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r nc p es o ommun ca ons
Meixia Tao
Shanghai Jiao Tong University
Chapter 6: Signal Space Representation
.
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performance of digital transmission is the
signals used in communications can be
Thus, we need to understand signal space
conce ts as applied to digital communications
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Traditional Bandpass Signal
epresentat ons Baseband si nals are the messa e si nal enerated at the
source
Passband signals (also called bandpass signals) refer tothe signals after modulating with a carrier. The bandwidthof these signals are usually small compared to the carrier
c
Passband signals can be represented in three forms
Magnitude and Phase representation
Complex Envelop representation
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Where a(t) is the magnitude ofs(t)
and is the phase of s(t)
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baseband signals called the in-phase
and uadrature com onents of s t .
Signal space is a more convenient way than
I/Q representation to study modulation scheme
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Consider an n-dimensional space with unity basis
vectors {e1, e2, , en} (think of the x-y-z axis in a
coordinate system) Any vectora in the space can be written as
Dimension = Minimum number of vectors that isnecessary and sufficient for
representation of any vector in space
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Inner Product
a and b are Orthogonal if
=
A set of vectors are orthonormal if they are mutually
orthogonal and all have unity norm
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chosen such that:
Should be com lete ors an the vector s ace: an vectora
can be expressed as a linear combination of these vectors.
Each basis vector should be orthogonal to all others
A set of basis vectors satisfying these properties is alsosaid to be a complete orthonormal basis
In an n-dim space, we can have at most n basis vectors
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Basic Idea: If a si nal can be re resented b n-tuple, then it can be treated in much the same
way as a n-dim vector. Let 1(t), 2(t),., n(t) be n signals
~ ~ basis functions and we have an-dim si nal s ace
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n
0 j k
=jc j k =
j k
.
In this case,
( ) ( )kk dtttxx =
( ))(1
n
i
ii txtx ==
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),...,,(21 nxxx=
x
10
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Let and be represented as
with
Then the inner roduct of and is
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( ) ( )
=ji
dttt ji0
Since
Ex = Energy of x(t) =
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i=1,2,,M} with finite energy. That is
Then, we can express each of these waveforms as
where N M is the dimension of the signal space
and are called the orthonormal basis functions
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Consider the following signal set:
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By inspection, the signals can be expressed in terms of
the following two basis functions:
1 2
+1
1 2
+1
Not that the basis is orthogonal Also note that each these functions have unit energy
2009/2010 Meixia Tao @ SJTU We say that they form an orthonormal basis 15
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A representation of a digital
1
signal space 1-1
1
and 2(t) -1
as points, called
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following form
w an
We can choose the basis functions as follows
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Since
and
The basis functions are thus orthogonal and they are alsonormalized
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describe various modulation schemes
.
diagram is identical to Example 1
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geometric representation.
performance and the receiver structure for a signal set
In previous examples, we were able to guess thecorrect basic functions
In general, is there any method which allows us to find
a complete orthonormal basis for an arbitrary singalset?
GramGram--Schmidt Orthogonalization (GSO) ProcedureSchmidt Orthogonalization (GSO) Procedure
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Gram Schmidt Orthogonalization (GSO)
roce ure
set
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ofs1(t)
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Step 2: Construct the Second Basis
unct on
u rac o e corre a on por on
Compute the energy in the remaining portion
Normalize the remaining portion
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Step 3: Construct Successive Basis
unct ons ,
Define
Energy of :
k-th basis function: In general
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1st basis function is normalized version of the firstsignal
portions of signals that are correlated to previousbasis functions and normalizing the result
This procedure is repeated until all basis functions arefound
If , no new basis functions is added
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orthonormal basis functions corresponding to the signals
show below
2) Express x1, x2, and x3 in terms of the orthonormal basisfunctions found in part 1)
3) Draw the constellation diagram for this signal set
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Step 2:
and
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=> No more new basis functionsProcedure completes
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,
Constellation diagram
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,
Find the orthonormal basis functions using Gram
What is the dimension of the resulting signal space ?
Draw the constellation diagram of this signal set
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functions
of the signal points around the origin.
procedure affect the resulting basis functions
e c o ce o as s unc ons
does not affect the performance
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