vector space.ppt

8/18/2019 Vector Space.ppt

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Signal-Space Analysis


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Digital Communication

System


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Representation of Bandpass Signal

Bandpass real signal x (t ) can be written as:

( ) ( ) ( )cos 2 c x t s t f t π =

( ) ( ) ( )22 Re where is complex envelopc j f t x t x t e x t π = % %

Note that ( ) ( ) ( ) I Q x t x t j x t = + ×% % %

In-phase Quadrature-phase


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Representation of Bandpass Signal

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

22 Re

2 Re cos 2 sin 2

2 cos 2 2 sin 2

c j f t

I Q c c

I c Q c

x t x t e

x t j x t f t j f t

x t f t x t f t

π

π π

π π

= = + × +

= + −

%

% %

% %

(1)

(2) Note that ( ) ( ) ( ) j t

x t x t eθ =% %

( ) ( ) ( ) ( )

( ) ( )( )

2 22 Re 2 Re

2 cos 2

c c j t j f t j f t

c

x t x t e x t e e

x t f t t

θ π π

π θ

= = ×

= +

% %

%


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Relation between and

2

2

( ) x t ( ) x t %

f

x

2 c j f t eπ −

f c

-f c

f f c

f f

( ) x t ( ) x t %

( ) ( ) ( )( )

( ) ( ) ( )

*1

2

( ), 0

,0, 0

c c

c

X f X f f X f f

X f f

X f X f X f f f + +

= − + − +

>= = +


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Energy of s(t )

( )

( )

( )

( )

2

2

2

0

2

0

(Rayleigh's energy theorem)

2 (Conjugate symmetry o real ( ) )

E s t dt

S f df

S f df s t

S f df

∞

−∞

∞

−∞

∞

∞

=

=

=

=

∫

∫

∫

∫ %


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Representation of bandpass LTI System

( )h t

( )h t %

( )s t

( ) s t %

( )r t

( )r t %

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) !ecause ( ) is !an"#limite"$c

r t s t h t

R f S f H f

S f H f f s t

= ∗

=

= +

%% %

%% %

%

( ) ( ) ( )( )

( )

( ) ( )

*

( ), 0

0, 0

c c

c

H f H f f H f f

H f f H f

f

H f H f f

+

+

= − + − + >

=


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Key Ideas


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Examples (1): BPSK


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Examples (): !PSK


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Examples ("): !A#


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$eomet%ic Inte%p%etation

(I)


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$eomet%ic Inte%p%etation

(II) %& representation is very convenient or somemo"ulation types$

e will examine an even more general way o

looing at mo"ulations, using signal space concept,which acilitates esigning a mo"ulation scheme with certain "esire"

properties

Constructing optimal receivers or a given mo"ulation

+nalying the perormance o a mo"ulation$

-iew the set o signals as a vector space.


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Basic Alge&%a: $%oup

+ group is "eine" as a set o elements G an" a !inary operation, "enote" !y / or which theollowing properties are satisie"

or any element a, b, in the set, a/b is in the set$ he associative law is satisie" that is or a,b,c in

the set (a/b)/c= a/(b/c)

here is an i"entity element, e, in the set such thata/e= e/a=a or all a in the set$

or each element a in the set, there is an inverseelement a#1 in the set satisying a/ a#1 = a#1 /a=e.


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$%oup: example

+ set o non#singular n3n matrices o

real num!ers, with matrix multiplication

4ote the operation "oes not have to !ecommutative to !e a 5roup$

6xample o non#group7 a set o non#

negative integers, with 8


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'niue identity 'niuein*e%se +%o eac, elementa/ x=a. hen , a#1·a·x=a#1·a=e, so x=e.

x·a=a

a/ x=e. hen, a#1·a·x=a#1·e=a#1 , so x=a#1.


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A&elian g%oup

% the operation is commutative, the group is

an +!elian group$ he set o m3n real matrices, with 8 $

he set o integers, with 8 $


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Application

9ater in channel co"ing (or error correction or

error "etection)$


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Alge&%a: eld

+ iel" is a set o two or more elements F :;α ,β ,$$<close" un"er two operations, 8 (a""ition) an" *(multiplication) with the ollowing properties

F is an +!elian group un"er a""ition he set F−;0


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Immediately +ollo.ing

p%ope%tiesα ∗β =0 implies α= 0 or β =0or any non#ero α , α ∗0= ?

α ∗0 + α = α ∗0 + α ∗1= α ∗(0 +1)= α ∗1=α; thereore α ∗0 =00∗0 =?

or a non#ero α, its a""itive inverse is non#ero$ 0∗0=(α+(− α) )∗0 = α∗0+(− α)∗0 =0+0=0


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Examples:

the set o real num!ers

he set o complex num!ers

9ater, inite iel"s (5alois iel"s) will !estu"ie" or channel co"ing

6$g$, ;0,1< with 8 (exclusive =R), * (+4)


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/ecto% space

+ vector space V over a given iel" is a set oelements (calle" vectors) close" un"er an" operation 8calle" vector a""ition$ here is also an operation *calle" scalar multiplication, which operates on an

element o (calle" scalar) an" an element o V to pro"uce an element o V $ he ollowing properties aresatisie"7 - is an +!elian group un"er 8$ 9et 0 "enote the a""itive

i"entity$ or every ,! in V an" every α ,β in F, we have

(α∗β )∗= α∗ (β∗ ) (α +β )∗= α∗ +β∗ α∗ ( "!):α∗ + α ∗ ! 1#=


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Examples o+ *ecto% space

R n over R

Cn over C

92 over


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Su&space0

9et - !e a vector space$ 9et !e a vector space an" $

% is also a vector space with the same operations as ,

then > is calle" a su!space o $

> is a su!space i

,

V S V

S V

V

! S a b! S

⊂

∈ ⇒ + ∈


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inea% independence o+

*ecto%s

1 2

e)

+ set o vectors , , are linearly in"epen"ent i n V ∈


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Basis

0

Consi"er vector space - over (a iel")$

e say that a set (inite or ininite) is a !asis, i

* every inite su!set o vectors o linearly in"epen"ent, an"

* or every ,

it

$ V

$ $

x V

⊂⊂

∈1 1

1 1

is possi!le to choose , $$$, an" , $$$,

such that $$$ $

he sums in the a!ove "einition are all inite !ecause without

a""itional structure the axioms o a vector

n n

n n

a a F $

x a a

∈ ∈= + +

space "o not permit us

to meaningully spea a!out an ininite sum o vectors$


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2inite dimensional *ecto%

space 1 21 2

+ set o vectors , , is sai" to span i

every vector is a linear com!ination o , , $

6xample7

n

n

n

V V

% V

R

∈

∈


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2inite dimensional *ecto%

space+ vector space - is f&n&te d&mens&'na( i thereis a inite set o vectors %1, %2 , ), %n that span -$


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2inite dimensional *ecto%space

1 2

1 2

1 2

9et - !e a inite "imensional vector space$ hen

% , , are linearly in"epen"ent !ut "o not span , then

has a !asis with vectors ( ) that inclu"e , , $

% , , span an" !ut ar

m

m

m

V V

n n m

V

•

>

•

1 2

e linearly "epen"ent, then

a su!set o , , is a !asis or with vectors ( ) $

6very !asis o contains the same num!er o vectors$

imension o a iniate "imensional vector space$

m V n n m

V

<

•


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Example: 3 n and its Basis /ecto%s

•••


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Inne% p%oduct space: +o%lengt, and angle


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Example: 3 n

•••

•••

•••

•••


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4%t,ono%mal set andp%o5ection t,eo%em

e)

+ non#empty su!set o an inner pro"uct space is sai" to !e

orthonormal i

1) , , 1 an"

2) % , an" , then , 0$

S

x S x x

x * S x * x *∀ ∈ < >=

∈ ≠ < >=


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P%o5ection onto a nitedimensional su&space

Gallager Th !"1

#orollar$: nor bound

#orollar$: Bessel%s ine&ualit$


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$%am 6Sc,midto%t,ono%mali7ation

{ }

{ }

1

1

1 1

Consi"er linearly in"epen"ent , $$$, , an" inner pro"uct space$

e can construct an orthonormal set , $$$, so that

; , $$$, < , $$$,

n

n

n n

s s V

V

s+an s s s+an

φ φ

φ φ

∈

∈

=


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$%am-Sc,midt 4%t,og0P%ocedu%e


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Step 1 : Sta%ting .it,

s1(t)


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Step :


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Step 8 :


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Key 2acts


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Examples (1)


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cont 9 (step 1)


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cont 9 (step )


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cont 9 (step ")


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cont 9 (step )


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Example application o+p%o5ection t,eo%em9inear estimation


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(;?)

(is an inne% p%oduct space0)

( )

[ ]( )2

Consi"er an orthonormal set

1 2 exp 0, 1, 2,$$$ $

+ny unction ( ) in 0, is , $ ourier series$

or this reason, this orthonormal set is calle" complete

t t j

- -

% t - % %

π φ

φ φ ∞=−∞

= = ± ± ÷

= ∑

2

$

hm7 6very orthonormal set in is containe" in some

complete orthonormal set$

4ote that the complete orthonormal set a!ove is not uni?ue$


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Signicance I!-modulation and %ecei*ed

signal in ( ) ( ) ( ) [ ]( )

( ) { }

{ }

2

2

@ A

, , 0,

span 2 cos 2 , 2 sin 2

+ny signal in can !e represente" as ( )$

here exist a complete orthonormal set

2 cos 2 , 2 sin 2 , ( ), ( ),$$$

c c

& &&

c c

r t s t / t -

s t - f t - f t

r t

f t f t t t

ξ ξ

π π

φ

π π φ φ

= + ∈

∈ −

−

∑


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4n @il&e%t space o*e% C02o% special +ol8s (e0g0= mat,ematicians)

only 92 is a separa!le Bil!ert space$ e have very useulresults on

1) isomorphism 2)counta!le complete orthonormal set

hm

% B is separa!le an" ininite "imensional, then it isisomorphic to ( 0 (the set o s?uare summa!le se?uenceo complex num!ers)

% B is n#"imensional, then it is isomorphic to Cn$

he same story with Bil!ert space over R$ %n some sense there is only one real an" onecomplex ininite "imensional separa!le Bil!ert space$

9$ e!nath an" $ Diusinsi, H&(bert S+aces !&th 1++(&cat&'ns, @r" 6"$, 6lsevier, 200E$


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@il&e%t space

e)

+ complete inner pro"uct space$

e) + space is complete i every Cauchy

se?uence converges to a point in the space$

6xample7 92


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4%t,ono%mal set S in@il&e%t space @ is complete

i+

22

6?uivalent "einitions

1) here is no other orthonormal set strictly containing $ (maximal)

2) , ,

@) , , implies 0

A) , ,

Bere, we "o not nee" to assume B is separa!le$

& &

&

S

x H x x e e

x e e S x

x H x x e

∀ ∈ =

∀ ∈ =∀ ∈ =

∑

∑

>ummations in 2) an" A) mae sense !ecause we can prove the ollowing7


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4nly +o% mat,ematicians(e dont need

sepa%a&ility0){ }

{ }2 2

9et !e an orthonormal set in a Bil!ert space $

or each vector x , set , 0 is

either empty or counta!le$

roo7 9et , $

hen, (inite)

+lso, any element in (however small

n

n

2 H

H S e 2 x e

S e 2 x e x n

S n

e S

∈ = ∈ ≠

= ∈ >

<

1

, is)

is in or some (suiciently large)$

hereore, $ Counta!le$

n

nn

x e

S n

S S ∞

== U


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>,eo%em

6very orothonormal set in a Bil!ert space iscontaine" in some complete orthonormal set$

6very non#ero Bil!ert space contains a completeorthonormal set$ (rivially ollows rom the a!ove$)

( Fnon#eroG Bil!ert space means that the space has a non#ero element$

e "o not have to assume separa!le Bil!ert space$)

Reerence7 $ >omasun"aram, 1 f&rst c'%rse &n f%nct&'na( ana(*s&s, 2xf'rd, 3.4.5 1(+ha Sc&ence, 006.


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4nly +o% mat,ematicians0(Sepa%a&ility is nice0)

6uivalent "einitions

e) is separa!le i there exists a counta!le su!set

which is "ense in , that is, $

e) is separa!le i there exists a counta!le su!set such that

,

H 7

H 7 H

H 7

x H

=

∀ ∈ there exists a se?uence in convergeing to $

hm7 % has a counta!le complete orthonormal set, then is separa!le$

proo7 set o linear com!inations (loosely speaing)

7 x

H H

with ratioanl real an" imaginary parts$ his set is "ense (show se?uence)

hm7 % is separa!le, then every orthogonal set is counta!le$

proo7 normalie it$ istance !etween two orthonorma

H

l elements is 2$ $$$$$

i l


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Signal Spaces: o+ complex +unctions


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'se o+ o%t,ono%mal set

1 2

1 2

1 2 1 2

D#ary mo"ulation ; ( ), ( ),$$$, ( )<

in" orthonormal unctions ( ), ( ), $$, ( ) so that

; ( ), ( ),$$$, ( )< ; ( ), ( ),$$, ( )<

8

4

8 4

s t s t s t

f t f t f t

s t s t s t s+an f t f t f t ⊂


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Examples (1)

2

T

2

T


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Signal Constellation


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cont 9


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cont 9


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cont 9

QPSK


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Examples ()

E l ' +


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Example: 'se o+o%t,ono%mal set and &asiswo s?uare unctions


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Signal Constellation

$ t i I t t ti


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$eomet%ic Inte%p%etation(III)


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Key 4&se%*ations

/ t >#33C/3


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/ecto% >#33C/3#odel

⊗ ⊗

⊗ ⊗

⊗ ⊗

⊕φ ( )1 t φ ( )1 t

φ ( )2 t φ ( )2 t

r : s 8 n1 1 1s1

-ector RC-R

-ector HDR

aveorm channel & CorrelationReceiver

s(t)

n(t)

r(t)s2

s 4

r : s 8 n2 2 2

r : s 8 n 4 4 4

φ ( )Ν t φ ( )Ν t

∑

⊕s(t)

n( t)

r ( t ) : s( t) 8 n ( t)

0

z

0

z

0

z

vector space.ppt

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