Comm Ch06 Signal Space En

8/8/2019 Comm Ch06 Signal Space En

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r nc p es o ommun ca ons

Meixia Tao

Shanghai Jiao Tong University

Chapter 6: Signal Space Representation

.

2009/2010 Meixia Tao @ SJTU


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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

2009/2010 Meixia Tao @ SJTU 2


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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 ==

2009/2010 Meixia Tao @ SJTU

),...,,(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


Comm Ch06 Signal Space En

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