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Transcript of 2000Communication Fundamentals1 Dr. Charles Surya DE634 6220 [email protected].
2000 Communication Fundamentals 1
Communication Fundamentals
Dr. Charles SuryaDE634 6220
2000 Communication Fundamentals 2
Chapter 1 and 2
Introduction
and
Signals and Systems
2000 Communication Fundamentals 3
• 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
2000 Communication Fundamentals 4
2000 Communication Fundamentals 5
• 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
•
2000 Communication Fundamentals 6
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.
2000 Communication Fundamentals 7
• 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.
2000 Communication Fundamentals 8
• 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.
2000 Communication Fundamentals 9
• The full-duplex, as shown in Fig.1-3, allows simultaneous communication in both directions.
2000 Communication Fundamentals 10
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
2000 Communication Fundamentals 11
• 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.
2000 Communication Fundamentals 12
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
2000 Communication Fundamentals 13
• We define an energy signal to be one for which
•
)(
2dttf
2000 Communication Fundamentals 14
• 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
2000 Communication Fundamentals 15
• 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.
2000 Communication Fundamentals 16
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
2000 Communication Fundamentals 17
• 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
2000 Communication Fundamentals 18
• 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.
2000 Communication Fundamentals 19
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.
2000 Communication Fundamentals 20
• 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
2000 Communication Fundamentals 21
• 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
2000 Communication Fundamentals 22
• 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
2000 Communication Fundamentals 23
• 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
2000 Communication Fundamentals 24
• 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
2000 Communication Fundamentals 25
• 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
2000 Communication Fundamentals 26
• The following figure shows the approximation when the function is approximated with 1, 2 and 3 terms.
2000 Communication Fundamentals 27
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
2000 Communication Fundamentals 28
• 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
2000 Communication Fundamentals 29
• 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
2000 Communication Fundamentals 30
• 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)(
2000 Communication Fundamentals 31
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 )(
2000 Communication Fundamentals 32
• 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
2000 Communication Fundamentals 33
• 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
2000 Communication Fundamentals 34
• 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
2000 Communication Fundamentals 35
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*
2000 Communication Fundamentals 36
• 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
2000 Communication Fundamentals 37
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
2000 Communication Fundamentals 38
• 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
)(;
2000 Communication Fundamentals 39
• 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
2000 Communication Fundamentals 40
• 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
2000 Communication Fundamentals 41
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)(
2000 Communication Fundamentals 42
• 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
2000 Communication Fundamentals 43
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
2000 Communication Fundamentals 44
2000 Communication Fundamentals 45
• 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
2000 Communication Fundamentals 46
• 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)/()(
2000 Communication Fundamentals 47
• It is important to note that:– the amplitude of the spectrum decrease as 1/T
– the spacing between lines vary as T/2
2000 Communication Fundamentals 48
2000 Communication Fundamentals 49
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 /
2000 Communication Fundamentals 50
• 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
2000 Communication Fundamentals 51
• Thus,
22/3
02/)(
tt
tttf
M
mn mnmfa
MTtMT
1
)50/cos()50/(100
2
50//,100,2