communication system Chapter 2

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Communication System Communication System Ass. Prof. Ibrar Ullah

BSc (Electrical Engineering)

UET Peshawar

MSc (Communication & Electronics Engineering)

UET Peshawar

PhD (In Progress) Electronics Engineering

(Specialization in Wireless Communication)

MAJU Islamabad

E-Mail: ibrar@cecos.edu.pk

Ph: 03339051548 (0830 to 1300 hrs)

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Chapter # 2

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Chapter-2

• Signals and systems• Size of signal• Classification of signals• Signal operations• The unit impulse function• Correlation• Orthogonal signals• Trigonometric Fourier Series• Exponential Fourier Series

4

Signals and systems • A signal is a any time-varying quantity of information or data.

• Here a signal is represented by a function g(t) of the independent time variable t. Only one-dimensional signals are considered here.

• Signals are processed by systems.

• "A system is composed of regularly interacting or interrelating groups of activities/parts which, when taken together, form a new whole." (from Wikipedia)

• Here a system is an entity that processes an input signal g(t) to produce a new output signal h(t).

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Size of Signals

The energy Eg of a signal g(t) can be calculated by the formula

For complex valued signal g(t) it can be written as

The energy is finite only, if

• EnergyThe size of any entity is a number that indicates the largeness or strength of that entity

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Size of Signals (cont…)

The power Pg of a signal g(t) can be calculated by the formula

For complex valued signal g(t) it can be written as

The power represents the time average(mean) of the signal amplitude squared. It is finite only if the signal is periodic or has statistical regularity.

• Power

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Size of Signals (cont…)

• Examples for signals with finite energy (a) and finite power (b):

• Remark:

• The terms energy and power are not used in their conventional sense as electrical energy or power, but only as a measure for the signal size.

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Example 2.1 Page: 17• Determine the suitable measures of the signals given below:

• The signal (a) 0 as Therefore, the suitable measure for this

signal is its energy given by

•

t gE

gE 8444)2()(= 0

0

1

22

dtedtdttg t

9

Example 2.1 Page: 17 (Cont.)The signal in the fig. Below does not --- to 0 as t . However it is periodic, therefore its power exits.

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Example 2.2 page-18

(a)

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Example 2.2 (cont…)

Remarks:

A sinosoid of amplitude C has power of regardless of its frequency and phase .

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Example 2.2 (cont…)

13

Example 2.2 (cont…)

We can extent this result to a sum of any number of sinusoids with distinct frequencies.

And rms value

14

Example 2.2 (cont…)

Recall that

Therefore

The rms value is

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Classification of signals

1) Continuous-time and discrete-time signals

2) Analog and digital signals

3) Periodic and aperiodic signals

4) Energy and power signals

5) Deterministic and random signals

6) Causal vs. Non-causal signals

16

Classification of signals

Continuous time (CT) and discrete time (DT) signals

CT signals take on real or complex values as a function of an independent variable that ranges over the real numbers and are denoted as x(t).

DT signals take on real or complex values as a function of an independent variable that ranges over the integers and are denoted as x[n].

Note the use of parentheses and square brackets to distinguish between CT and DT signals.

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Classification of signalsAnalog continuous time signal x(t)

Analog discrete time signal x[n]

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Classification of signals

Digital continuous time signal

Digital discrete time signal

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Classification of signals

periodic and aperiodic signals

Examples:

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Classification of signals

21

Classification of signals

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Classification of signals

Energy and Power Signals

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Classification of signals

Remarks:

• A signal with finite energy has zero power.

• A signal can be either energy signal or power signal, not both.

• A signal can be neither energy nor power e.g. ramp signal

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Classification of signals

• A signal g(t) is called deterministic, if it is completely known and can be described mathematically

• A signal g(t) is called random, if it can be described only by terms of probabilistic description, such as

– distribution– mean value (The average or expected value)– squared mean value (The expected value of the squared

error)– standard deviation (The square root of the variance)

Deterministic and Random signals

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Classification of signals

Causal vs. Non-causal signals A causal signal is zero for t < 0 and an non-causal signal is

zero for t > 0 or

A causal signal is any signal that is zero prior to time zero. Thus, if x(n) denotes the signal amplitude at time (sample) n, the signal x is said to be causal if x(n)=0 for all n< 0

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Classification of signals

Right- and left-sided signals

A right-sided signal is zero for t < T and a left-sided signal is zero for t > T where T can be positive or negative.

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Classification of signals Even signals xe(t) and odd signals xo(t) are defined as

xe(t) = xe(−t) and xo(−t) = −xo(t).

If the signal is even, it is composed of cosine waves. If the signal is odd, it is composed out of sine waves. If the signal is neither even nor odd, it is composed of both sine and cosine waves.

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Signal operations

Time Shifting

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Signal operations (cont'd)

Time-Scaling

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Signal operations (cont'd)

Time- Inversion/ Time-Reversal

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Signal operations (cont'd)

Example: 2.4

For the dignal g(t), shown in the fig. Below , sketch g(-t)

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Unit Impilse Function

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Unit Impilse Function

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Unit Step Function

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Signals and Vectors

• Analogy between Signals and Vectors• --A vector can be represented as a sum of its components• --A Signal can also be represented as a sum of its components• Component of a vector:• A vector is represented by bold-face type• Specified by its magnitude and its direction. • E.g

Vector x of magnitude | x | and Vector g of magnitude | g |• Let the component of vector g along x be cx• Geometrically this component is the projection of g on x• The component can be obtained by drawing a perpendicular from the

tip of g on x and expressed as

g = cx + e

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Component of a Vector

• There are infinite ways to express g in terms of x

• g is represented in terms of x plus another vector which is called

the error vector e

• If we approximate g by cx

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Component of a Vector (cont..)

The error in this approximation is the vector e

e = g - cx

The error in the approximation in both cases for last figure are

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Component of a Vector (cont..)

• We can mathematically define the component of vector g along x• We take dot product (inner or scalar) of two vectors g and x as:

g.x = | g || x | cos

• The length of vector by definition is

| x | ² = x . x• The length of component of g along x is

c| x | = | g | cos Multiply both sides by | x |

c | x | ² = | g | | x | cos = g . x

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Component of a Vector (cont..)

Consider the first figure again and expression for c

• Let g and x are perpendicular (orthogonal)

• g has a zero component along x gives c = 0

• From equation g and x are orthogonal if the inner (scalar or dot) product of two vectors is zero i.e.

g . x = 0

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Component of a Signal

• Vector component and orthogonality can be extended to signals

• Consider approximating a real signal g(t) in terms of another real signal x(t)

The error e(t) in the approximation is given by

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Component of a Signal

• As energy is one possible measure of signal size.

• To minimize the effect of error signal we need to minimize its size-----which is its energy over the interval

This is definite integral with dummy variable t

Hence function of c (not t)

This is definite integral with dummy variable t

Hence function of c (not t)

For some choice of c the energy is minimum

Necessary condition Necessary condition

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Component of a Signal

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Component of a Signal

Recall equation for two vectors

• Remarkable similarity between behavior of vectors and signals. Area under the product of two signals corresponds to the dot product of two vectors

• The energy of the signal is the inner product of signal with itself and corresponds to the vector length squared (which is the inner product of the vector with itself)

• Remarkable similarity between behavior of vectors and signals. Area under the product of two signals corresponds to the dot product of two vectors

• The energy of the signal is the inner product of signal with itself and corresponds to the vector length squared (which is the inner product of the vector with itself)

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Component of a Signal

Consider the signal equation again:

• Signal g(t) contains a component cx(t)

• cx(t) is the projection of g(t) on x(t)

• If cx(t) = 0 c = 0

signal g(t) and x(t) are orthogonal over the interval

2,1 tt

dttxtgE

ct

tx2

1

)()(1

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Example 2.5 Component of a Signal (cont..)

For the square signal g(t), find the component of g(t) of the form sint or in other words approximate g(t) in terms of sint

tctg sin)( 20 t

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Example 2.5 (cont…)

ttx sin)(

From equation for signals

4sinsin

1sin)(

1

0

22

tdttdttdttgc

o

ttg sin4

)(

dttxtgE

ct

tx2

1

)()(1

and

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Orthogonality in complex signals

dttxEx

tcxtgt

t

22

1

)(

)()(

Coefficient c and the error in this case is

)()()( tcxtgte

For complex functions of t over an interval

22

1

)()( t

t

e tcxtgE

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Orthogonality in complex signals

uvvuvuvuvuvu222

2222

1

2

1

2

1

)()(1

)()(1

)( t

tx

x

t

tx

t

t

e dttxtgE

EcdttxtgE

dttgE

22

1

)()( t

t

e tcxtgE

We know that:

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Orthogonality in complex signals

dttxtgE

ct

tx

)()(1 2

1

0)()( 21

2

1

dttxtxt

t

0)()( 21

2

1

dttxtxt

t

So, two complex functions are orthogonal over an interval, if

or

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Energy of the sum of orthogonal signals

• Sum of the two orthogonal vectors is equal to the sum of the lengths of the squared of two vectors. z = x+y then

222yxz

• Sum of the energy of two orthogonal signals is equal to the sum of the energy of the two signals. If x(t) and y(t) are orthogonal signals over the interval, and if

z(t) = x(t)+ y(t) then

21, tt

yxz EEE

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Correlation

Consider vectors again:

• Two vectors g and x are similar if g has a large component along x OR

• If c has a large value, then the two vectors will be similar

c could be considered the quantitative measure of similarity between g and x

But such a measure could be defective. The amount of similarity should be independent of the lengths of g and x

But such a measure could be defective. The amount of similarity should be independent of the lengths of g and x

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Correlation

Doubling g should not change the similarity between g and x

However:

Doubling g doubles the value of c

Doubling x halves the value of c

However:

Doubling g doubles the value of c

Doubling x halves the value of cc is faulty measure for similarity

• Similarity between the vectors is indicated by angle between the vectors.

• The smaller the angle , the largest is the similarity, and vice versa

• Thus, a suitable measure would be , given bycosnc

xg

xgcn

.cos Independent of the lengths of g

and x

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Correlation

This similarity measure is known as correlation co-efficient.nc

nc 11 ncThe magnitude of is never greater than unity

•Same arguments for defining a similarity index (correlation co-efficient) for signals

• consider signals over the entire time interval

• normalize c by normalizing the two signals to have unit energies. dttxtg

EEc

xg

n

)()(1

xg

xgcn

.cos

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Correlation

consider )()( tkxtg

1nc

1nc

0nc

If k is positive then:

Negative then:

If g(t) and x(t) are orthogonal then Unrelated signals-------StrangersUnrelated signals-------Strangers

Related signals-------Best friendsRelated signals-------Best friends

Dissimilarity worst enemiesDissimilarity worst enemies

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Example 2.6

Find the correlation co-efficient between the pulse x(t) and the pulses

nc6,5,4,3,2,1,)( itgi

5)(5

0

5

0

2 dtdttxEx5

1gE

155

1 5

0

dtcndttxtgEE

cxg

n

)()(1

Similarly

Maximum possible similarityMaximum possible similarity

56

Example 2.6 (cont…)

5)(5

0

5

0

2 dtdttxEx25.1

2gE

1)5.0(525.1

1 5

0

dtcndttxtg

EEc

xg

n

)()(1

Maximum possible similarity……independent of amplitudeMaximum possible similarity……independent of amplitude

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Example 2.6 (cont…)

5)(5

0

5

0

2 dtdttxEx 51gE

1)1)(1(55

1 5

0

dtcndttxtgEE

cxg

n

)()(1

Similarly

58

Example 2.6(cont…)

)1(2

1)( 2

0

2

2

0

aTT

atT

at ea

dtedteE

5

1a 5T

1617.24 gE

5)(5

0

5

0

2 dtdttxEx

961.01617.25

1 5

0

5

dtect

n

Here

Reaching Maximum similarityReaching Maximum similarity

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Orthogonal Signal Space

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Orthogonal Signal Space

61

Trigonometric Fourier series

Consider a signal set:

,....sin....2sin,sin,....cos........2cos,cos,1 tnwtwtwtnwtwtw oooooo

2

0coscos

oTo

oo Ttdtmwtnwomn

mn

omn

mn

2

0sinsin

oTo

oo Ttdtmwtnw

•A sinusoid function with frequency is called the nth harmonic of the sinusoid of frequency when n is an integer.

• A sinusoid of frequency is called the fundamental

•This set is orthogonal over any interval of duration

because:

ow

ow

oo wT 2

onw

62

Trigonometric Fourier series

0cossin tdtmwtnwTo

oofor all n and m

and

The trigonometric set is a complete set.

Each signal g(t) can be described by a trigonometric Fourier series over the interval To :

...2sinsin 21 twbtwb oo

...2coscos)( 21 twatwaatg ooooTttt 11

1

sincos)(n

onono tnwbtnwaatg oTttt 11

or

on T

w2

63

Trigonometric Fourier series

We determine the Fourier co-efficient as:nno baa ,,

o

o

Tt

t o

o

Tt

t

n

tdtnw

tdtnwtgC

1

1

1

1

2cos

cos)(

tdtnwtgT

a o

Tt

ton

o

cos)(2 1

1

tdtnwtgT

b o

Tt

ton

o

sin)(2 1

1

,......3,2,1n

,......3,2,1n

dttgT

aoTt

to

1

1

)(1

0

64

Compact Trigonometric Fourier series

...2sinsin 21 twbtwb oo

...2coscos)( 21 twatwaatg ooooTttt 11

Consider trigonometric Fourier series

It contains sine and cosine terms of the same frequency. We can represents the above equation in a single term of the same frequency using the trigonometry identity

)cos(sincos nononon tnwCtnwbtnwa

22nnn baC

n

nn a

b1tan

oo aC

65

Compact Trigonometric Fourier series

1

0 )cos()(n

non tnwCCtg oTttt 11

66

Example 2.7

Find the compact trigonometric Fourier series for the following function

67

Example 2.7Solution:We are required to represent g(t) by the trigonometric Fourier series over the interval and t0 oT

sec22 radT

wo

o

ntbntaatg nn

no 2sin2cos)(1

t 0

Trigonometric form of Fourier series:

??,?,0 nn baa

68

Example 2.7

50.01

0

20

dtea

t

20

2

161

2504.02cos

2

ndtntea

t

n

20

2

161

8504.02sin

2

n

nntdteb

t

n

)cos()(1

0 non

n tnwCCtg

t 0

Compact Fourier series is given by

22nnn

oo

baC

aC

69

Example 2.7

)161

2(504.0

)161(

64

161

4504.0

504.0

222

2

22

22

nn

n

nbaC

aC

nnn

oo

nna

b

n

nn 4tan4tantan 11

.......)42.868cos(063.0)24.856cos(084.0

)87.824cos(25.1)96.752cos(244.0504.0

4tan2cos161

2504.0504.0)( 1

12

oo

oo

n

tt

tt

nntn

tg t 0

t 0

70

Example 2.7

n 0 1 2 3 4 5 6 7

Cn 0.504 0.244 0.125 0.084 0.063 0.054 0.042 0.063

Өn 0 -75.96 -82.87 -85.24 -86.42 -87.14 -87.61 -87.95

Amplitudes and phases for first seven harmonics

71

Periodicity of the trigonometric Fourier series

The co-efficient of the of the Fourier series are calculated for the interval oTtt 11,

100

1

])([cos()(

)cos()(

nnono

nnono

TtnwCCTt

tnwCCt

)(

)cos(

)2cos(

1

1

t

nwtCC

nnwtCC

no

n

no

no

n

no

for all t

for all t

72

Periodicity of the trigonometric Fourier series

tdtnwtgT

b

tdtnwtgT

a

o

Ton

o

Ton

o

o

sin)(2

cos)(2

dttgT

aTo

o

o

)(1

n= 1,2,3,……

n= 1,2,3,……

Means integration over any interval of To Means integration over any interval of TooT

73

Fourier Spectrum

)cos()(1

0 non

n tnwCCtg

Consider the compact Fourier series

This equation can represents a periodic signal g(t) of frequencies:

Amplitudes:

Phases:

oooo nwwwwdc ,.....,3,2,),(0

nCCCCC ,......,3210 ,,,

n ,.....,,,0 321

74

Fourier Spectrum

nc vs w (Amplitude spectrum)

wvs (phase spectrum)

Frequency domain description of )( t

Time domain description of )( t

75

Fourier Spectrum

)cos()(1

0 non

n tnwCCtg

Consider the compact Fourier series

This equation can represents a periodic signal g(t) of frequencies:

Amplitudes:

Phases:

oooo nwwwwdc ,.....,3,2,),(0

nCCCCC ,......,3210 ,,,

n ,.....,,,0 321

76

Fourier Spectrum

nc vs w (Amplitude spectrum)

wvs (phase spectrum)

Frequency domain description of )( t

Time domain description of )( t

77

Example 2.8

Find the compact Fourier series for the periodic square wave w(t) shown in figure and sketch amplitude and phase spectrum

1

sincos)(n

onono tnwbtnwaatw

Fourier series:

dttgT

aoTt

to

1

1

)(1

0 2

11 4

4

0 dtT

a

o

o

T

To

W(t)=1 only over (-To/4, To/4) and

w(t)=0 over the remaining segment

W(t)=1 only over (-To/4, To/4) and

w(t)=0 over the remaining segment

78

Example 2.8

2sin

2cos

2 4

4

n

ndttnw

Ta

o

o

T

T

oo

n

n

n2

2

0

...15,11,7,3

...13,9,5,1

n

n

evenn

0sin2 4

4

ntdtT

b

o

o

T

Ton 0 nb

All the sine terms are zeroAll the sine terms are zero

79

Example 2.8

....7cos

7

15cos

5

13cos

3

1cos

2

2

1)( twtwtwtwtw oooo

The series is already in compact form as there are no sine terms

Except the alternating harmonics have negative amplitudes

The negative sign can be accommodated by a phase of radians as

The series is already in compact form as there are no sine terms

Except the alternating harmonics have negative amplitudes

The negative sign can be accommodated by a phase of radians as)cos(cos xx

Series can be expressed as:

....9cos

9

1)7cos(

7

15cos

5

1)3cos(

3

1cos

2

2

1)( twtwtwtwtwtw ooooo

80

Example 2.8

2

1oC

n

C n 2

0

oddn

evenn

0

nfor all n 3,5,7,11,15,…..

for all n = 3,5,7,11,15,…..

We could plot amplitude and phase spectra using these values….

In this special case if we allow Cn to take negative values we do not need a phase of to account for sign.

Means all phases are zero, so only amplitude spectrum is enough

We could plot amplitude and phase spectra using these values….

In this special case if we allow Cn to take negative values we do not need a phase of to account for sign.

Means all phases are zero, so only amplitude spectrum is enough

81

Example 2.8

Consider figure

)5.0)((2)( twtwo

....7cos

7

15cos

5

13cos

3

1cos

4)( twtwtwtwtw oooo

82

Exponential Fourier series

83

Exponential Fourier series

84

Example

Consider example 2.7 again, calculate exponential Fourier series

sec22 radT

wo

o

oT

n

ntjn eDt 2)(

dteedtetT

D ntjtntj

To

no

2

0

22 1)(

1

0

)22

1(1

dtetn

85

Example

nj 41

504.0

ntj

n

enj

t 2

41

1504.0)(

and

...121

1

81

1

41

1

...121

1

81

1

41

11

504.0642

642

tjtjtj

tjtjtj

ej

ej

ej

ej

ej

ej

Dn are complex

Dn and D-n are conjugates

Dn are complex

Dn and D-n are conjugates

86

Example

nnn CDD2

1

nnD nnD and

thusnj

nn eDD njnn eDD

and

o

o

j

j

o

ej

D

ej

D

D

96.751

96.751

122.041

504.0

122.041

504.0

504.0

o

o

D

D

96.75

96.75

1

1

87

Example

o

o

j

j

ej

D

ej

D

87.822

87.822

625.081

504.0

625.081

504.0

o

o

D

D

87.82

87.82

1

1

And so on….

88

Exponential Fourier Spectra