' $ 6.962 Gr aduate Seminar in Communic ation Linear Mul - - PowerPoint PPT Presentation

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' $ 6.962 Gr aduate Seminar in Communic ation Linear Mul - - PowerPoint PPT Presentation

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' $ 6.962 Gr aduate Seminar in Communic ation Linear Mul tiuser Receivers: Effective Interference, Effective Band width and User Cap a city 6.962 Gradua te Seminar in Communica tion No vember 9, 2000 Presenter: C.

slide-1
SLIDE 1 6.962 Gr aduate Seminar in Communic ation ' & $ % Linear Mul tiuser Receivers: Effective Interference, Effective Band width and User Cap a city 6.962 Gradua te Seminar in Communica tion No vember 9, 2000 Presenter: C. Emre K
  • ksal
slide-2
SLIDE 2 6.962 Gr aduate Seminar in Communic ation ' & $ % Outline 1. In tro duction 2. Linear m ultiuser receiv ers 3. P erformance under random spreading sequences 4. User capacit y under p
  • w
er con trol 5. Multiple classes and eectiv e bandwidths 6. Summary and nal remarks
slide-3
SLIDE 3 6.962 Gr aduate Seminar in Communic ation ' & $ % In tro duction Motiv ation
  • High
demand for all kinds
  • f
applications
  • v
er wireless { V arious qualit y
  • f
service (bit rate, probabilit y
  • f
error) requiremen ts { Can the system accomo date another user with a QoS constrain t?
  • Ho
w to tak e adv an tage
  • f
the additional degrees
  • f
freedom pro vided b y spread-sp ectrum tec hniques.
  • A
t the ph ysical la y er, signal to in terference ratio (SIR) is the k ey parameter.
  • Previous
w
  • rk:
Not m uc h insigh t
  • n
ho w a user aects the system except in the w
  • rst
case.
slide-4
SLIDE 4 6.962 Gr aduate Seminar in Communic ation ' & $ % System Mo del W e consider a sym b
  • l-sync
hronous m ulti-access spread-sp ectrum system

. . .

X1 s1 X2 s2 XKsK demodulators K

Receiv ed v ector, Y: Y = K X i=1 X i s i + W User i: X i 2 < is the transmitted signal (E [X ] = 0; E
  • X
2
  • =
P i ; X i 's are iid) s i 2 < N is the random spreading sequence and W
  • N
(0;
  • 2
I ) Demo dulators : Mak e a go
  • d
estimate (soft)
  • n
the transmitted sym b
  • ls.
slide-5
SLIDE 5 6.962 Gr aduate Seminar in Communic ation ' & $ % Mo del
  • Con
tin ued
  • W
e are in terested in the SIR $ rates (bits p er sym b
  • l).
e.g., Gaussian input distribution ) 1 2 log (1 + SIR i )
  • Successiv
e cancellation is another p
  • ssibilit
y Linear receiv ers ! Receiv er 1: ^ X 1 = c T 1 Y = X 1 c T 1 s 1 + K X i=2 X i c T 1 s i + c T 1 W SIR 1 =
  • 1
= E
  • (X
1 c T 1 s 1 ) 2
  • (c
T 1 c 1 ) 2 + P K i=2 E
  • (X
i c T 1 s i ) 2
  • =
P 1 (c T 1 s 1 ) 2 (c T 1 c 1 ) 2 + P K i=2 P i (c T 1 s i ) 2
slide-6
SLIDE 6 6.962 Gr aduate Seminar in Communic ation ' & $ % Linear Multiuser Receiv ers
  • Matc
hed lter { The lter c i = s i : ^ X mf ;1 (Y ) = s T 1 Y s T 1 s 1 ; SIR 1 = P 1 (s T 1 s 1 ) 2 (s T 1 s 1 ) 2 + P K i=2 P i (s T 1 s i ) 2 { ^ X mf ;i is the pro jection
  • f
Y
  • n
s i .
  • Decorrelator
{ In the matrix form, Y can b e written as Y = S X + W where X = [X 1
  • X
K ] T and S = [s 1
  • s
K ] { matc hed lter
  • utputs
R form sucien t statistic for X: R = S T S X + S T W
slide-7
SLIDE 7 6.962 Gr aduate Seminar in Communic ation ' & $ % { Decorrelating lter is (S T S ) 1 in addition to matc hed lter: U = (S T S ) 1 R = X + (S T S ) 1 S T W { Decorrelating receiv er for user i is tak es the pro jection
  • f
Y
  • n
to (spanf(s j ) j 6=i g) ? (Do es not exploit correlation b et w een the terms
  • f
the in terference v ector ) sub
  • ptimal),
SIR 1 = P 1 P ii
  • Minim
um mean square error (MMSE) receiv er { The total in terference for user 1 is: Z = K X i=2 X i s i + W { The co v ariance matrix
  • f
Z is: K Z = S 1 D 1 S T 1 +
  • 2
I where S 1 is the N
  • (K
  • 1)
matrix
  • f
signature sequences
  • f
in terferers and D is the co v ariance matrix
  • f
[X 2
  • X
K ] T .
slide-8
SLIDE 8 6.962 Gr aduate Seminar in Communic ation ' & $ % { Eigen v alue decomp
  • sition
! K Z = Q T Q. K Z > and the whitening lter for the in terference is
  • 1
2 Q:
  • 1
2 QY = X 1
  • 1
2 Qs 1 +
  • 1
2 QZ { No w that in terference is white, apply matc hed lter to get scalar sucien t statistic for X 1 . Pro ject
  • 1
2 QY along
  • 1
2 Qs 1 : R = s T 1 K 1 Z Y = (s T 1 K 1 Z s 1 )X 1 + s T 1 K 1 Z Z { Finally , the MMSE estimate is the linear least squares estimate (LLSE)
  • f
X 1 giv en the
  • bserv
ation R : X mmse (Y ) = co v (X 1 ; R ) v ar(R ) R = P 1 R 1 + P 1 R = P 1 s T 1 K 1 Z Y 1 + P 1 s T 1 K 1 Z Y { The signal to in terference ratio for user 1 is: S I R 1 = (s T 1 K 1 Z s 1 ) 2 P 1 s T 1 K 1 Z s 1 = P 1 s T 1 K 1 Z s 1
slide-9
SLIDE 9 6.962 Gr aduate Seminar in Communic ation ' & $ % P erformance Under Random Spreading Sequences
  • Spreading
sequences: s i = 1 p N [V i1
  • V
iN ] T , where V ik 's are iid mean and v ariance 1 ) E
  • ks
i k 2
  • =
1. e.g.,

. . . . . . . . . . . .

degrees

  • f freedom

1

  • 1

1

  • 1

user 1 user 2 user K 1 2 3 N

  • W
e are in terested in the case K ; N ! 1; K N =
  • .
Assume that asymptotically empirical distribution
  • f
the p
  • w
ers
  • f
users (i.e., 1 K P X 2 i = P i ) con v erge to F (P )
slide-10
SLIDE 10 6.962 Gr aduate Seminar in Communic ation ' & $ % Matc hed Filter Pr
  • p
  • sition
3.3: L et
  • (N
) 1;M F b e the (r andom) SIR
  • f
the c
  • nventional
matche d lter r e c eiver for user 1. Then, with pr
  • b
ability 1:
  • (N
) 1;M F !
  • 1;M
F = P 1
  • 2
+ E F [P ] Sketch
  • f
the Pr
  • f:
By denition,
  • 1;M
F = P 1 (s T 1 s 1 ) 2 (s T 1 s 1 ) 2 + P K i=2 P i (s T 1 s i ) 2 Note that s T 1 s 1 ! 1 w.p. 1 and expanding s T 1 s i , it w as sho wn that v ar h P K i=2 P i (s T 1 s i ) 2 jP 1 ; P 2 ; : : : i = 0; 8 realizations
  • f
P i 's. Th us, K X i=2 P i (s T 1 s i ) 2 = E " K X i=2 P i (s T 1 s i ) 2 # = K N 1 K K X i=2 P i ! E F [P ] with probabilit y 1. Hence, N ! 1, P( P in terferer i ) = P P(in terferer i )
  • 1;M
F = P 1
  • 2
+ 1 N P K i=2 P i
slide-11
SLIDE 11 6.962 Gr aduate Seminar in Communic ation ' & $ %
slide-12
SLIDE 12 6.962 Gr aduate Seminar in Communic ation ' & $ % MMSE Receiv er The
  • r
em 3.1: L et
  • (N
) 1;M M S E b e the (r andom) SIR
  • f
the MMSE r e c eiver for user 1. Then, with pr
  • b
ability 1:
  • (N
) 1;M M S E !
  • 1;M
M S E = P 1
  • 2
+ E F [I (P ; P 1 ;
  • 1;M
M S E )] where I (P ; P 1 ;
  • 1;M
M S E ) = P P 1 P 1 + P
  • 1;M
M S E Notes
  • n
the Pr
  • f:
W e will use the follo wing theorem due to Silv erstein and Bai ab
  • ut
the limiting eigen v alue distribution
  • f
large matrices. Let A nm b e a n
  • m
matrix whose (i; j ) th en try is X ij p n where X ij 's are iid with unit v ariance. Let T m b e a m
  • m
diagonal matrix whose en tries are real v alued random v ariables. The matrix A nm T m A T nm has real non-negativ e eigen v alues with empirical distribution G n (). Note that G n () is a random v ariable. As n; m ! 1; m n = , G n () approac hes to a deterministic function, G().
slide-13
SLIDE 13 6.962 Gr aduate Seminar in Communic ation ' & $ % Some Observ ations
  • The
co v ariance matrix K Z = S 1 D 1 S T 1 +
  • 2
I has exactly the desired form.
  • The
asymptotic eigen v alue distribution, G() is not degenerate ) in this asymptotic regime, in terference is not white. { If K w ere nite and N ! 1, in terference w
  • uld
b e white w.p. 1. { If N w ere nite and K is increased, in terference will b e increasingly white. { If the in terference is white, matc hed lter and MMSE receiv er are iden tical. Th us,
  • nly
if K N is constan t, the MMSE receiv er
  • utp
erfroms the matc hed lter.
slide-14
SLIDE 14 6.962 Gr aduate Seminar in Communic ation ' & $ % Pro
  • f
  • con
tin ued
  • One
last complication left: Just the eigen v alue distribution ma y not b e sucien t for SIR c haracterization: { K Z = U T U where
  • is
diagonal and U is
  • rthogonal
for ev ery realization
  • f
Z. { Recall that
  • 1:M
M S E = P 1 s T 1 K 1 Z s 1 = P 1 (U s 1 ) T
  • 1
(U s 1 ) th us the relativ e p
  • sition
  • f
s 1 wrt. eigen v ectors
  • f
K Z also matters. { It is sho wn in Lemma 4.2 that as N ! 1; s 1 is white in an y co
  • rdinate
system and kU sk is constan t for an y realization
  • f
Z .
  • Th
us,
  • 1;M
M S E can b e c haracterized using
  • nly
the eigen v alue distribution K Z .
slide-15
SLIDE 15 6.962 Gr aduate Seminar in Communic ation ' & $ % Comparison
  • f
MF and MMSE Receiv ers
  • The
p erformances
  • f
the t w
  • receiv
ers for large N :
  • 1;M
F = P 1
  • 2
+ 1 N P K i=2 P i ;
  • 1;M
M S E = P 1
  • 2
+ 1 N P K i=2 I (P i ; P 1 ;
  • 1;M
M S E ) where I (P ; P 1 ;
  • 1;M
M S E ) = P P 1 P 1 +P
  • 1;M
M S E { I (P i ; P 1 ;
  • 1
) < P i since MMSE maximizes SIR {
  • 1;M
F is indep enden t
  • f
  • ther
signature sequences.
  • 1;M
F ! as P i ! 1. { I (P i ; P 1 ;
  • 1
) ! P 1
  • 1
ev en as P i ! 1 (near-far resistance) since MMSE receiv er exploits the structure
  • f
the in terference. { Decoupling
  • f
in terfering eects
  • f
  • ther
users. Eac h user con tributing an amoun t called \eectiv e in terference"
slide-16
SLIDE 16 6.962 Gr aduate Seminar in Communic ation ' & $ % F urther In tuition
  • n
MMSE SIR F
  • rm
ula
  • The
equation
  • =
P 1
  • 2
+ 1 N P K i=2 I (P i ; P 1 ;
  • )
has a unique solution since 1
  • P
1
  • 2
+ 1 N P K i=2 I (P i ;P 1 ; ) is a monotonically decreasing function
  • f
  • (it
cuts 1 at a single p
  • in
t). Let
  • b
e the unique ro
  • t
  • f
the equation.
  • In
general it is hard to ev aluate
  • .
But it can still b e v ery useful:
  • T
6
  • 1
,
  • T
6 P 1
  • 2
+ 1 N P K i=2 I (P i ; P 1 ;
  • T
)
  • Th
us, to c hec k whether the SIR requiremen t
  • T
  • f
a user can b e met, c hec k the second condition.
slide-17
SLIDE 17 6.962 Gr aduate Seminar in Communic ation ' & $ % P erformance
  • f
Decorrelator
  • If
W = then the input-output relation for decorrelator is as follo ws: S X ! X
  • It
is clear that decorrelator for user 1, r 1 , lies
  • n
(span f(s j ) j 6=1 g) ? .
  • It
is further sho wn that (Prop
  • sition
7.1) r 1 is the
  • rthogonal
pro jection
  • f
Y along (span f(s j ) j 6=1 g) ? . Giv en r 1 , S I R 1 = P 1 (r T 1 s 1 )
  • 2
(r T 1 r 1 )
  • In
  • ur
asymptotic regime, what happ ens to r T 1 s 1 ?
slide-18
SLIDE 18 6.962 Gr aduate Seminar in Communic ation ' & $ % Random SIR
  • f
the Decorrelator
  • A
theorem b y Bai and Lin sho w that the smallest eigen v alue
  • f
the random matrix S T 1 S 1 con v erges almost surely to a strictly p
  • sitiv
e n um b er. { Hence S 1 is almost surely
  • f
full rank, min (K
  • 1;
N ) { Note that dim (span f(s j ) j 6=1 g) ? = N
  • rank(S
1 ) { If K > N ) r T 1 s 1 ! 0. If K < N ) r T 1 s 1 !
  • 1
  • K
N
  • r
T 1 r 1
  • Th
us,
  • (N
) 1;D E C !
  • 1;D
E C = 8 < : P 1 (1)
  • 2
  • <
1
  • >
1 where
  • =
K N
slide-19
SLIDE 19 6.962 Gr aduate Seminar in Communic ation ' & $ % User Capacit y under P
  • w
er Con trol
  • If
the p
  • w
er lev els
  • f
the users can b e con trolled, in terference lev el can b e con trolled and { The user capacit y can b e increased (What is the maxim um n um b er
  • f
users p er degree
  • s
freedom?) giv en a maxim um p
  • w
er constrain t { The p
  • w
er consumption p er user can b e reduced giv en a constan t n um b er
  • f
users
slide-20
SLIDE 20 6.962 Gr aduate Seminar in Communic ation ' & $ % Matc hed Filter
  • The
asymptotic SIR relation for matc hed lter is:
  • 1;M
F = P 1
  • 2
+ 1 N P K i=2 P i
  • T
  • meet
the SIR requiremen t
  • ,
set all the receiv ed p
  • w
er lev els to: P mf (
  • )
=
  • 2
1
  • Without
a p
  • w
er constrain t { User capacit y is b
  • unded
b y
  • max
< 1
  • users
/ degree
  • f
freedom { As
  • !
1
  • ,
p
  • w
er requiremen t ! 1
  • With
p
  • w
er constrain t, P max , set P i = P max ; 8i to maximize :
  • max
= 1
  • 2
P max users/degree
  • f
freedom
slide-21
SLIDE 21 6.962 Gr aduate Seminar in Communic ation ' & $ % MMSE Receiv er Supp
  • se
the system supp
  • rts
an SIR
  • f
  • for
all users. Th us 8i P i
  • 2
+ 1 N P j 6=i I (P j ; P i ;
  • )
>
  • Let
P
  • =
inf i P i , the p
  • w
er
  • f
the w eak est user, k . But still, P
  • 2
+ 1 N P j 6=k I (P j ; P
  • ;
  • )
>
  • Since
I
  • nly
decreases as the in terferer p
  • w
ers decrease, w e ha v e P
  • 2
+ 1 N P j 6=k I (P
  • ;
P
  • ;
  • )
= P
  • 2
+
  • I
(P
  • ;
P
  • ;
  • )
>
  • With
I (P
  • ;
P
  • ;
  • )
= P
  • 1+
  • ,
w e ha v e:
  • 6
1 +
  • (1
+
  • )
  • 2
P
slide-22
SLIDE 22 6.962 Gr aduate Seminar in Communic ation ' & $ % MMSE Receiv er
  • con
tin ued
  • W
e can summarize this result as follo ws: 1. Giv en a maxim um p
  • w
er constrin t P
  • and
an SIR requiremen t
  • ,
the maxim um ac hiev able rate (P i = P
  • ;
8i):
  • 6
1 +
  • (1
+
  • )
  • 2
P
  • 2.
Without the p
  • w
er constrain t, the b
  • und
relaxes to:
  • <
1 +
  • =
1 + 1
  • 3.
The minimal solution is the degenerate p
  • w
er assignmen t. Otherwise,
  • ne
can decrease the p
  • w
er lev els
  • f
ev ery
  • ther
user to that
  • f
the w eak est user and still meet
  • Giv
en an SIR,
  • ,
the minim um p
  • w
er necessary at rate
  • is:
P mmse (
  • )
=
  • 2
1
  • 1+
slide-23
SLIDE 23 6.962 Gr aduate Seminar in Communic ation ' & $ % Remarks
  • n
P
  • w
er Con trol and User Capacit y
  • With
the con v en tional receiv er, arbitrarily high
  • is
not ac hiev able:
  • "
1
  • )
saturation (P
  • !
1).
  • MMSE
receiv er supp
  • rts
an extra user p er degree
  • f
freedom ( max = 1 + 1
  • ).
{ A t
  • =
1 user/degree
  • f
freedom, arbitrarily high
  • can
b e ac hiev ed with P mmse = O( 2 ). { A t
  • <
1 users/degree
  • f
freedom, arbitrarily high
  • can
b e ac hiev ed with P mmse = O(
  • ).
  • With
a similar analysis for the decorrelator, { W e get
  • max
< 1. The b
  • und
is indep enden t
  • f
  • (mak
es sense b ecause pro jection along (spanf(s j ) j 6=1 g) ? ) is indep enden t
  • f
in terferer p
  • w
ers. { A t
  • <
1 users/degree
  • f
freedom, arbitrarily high
  • can
b e ac hiev ed with P dec = O(
  • ).
slide-24
SLIDE 24 6.962 Gr aduate Seminar in Communic ation ' & $ % Multiple Classes and Eectiv e Bandwidths
  • Conserv
ation la ws { W
  • rk
conserv ation ) total resource consumption is constan t. e.g., queueing systems { Eectiv e bandwidth
  • f
a user is a measure
  • f
what p
  • rtion
  • f
the total resource that user consumes. { P eectiv e bandwidths is constan t
  • Matc
hed lter with J m ultiple classes eac h with K j =
  • j
N users { Supp
  • se
class j users require
  • j
. The minim um p
  • w
er required b y class j users is: P mf (j ) =
  • j
  • 2
1
  • P
J k =1
  • k
  • k
{ Th us, P J j =1
  • j
  • j
< 1 giv es us the feasible rate region.
slide-25
SLIDE 25 6.962 Gr aduate Seminar in Communic ation ' & $ % Multiple Classes
  • con
tin ued
  • Using
a similar approac h for the MMSE receiv er, { The minim um p
  • w
er necessary to meet the SIR requiremen t
  • j
; 8j is P mmse (j ) =
  • j
  • 2
1
  • P
K k =1
  • k
  • k
1+ k { Th us, P J j =1
  • j
  • j
1+ j < 1 giv es us the feasible rate region.
  • Similarly
for the decorrelator, the feasible rate region is sp ecied b y P J j =1
  • j
< 1
slide-26
SLIDE 26 6.962 Gr aduate Seminar in Communic ation ' & $ % Remarks
  • n
Eectiv e Bandwidths
  • The
necessary p
  • w
er to meet
  • j
is constan t
  • v
er users
  • f
t yp e j (otherwise decrease the p
  • w
ers do wn to that
  • f
the w eak est t yp e j user without violating the SIR constrain t).
  • Dene
e mf ;j =
  • j
; e mmse;j =
  • j
1+ j ; e dec;j = 1. F easible rate regions are dened as: X j e ;j
  • j
< 1 { It seems reasonable to dene e ;j degrees
  • f
freedom/user as the eectiv e bandwidth
  • f
t yp e j users ( j is in v ersely prop
  • rtional
to e ;j ). { Boundary
  • f
the feasible rate region is linear (con v ex p
  • lyhedral)
in all receiv ers. Mak es sense in MF. In MMSE, a consequence
  • f
asymptotic decoupling
  • f
in terference due to
  • ther
users). { P erformance with p
  • w
er con trol: lo w SIR ) MMSE
  • MF;
high SIR ) MMSE
  • decorrelator
slide-27
SLIDE 27 6.962 Gr aduate Seminar in Communic ation ' & $ % An tenna Div ersit y
  • The
results can b e applied to a m ultiple-an tenna scenario: { The mo del for a sync hronous m ulti-access an tenna-arra y system: Y = K X m=1 X m h m + W where X m is the sym b
  • l
  • f
m th user { Supp
  • se
fading v ectors, h m , are slo wly v arying and iid. { W e ha v e exactly the same mo del:
  • Asymptotically
in terfering users con tribute additiv ely to eectiv e in terference.
  • User
capacit y is c haracterized b y sharing the N degrees
  • f
freedom among users according to their eectiv e bandwidths.
slide-28
SLIDE 28 6.962 Gr aduate Seminar in Communic ation ' & $ % Final Remarks
  • Asymptotically
, a decoupling
  • f
in terfering eects is indeed p
  • ssible
for linear receiv ers { Eac h user can b e ascrib ed a lev el
  • f
eectiv e in terference that it pro vides to ev ery
  • ther
user. { Under p
  • w
er con trol, the user capacit y region has a linear b
  • undary
and eac h user can b e assigned an eectiv e bandwidth represen ting its resource consumption whic h is a non-decreasing function
  • f
their SIR requiremen t. Discussion : TDMA
  • r
FDMA systems ha v e a user capacit y
  • f
1 user/degree
  • f
freedom whic h is indeed equal to that
  • f
the decorrelator and almost iden tical to MMSE at high SIR. They are m uc h simpler to implemen t (matc hed lters
  • MMSE
receiv er). What do es this imply ab
  • ut
direct sequence CDMA systems?