Advance Communication System Lectures Part 2

8/2/2019 Advance Communication System Lectures Part 2

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INTRODUCTION TO SIGNALS

(Reference Text Book)


2/22

A Signal: is a function that specifies how aspecific variable changes versus anindependent variable such as time. Usuallyrepresented as an X-Y plot.


3/22

Analog vs. Digital signals:

Analog signals are signals with magnitudes

that may take any value in a specific rang

Digital signals have amplitudes that take onlya finite number of values.


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Continuous-time vs. discrete-time:

Continuous-time signals have theirmagnitudes defined for all values of t. Theymay be analog or digital.

Discrete-time signals have their magnitudesdefined at specific instants of time only. Theymay be analog or digital.


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Periodic vs. aperiodic signals:Periodic signals are signals constructed froma shape that repeats itself regularly after aspecific amount of time T0, that is:

f(t) = f(t+nT0) for all integer n

Aperiodicsignals do not repeat regularly.


6/22

2| ( ) |

fE f t dt

/ 2

2

/ 2

1lim | ( ) |

T

fT

T

P f t dt T

0

0

2|)(|1

tT

t

fPeriodic dttfT

P


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Energy Signals: an energy signal is a signalwith finite energy and zero average power

(0 E< , P= 0)

Power Signals: a power signal is a signal withinfinite energy but finite average power

(0 < P< , E).


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A signal cannot be both an energy and power signal.

A signal may be neither energy nor power signal.

All periodic signals are power signals (but not all nonperiodic signals are energy signals).

Any signal fthat has limited amplitude (|f|< ) and istime limited (f= 0 for |t |> t0) is an energy signal.

The square root of the average power of a power signalis called the RMS value.


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It is a Power signal

( ) 3sin(2 ),a t t t

2 2| ( ) | | 3sin(2 ) |

19 1 cos(4 )

2

19 9 cos(4 )

2J

aE a t dt t dt

t dt

dt t dt

1 1

2 2

0 0

1

0

0 1

0 0

1

0

1| ( ) | | 3sin(2 ) |

1

19 1 cos(4 )

2

19 9 cos(4 )

2

9 9 sin(4 )2 4

9W

2

aP a t dt t dt

t dt

dt t dt

t

1

2

8


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It is an energy signal

2| |( ) 5 ,

tb t e t

22 2| |

0

4 4

0

04 4

0

| ( ) | 5

25 25

25 25

4 4

25 25 50J

4 4 4

t

b

t t

t t

E b t dt e dt

e dt e dt

e e

/ 2 / 22

2 2| |

/ 2 / 2

0 / 24 4

/ 2 0

0 / 24 4

/ 2 0

2 2

1 1lim | ( ) | lim 5

1 125 lim 25 lim

25 1 25 1lim lim

4 4

25 1 25 1

lim 1 lim 14 4

0 0 0

T T

t

bT T

T T

T

t t

T TT

Tt t

TT T

T T

T T

P b t dt e dt T T

e dt e dt T T

e eT T

e eT T


11/22

Time Shifting: given the signal f(t), the signalf(tt0) is a time-shifted version of f(t) that isshifted to the leftif t0 is positive and to therightif t0 is negative.


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Magnitude Shifting:Given the signal f(t), the signal c+f(t) is amagnitude-shifted version of f(t) that isshifted upifcis positive and shifted downif

cis negative.


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Time Scaling and Time Inversion: Given f(t),the signal f(at) is a time-scaled version off(t), where ais a constant, such that f(at) isan expandedversion off(t) if 0


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Magnitude Scaling and Mag. Inversion:Given f(t), the signal bf(t) is a magnitude-scaled version off(t), where bis a constant,

such that bf(t) is an attenuatedversion of

f(t) if 0


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-2

2

-1

6

f(t)


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Graphical Definition:The rectangular pulseshape approaches the

unit impulse functionas approaches 0(notice that the areaunder the curve is

always equal to 1).

(t)

t


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Mathematical Definition:The unit impulse function (t) satisfies thefollowing conditions:

1. (t) = 0 if t 0,

2.

1)(

dtt


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f(t)(t) = f(0)(t)

)()(

therefore, tdt

tdu)(

0,1

0,0)( tu

t

td

t

0 0 0( ) ( ) ( ) ( )t t t f t t t

)()()()()()()( 0000 tfdttttfdtttTfdtTttf

)0()()( fdtttf


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A signal g(t) in the intervalt1tt1+T0 can be represented by

1

000 )sin()cos()(n

nn tnbtnaatg 011 Tttt

01

1

)(1

0

0

Tt

t

dttgT

a

01

1

)cos()(2

0

0

Tt

t

n dttntgT

a

01

1

)sin()(2

0

0

Tt

t

n dttntgT

b T0 = 2 /0


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Or, in the compact form

Ifg(t) is even then bn = 0 for all n

Ifg(t) is odd then an=0 for all n.

1

00 )cos()(

n

nn tnCCtg

;22

nnn baC

n

nn

a

b1tan

011 Tttt

C0 = a0 ;


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The frequency 0= 2/T0 is called thefundamental frequency and the multiple of thisfrequency n0 is called the nth harmonic.

FS ofg(t) is equal to g(t) over the interval t1t

t1+T0 only. The FS for all tis a periodic function of period

T0 in which the segment ofg(t) over the intervalt1tt1+T0 repeats periodically.

If the function g(t) itself is periodic with periodT0 then the FS represents g(t) for all t.


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Dn is related to Cn and n as

| Dn | is called the amplitude spectrum of the signal.

Dn is called the phase spectrum of the signal.

They provide a frequency-domainrepresentation of thesignal.

)0(

000)(

nn

tjn

n

n

tjn

n eDDeDtg

dtetg

T

DT

tjn

n

0

0)(1

0

nnn CDD2

1||||

011 Tttt

nnnDD

Advance Communication System Lectures Part 2

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