Multiple Access Techniques Using Superframes
Radu Balan
Siemens Corporate Research
Princeton , NJ 08540
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
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