dc digital communication module i
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8/7/2019 DC Digital Communication MODULE I
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DIGITAL COMMUNICATION
Module I
variables and random processes Detection and -
interpretation of signals Response of a bank ofcorrelators to noisy input Detection of known
signals in noise probability of error correlationand matched filter receiver detection of signals.
Estimation concepts criteria: MLE estimator
filter for wave form estimation Linear
rediction.
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Ortho onal functions Consider a set of functions defined
over the interval1 2 3( ), ( ), ( ),........, ( ),........nt t t t
Let these functions satisfy the condition
2t
A set of functions which has this property is said to be orthogonal
1
i jt
1 2 .
Suppose we have an arbitrary function s(t) and we are interested in s(t)
only in the interval t1 tot2 .where the set of functions (t) are
or ogona .
Now we can express s(t) as a linear sum of the functions
n(t).
where s1,s2,s3 ,,sn are constants.
( ) ( ) ( )1 1 2 2( ) ...... ...... (1)n n s t s t s t s t = + + + +
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Ortho onal functions If such an expansion is possible, the orthogonality of the s makes it
eas to com ute the coefficients s .
To evaluate sn we multiply both sides of equation (1) by n(t) and
integrate over the interval of orthogonality.
( ) ( )2 2 2 2
1 1 1 1
2
1 1 2 2( ) ( ) ( ) ( ) ...... ( ) ......
t t t t
n n n n nt t t t
s t t s t t dt s t t dt s t dt = + + + +
equation become zero except a single term.
2 2 2t t
1 1n n n
t t
2 ( ) ( )t s t t dt 1
2
1
2,
( )
t
n t
nt
st dt
=
Then
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Ortho onal functions If we select the functions n(t) such that
2
1
2( ) 1t
nt
t dt =
When orthogonal functions are selected with the condition,
1
, ( ) ( )n nt
s s t t dt =Then2
2( ) 1t
n t dt =they are said to be normalized to have unit energy.
The use of normalized functions has the advantage that sns can be
1t
.
A set of functions which are both orthogonal and normalized is
.
We can represent any real valued signals as linear combinations of
orthonormal basis functions.
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Gram-Schmidt Orthogonalization Procedure
We may represent a given set of real-valued energy signals
s t s t s t .. s t each of duration T seconds usin
orthonormal basis functions as given below.
0N t T 1
( ) ( ) (1)1,2,.....,
i jj ijs
i M s t t
=
==
ij
1,2,.....,T i M==
The functions are orthonormal which
0 1,2,....,j N=1 2 3( ), ( ), ( ),........, ( )Nt t t t
,
0
0
1( ) ( )
T
i j
for i j
ort t d
it
==
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Gram-Schmidt Orthogonalization Procedure
The first condition states that the basis functionsare orthogonal with respect to each other over the interval 0 to T.
1 2( ), ( ),....., ( )Nt t t
The second condition states that the basis functions are normalized tohave unit energy.
Given the set of coefficients {sij}, j=1,2,,N .we can generate thesignal si(t), i=1,2,.M using a scheme as shown in figure (1).
,its own basis function, followed by a summer.
Conversely, given the set of signals si(t), i=1,2,. , operating asinput, we can generate the coefficients {sij}, j=1,2,,N using the
scheme given in figure (2)
It consists of a set of N product-integrators or correlators with acommon input, and with each supplied with its own basis function.
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Gram-Schmidt Orthogonalization Procedure
sT
( )1 t
0
2is0
T
d t2 tis t
iNs
t
0d t
(2)Figure0
1, 2, ,( ) ( ) ,
T
ij i j
i M s s t t dt
==
=
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, , ,
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Gram-Schmidt Orthogonalization Procedure
If the given set of signals s1(t), s2(t), s3(t),.. sM(t) are linearly
dependent, then there exists a set of coefficients a1,a2,..,a not all
zero, such that we may write
1 1 2 2
( ) ( ) ( ) 0M M
a s t a s t a s t + + + = aM sM
1 2 11 2 1( ) ( ) ( ) ( )
MM M
a a a s t s t s t s t
= + + +
It implies that sM(t) may be expressed in terms of the remaining (M-1)signals.
M M M
Next consider the set of signals s1(t), s2(t), s3(t),.. sM-1(t) . If this set
is linearly dependent there exists a set of numbers b1,b2,..,bM-1 not alle ual to zero such that
1 1 2 2 1 1( ) ( ) ( ) 0M Mb s t b s t b s t + + + =
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Gram-Schmidt Orthogonalization Procedure
Suppose that bM-1 0 . Then we may express sM-1(t) as a linear combination of the remaining M-1 signals.
1 2 21 1 2 2
1 1 1
( ) ( ) ( ) ( )MM MM M M
b b b s t s t s t s t
b b b
= + + +
es ng e se o s gna s s1 t , s2 t , s3 t ,.. sM-2 t or near
dependencies, and continuing in this fashion, we will eventually end
up with a linearly independent subset s1(t), s2(t), s3(t),.. sN(t) , NM
of the original signal. It is important to note that each member of the original set of signals
s t s t s t .. s t ma be ex ressed as a linear combination of
this subset of N signals s1(t), s2(t), s3(t),.. sN(t).
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Gram-Schmidt Orthogonalization Procedure
Now we may express the linearly independent functionss1(t), s2(t), s3(t),.. sN(t) in terms of orthonormal functions
using1 2 3( ), ( ), ( ), ........, ( )Nt t t t
0N t T 1 1,2,. . .,. .
i jj ij i N= =
1 11 1 12 2 1( ) (t)+ (t)+ + (t) (1 .)N N s t s s s a =
2 21 1 22 2 2.N N
1 1 2 2( ) (t)+ (t)+ + (t) (1 .) N N N NN N s t s s s n =
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Gram-Schmidt Orthogonalization Procedure
In equation (1 a.), set to zero all coefficients except s11. We then have
Since is to be a normalized function,1(t)
1 11 1
2
11 10
( )T
s s t dt = 1E=
11
11
( )(t) s ts
=1
1
( )s tE
=
In the next step we set to zero all the coefficients except the first two
s21
and s22
in equation (1b)
2 21 1 22 2( ) (t)+ (t) (3) s t s s =
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Gram-Schmidt Orthogonalization Procedure
Multiply both sides of equation (3) by and integrating over theinterval 0-T
1(t)
2
2 1 21 1 22 1 20 0 0
( ) (t) (t)+ (t) (t)T T T
s t s s =
2 1 21
0( ) (t)
T
s t s = 21 2 10 ( ) (t)T
s s t =
To evaluate s22, we rewrite equation (3) as
2 21 1 22 2( ) (t) (t) (4) s t s s =
[ ]2 2 2 2
2 21 1 22 2 22( ) (t) (t)dt = (5)T T
s t s dt s s = Compiled by MKP for CEC S6-EC, DC, Dec 2008
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Gram-Schmidt Orthogonalization Procedure
2( ) (t)
T
s s t s dt =
Again re-writing equation (3)
0
[ ]2 2 21 122
(t) = ( ) (t) s t ss
21 12
22 11
1 (t)= ( ) s ss ts s
21 12 2
22 11(t)= ( )
s ss ts s
Continuing in this manner we re-write equation(1c) as
3 31 1 32 2 32 2( ) (t)+ (t)+ (t) s t s s s =
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Gram-Schmidt Orthogonalization Procedure
Manipulating the equation as above we getT
31 3 10
=
T
32 3 20
t s s t =
[ ]33 3 31 1 32 20 ( ) (t) (t) s s t s s dt = 3 31 1 32 2
3
( ) (t) (t)
( )
s t s s
t
=
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Gram-Schmidt Orthogonalization Procedure
We continue in this manner until we have used all the N equationsand obtained all the orthonormal functions 1 2 3( ), ( ), ( ), ........, ( )Nt t t t
an a e coe c en s sij nee e o express e unc ons
s1(t), s2(t), s3(t),.. sN(t) in terms of 1 2 3( ), ( ), ( ), ........, ( )Nt t t t
Since all of the derived subset of linearly independent signals
s1(t), s2(t), s3(t),.. sN(t) may be expressed as a linear combination of
,
that each one of the original set of signals s1(t), s2(t), s3(t),.. sM(t)may be expressed as a linear combination of the same set of basis
1 2 3, , ,........, N
.
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Gram-Schmidt Orthogonalization Procedure
1
11
)()(
E
tst =
)()()( 111111 tstEts ==
T
)()()(
)()(
12122
01221
tststg
tttss
=
=
= T
dttg
tgt
0
2
2
22
)(
)()(
T
)()()()( 12120
2
22 tstdttgtsT
+=
=
1
0)()(
i
jiij dtttss )()( 121222 tsts +=
=T
ii
dtt
tgt
2
)()(
=
1j
jijii
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Geometric inter retation of si nals
0N t T
1
1,2,.....,i jj ij i M= =
=
Each si nal in the set S t is com letel determined b the vector of
0
, ,....( )
.,
1,2,....,( )ij i j s s t t dt
j N
==
its coefficients as given by
1iS
2
1, 2, 3, ,
i
i
S
s i M
= = i
S
i
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Geometric inter retation of si nals
The vectorSi is called signal vector. We ma now ex and the conventional idea of two and
three dimensional Euclidean spaces to an N-dimensional
Euclidean space. We can visualize the set of signal vectors { Si },
i=1,2,3,,M as defining a corresponding set of M points
n an - mens ona uc an space.
The N mutually perpendicular axes are labeled 1, 2,
3,.. N
This N-dimensional Euclidian space is called signal space.
-
three signals, that is, N=2 and M=3.
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Geometric inter retation of si nals
2 32, 3N M= =
2
s
1
1s1
2
0
3
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Geometric inter retation of si nals The length of a vectorsi is defined by the symbol The dot product of any vector with itself gives the squared length.
iS
( )
2
i i i s s s=
2 (4)N
ijs=
The cosine of the angle between the vectors Si and Sj is given by
1j=
( )cos i js s =
The two vectors are orthogonal if their dot product is zero.
i js s
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Geometric inter retation of si nals The energy of a signalsi (t) of duration T is defined as
2
0( )i i E s t dt =
1( ) ( )
N
i jj ijt tss
==
2
0( )i s t dt =
T
0 i is t s t t =
NT N
Interchanging the order of summation and integration,
110ik kkjj i j ==
1 1 0( ) ( )
N N
ij ik j
T
j ki kts s t dtE
= ==
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Geometric inter retation of si nals
=T
0 j k
N N2
N
s= 1 1i ij ik j k= = 1 j=
i
is equal to the squared length of the signal vectorsi representing it.
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Geometric inter retation of si nals
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Geometric inter retation of si nals
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Geometric inter retation of si nals
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Geometric inter retation of si nals
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