Multiple Access Techniques Using Superframes

Radu Balan

Siemens Corporate Research

Princeton , NJ 08540

rvbalan@scr.siemens.com

SPIE Conference, San Diego

August 2000

Overview

1. Superframe – what is it?

2. Multiplexing of Signals using Superframes

3. Intermezzo – Band Limited Signals

4. TDMA, FDMA within this framework

5. FHMA as an extension on FDMA

1. Superframes - Introduction

Let H be a Hilbert space (e.g. , the space of -band limited signals) embedded in K (e.g. with ). Let I be a countable index set.

0

2B 02B 0

A subset of K is a frame for H with bounds A,B>0 if:{ , }if i I F

2 22| , | , ii I

A h h f B h h H

Property. There is a dual frame such that:{ , }if i I F

, i ii I

h h f f

, if i ii

d fic id

i

h h

Encoder Channel Decoder

T

*T

icid

h

hH 2() lI

i

Single Access Encoding – Decoding Scheme

Formal Embedding in the Coefficients Space

1H

2H

1h

2h

1T

2T

1E

2E1c

2c

c

2 ( )l I

The signals can be reconstructed from provided:1 2( , )h h 1 2c c c

1 2

21 2 1 2

1. , are frames;

2. (i) 0 , and (ii) is closed in ( ).E E E E l I F F

Or, equivalently:

1 21 2 1 2, is a frame for .i if f i I H H F F F

Definition. A collection of indexed sets of vectors is called a superframe if the direct sum set

is a frame for

1 2( , , , )LF F F

1 .LH H 1

1 ,LL i if f i I F F F

Properties.1. Geometric Picture: Equivalent conditions in term of componentsets (frames and transversal intersection of coeffs ranges);2. Duality: The existence of dual superframe -> Reconstruction formula

1

( , ) , ,1L kl

k l i l lii I l

h h f f h H l L

2. Multiplexing with Superframes

1, if

, Lif

1h

Lh

1ic

Lic

ic

i1Encoder

LEncoder

Channel

id

1

i ii

d f 1h

L

i ii

d f Lh

1Decoder

LDecoder

3. Intermezzo – Band Limited Signals

0

2

2 2

20

20

Denote by the space of -band limited signals:

ˆ{ ( ) , supp [ , ]}

1ˆConsider < and choose such that ( )

2

for all . Denote . Then for every , and ,

1. (

B

B f L R f

g B g

T f h B a R

f

Lemma

0

2

) , ( ) ; in particular, ( ) , ( )

22. ( ) ( ) ( )

2

23. { ( ), } is a tight frame for with bound

22

4. ( ) ( ) ,2

n

n

x f g x f nT f g nT

f x f nT a g x nT a

g nT a n B

f nT a h nT a f h

Z

4. The TDMA and FDMA Superframes

0 0 1

0

2 0 2 1 1

Consider two -band limited signals , sampled at a period

, for some . The

scheme uses the following encoding sequence:

( ) ( ) , ((n n k k

f f

T Time - Division with Multiple Access

c c c f kT c f k

1

) )2

T

Note the following:

0 1 0 10 1 0 1

mod 2,

( ) ( ) , ( ) , ( )2 2 2 2

where with the Kronecker symbol.

n n n n n

kn n k

T T T Tc a f n a f n f a g n f a g n

a

This suggests the use of the following frames:

0 0 00 2 2 1

1 1 11 2 2 1

{ ( ), } , 1, 02

{ ( ), } , 0, 12

TDn p p

TDn p p

Ta g n n a a

Ta g n n a a

Z

Z

G

G

0

0 1

2

The pair ( , ) is a normalized tight superframe

for .

TD TD

B

Theorem G G

00

11

Hence, the reconstruction formula is:

( ) ( )2

( ) ( )2

n nn

n nn

Tf x c a g x n

Tf x c a g x n

In the Frequency-Division Multiple Access (FDMA) case, theencoding scheme has the following structure:

0 0

0

0

( 2 )0 1 0 1

2

20 1

0 0 20

11

( , ) [ ( ) ( ) ( )] |

Thus the encoded sequence is: ( ( ) ( 1) ( ))2 2

and the associated frames are:

{ ( ), } , 2

{ ( ),2

i t i tT

t n

Tin n

n

TinFD

n n

FDn

f f f t e f t e f t

T Tc e f n f n

Tb g n n b e

Tb g n n

ZG

G01 2} , ( 1)

Tinn

nb e

Z

0

0 1

2

The pair ( , ) is a normalized tight superframe

for .

FD FD

B

Theorem G G

5. FHMA as an Extension of FDMA

The Frequency-Hoping Multiple Access is a variation of FDMA. TheEncoding scheme uses the following relation:

0 0( 2 (0)) ( 2 (1))0 1 0 1

2

( , ) [ ( ) ( ) ( )] |t ti t i tT

t nf f f t e f t e f t

0 (0) (1)20 1

where is a permutation of {0,1}. The encoded sequence is then:

(( 1) ( ) ( 1) ( ))2 2

n n

t

Tin n n

n

T Tc e f n f n

and the associated frames are:

0

0

(0)0 0 20

(1)1 1 21

{ ( ), } , ( 1)2

{ ( ), } , ( 1)2

n

n

TinnFH

n n

TinnFH

n n

Tc g n n c e

Tc g n n c e

Z

Z

G

G

0 1

2

In general, ( , ) may not be a superframe! And even if

it is, its dual may be hard to compute!

One case of interest: When there is a 0 such that:

In this case, the superframe condition

FH FH

n R nR

G G

0 1

0

2 1 ( ( ) ( ))

,0

reduces to a simple test:

The pair ( , ) is a superframe iff det( ) 0,

where is the 2 2 complex matrix ( ( / )) :

, 0 , 1 , 0 ,r r

FH FH

n mR i r k lR

l m k nr

A

A ceil R

A e n m l

Theorem G G

0 22

1

Moreover, the largest and smallest eigenvalues of give the frame

bounds, and the standard dual superframe is defined by 4 windows:

{ ( ), } with 2

TinFH k k k

k n Rp r r

k

A

R

Te g n n g g

ZG

For 2 and the following permutations:

1 2 2 1 we obtained 8 windows.

2 1 1 2

R