Chapter 2

2/9/2014
Communication Systems 1
Communication Systems
Instructor: Engr. Dr. Sarmad Ullah Khan
Assistant ProfessorAssistant Professor
Electrical Engineering Department
CECOS University of IT and Emerging Sciences
Sarmad@cecos.edu.pk
Chapter 2
Dr. Sarmad Ullah Khan
Signals and Signal Space
2

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Outlines
• Signals and systems
• Size of signal
Size of signal
• Classification of signals
• Signal operations
• The unit impulse function
• Signals versus Vectors
• Correlation
O th l i l• Orthogonal signals
• Trigonometric Fourier Series
• Exponential Fourier Series
3
Outlines
• Size of signal
Size of signal
• Correlation
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Signal and system
• Signal
Signal is a set of Information / Data
Signal is a set of Information / Data
For example, Telephone, Telegraph, Stock Market
Price
Signals are function of independent variable time
In electric charge distribution over surface signalIn electric charge distribution over surface, signal
is function of space rather than time
5
Signal and system
• System
System process the received signal
System process the received signal
System might modify or extract information from
signal
For example, Anti aircraft missile launcher may
want to know the future location of targetwant to know the future location of target
Anti aircraft missile launcher gets information
from radar (INPUT)
Radar provides target past location and velocity
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Signal and system
• System
Anti aircraft missile launcher calculates the future
Anti aircraft missile launcher calculates the future
location (OUTPUT) using the received
information
• Definition
System gets a set of signals as INPUT and yield aSystem gets a set of signals as INPUT and yield a
set of signals as OUTPUT after proper processing
System might be a physical device or it might be
an algorithm
7
Signal and system
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Outlines
• Size of signal
Size of signal
• Correlation
9
Size of Signal
• Size of an entity is a quantity that shows
largeness or strength of entity
g g y
• It is a single value/number measure
• How signal (amplitude and duration) can be
represented by a single number measure?
• For example, person’s width ‘r’ and height ‘h’
• To be more precise, single value measure of a
person is its volume
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Size of Signal
• Signal Energy
 Area under a signal g(t) is its SIZE
g g( )
 Signal Size takes two values “Amplitude” and
“Duration”
 This measuring approach is defective for large
signals having positive and negative portions
11
Size of Signal
• Signal Energy
 Positive portion is cancelled by negative portion
p y g p
resulting in small signal
 This can be solve by calculating area under g2(t)
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Size of Signal
• Signal Energy
 If signal is complex signal, then
g p g ,
h ibl h i d | ( )| Other possible approach is area under |g(t)|
 Energy measure is more desirable
13
Size of Signal
• Signal Power
 Signal energy must be finite for it to be
g gy
meaningful
 Necessary conditions to make it finite
 Amplitude goes to zero as time approaches infinity
 Signal must converge
 IfIf
 Amplitude does not go to zero
 Then
 Energy is infinite
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Size of Signal
 Does this mean that a 60 hertz sine wave feeding into
your headphones is as strong as the 60 hertz sine wave
coming out of your outlet? Obviously not. This is
what leads us to the idea of signal power.
15
Size of Signal
• Signal Power
 To make energy finite and meaningful, time
gy g ,
average of signal energy is taken into consideration
 For comple signals For complex signals
 Signal power is time average of signal amplitude
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Size of Signal
• Square root of signal power is the Root Mean
Square (RMS) value of signal
17
Size of Signal
• Are all energy signals also power signals?
• No. In fact, any signal with finite energy will have
zero power.
• Are all power signals also energy signals?
• No, any signal with non-zero power will have infinite
energy.
• Are all signals either energy or power signals?
• No. Any infinite-duration, increasing-magnitude
function will not be either. (e.g. f(t)=t is neither)
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Size of Signal
• 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.
19
Example 2.1
• Determine the suitable measures of the signals given
below:
• The signal (a) amp 0 as t infinity Therefore• The signal (a) amp. 0 as t infinity .Therefore,
the suitable measure for this signal is its energy, given
by
gE 8444)2()(=
0
0
1
22
=+=+= 
∞
−
−
∞
∞−
dtedtdttg t
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Example 2.1
The signal in the fig. Below does not --- 0 as t 
. However it is periodic, therefore its power exits.
∞
21
Example 2.2
(a)
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Example 2.2
Remarks:
A sinusoid of amplitude C has power of regardless of its
frequency
and phase .
23
Example 2.2
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Example 2.2
We can extent this result to a sum of any number of sinusoids with distinct frequencies.
25
Example 2.2
Recall that
ThereforeTherefore
The rms value is
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Outlines
• Size of signal
Size of signal
• Correlation
27
Classification of signals
• Continuous-time and discrete-time signals
A l d di it l i l
• Analog and digital signals
• Periodic and Aperiodic signals
• Energy and power signals
• Deterministic and random signals
• Causal vs Non-causal signals• Causal vs. Non-causal signals
• Right Sided and Left Sided Signals
• Even and Odd Signals
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• Continuous-time Signal
 A signal that is specified for every value of time t
e.g. Audio and video signals
29
• Discrete-time signal
 A i l h i ifi d f di l f
 A signal that is specified for discrete value of
time t = nT, e.g. Stock Market daily average
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• Analog signal
 A l i l d i i i l
 Analog signal and continuous-time signal are
two different signals
 Analog signal whose amplitude can have any
value over a continuous range
Analog continuous time signal x(t) Analog discrete time signal x[n]
31
• Digital signal
 Di i l i l d di i i l
 Digital signal and discrete-time signal are two
different signals
 Discrete signal whose amplitude can have only a
finite number of value
Digital continuous time signal Digital discrete time
signal 32

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• Periodic signal
 A i l ( ) i id b i di if h i
 A signal g(t) is said to be periodic if there exist a
positive constant T0, such that
g(t) = g(t+T0) for all t
 Oth i i di i l Otherwise aperiodic signal
33
• Properties of Periodic Signal
 P i di i l i
 Periodic signal must start at time t = -
 Periodic signal shifted by integral multiple of T0
remains unchanged
 A periodic signal g(t) can be generated by
periodic extension of any segment of g(t) with
duration T0duration T0
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• Energy Signal
 A i l i h fi i i ll d i l
 A signal with finite energy is called energy signal
• Power Signal
 A signal with finite power is called power signal
35
• Remarks:
A signal with finite energy has zero power.
A signal can be either energy signal or power signal, not
both.
Every signal in daily life is energy signal, NOT power
signal
Power signal in practice is not possible because of infinite
duration and infinite energy
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• Deterministic Signal:
A signal whose physical description is know completely,
either mathematically or graphically is called deterministic
signal
• Random Signal:
A signal which is known in terms of probabilistic
description such as mean value, mean squared value and
distribution
37
• Casual Signal:
A signal which is zero prior to zero time
Signal amplitude A=0 for T = -t
• Non Casual Signal:
A signal which is zero after zero time
Signal amplitude A=0 for T = +t
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• Right sided and Left sided Signal:
A i ht id d i l i f t < T d l ft id d i l
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.
39
• Even and Odd Signal:
E i l ( ) d dd i l ( ) d fi d
 Even signals xe(t) and odd signals xo(t) are defined as
xe(t) = xe(−t) and xo(−t) = −xo(t).
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• Even and Odd Signal:
If h i l i i i d f i If h
 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.
41
Outlines
• Size of signal
Size of signal
• Correlation
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Signal Operation
Time Shifting
A signal g(t) is said to be time shifted if g(t) is
delayed or advanced by time T
If signal g(t) is delayed by time T, then
43
If signal g(t) is advanced by time T, then
Signal Operation
Time Scaling
The compression or expansion of signal g(t) in
time is known as Time Scaling
Compression
g(t)=g(at)
Expansion
g(t)=g(t/a)
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Signal Operation
Time Inversion
In time inversion, signal g(t) is multiplied by a
factor a = -1 in time domain
g(t) = g(at) if a=1
Time inverted signal
g(t) = g(at) if a=-1
45
Signal Operation
• Example: 2.4
F th i l (t) h i th fi B l k t h
• For the signal g(t), shown in the fig. Below , sketch
g(-t)
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Outlines
• Size of signal
Size of signal
• Correlation
47
Unit Impulse Signal
• The Dirac delta function or unit impulse or often
referred to as the delta function is the function that
referred to as the delta function, is the function that
defines the idea of a unit impulse in continuous-time
• It is infinitesimally narrow, infinitely tall, yet
integrates to one
• simplest way to visualize this as a rectangular pulsep y g p
from a -D/2 to a +D/2 with a height of 1/D
• The impulse function is often written as
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Unit Impulse Signal
=0 for al t ≠ 0
49
Unit Impulse Signal
• Multiplication of a Function by an Impulse
If a function g(t) is multiplied by impulse function we get
impulse value of g(t)
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Unit Impulse Signal
51
Unit Impulse Signal
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Unit Impulse Signal
• Unit Step Function u(t)
53
Outlines
• Size of signal
Size of signal
• Correlation
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Signals versus Vectors
• A Vector can be represented as a sum of its
components
components
• A Signal can also be represented as a sum of its• A Signal can also be represented as a sum of its
components
55
• A Signal defined over a finite number of time instants
can be written as a Vector
can be written as a Vector
• Consider a signal g(t) defined over interval [a,b]
• Uniformly divide interval [a,b] in N points
T1 = a, T2 = a+ϵ, T3 = a+2ϵ, …. TN = a+(N-1)ϵ
• Where
Step Sizeϵ =
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• Signal vector g can be written a N-dimensional vector
• Signal vector g grows as N increases
g = [g(t1) g(t2) … g(tN)]
• Signal transforms into continuous time signal g(t)
57
• Signal transforms into continuous time signal g(t)
• Continuous time signals are straightforward
generalization of finite dimension vectors
• Vector properties can be applied to signals
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• A vector is represented by bold-face type
• Specified by its magnitude and its direction
• For example, Vector x have magnitude | x | and
Vector g have magnitude | g |
• Inner product (dot or scalar) of two real valued
vectors ‘g’ and ‘x’ is
59
• Component of a Vector
Consider two vectors ‘x’ and ‘g’
‘cx’ (projection) is component of ‘g’ along ‘x’
What is the mathematical significance of a vector along
another vector?
g = cx + eg
However, this is not a unique way of vector decomposition
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Oth t ‘ ’ i
Other ways to express ‘g’ is
g is represented in terms of x plus another vector
which is called the error vector e
61
If we approximate
e = g – cx
Geometrically component of g along x is
Hence
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Based on definition of inner product, multiply both side by
|x|
63
If g and x are orthogonal, then
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• Component of Signal
Vector component and orthogonality can be extended to
continuous time signals
Consider approximating a real signal g(t) in terms of
another real signal x(t)
And
65
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 [t1 , t2]
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But ‘e’ is a function of ‘c’, not ‘t’, hence
67
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69
Two signals g(t) and x(t) are said to be orthogonal if there
is zero contribution from one signal to other signal
For N dimensional vectors ‘g’ and ‘x’
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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
Example 2.5
sint or in other words approximate g(t) in terms of sint
tctg sin)( ≅ π20 ≥≤ t
71
ttx sin)( = and
Example 2.5
)(
From equation for signals

dttxtg
E
c
t
tx
=
2
1
)()(
1
72
πππ
π π
π
π
4
sinsin
1
sin)(
1
0
22
=





−+==   tdttdttdttgc
o
ttg sin
4
)(
π
≅

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• Orthogonality in Complex Signals
For complex function g(t), its approximation by another
complex function x(t) over finite interval
‘c’ and ‘e’ are complex functions
73
Energy of a complex signal x(t) over finite interval
Choose ‘c’ such that it reduces ‘Ee’
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We know that
After certain manipulations
( )( ) ∗∗∗∗
++=++=+ uvvuvuvuvuvu
222
222 222
2
1
2
1
2
1
)()(
1
)()(
1
)( 
∗∗
−+−=
t
tx
x
t
tx
t
t
e dttxtg
E
Ecdttxtg
E
dttgE
dttxtg
E
c
t
tx
)()(
1 2
1
∗
=
75
So, two complex functions are orthogonal over an interval,
if
0)()( 21
2
1
=∗
 dttxtx
t
t
0)()( 21
2
1
=
∗
dttxtx
t
t
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
Sum of the energy of two orthogonal signals is equal to the
222
yxz +=
gy g g q
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
yxz EEE += 77
Outlines
• Size of signal
Size of signal
• Correlation
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Correlation of Signals
• Correlation addresses the question: “to what degree is
signal A similar to signal B”
signal A similar to signal B
• Two vectors ‘g’ and ‘x’ are similar if ‘g’ has a large
component along ‘x’
• If ‘c’ has a large value, then the two vectors will be
similar
‘ ’ ld b id d h i i f• ‘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 79
• Doubling g should not change the similarity between g
and x
and x
• Similarity between the vectors is indicated by angle
However:
Doubling g doubles the value of c
Doubling x halves the value of c
c is faulty measure
for similarity
y y g
between the vectors.
• The smaller the angle , the largest is the similarity, and
vice versa
• Thus, a suitable measure would be , given by
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• Where
• This similarity measure is known as correlation
co-efficient
Independent of the lengths of g
and x
• And
81
• Same arguments for defining a similarity index
(correlation co efficient) for signals
(correlation co-efficient) for signals
• Consider signals over the entire time interval
• To establish a similarity index independent of
energies (sizes) of g(t) and x(t), normalize c by
normalizing the two signals to have unit energiesg g g
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• Best Friends
• Opposite personalities (Enemies)
• Complete Strangers
83
Example 2.6
Find the correlation co-efficient between the pulse x(t) and the
pulses
nc
6,5,4,3,2,1,)( == itgi
p i
84
5)(
5
0
5
0
2
===  dtdttxEx
51
=gE
1
55
1
5
0
=
×
= dtcndttxtg
EE
c
xg
n 
∞
∞−
= )()(
1
Similarly
Maximum possible similarity

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Example 2.6 (cont…)
5)(
5
0
5
0
2
===  dtdttxEx
25.12 =gE
85
1)5.0(
525.1
1
5
0
=
×
=  dtcn
dttxtg
EE
c
xg
n 
∞
∞−
= )()(
1
Maximum possible similarity……independent of amplitude
Example 2.6 (cont…)
5)(
55
2
===  dtdttxEx 51
=gESimilarly
86
00
 1g
1)1)(1(
55
1
5
0
−=−
×
=  dtcndttxtg
EE
c
xg
n 
∞
∞−
= )()(
1
y

2/9/2014
Example 2.6(cont…)
)1(
2
1
)( 22
2
aT
T
at
T
at
e
a
dtedteE −−−
−=== 
5)(
5
0
5
0
2
===  dtdttxEx
87
200
a
5
1
=a 5=T
1617.24 =gE
961.0
1617.25
1
5
0
5
=
×
= 
−
dtec
t
n
Here
Reaching Maximum similarity
Outlines
• Size of signal
Size of signal
• Correlation
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Orthogonal Signal Sets
• A signal can be represented as a sum of orthogonal
set of signals
set of signals
• Orthogonal set of signals form a basis for specific
signal space
• For example, a vector is represented as a sum of
orthogonal set of vectors
I f di f• It forms a coordinate system for vector space
89
• Orthogonal Vector Space
C t i i d ib d b th t ll th l
Cartesian space is described by three mutually orthogonal
vectors x1, x2, and x3
If a three dimensional vector g is approximated by two
orthogonal vectors x1 and x2, then
And
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If th di i l t i t d b th
If a three dimensional vector g is represented by three
orthogonal vectors x1 , x2 and x3, then
And e = 0 in this case
x1,x2 and x3 is complete set of orthogonal space, No x4 exist
91
Th th l t ll d b i t
These orthogonal vectors are called basis vectors
Complete set of vectors is called complete orthogonal basis
of a vector
A set of vector {xi} is mutually{ i} y
orthogonal if
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• Orthogonal Signal Space
Lik t th lit f i l t (t) (t)
Like vector, orthogonality of signal set x1(t), x2(t), …..
xN(t) over time interval [t1, t2] is defined as
If all signal energies are equal En = 1 then set is normalized
and is called an orthogonal set
An orthogonal set can be normalized by dividing xN(t) by
93
• Orthogonal Signal Space
N i l (t) [t t ] b t f N th l
Now signal g(t) over [t1, t2] by a set of N-orthogonal
signals x1(t), x2(t), ….. xN(t) is
Energy of error signal e(t) can be minimized if
94

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Outlines
• Size of signal
Size of signal
• Correlation
95
Trigonometric Fourier Series
• Like vector, signal can be represented as a sum of its
orthogonal signal (Basis signals)
g g ( g )
• There are number of such basis signals e.g. trigonometric
function, exponential function, Walsh function, Bessel
function, Legendre polynomial, Laguerre functions,
Jaccobi polynomial
• Consider a periodic signal of period T0
• Consider a signal set• Consider a signal set
{1+Cos w0t+Cos 2w0t+……Cos nw0t…. Sin w0t+Sin 2w0t……Sin
nw0t….}
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• nw0 is called the nth harmonic of sinusoid of angular
frequency w where n is an integer
frequency w0 where n is an integer
• A sinusoid of frequency w0 is called the fundamental
tone/anchor of the set
97
• This set is orthogonal over any interval of duration
because:
o
o w
T π2=
because:




=
2
0
coscos
o
To
oo Ttdtmwtnw
omn
mn
≠=
≠
mn≠




=
0
sinsin oo Ttdtmwtnw
omn ≠=

2
o
To
oo T
0cossin = tdtmwtnw
To
oo
for all nand m
and
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• 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 ooo
oTttt +≤≤ 11

∞
=
++=
1
sincos)(
n
onono tnwbtnwaatg oTttt +≤≤ 11
or
o
n
T
w
π2
=
99
• We determine the Fourier co-efficient as:


+
+
= o
o
Tt
t
o
o
Tt
t
n
tdtnw
tdtnwtg
C 1
1
1
1
2
cos
cos)(
d
oTt

+1
)(
1
,......3,2,1=n 100
dttg
T
a
to
=
1
)(
1
0
tdtnwtg
T
a o
Tt
to
n
o
cos)(
2 1
1

+
=
tdtnwtg
T
b o
Tt
to
n
o
sin)(
2 1
1

+
=

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Compact Trigonometric Fourier Series
• Consider trigonometric Fourier series
• It contains sine and cosine terms of the same
frequency. We can represents the above equation in a
i l t f th f i th
...2sinsin 21 +++ twbtwb oo
...2coscos)( 21 +++= twatwaatg ooo oTttt +≤≤ 11
single term of the same frequency using the
trigonometry identity
)cos(sincos nononon tnwCtnwbtnwa θ+=+
22
nnn baC +=





 −
= −
n
n
n
a
b1
tanθ
oo aC =
oTttt +≤≤ 11
101

∞
=
++=
1
0 )cos()(
n
non tnwCCtg θ
Example 2.7
Find the compact trigonometric Fourier series for the following function
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Example 2.7
Solution:
We are required to represent g(t) by the trigonometric Fourier
series over the interval andπ≤≤ t0 π=T
series over the interval andπ≤≤ t0 πoT
sec
2
2 rad
T
w
o
o ==
π
T i i f f F i i
103
Trigonometric form of Fourier series:
??,?,0 nn baa
ntbntaatg n
n
no 2sin2cos)(
1
++= 
∞
=
π≤≤ t0
Example 2.7
500
1 2
== 
−
dtea
t
π
50.0
0
0 ==  dtea
π






+
== 
−
2
0
2
161
2
504.02cos
2
n
dtntea
t
n
π
π






+
== 
−
2
0
2
161
8
504.02sin
2
n
n
ntdteb
t
n
π
π
22
nnn
oo
baC
aC
+=
=
104
0
Compact Fourier series is given by
)cos()(
1
0 no
n
n tnwCCtg θ++= 
∞
=
π≤≤ t0

2/9/2014
Example 2.7
2644
504.0
2
n
aC oo ==
( )
)
161
2
(504.0
)161(
64
161
4
504.0
22222
22
nn
n
n
baC nnn
+
=
+
+
+
=+=
( ) nn
a
b
n
n
n 4tan4tantan 11
−=−=




 −
= −−
θ
( )4t2
2
50405040)( 1−
∞
 tt π≤≤ t0
105
( )
.......)42.868cos(063.0)24.856cos(084.0
)87.824cos(25.1)96.752cos(244.0504.0
4tan2cos
161
504.0504.0)( 1
1
2
+−+−+
−+−+=
−
+
+=
=

oo
oo
n
tt
tt
nnt
n
tg π≤≤ t0
π≤≤ t0
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
l d d h f f h
106
Amplitudes and phases for first seven harmonics

2/9/2014
• Periodicity of the trigonometric Fourier series
The co-efficient of the of the Fourier series are calculated
for the interval [t1, t1+T0]


∞
=
∞
=
+++=+
++=
1
00
1
])([cos()(
)cos()(
n
nono
n
nono
TtnwCCTt
tnwCCt
θφ
θφ for all t
)(
)cos(
)2cos(
1
1
t
nwtCC
nnwtCC
no
n
no
no
n
no
φ
θ
θπ
=
++=
+++=


∞
=
∞
=
for all t
107
• Periodicity of the trigonometric Fourier series
tdtnwtg
T
a o
To
n
o
2
cos)(
2
=
dttg
T
a
To
o
o
)(
1
=
n= 1,2,3,……
tdtnwtg
T
b o
To
n
o
sin)(
2
= n= 1,2,3,……
108

2/9/2014
• Fourier Spectrum
Consider the compact Fourier series
This equation can represents a periodic signal g(t) of
f i
)cos()(
1
0 no
n
n tnwCCtg θ++= 
∞
=
d )(frequencies:
Amplitudes:
Phases:
oooo nwwwwdc ,.....,3,2,),(0
nCCCCC ,......,3210 ,,,
nθθθθ ,.....,,,0 321
109
• Fourier Spectrum
F d i d i ti f )(
nc vs w (Amplitude spectrum)
wvsθ (phase spectrum)
Frequency domain description of )( tφ
Time domain description of )( tφ
110

2/9/2014
Example 2.8
Find the compact Fourier series for the periodic square wave
w(t) shown in figure and sketch amplitude and phase spectrum( ) g p p p
F i i
111

∞
=
++=
1
sincos)(
n
onono tnwbtnwaatw
Fourier series:
2
11 4
4
0 ==  dt
T
a
o
o
T
To
W(t)=1 only over (-To/4, To/4)
and
w(t)=0 over the remaining
segmentdttg
T
a
oTt
to

+
=
1
1
)(
1
0
Example 2.8






==  2
sin
2
cos
2 4
π
π
n
n
dttnw
T
a
oT
on

−
2
4
πnT oTo









−
=
π
π
n
n
2
2
0
...15,11,7,3
...13,9,5,1
=
=
−
n
n
evenn
112
 πn ,,,
0sin
2 4
4
==  ntdt
T
b
o
o
T
To
n 0= nb
All the sine terms are zero

2/9/2014
Example 2.8






+−+−+= ....7cos
7
1
5cos
5
1
3cos
3
1
cos
2
2
1
)( twtwtwtwtw oooo
π  7532 π
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 asπ
)cos(cos π−=− xx
113
Series can be expressed as:






++−++−++= ....9cos
9
1
)7cos(
7
1
5cos
5
1
)3cos(
3
1
cos
2
2
1
)( twtwtwtwtwtw ooooo ππ
π
Example 2.8
2
1
=oC




=
πn
C n 2
0
oddn
evenn
−
−



−
=
π
θ
0
n
for 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.π−
114
Means all phases are zero, so only
amplitude spectrum is enough

2/9/2014
Example 2.8
Consider figure
)5.0)((2)( −= twtwo
115






+−+−= ....7cos
7
1
5cos
5
1
3cos
3
1
cos
4
)( twtwtwtwtw oooo
π
Outlines
• Size of signal
Size of signal
• Correlation
116

2/9/2014
Exponential Fourier Series
• According to Euler’s theorem, each Sin function can
be represented as a sum of ejwt and e-jwt
be represented as a sum of ejwt and e jwt
• Also a set of exponentials ejnwt is orthogonal over any
time interval T=2*pi/w
A i l (t) l b t d• A signal g(t) can also be represented as an
exponential Fourier series over an interval T0
117
• Where the coefficient Dn can be calculated as
• Exponential Fourier series is an another form of
trigonometric Fourier series
118

2/9/2014
• Exponential Fourier series is an another form of
119
• The compact trigonometric Fourier series of a
periodic signal g(t) is given by
periodic signal g(t) is given by
120

2/9/2014
• Exponential Fourier Spectra
To draw Dn we need to find its spectra
As Dn is a complex quantity having real and imaginary
value, thus we need two plots (real and imaginary parts OR
amplitude and angle of Dn)
Amplitude and phase is prefered because of its close
ti ith th di t fconnection with the corresponding components of
We plot |Dn| vs ω and ∟Dn vs ω
121
Comparing exponential with trigonometric Fourier
spectrum yields
For real periodic signal, Dn and D n are conjugate, thusp g , n -n j g ,
122

2/9/2014
123
sec
2
2 rad
T
w
o
o ==
π

∞
−∞=
=
n
ntj
n eDt 2
)(ϕ
dteedtet
T
D ntj
t
ntj
T
o
n
o
2
0
22 1
)(
1 −
−
−
 ==
π
π
ϕ
== 
+−
π
)2
2
1
(1
dte
tn
π=oT
124
π 0
nj 41
504.0
+
=

2/9/2014
sec
2
2 rad
T
w
o
o ==
π
π=oT
ntj
n
e
nj
t 2
41
1
504.0)( 
∞
−∞= +
=ϕ
and








+
+
+
+
+
+
+
=
111
...
121
1
81
1
41
1
1
504.0
642 tjtjtj
e
j
e
j
e
j
Dn are complex
Dn and D-n are conjugates
125






+
+
+
−
+−
−
−−
...
121
1
81
1
41
1 642 tjtjtj
e
j
e
j
e
j
sec
2
2 rad
T
w
o
o ==
π
π=oTnnn CDD
2
1
== −
nnD θ=< nnD θ−=< −and
thus
nj
nn eDD θ
= nj
nn eDD θ−
− =
126
o
o
j
j
o
e
j
D
e
j
D
D
96.75
1
96.75
1
122.0
41
504.0
122.0
41
504.0
504.0
−
−
−

−
=

+
=
=
o
o
D
D
96.75
96.75
1
1
=<
−=<
−

2/9/2014
sec
2
2 rad
T
w
o
o ==
π
π=oT
o
o
j
j
e
j
D
e
j
D
87.82
2
87.82
2
625.0
81
504.0
625.0
81
504.0
−
−
−

−
=

+
=
o
o
D
D
87.82
87.82
1
1
=<
−=<
−
127
And so on….
128

Chapter 2

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Chapter 2