

−=−




Any signal x(t) or x[n] can be expressed as asum of two signals, one of which is even andone of which is odd.That is,



)()()( txtxtx oe +=


][][][ nxnxnx oe

oe

+=

][][21

][

)()(21

)(

nxnxnx

txtxtx

e

e

−+=

−+=

][][21

][

)()(21

)(

nxnxnx

txtxtx

o

o

−−=

−−=

even part odd part



Example 4 Find the even and odd components ofthe signals shown in figure below


Solve even part )()()(2 tftftfe −+=

2fe(t)


Odd part



)()()(2 tftftfo −−=

2fo(t)


2fo(t)


Check

)()()( tftftf oe +=



)()(2

1)( tftftfo −−= ,)()(

2

1)( tftftfe −+=

32

)()()( tftftf oe +=

Sopapun Suwansawang

2

fo(t)

fe(t)

Note that the product of two even signals or oftwo odd signals is an even signal and that theproduct of an even signal and an odd signal is anodd signal.

(even)(even)=even



(even)(even)=even

(even)(odd)=odd

(odd)(even)=odd

(odd)(odd)=even


Example 5 Show that the product of two evensignals or of two odd signals is an even signaland that the product of an even and an oddsignaI is an odd signal.



Let )()()( txtxtx =


Let )()()( 21 txtxtx =

If x1(t) and x2(t) are both even, then

)()()()()()( 2121 txtxtxtxtxtx ==−−=−

If x1(t) and x2(t) are both even, then

)()()())()(()()()( 212121 txtxtxtxtxtxtxtx ==−−=−−=−

A deterministic signal is a signal in which eachvalue of the signal is fixed and can bedetermined by a mathematical expression, rule,or table. Because of this the future values of thesignal can be calculated from past values with


Deterministic and Random Signals: Deterministic and Random Signals:

signal can be calculated from past values with

complete confidence.

A random signal has a lot of uncertainty aboutits behavior. The future values of a randomsignal cannot be accurately predicted and canusually only be guessed based on the averagesof sets of signals



Deterministic and Random Signals: Deterministic and Random Signals:

Deterministic


Random

RightRight--Handed and LeftHanded and Left--Handed SignalsHanded Signals

A right-handed signal and left-handed signal arethose signals whose value is zero between agiven variable and positive or negative infinity.

Mathematically speaking,

A right-handed signal is defined as any signal


A right-handed signal is defined as any signal where f(t) = 0 for

A left-handed signal is defined as any signal where f(t) = 0 for


∞<< 1tt

−∞>> 1tt

RightRight--Handed and LeftHanded and Left--Handed SignalsHanded Signals


Right-Handed

1t


Left-Handed

1

1t

Causal vs. Anticausal vs. NoncausalCausal vs. Anticausal vs. Noncausal

Causal signals are signals that are zero forall negative time.

Anticausal signals are signals that are zerofor all positive time.


for all positive time.

Noncausal signals are signals that havenonzero values in both positive andnegative time.


Causal vs. Anticausal vs. NoncausalCausal vs. Anticausal vs. Noncausal


Causal


Anticausal

Noncausal

Energy and Power SignalsEnergy and Power Signals


Consider :


Rti

R

tv

titvtp

)(

)(

)()()(

2

2

=

=

⋅=

∫∫

∫

∞

∞−

∞

∞−

∞

∞−

==

=

)()()()(1

)(

22 tdtitdtvR

dttpE

Power Energy

Total energy E and average power P on a per-ohm basis are



dttiE ∫∞

∞−

= )(2 Joules


dttiT

PT

TT∫

−∞→

∞−

= )(21 2lim Watts

For an arbitrary continuous-time signal x(t), the normalized energy content E of x(t) is defined as

∫∫−∞→

∞

∞−

==T

TTdttxdttxE 22 )()( lim



The normalized average power P of x(t) is defined as

43

−∞→∞− TT

∫−∞→

=T

TTdttx

TP 2)(

21

lim

Sopapun Suwansawang

Similarly, for a discrete-time signal x[n],

the normalized energy content E of x[n] is defined as



∑∑−=∞→

∞

−∞===

N

NnNnnxnxE 22 ][lim][

The normalized average power P of x[n] is defined as


−=∞→−∞= NnNn

∑−=∞→ +

=N

NnNnx

NP 2][

121

lim

x(t) (or x[n]) is said to be an energy signal (or sequence) if and only if 0 < E < ∞∞∞∞, and P = 0.

x(t) (or x[n]) is said to be a power signal (or sequence) if and only if 0 < P < ∞∞∞∞, thusimplying that E = ∞∞∞∞.



implying that E = ∞∞∞∞. Note that a periodic signal is a power signal if its energy content per period is finite, and then the average power of this signal need only be calculated over a period.


∫=0

0

2

0)(

1T

dttxT

P

Exercise Determine whether the followingsignals are energy signals, power signals, orneither.

1.



)cos()( 0 θω += tAtx1.

2.

3.


)cos()( 0 θω += tAtxtjeAtx 0)( ω=

)()( 3 tuetv t−=

Solve Ex.1

The signal x(t) is periodic with T0=2π/ω0.



dttAT

dttxT

PTT

∫∫ +==00

00

22

00

2

0)(cos

1)(

1θω

)cos()( 0 θω += tAtx


0000

dttT

AP

T

∫ ++=0

00

0

2)22cos(1(

21

θω

++= ∫ ∫ dttdt

T

AP

T T0 0

00

00

2)22(cos1

2θω0

2

2A= ∞<

Thus, x(t) is power signal.



Solve Ex.2 tjAtAeAtx tj00 sincos)( 0 ωωω +==

The signal x(t) is periodic with T0=2π/ω0.

Note that periodic signals are, in general, power signals.

∫T

21∫T

21


∫=T

x dttxT

P0

2)(

1∫=T

tj dtAeT 0

20

1 ω

2 22

0 0

2

1 T T

x

A AA dt dt T

T T T

P A W

= = = ⋅

=

∫ ∫

Documents

Lecture # 1 - NPRUhome.npru.ac.th/sopapun/Lecture1.pdfSignals Forexample: Electrical signals voltages and currents in a circuit Acoustic signals audio or speech 6552111 Signals and