Block diagram reduction techniques in control systems.ppt
Chapter 2
1. 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
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Communication Systems 2
Outlines
• Signals and systems
• Size of signal
Dr. Sarmad Ullah Khan
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
• Signals and systems
• Size of signal
Dr. Sarmad Ullah Khan
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
4
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Communication Systems 3
Signal and system
• Signal
Signal is a set of Information / Data
Dr. Sarmad Ullah Khan
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 signalIn electric charge distribution over surface, signal
is function of space rather than time
5
Signal and system
• System
System process the received signal
Dr. Sarmad Ullah Khan
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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Communication Systems 4
Signal and system
• System
Anti aircraft missile launcher calculates the future
Dr. Sarmad Ullah Khan
Anti aircraft missile launcher calculates the future
location (OUTPUT) using the received
information
• Definition
System gets a set of signals as INPUT and yield aSystem 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
Dr. Sarmad Ullah Khan
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Communication Systems 5
Outlines
• Signals and systems
• Size of signal
Dr. Sarmad Ullah Khan
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
9
Size of Signal
• Size of an entity is a quantity that shows
largeness or strength of entity
Dr. Sarmad Ullah Khan
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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Communication Systems 6
Size of Signal
• Signal Energy
Area under a signal g(t) is its SIZE
Dr. Sarmad Ullah Khan
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
Dr. Sarmad Ullah Khan
p y g p
resulting in small signal
This can be solve by calculating area under g2(t)
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Communication Systems 7
Size of Signal
• Signal Energy
If signal is complex signal, then
Dr. Sarmad Ullah Khan
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
Dr. Sarmad Ullah Khan
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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Communication Systems 8
Size of Signal
Dr. Sarmad Ullah Khan
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
Dr. Sarmad Ullah Khan
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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Communication Systems 9
Size of Signal
Dr. Sarmad Ullah Khan
• Square root of signal power is the Root Mean
Square (RMS) value of signal
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Size of Signal
• Are all energy signals also power signals?
• No. In fact, any signal with finite energy will have
Dr. Sarmad Ullah Khan
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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Communication Systems 10
Size of Signal
• Remark:
Dr. Sarmad Ullah Khan
• 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:
Dr. Sarmad Ullah Khan
• 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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Communication Systems 11
Example 2.1
The signal in the fig. Below does not --- 0 as t
. However it is periodic, therefore its power exits.
∞
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21
Example 2.2
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(a)
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Communication Systems 12
Example 2.2
Dr. Sarmad Ullah Khan
Remarks:
A sinusoid of amplitude C has power of regardless of its
frequency
and phase .
23
Example 2.2
Dr. Sarmad Ullah Khan
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Communication Systems 13
Example 2.2
Dr. Sarmad Ullah Khan
We can extent this result to a sum of any number of sinusoids with distinct frequencies.
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Example 2.2
Dr. Sarmad Ullah Khan
Recall that
ThereforeTherefore
The rms value is
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Communication Systems 14
Outlines
• Signals and systems
• Size of signal
Dr. Sarmad Ullah Khan
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
27
Classification of signals
• Continuous-time and discrete-time signals
A l d di it l i l
Dr. Sarmad Ullah Khan
• 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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Classification of signals
• Continuous-time Signal
Dr. Sarmad Ullah Khan
A signal that is specified for every value of time t
e.g. Audio and video signals
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Classification of signals
• Discrete-time signal
A i l h i ifi d f di l f
Dr. Sarmad Ullah Khan
A signal that is specified for discrete value of
time t = nT, e.g. Stock Market daily average
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Communication Systems 16
Classification of signals
• Analog signal
A l i l d i i i l
Dr. Sarmad Ullah Khan
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]
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Classification of signals
• Digital signal
Di i l i l d di i i l
Dr. Sarmad Ullah Khan
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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Communication Systems 17
Classification of signals
• Periodic signal
A i l ( ) i id b i di if h i
Dr. Sarmad Ullah Khan
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
Classification of signals
• Properties of Periodic Signal
P i di i l i
Dr. Sarmad Ullah Khan
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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Communication Systems 18
Classification of signals
• Energy Signal
A i l i h fi i i ll d i l
Dr. Sarmad Ullah Khan
A signal with finite energy is called energy signal
• Power Signal
A signal with finite power is called power signal
35
Classification of signals
• Remarks:
Dr. Sarmad Ullah Khan
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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Communication Systems 19
Classification of signals
• Deterministic Signal:
Dr. Sarmad Ullah Khan
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
Classification of signals
• Casual Signal:
Dr. Sarmad Ullah Khan
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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Communication Systems 20
Classification of signals
• Right sided and Left sided Signal:
A i ht id d i l i f t < T d l ft id d i l
Dr. Sarmad Ullah Khan
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
Classification of signals
• Even and Odd Signal:
E i l ( ) d dd i l ( ) d fi d
Dr. Sarmad Ullah Khan
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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Communication Systems 21
Classification of signals
• Even and Odd Signal:
If h i l i i i d f i If h
Dr. Sarmad Ullah Khan
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
• Signals and systems
• Size of signal
Dr. Sarmad Ullah Khan
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
42
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Communication Systems 22
Signal Operation
Dr. Sarmad Ullah Khan
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
Dr. Sarmad Ullah Khan
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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Communication Systems 23
Signal Operation
Time Inversion
Dr. Sarmad Ullah Khan
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
Dr. Sarmad Ullah Khan
• For the signal g(t), shown in the fig. Below , sketch
g(-t)
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Communication Systems 24
Outlines
• Signals and systems
• Size of signal
Dr. Sarmad Ullah Khan
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
47
Unit Impulse Signal
• The Dirac delta function or unit impulse or often
referred to as the delta function is the function that
Dr. Sarmad Ullah Khan
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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Communication Systems 25
Unit Impulse Signal
Dr. Sarmad Ullah Khan
=0 for al t ≠ 0
49
Unit Impulse Signal
• Multiplication of a Function by an Impulse
Dr. Sarmad Ullah Khan
If a function g(t) is multiplied by impulse function we get
impulse value of g(t)
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Communication Systems 27
Unit Impulse Signal
• Unit Step Function u(t)
Dr. Sarmad Ullah Khan
53
Outlines
• Signals and systems
• Size of signal
Dr. Sarmad Ullah Khan
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
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Communication Systems 28
Signals versus Vectors
• A Vector can be represented as a sum of its
components
Dr. Sarmad Ullah Khan
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
Signals versus Vectors
• A Signal defined over a finite number of time instants
can be written as a Vector
Dr. Sarmad Ullah Khan
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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Communication Systems 29
Signals versus Vectors
• Signal vector g can be written a N-dimensional vector
Dr. Sarmad Ullah Khan
• Signal vector g grows as N increases
g = [g(t1) g(t2) … g(tN)]
• Signal transforms into continuous time signal g(t)
57
Signals versus Vectors
• Signal transforms into continuous time signal g(t)
Dr. Sarmad Ullah Khan
• Continuous time signals are straightforward
generalization of finite dimension vectors
• Vector properties can be applied to signals
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Communication Systems 30
Signals versus Vectors
• A vector is represented by bold-face type
Dr. Sarmad Ullah Khan
• 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
Signals versus Vectors
• Component of a Vector
Dr. Sarmad Ullah Khan
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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Communication Systems 31
Signals versus Vectors
• Component of a Vector
Oth t ‘ ’ i
Dr. Sarmad Ullah Khan
Other ways to express ‘g’ is
g is represented in terms of x plus another vector
which is called the error vector e
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Signals versus Vectors
• Component of a Vector
Dr. Sarmad Ullah Khan
If we approximate
e = g – cx
Geometrically component of g along x is
Hence
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Communication Systems 32
Signals versus Vectors
• Component of a Vector
Dr. Sarmad Ullah Khan
Based on definition of inner product, multiply both side by
|x|
63
Signals versus Vectors
• Component of a Vector
Dr. Sarmad Ullah Khan
If g and x are orthogonal, then
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Communication Systems 33
Signals versus Vectors
• Component of Signal
Dr. Sarmad Ullah Khan
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
Signals versus Vectors
• Component of Signal
Dr. Sarmad Ullah Khan
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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Communication Systems 34
Signals versus Vectors
• Component of Signal
Dr. Sarmad Ullah Khan
But ‘e’ is a function of ‘c’, not ‘t’, hence
67
Signals versus Vectors
• Component of Signal
Dr. Sarmad Ullah Khan
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Communication Systems 35
Signals versus Vectors
• Component of Signal
Dr. Sarmad Ullah Khan
69
Signals versus Vectors
• Component of Signal
Dr. Sarmad Ullah Khan
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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Communication Systems 36
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
Dr. Sarmad Ullah Khan
sint or in other words approximate g(t) in terms of sint
tctg sin)( ≅ π20 ≥≤ t
71
ttx sin)( = and
Example 2.5
Dr. Sarmad Ullah Khan
)(
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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Communication Systems 37
Signals versus Vectors
• Orthogonality in Complex Signals
Dr. Sarmad Ullah Khan
For complex function g(t), its approximation by another
complex function x(t) over finite interval
‘c’ and ‘e’ are complex functions
73
Signals versus Vectors
• Orthogonality in Complex Signals
Dr. Sarmad Ullah Khan
Energy of a complex signal x(t) over finite interval
Choose ‘c’ such that it reduces ‘Ee’
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Communication Systems 38
Signals versus Vectors
• Orthogonality in Complex Signals
Dr. Sarmad Ullah Khan
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
Signals versus Vectors
• Orthogonality in Complex Signals
Dr. Sarmad Ullah Khan
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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Communication Systems 39
Signals versus Vectors
• Energy of the Sum of Orthogonal Signals
Dr. Sarmad Ullah Khan
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
• Signals and systems
• Size of signal
Dr. Sarmad Ullah Khan
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
78
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Communication Systems 40
Correlation of Signals
• Correlation addresses the question: “to what degree is
signal A similar to signal B”
Dr. Sarmad Ullah Khan
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
Correlation of Signals
• Doubling g should not change the similarity between g
and x
Dr. Sarmad Ullah Khan
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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Correlation of Signals
• Where
Dr. Sarmad Ullah Khan
• This similarity measure is known as correlation
co-efficient
Independent of the lengths of g
and x
• And
81
Correlation of Signals
• Same arguments for defining a similarity index
(correlation co efficient) for signals
Dr. Sarmad Ullah Khan
(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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Communication Systems 42
Correlation of Signals
• Best Friends
Dr. Sarmad Ullah Khan
• 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
Dr. Sarmad Ullah Khan
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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Communication Systems 43
Example 2.6 (cont…)
Dr. Sarmad Ullah Khan
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…)
Dr. Sarmad Ullah Khan
5)(
55
2
=== dtdttxEx 51
=gESimilarly
86
00
1g
1)1)(1(
55
1
5
0
−=−
×
= dtcndttxtg
EE
c
xg
n
∞
∞−
= )()(
1
y
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Communication Systems 44
Example 2.6(cont…)
Dr. Sarmad Ullah Khan
)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
• Signals and systems
• Size of signal
Dr. Sarmad Ullah Khan
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
88
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Communication Systems 45
Orthogonal Signal Sets
• A signal can be represented as a sum of orthogonal
set of signals
Dr. Sarmad Ullah Khan
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 Signal Sets
• Orthogonal Vector Space
C t i i d ib d b th t ll th l
Dr. Sarmad Ullah Khan
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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Communication Systems 46
Orthogonal Signal Sets
• Orthogonal Vector Space
If th di i l t i t d b th
Dr. Sarmad Ullah Khan
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
Orthogonal Signal Sets
• Orthogonal Vector Space
Th th l t ll d b i t
Dr. Sarmad Ullah Khan
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
92
47. 2/9/2014
Communication Systems 47
Orthogonal Signal Sets
• Orthogonal Signal Space
Lik t th lit f i l t (t) (t)
Dr. Sarmad Ullah Khan
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 Sets
• Orthogonal Signal Space
N i l (t) [t t ] b t f N th l
Dr. Sarmad Ullah Khan
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
48. 2/9/2014
Communication Systems 48
Outlines
• Signals and systems
• Size of signal
Dr. Sarmad Ullah Khan
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
95
Trigonometric Fourier Series
• Like vector, signal can be represented as a sum of its
orthogonal signal (Basis signals)
Dr. Sarmad Ullah Khan
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….}
96
49. 2/9/2014
Communication Systems 49
Trigonometric Fourier Series
• nw0 is called the nth harmonic of sinusoid of angular
frequency w where n is an integer
Dr. Sarmad Ullah Khan
frequency w0 where n is an integer
• A sinusoid of frequency w0 is called the fundamental
tone/anchor of the set
97
Trigonometric Fourier Series
• This set is orthogonal over any interval of duration
because:
o
o w
T π2=
Dr. Sarmad Ullah Khan
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
98
50. 2/9/2014
Communication Systems 50
Trigonometric Fourier Series
• The trigonometric set is a complete set.
Dr. Sarmad Ullah Khan
• 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
Trigonometric Fourier Series
• We determine the Fourier co-efficient as:
Dr. Sarmad Ullah Khan
+
+
= 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
+
=
51. 2/9/2014
Communication Systems 51
Compact Trigonometric Fourier Series
• Consider trigonometric Fourier series
Dr. Sarmad Ullah Khan
• 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
Dr. Sarmad Ullah Khan
102
52. 2/9/2014
Communication Systems 52
Example 2.7
Solution:
We are required to represent g(t) by the trigonometric Fourier
series over the interval andπ≤≤ t0 π=T
Dr. Sarmad Ullah Khan
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
Dr. Sarmad Ullah Khan
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
53. 2/9/2014
Communication Systems 53
Example 2.7
2644
504.0
2
n
aC oo ==
Dr. Sarmad Ullah Khan
( )
)
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
Dr. Sarmad Ullah Khan
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
54. 2/9/2014
Communication Systems 54
Trigonometric Fourier Series
• Periodicity of the trigonometric Fourier series
Dr. Sarmad Ullah Khan
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
Trigonometric Fourier Series
• Periodicity of the trigonometric Fourier series
Dr. Sarmad Ullah Khan
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
55. 2/9/2014
Communication Systems 55
Trigonometric Fourier Series
• Fourier Spectrum
Dr. Sarmad Ullah Khan
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
Trigonometric Fourier Series
• Fourier Spectrum
F d i d i ti f )(
Dr. Sarmad Ullah Khan
nc vs w (Amplitude spectrum)
wvsθ (phase spectrum)
Frequency domain description of )( tφ
Time domain description of )( tφ
110
56. 2/9/2014
Communication Systems 56
Example 2.8
Dr. Sarmad Ullah Khan
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
Dr. Sarmad Ullah Khan
== 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
57. 2/9/2014
Communication Systems 57
Example 2.8
Dr. Sarmad Ullah Khan
+−+−+= ....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
Dr. Sarmad Ullah Khan
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
58. 2/9/2014
Communication Systems 58
Example 2.8
Dr. Sarmad Ullah Khan
Consider figure
)5.0)((2)( −= twtwo
115
+−+−= ....7cos
7
1
5cos
5
1
3cos
3
1
cos
4
)( twtwtwtwtw oooo
π
Outlines
• Signals and systems
• Size of signal
Dr. Sarmad Ullah Khan
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
116
59. 2/9/2014
Communication Systems 59
Exponential Fourier Series
• According to Euler’s theorem, each Sin function can
be represented as a sum of ejwt and e-jwt
Dr. Sarmad Ullah Khan
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
Exponential Fourier Series
Dr. Sarmad Ullah Khan
• Where the coefficient Dn can be calculated as
• Exponential Fourier series is an another form of
trigonometric Fourier series
118
60. 2/9/2014
Communication Systems 60
Exponential Fourier Series
• Exponential Fourier series is an another form of
trigonometric Fourier series
Dr. Sarmad Ullah Khan
trigonometric Fourier series
119
Exponential Fourier Series
• The compact trigonometric Fourier series of a
periodic signal g(t) is given by
Dr. Sarmad Ullah Khan
periodic signal g(t) is given by
120
61. 2/9/2014
Communication Systems 61
Exponential Fourier Series
• Exponential Fourier Spectra
Dr. Sarmad Ullah Khan
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
trigonometric Fourier series
We plot |Dn| vs ω and ∟Dn vs ω
121
Exponential Fourier Series
• Exponential Fourier Spectra
Dr. Sarmad Ullah Khan
Comparing exponential with trigonometric Fourier
spectrum yields
For real periodic signal, Dn and D n are conjugate, thusp g , n -n j g ,
122
62. 2/9/2014
Communication Systems 62
Exponential Fourier Series
• Exponential Fourier Spectra
Dr. Sarmad Ullah Khan
123
Exponential Fourier Series
• Exponential Fourier Spectra
Dr. Sarmad Ullah Khan
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
+
=
63. 2/9/2014
Communication Systems 63
Exponential Fourier Series
• Exponential Fourier Spectra
Dr. Sarmad Ullah Khan
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
Exponential Fourier Series
• Exponential Fourier Spectra
Dr. Sarmad Ullah Khan
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
=<
−=<
−
64. 2/9/2014
Communication Systems 64
Exponential Fourier Series
• Exponential Fourier Spectra
Dr. Sarmad Ullah Khan
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….
Exponential Fourier Series
• Exponential Fourier Spectra
Dr. Sarmad Ullah Khan
128