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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: [email protected]
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
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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
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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…)
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Example 2.2 (cont…)
We can extent this result to a sum of any number of sinusoids with distinct frequencies.
And rms value
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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
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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
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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
53
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
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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
57
Example 2.6 (cont…)
5)(5
0
5
0
2 dtdttxEx 51gE
1)1)(1(55
1 5
0
dtcndttxtgEE
cxg
n
)()(1
Similarly
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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
59
Orthogonal Signal Space
60
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
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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
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Compact Trigonometric Fourier series
1
0 )cos()(n
non tnwCCtg oTttt 11
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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
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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
