2000communication fundamentals1 dr. charles surya de634 6220 [email protected]

2000 Communication Fundamentals 1

Communication Fundamentals

Dr. Charles SuryaDE634 6220

[email protected]


Chapter 1 and 2

Introduction

and

Signals and Systems


• Communication refers to the conveying of information from one point to another

• It is a crucial component of information technology, which consists of: generation, transmission, reception, manipulation, storage and display of information

• Electromagnetic signals are typically used in the transmission of information

• In the process of transmitting the information, some alterations need to be done on the information-bearing signal to facilitate the transmission process. Upon reception, the inverse operation process needs to be done to retrieve to original signal


• The encoder chooses the best representation of the information to optimize its detection

• The decoder performs the reverse operation for the retrieval of the information

• The modulator produces a varying signal at its output which is proportional in some way to the signal appearing across it input terminals. A sinusoidal modulator may vary the amplitude, frequency or phase of a sinusoidal signal in direct proportion to the voltage input.

• The encoder and modulator both serve to prepare the signal for more efficient transmission. However, the process of coding is designed to optimize the error-free detection, whereas the process of modulation is designed to impress

•


the information signal onto the waveform to be transmitted.

• The demodulator performs the inverse operation of the modulator to recover the signal in its original form.

• The transmission medium is the crucial link, which may include the ionosphere, troposphere, free space, or simply a transmission line. Here attenuation, distortion, and noise in the medium are introduced.

• Noise is any electrical signals that interfere with the error-free reception of the message-bearing signal.


• The 3 basic subsystems of a communication system are indicated by the dashed lines in Fig. 1-1.

• The transmitter is to prepare the information to be sent in a way that best cope with the restrictions imposed by the channel.

• The receiver is to perform the inverse of the transmitter operation. The transmitter and the receiver as a pair are specifically designed to combat the deleterious effects of the channel on the information transmission.


• Fig. 1-1 is a simplex system. In many cases it is desirable to maintain 2-way communication. One way to accomplish this is to use the same channel alternately for transmission in each direction. This is called half-duplex.


• The full-duplex, as shown in Fig.1-3, allows simultaneous communication in both directions.


Signals and Systems

• A signal is an event capable of starting an action.

• For our purposes, a signal is defined to be a single-valued function of time and may be real or complex.

• The complex notation can be used to describe signals in terms of 2 independent variables. Thus, it is convenient for describing 2-D phenomena such as circular motion, plan wave propagation etc.

• Sinusoids play a major role in the analysis of communication systems e.g.

)cos()( tAtf


• Where A is the amplitude, is the phase and is the rate of phase change or frequency of the sinusoid in radians/s.

• The principle of Fourier methods of signal analysis is to break up all signals into summations of sinusoids. This provides a description of a given signal in terms of how the energy and power are distributed in these sinusoidal frequencies.


Classification of Signals Energy Signals and Power Signals

• An energy signal is a pulse-like signal that usually exists for only a finite interval of time, or even for an infinite amount of time, at least has a manor portion of its energy concentrated in a finite time interval.

• The instantaneous power of an electrical signal e(t) is

• In each case the instantaneous power is proportional to the squared magnitude of the signal. For a 1-Ohm resistance, these equations assume the same form. Thus, it is customary in signal analysis to speak of the instantaneous power associated with a given signal as Watts.

• The energy dissipated by the signal during a time interval (t1,t2) is

Rtep /)(2

2)(tfp

2

1

Joules )(2t

tdttfE


• We define an energy signal to be one for which

•

)(

2dttf


• The average power dissipated by the signal f(t) is

• A signal with the following property is defined as power signal

•

• A periodic signal is one that repeats itself exactly after a fixed period of time

• otherwise it is an aperiodic signal

2

1

2

12

)(1 t

tdttf

ttp

2/

2/

2)(

10 lim

T

TT

dttfT

)()( tfTtf


• A random signal is one about which there is some degree of uncertainty before it actually occurs, whereas a deterministic signal is one that has no uncertainty in its values.


Classification of Systems

• A system is a rule used for assigning an output, g(t), to an input, f(t) i.e.

• This rule can be in terms of an algebraic operation, a differential and/or integral equation, etc. For two systems connected in cascade the output of the first system forms the input to the second, thus forming a new overall system

• If a system is linear then superposition applies

)()( tftg

)()()( 12 tftftg

)()()()( 22112211 tgatgatfatfa


• A system is time-invariant if a time shift in the input results in a corresponding time shift in the output so that

• The output of a time-invariant system depends on time differences and not on absolute values of time. Any system not meeting this requirement is said to be time-varying.

000 any for )()( tttfttg


• A physically realizable or causal system cannot have an output response before an arbitrary input function is applied, otherwise it is a physically nonrealizable or noncausal system.


Orthogonal Functions

• If we wish to express a function f(t) as a set of numbers, fn, which, when expressed in terms of a properly chosen coordinate space, n, will specify the function uniquely.

• It is highly desirable that the set so chosen be a linearly independent set. That is the individual terms are not dependent on each other and that the set is formed by the totality of these terms.

• Such complete set of orthogonal functions are capable of uniquely representing the function of interest. This is known as the basis function.


• Two complex-valued functions 1 and 2 are orthogonal over the interval (t1, t2) if

• Thus if members of a set of complex-valued functions are mutually orthogonal over (t1, t2) then

0)()()()(2

1

2

12

*1

*21

t

t

t

tdtttdttt

2

1

0)()(

t

tn

mn mnK

mndttt


• The set of basis functions is said to be “normalized” if

• If the set is both orthogonal and normalized it is called an orthonormal set.

• The integral of the product of 2 functions over a given interval is called the inner product of the 2 functions. The square root of the inner product of a function with itself is called the norm.

2

1

n allfor 1)(2t

t nn dttK


• f(t) can be approximated by summation of a finite number of term n

• The integral-squared error remaining in this approximation after N terms is

n(t) is said to be complete over (t1, t2) if

N

nnn tftf

1

)()(

dttftfdttt

t

N

nnn

t

t N

2

1

2 2

1

2

1

)()()(

2

1

0)(2t

t NN

dttLim


• For a complete orthogonal set

• This relationship is known as the Parseval’s Theorem.

• Example: A given rectangular function is shown below:

1

)()(n

nn tftf


• First we can easily show that sin(nt) are orthonormal over the interval (0,2)

• Thus we have

• where fn is defined as

2

0 0

1sinsin

mn

mntdtmtn

1

sin)(n

n tnftf

2

02

0

2

2

0 sin)(sin

sin)(tdtntf

tdtn

tdtntffn


• Substituting into the equation above we obtain

• Thus f(t) can be represented by the following series

)2cos1(2

1sin 2 tntn

evenn for 0

oddn for /4sinsin

sin)(

2

1

1

0

2

0

ntdtntdtn

tdtntffn

...5sin

5

13sin

3

1sin

4)( ttttf


• The following figure shows the approximation when the function is approximated with 1, 2 and 3 terms.


The exponential Fourier Series

• For a set of complex-valued exponential functions

• where n is an integer and 0 is a constant. The value of n is referred to as the harmonic number or harmonic. Consider the following operation on n(t)

tjnn et 0)(

1)(

1

m,n ,)(

1

)()(

)()()(

0

)()(

0

*

12010

1020

2

1

002

1

ttmnjtmnj

tmnjtmnj

t

t

tjmtjnm

t

t n

eemnj

eemnj

dteedttt


• Excluding the trivial case where t2 = t1, we can force the term within the brackets to zero if we choose

• in which (n-m) is an integer. Thus,

• forms an orthogonal set of basis over the interval (t1, t2) if

2)( 120 tt

)(

2

if

,0

)(

120

122

1

00

tt

mn

mnttdtee

t

t

tjmtjn

tjnn et 0)(

)/(2 120 tt


• An arbitrary signal f(t) can be expressed as

• where the coefficients Fn are to be determined. It can be shown that the error energy between f(t) and its approximations decreases to zero as the number of terms taken approaches infinity. When a set of n(t) meets this condition, it is said to be complete

• It is therefore possible to represent any arbitrary complex-

N

Nn

tjnn ttteFtf )()( 21

0

n

tjnn ttteFtf )()( 21

0


• valued function with finite energy by a linear combination of complex exponential functions over an interval (t1, t2). Such representation is known as the exponential Fourier series representation. The coefficients in this series can be found by multiplying both sides by and integrating with respect to t over the interval. As a result of orthogonality, all terms on the right-hand side vanish except for m = n

tjmm et 0)(*

dtetftt

F

ttFdteFdtetf

tjnt

tn

mn

t

t

tmnjn

tjmt

t

02

1

2

1

002

1

)()(

1

)()(

12

12)(


Representation of periodic signal by Fourier Series

• A periodic signal is such that

• It is assumed that the signal has finite energy over an interval (t0, t0+T). We further assume that the energy content is constant over any interval of T seconds long. The power of the signal is, therefore, constant. The series representation

• will represent a periodic function over the infinite interval and the representation converges in a mean-square sense.

)()( tfTtf

n

tjnn

oeFtf )(


• The interval of integral to determine the Fourier series is taken over the complete period T.

• Example:

• Determine the Fourier series expression for the above waveform.

dtetfT

F tjnT

Tn0

4/3

4/)(

1


• To derive the trigonometric terms of the Fourier series we first consider n=1 and -1

2/2/32/2/

0

4/3

4/0

4/

4/0

4/

4/

4/3

4/

2

1

have we/2for

111

1

00

00

jnjnjnjnn

T

T

tjn

T

T

tjn

T

T

T

T

tjntjnn

eeeenj

F

T

ejn

ejnT

dtedteT

F


• For n=1

• Thus, the Fourier series term for 0 is

2

2

1

2

2

1

1

1

jjjjj

F

jjjjj

F

teFeF tjtj011 cos

400


Parseval’s Theorem For Power Signals• The average power developed across a 1-Ohm resistance is

• where

• We may interchange the order of summation and integration

• Since the complex exponential functions are orthogonal over T, the integral will be zero except for m=n. For this case, the double summation reduces to a single summation giving --- Parseval’s Theorem

dteFeFT

dttftfT

Pn

tjnn

m

tjmm

T

T

00 *2/

2/

* 1)()(

1

T/20

2/

2/

)(* 01 T

T

tnmj

nn

mm dte

TFFP

n

nn

nn FFFP2*


• Determine the average power of ttf 100sin2)(

WFPFF

jFdtteF

FFjF

tjtI

yxyxyxxx

tjttI

dtteT

dtteT

F

T

dtetT

dtetfT

F

nn

T

T

tT

j

T

T

tT

jT

T

tT

j

tT

jnT

T

T

T

tjnn

2.1

equals that shows analysissimilar ,100sin2

1

)2cos1(2sin

)sin(2

1)sin(

2

1cossin;2cos1

2

1sin since

sin2cossin2 integrand The

100sin21

100sin21

1nfor except zero be willintegralsFourier all 2

rads200 where

100sin21

)(1

2*11

1

2/

2/

2

1

*111

00

2

02

00

2/

2/

22/

2/

2

1

10

22/

2/

2/

2/

0


Transfer Function

• A system can be characterized in both time and frequency domains. In both approaches, linear superposition is assumed to add up the responses of the system for combinations of elemental functions. In this course we will focus on representation in the frequency domain.

• H() is known as the frequency transfer function of the system, which, in general, can be expressed as

)()()(

)()(

j

m

mm

k

kk

eHja

jbH

|H()| is the magnitude

response and () is the phase shift of the system


• The system response to a periodic signal

• is

• The output power can be determined by using Parseval’s theorem

n

tjnneFtf 0)(

n

tjnn

n

tjnn eFnHeGtg 00 )()( 0

n

non

nn

gnf FnHGPFP2222

)(;


• Example: Determine the output, g(t), of a linear time-invariant system whose input and frequency transfer function are as shown

• The Fourier representation for the input is

t

en

nnHtge

n

ntf tjn

n

tjn

n

2cos4

1

2/

)2/sin()2()( and

2/

)2/sin()( 22


• The average input power is

• The output power is

•

2/

2/

4/1

4/1

224)(

1 T

Tf WdtdttfT

P

W811.12

122

,2

.)(22

11

22

0

gn

ng PFFFnHP


Harmonic Generation

• An important application of the Fourier series representation is in the measurement of the generation of harmonic content. A device with a nonlinear output-input gain characteristics can be used to accomplished this, e.g.

• If we let , then

• The nonlinear output-input characteristic has therefore resulted in the generation of a second-harmonic term. This device is known as the frequency doubler. Similarly, a third-order nonlinearity results in generation of third-harmonic content, etc.

)()()( 2210 teateate ii

ttAtei 0cos)(

tAatAaAatAatAate 02

2012

2022

2010 2cos2

1cos

2

1coscos)(


• The presence presence of harmonic content in the output when only a single-frequency sinusoid is applied to the input represents distortion resulting from nonlinearities in the amplifier. A convenient way to measure this distortion is to take the ratio of the mean-square harmonic distortion terms to the mean-square of the first harmonic. This is known as the total harmonic distortion

21

21

2

22 )(

ba

baTHD n

nn

where an and bn are the coefficients for the cosine and sine Fourier series components respectively


The Fourier Spectrum

• It is the plot of the Fourier coefficients as a function of the frequency. In general, Fn are complex-valued. To describe the coefficients will require 2 graphs, the magnitude spectrum and phase spectrum.

• Example: Sketch the amplitude spectrum and the magnitude and phase spectrum of

• From our previous analysis

• Thus

• The solutions are shown in the next figure

ttf 100sin2)( jFjF 11 ;

)1 i.e.(100for 1 equals and )1 i.e.(100for 1 22 njnjjFn


• Example: Find the Fourier spectrum for the periodic function shown below:

• The Fourier coefficients are

)2/(

)2/sin(2/sin2

1)(

1

0

0

0

02/2/

0

2/

2/

2/

2/

00

00

n

n

T

A

Tn

nAee

Tjn

A

dtAeT

dtetfT

F

jnjn

tjnT

T

tjnn


• For , we have

• Thus, the exponential Fourier representation of the periodic gate function is

20n

x

x

x

T

AFn

sin

n

tjneTnSaT

Atf 0)/()(


• It is important to note that:– the amplitude of the spectrum decrease as 1/T

– the spacing between lines vary as T/2


Numerical Computation of Fourier Coefficients

• Fourier series coefficients may be approximated numerically. The trigonometric Fourier coefficients are

• Approximating the integration we have

• where

TT

nn tntfT

btntfT

a0 00 0 sin)(

2;cos)(

2

ttmTntmfT

b

ttmTntmfT

a

M

mn

M

mn

1

1

))(/2(sin)(2

))(/2(cos)(2

MTt /


• Example: Using 100 equally spaced sample points per period, compute the coefficients of the first 10 harmonic terms of the trigonometric Fourier series for the triangular waveform

M

mn

M

mn

MmntmfM

b

MmntmfM

a

1

1

)/2sin()(2

)/2cos()(2

obtain weequations above theinto ngSubstituti


• Thus,

22/3

02/)(

tt

tttf

M

mn mnmfa

MTtMT

1

)50/cos()50/(100

2

50//,100,2

2000communication fundamentals1 dr. charles surya de634 6220 [email protected]

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