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Applied Ca alysis A: Gene al 897 (2010), a ailable
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1
Hyb id Highe O de S a is ics Lea ning in
Mul iuse De ec ion
An onio J. Caama˜
no Membe , IEEE,, Ra ael Boloix-To osa, Ja ie Ramos Membe , IEEE,,
Juan J. Mu illo-Fuen es Membe , IEEE,
Abs ac —In his wo k we explo e he signi icance o second
and highe o de s a is ics lea ning in communica ions sys ems.
The inal goal in sp ead spec um communica ion sys ems is o e-
cei e a signal o in e es comple ely ee om in e e ence caused
by o he concu en signals. To achie e his end we exploi he
s uc u e o he in e e ence, designing Second O de S a is ics
(SOS) de ec o s, such as he Minimum Squa e E o (MMSE)
(known o be op imum linea de ec o s in noisy en i onmen s)
in conjuc ion wi h Highe O de S a is ics (HOS) echniques,
such as he Blind Sou ce Sepa a ion (BSS) and he Independen
Componen Analysis (ICA) (known o exhibi such ema kable
p ope ies in lea ning such as scale in a iance and supe e iciency).
Such Hyb id Highe O de S a is ics (HyHOS) app oach enables
us o alle ia e BSS/ICA algo i hms o one o hei main p oblems,
ha is, hei sensi i iness o high le els o noise as we bene i om
hei speed o lea ning and independence o he ini ial se ings o
he p oblem. We success ully applied he esul s o his app oach
o he design o mul iuse de ec o s in Code Di ision Mul iple
Access (CDMA) channels.
Index Te ms—A ay Signal P ocessing, blind sou ce sepa a-
ion, highe o de s a is ics, independen componen analysis,
unsupe ised lea ning, CDMA.
I. INTRODUCTION
INTERFERENCE limi a ion due o he simul aneous access
o mul iple use s sys ems has been he s imulus o he
de elopmen o a powe ul amily o Signal P ocessing ech-
niques, namely Mul iuse De ec ion (MUD). This echniques
ha e been ex ensi ely applied o Di ec Sequence Code Di-
ision Mul iple Access (DS-CDMA) sys ems. Thus, mos o
las gene a ion digi al communica ion sys ems such as GPS,
wi eless 802.11b, UMTS, e c, may ake ad an age o any
imp o emen on his opic. In he blind case he algo i hms
should cope no only wi h noise and he nea - a p oblem
bu wi h no aining sequence a ailable. In his sense, he
minimum mean squa e e o (MMSE) c i e ia [1] p o ides
a good blind linea solu ion o he p oblem. Due o i s
compu a ional complexi y, some o he al e na i e algo i hms
ha e been p oposed [2], [3], [4].
On he o he hand, blind sou ce sepa a ion (BSS) and
independen componen analysis (ICA) ha e la ely been main
ields o esea ch due o hei g ea numbe o applica ions
A.J. Caama˜
no and Ja ie Ramos a e wi h he DTSC, EPS-
Telecomunicaciones, Uni e sidad Ca los III de Mad id. Bu a que, 15,
28911 Legan´
es, Mad id, Spain. Tl: +34 916248736. Fax: +34 916248749.
E-mail: [email p o ec ed]
J.J. Mu illo-Fuen es and R. Boloix-Tolosa a e wi h ATSC. Escuela Supe io
de Ingenie os. Uni e sidad de Se illa. Paseo de los Descub imien os sn.
41092 Se illa. Spain. Tl: +34 954487333. Fax: +34 954487372. E-mail:
[email p o ec ed].
in di e en a eas. ICA and BSS ha e been usually applied
(see [5] and e e ences he ein) o communica ions signals
(mainly in a ay p ocessing), biomedical signals such as ECG
o EEG, moni o ing, as an al e na i e o p incipal componen
analysis in image o inancial da a p ocessing, in moni o ing,
e c... Besides, hey ha e been la ely applied o o he p oblems
such as sp ead-spec um based communica ions [6], [7], digi al
wa e ma king o images [8], audio spec um basis unc ions
compu a ion [9], image classi ica ion, encoding o comp es-
sion [10],...
Se e al me hods ha e been p oposed as solu ions o he
BSS/ICA p oblem. We b ing ou he e wo main app oaches.
On he one hand, we ha e echniques based on he cancella ion
o es ima ion equa ions by using he s eepes descen algo i hm
[11], [12], [13], [7]. These me hods usually ha e he maximum
likelihood (ML) app oach as s a ing poin and a e no usually
obus as he p obabili y densi y unc ions (p.d. .) a e gi en a
p io i h ough he sepa a ion [14], also sco e [12], unc ions.
He e obus ness s ands o he me hods o be a ailable o
mix u es o any s a is ical dis ibu ions. On he o he hand,
we ha e con as unc ions. Con as a e cos unc ions whose
minimiza ion yields he solu ion o he BSS/ICA. These me h-
ods usually ace he BSS/ICA p oblem s a ing om lowe o
highe o de momen s analysis. In his sense he mix u es a e
i s educed o ze o mean signals. Then a whi ening p ocess
is ca ied ou o achie e deco ela ion. Finally, highe o de
momen s a e used o compu e a uni a y ans o ma ion o
diagonalize he associa ed highe o de enso o he whi ened
ou pu s. I a mos one ma ginal ou h-o de cumulan is
null, his uni a y ma ix may be compu ed by diagonalizing
he ou h o de enso . This enso may be diagonalized
by ei he canceling he whole se o cumulan s ou o he
diagonal (minimiza ion o he mu ual in o ma ion, MI), o by
he maximiza ion o he diagonal en ies (minimiza ion o he
ma ginal en opy, ME [15]) as he SICA me hod used in his
pape .
Some blind sou ce sepa a ion (BSS) algo i hms ha e been
p oposed as ully blind MUD’s, in he sense ha sp eading
codes a e unknown [16], [17], [7], [18]. Howe e his blind
echniques a e no usually noise obus . We p opose in his
pape o in oduce a simila echnique o BSS such as inde-
penden componen analysis (ICA) [19], [5] in o he s uc u e
o he MMSE MUD o exploi i s p o en noise obus ness. We
will compa e he pe o mance o a block ICA algo i hm such
as SICA [19] o an adap i e me hod o compu e he ma ix
such as M-EASI algo i hm [16].
The pape is o ganized as ollows. Sec ion II summa izes
2
he ma ix model, main assump ions and de ini ions o a DS-
CDMA communica ion sys em. The MMSE de ec o will be
also add essed. Then, BSS/ICA echniques a e in oduced in
Sec ion III, whe e we will ocus on he SICA algo i hm.
Sec ion IV is de o ed o ela e bo h o hese app oaches
BSS/ICA and he MMSE o p opose a new HOS based
solu ion, he so called BH MUD de ec o . These a e e e ed o
h oughou his pape as HyHOS I and II espec i ely. Sec ion
V includes expe imen s o compa e MMSE wi h he Hyb id
High O de S a is ics de ec o s de eloped in his wo k and o
show noise esis ance en supe e icien lea ning. Sec ion VI is
de o ed o conclusions.
II. COMMUNICATION SYSTEM MODEL:THE MMSE
Conside a CDMA ansmi e [1].The ansmi ed baseband
signal, assuming ha he p opaga ion channel o each use is
memo yless, can be ep esen ed as
x( ) =
L
X
l=1
n
X
i=1
χisici(l−τi)g( −lTc−τi),0≤ ≤Ts(1)
whe e Lis he leng h o he sp eading code, nis he numbe
o ac i e use s, χnis a scaling ac o speci ied by he powe
con ol loop, snis he symbol ansmi ed by use n,g( )
ep esen s he ansmi e pulse shaping il e , Tcis he chip
pe iod, and τnis he delay o each use . The sp eading code
o use nis deno ed by cn. No e ha indi idual chips in cn
a e deno ed by he elemen s in [cn(1),...,cn(L)]Twi h each
cn(·)∈ ±1. Equa ion 1 usually is exp essed as:
x( ) =
N
X
n=1
χnsncn( −τn)(2)
whe e cn( )is he no malized signa u e wa e o m assigned o
he n- h use . I he ansmi ed pulse-shape il e s a e Nyquis ,
we assume he synch onous case and he ecei e uses a chip-
a e sampled ma ched il e ollowed by a se ial o pa allel
con e e , he ma ix o m o he sys em equa ion is as ollows:
x =A s +n (3)
whe e x is a L×1 ec o , A is a L×nma ix ep e-
sen ing he channel esponse and n is a noise ec o . The
memo yless easies solu ion is a synch onized ma ched il e
(MF). Howe e , i he di e ence be ween powe s o use s is
high enough and use ’s codes a e no o hogonal, we ha e he
nea - a p oblem and o he well-known SOS based de ec o s
such as he MMSE ha e much be e a pe o mance.
A. Linea Mul iuse De ec ion
A linea mul iuse de ec o Cgi es an es ima ion o he
o iginal ansmi ed signals as
y( ) = Cx( ) = CAs( ) + Cn( )(4)
The cen alized de ec o minimizing he mean-squa e-e o
(MSE) o each use k, MSEk=E|yk( )−sk( )|2, is he
CMMSE
CMMSE =R−1
H∗=H∗HW ∗
W H∗(5)
whe e ( ) = H∗x( )and W is he deco ela ing ma ix
o ec o . He e, as in he ollowing, ∗deno es anspose-
conjuga e.
In [17] he au ho s de ine a whi ening– o a ion de ec o
(WR) as he CW R ma ix ha minimizes he MSE sum o
MSEk,k= 1, ..., n. A posible s uc u e o his ecei e was
gi en as ollows
CW R =JQW (6)
whe e Wis a m×mwhi ene , Qam×m o a ion ma ix,
and J = [I0] is n×m. In he con ex o he BSS, he p oblem
consis o compu ing he ma ix WB ha minimizes s a is ical
dependence a he ou pu .
O he p e ious BSS app oaches o ully blind MUD’s use a
SVD decomposi ion. Howe e , his in ol es a highe compu-
a ional complexi y. The na u al o ela i e g adien (NG) has
been also used as a s eepes descen algo i hm o adap i e
app oach in bo h BSS and MUD [20]. We ace in his pape
he de elopmen o a HOS based de ec o . Le ’s i s in oduce
he HOS-BSS/ICA ools we a e going o use.
III. BSS AND SICA
A. Main assup ions and model
Blind sepa a ion o sou ces (BSS) in ol e he ask o ob-
aining a non-obse able se o signals, he so-called sou ces,
om ano he se o obse able signals ega ded as mix u es.
He e, he adjec i e ”blind” s ands o he ac ha nei he he
o iginal sou ces no he mix u e i sel a e known. Usually, and
in he con ex o his pape , he assump ion o spa ial s a is ical
independence is he key o achie e sepa a ion. In his sense,
he ela ed [21] and mo e gene al echnique o independen
componen analysis (ICA) compu es he p ojec ion o a se o
componen s (mix u es in BSS) ha minimizes he s a is ical
dependence o he oupu s [15].
In i s simples o m, he BSS educes o he ollowing
ma ix o m. The m=nmix u es sampled a ime ,x ,
a e ins an aneous linea combina ions o he nsou ces s , i.e.,
x( ) = As( )(7)
Usually, and in he con ex o his pape , s a is ical indepen-
dence o he inpu s is assumed. Thus, i x is a s a iona y
e godic andom sequence and he mixing ma ix Ais non-
singula , i is possible o es ima e a sepa a ion ma ix B o
ob ain he sou ces as y=Bx =BAs.
y =Bx =BAs =Cs = 1,2,... (8)
In Blind Sou ce Sepa a ion (BSS) we compu e ma ix Bso
ha Cis ideally he iden i y ma ix. Howe e , he o iginal
scaling and a angemen canno be es ima ed om he inde-
pendence assump ion. In his sense, Cis a non mixing ma ix
i i has one and only one non-ze o en y in each column
and each ow. The ela ed [21] and mo e gene al p oblem
o independen componen analysis (ICA) [15] consis s o
ob aining he change o basis B ha p ojec s componen s x
in o a se y as s a is ically independen as possible (sou ces).
As ma ix Bcan be decomposed in o he p oduc o a
3
whi ening Wand a uni a y Vma ix, his sphe ing s age
gi es us signals zi( ):
y =Bx =V W As =V z = 1,2,... (9)
In addi ion o he sou ce independence hypo hesis, he e
a e o he mino conside a ions. Fi s , al hough he ins an a-
neous mix u e model conside ed he e applies o many o he
applica ions e e ed a he In oduc ion, i may be ex ended
o he con olu i e case [22], [23] when needed. Besides, we
may conside he e he case we e he numbe o mix u es m
is g ea e o equal o he numbe o sou ces n, i.e. m≥n.
In he case m>n, a subspace app oach such as a singula
alue decomposi ion may be used o p ojec he mix u e space
on o he signal subspace, educing he e ec o weak noises.
Fo noisy mix u es see [24], [25], [26]. We hus assume
he noiseless case wi h m=n. In addi ion, a necessa y
and su icien condi ion o he wa e o m-p ese ing sou ce
es ima ion o be easible is ha no mo e han one gaussian
dis ibu ed sou ce be p esen in he mix u e [24], [15], [27].
This model in (7) may be iden i ied wi h he na owband m-
senso linea -a ay applica ion, he synch onous-CDMA case
wi h sp eading ac o L=m, o he gene al ins an aneous
BSS/ICA model. In synch onous CDMA communica ions,
ma ix Amay be decomposed in o A=Hχ, whe e χis
a diagonal ma ix wi h he ampli udes o each use and His
a ma ix whose columns a e he sp eading codes. No ice ha
BSS/ICA applied o CDMA would lead o a de ec o whe e
nei he a aining sequence no he sp eading codes hemsel es
a e needed. Howe e , noise ec o in (3) is no included in
he BSS/ICA model in (7). Thus, i noise is high enough, he
pe o mance o he BSS/ICA model de e io a es.
Adap i e solu ions o his p oblem a e usually based on
maximum likelihood and he na u al (o ela i e) g adien [11],
[28], [29]. On he con a y, mos o he o -line solu ions
o ICA [15], [30] minimize one c i e ion, con as unc ion,
o cancella ion o mul iple c i e ia. Fo ins an aneous linea
mix u es o wo sou ces, di ec me hods, which consis o
di ec ly es ima ing he mixing ma ix om he mix u es, a e
also possible. Di ec me hods sugges ed in [31], [32], [33]
sol e polynomial equa ions based on cumulan s up o ou h-
o de . The solu ions o he equa ions p o ide he en ies o he
mixing ma ix. Howe e , me hods based on he op imiza ion
o c i e ia usually ha e a be e pe o mance. In his sense we
conside he e he con as unc ion based on he minimiza ion
o he ma ginal en opies.
B. Minimum ma ginal en ophy based con as
A con as unc ion φ(·)maps Y, he se o andom ec o s
y(mul i a ia e p.d. .’s), on R. I has a minimum when he
en ies o ya e s a is ically independen , i.e, i yhas inde-
penden componen φ(y)≤φ(Ay)∀Anon-singula . Thus,
he minimiza ion o a con as unc ion yields he solu ion o
he BSS/ICA p oblem. I he numbe o mix u es m > 2, hese
con as s a e complex unc ions o minimize. The ‘Jacobi op-
imiza ion’ [15] consis o sol ing he n-dimensional p oblem
by decomposing i in a se o 2-dimensional op imiza ions. The
o a ion ma ix Vis decomposed in o g=m(m−1)/2Gi ens
o a ions. Thus, minimiza ion o he con as s o each angle
θh,h= 1, ..., g, d i es he sys em o he solu ion. We will
i s ocus on a 2-dimenisonal con as , he SICA. Unde he
whi eness cons ain , i.e. E[yyT] = I, i yields he ollowing
o hogonal (deno ed by φ2) con as [21],
φME
2(y)c
=
n
X
i=1
H[yi](10)
whe e he c i e ia o minimize is he ma ginal en opy H[y].
He e, as in he ollowing, c
=means equali y up o an addi i e
cons an c. Besides, we will deno e by µy
ijkl =E[yiy∗
jyky∗
l]
and µy
ij =E[yiy∗
j] he ou h o de and second-o de momen s.
I we app oxima e he possible dis ibu ions o sby and Edge-
wo h expansion, ew i e he esul in e ms o second-o de
and ou h-o de cumulan s, Cy
ijkl =µy
ijkl −µy
ijµy
kl −µy
ikµy
jl −
µy
ilµy
jk and hen minimize i o all possible dis ibu ions [15],
i ollows ha
φME
2(y)≈1
48φME
24 (y) = −1
48 X
i
(Cy
iiii)2(11)
whe e Cy
iiii =µy
iiii −3µ2
ii is he au ocumulan o ou pu yi,
symbol ∗deno ing complex conjuga ion. Thus, he minimiza-
ion o he ou h-o de cumulan based ME con as yields
he diagonaliza ion o he associa ed enso when a mos
one ma ginal ou h-o de cumulan is null. in he ollowing,
we will ace he eal case. The ex ension o complex- alued
sou ces is somehow immedia e. By using he complex no a ion
and unde he whi ening cons ain , he independen compo-
nen s yp( )and yq( )in he wo-dimensional app oach yield
y∗( ) = ( )ejρpq ( )= ( )ej(θpq +βpq ( )) =ejθpq z∗( )(12)
whe e j=√−1. The whi enned mix u e ec o z∗( ) =
zp( ) + jzq( )a e a simple o a ion o he no malized (uni
a iance) sou ces ¯s∗( ) = ¯sp( ) + j¯sq( ) = ( )ejα. No ice
ha a he solu ion ρ( ) = θ+β( ) = α( ) + kπ/2,k=
0,1,2, ... The associa ed es ima ion o he o a ion angle θ
in he op imiza ion o 2-dimensional con as s may be easily
exp essed as a closed unc ion o he ollowing complex-
alued linea combina ions (cen oids) o he s a is ics o he
ou pu s [34]
ξγ=E[ 4( )ej4βpq( )](13)
ξη=E2[ 4( )ej2βpq( )](14)
γ=κ −8(15)
Based on he so called ‘weigh ed es ima o s’ (WE) o
WAML [35], [36] a gene al 2-dimensional es ima o yields
ˆ
θGW E(ωγ, ωξ) = 1
4∠(ωξωγξγ+ (1 −ωξ)ξη)(16)
0< ωξ<1, ωγ=±1, γ
This es ima es yields EML es ima o in [34] ˆ
θEML =
ˆ
θGW E(sign(γ),1) o he [37], MK [38], [30], SKSE and
SKDE [39], ML [40] es ima es ˆ
θGW E(±1,1),ˆ
θMaSSF OC =
ˆ
θGW E(γ, 1/2) in [39], o he ˆ
θAML =ˆ
θGW E(γ, 1/3) [35].
The discussion on he op imum wξ[36] is open as i is di icul
4
o compu e he s a is ics o hese es ima es and analyze hem
o all possible p.d. ’s. In his pape , we p opose o use
θSICA =ˆ
θGW E(γ, 3/7) (17)
as i may be p o ed ha he SICA app oach is as obus as
he φME
24 (y), i.e, i p o ides a solu ion o he ICA p oblem
wi h he only condi ion o no mo e han one sou ce wi h null
ou h o de s a is ics in he mix u e. The minimiza ion o
φSICA(θ)is immedia e as he solu ion is minus ou imes
he phase o he esul ing unc ion, a sinusoid. This may be
easily implemen ed by a look-up able so ha we minimize
he numbe o ope a ions needed.
C. n-dimensional case
The 2-dimensional case may be easily ex ended o ndimen-
sions by using he ’Jacobi Op imiza ion’ [15]. We ha e he e
ew i en he algo i hm using φSICA(θ). Such an algo i hm
can be summa ized as ollows.
Algo i hm 1: n-dimensional SICA using Jacobi Op imiza-
ion.
1) Whi enning. Compu e a whi ening ma ix Wand he
ou pu ec o z=W x. Se c= 1 and y=z.
2) Sweep c. Fo all g=n(n−1)/2pai s, i.e., o 1≤p <
q≤n, do
(a) Compu e he Gi ens angle θpq θSICA(y)in (17)
wi h [zp, zq]T= [yp, yq]T.
(b) i θpq > θmin, do o a e he pai (yp, yq)by θpq
acco ding o (12).
3) End? I he numbe o sweeps csa is ies c≥K= 1 +
√no no angle θpq has been upda ed, s op. O he wise
go o s ep 2 o ano he sweep wi h c=c+ 1.
D. The Na u al G adien
The s eepes descen me hod upda es Cacco ding o he
di ec ion o he g adien ∇Lo a loss unc ion L(C). The
na u al [23] o ela i e g adien [5] p oposes o use e
∇L(C) =
e
∇E[l(C)] = ∇L(C)CTC. The s ochas ic e sion uses he
in an aneous alue e
∇l(C). The lea ning law yields
C←− C−λ∇l(C)CTC(18)
In he BSS p oblem i is usually assumed s a is ical indepen-
dence a he ou pu s y( ). Maximum likelihood (ML) is an
ex ended echnique o de i e a loss unc ion o his c i e ia
[23], [5]. I can be shown ha by o cing he es ima ing
unc ion K(y) = ∇l(C)·CT=ϕ(y)yT−I o cancel, we
achie e independence a he ou pu . The lea ning law yields
C←− C−λK(y)C(19)
whe e in he ML app oach ϕi(yi) = −q0
i(yi)/qi(yi), being
qi(·) he p obabili y densi y unc ion o sou ce bi. Howe e ,
as sou ce dis ibu ions a e supposed unknown, each au ho
in oduces his own ac i a ion unc ion ϕi(yi). A amily o
hem may be ound, o sou ces wi h nega i e o posi i e
ku oses, in [41]. By no malizing (19) he lea ning law may
be w i en as ollows
C←− C−λϕ(y)yT−I
1 + λ|ϕT(y)y|C(20)
In [16] he au ho s p oposed a NG based algo i hm o sep-
a a e signals in digi al communica ion, he M-EASI (Median-
Equi a ian Adap i e Sepa a ion ia Independence). This me-
hod assumes ze o-mean, symme ic, ‘ci cula ly dis ibu ed’
signals and in oduces he sign unc ion o educe he bias
in oduced by he noise. I es ima ing unc ion yields
K(y) = ysgn(y)∗−I
1 + λsgn(y)∗y+1
α
ϕsgn(y)∗−sgn(y)ϕ∗
1 + λ|y∗ϕ|(21)
whe e sgn(y) = sgn(<(y)) + jsgn(=(y)),y∈C. Wi h his
me hod we imp o e he s abili y o he algo i hm [29], p o ide
he me hod wi h phase eco e ing p ope ies and make he
me hod mo e obus agains noise.
IV. HYBRID HIGH ORDER STATISTICS MULTIUSER
DETECTORS
A. Na u al G adien Based HyHOS
We could use he na u al g adien o compu e a de ec o
Cby imposing some c i e ia such as he MSE. Bu we may
di ec ly exploi he MMSE es uc u e in 5 o he cen alized
(i.e., base s a ion, all codes a ailable) and non-cen alized case
(i.e., use equipmen , only use code a ailable).
Fi s , we p opose he WWH, cen alized de ec o by com-
pu ing W in
CWWH =H∗HW ∗
W H∗(22)
by using (20) wi h y= =W H∗xand ac i a ion unc ions
ϕ( ) = . The lea ning law yields he deco ela ing algo i hm
W ←− W +λI−(W )(W )T
1 + λ|(W )T(W )|W (23)
Now, we p opose a non-cen alized MUD, he HWW de ec-
o . The non-cen alized MMSE-MUD was gi en in (5) as
CHW W =H∗W∗
xWx(24)
I we ede ine y=Wxxand ew i e he na u al g adien
blind sou ce sepa a o in (18) as we did in (23), i ollows
ha
Wx←− Wx+λI−(Wxx)(Wxx)T
1 + λ|(Wxx)T(Wxx)|Wx(25)
Following he same s uc u e as be o e, we ha e cen alized
and non-cen alized BSS based de ec o algo i hms. In he
case o he BSS cen alized MUD, he BH de ec o , we can
p oceed as ollows I is posible o subs i u e he ma ix p oduc
H∗HW ∗
W in (22) by a sepa a ing ma ix B. The new
de ec o yields
CBH =BH∗=QW H∗
(26)
whe e Bis compu ed as a blind sou ce sepa a o , ha is,
a whi ening- o a o . No ice ha in he cen alized case he
dimensional educ ion is ca ied ou by he ma ched il e H∗,

5
hus J=Iin (6). Ma ix in B(31) is compu ed by using he
M-EASI algo i hm
B←− B−λKB(y)·B(27)
whe e
KB(y) = ysgn(y)∗−I
1 + λsgn(y)∗y+1
α
ϕsgn(y)∗−sgn(y)ϕ∗
1 + λ|y∗ϕ|,
(28)
y=B =BH∗x, and ϕ(y)may be chosen as desc ibed
in [41]. Thus, he MMSE s uc u e is u he enhanced by he
p ope ies o he M-EASI algo i hm.
On he o he hand, i is no possible o de ine an a chi ec u e
o he non-cen alized MUD as in (31). Suppose he candida e
now o be he de ec o
CHB =H∗B=H∗QW x(29)
He e, ma ix CB=Bis al eady a solu ion, as independence
is imposed a he ou pu s. Thus CHB is no a de ec o as
H6=I. Compu ing ma ix CBis basically a blind sepa a ion
p oblem and i is ou o he scope o his pape . No ice ha
his de ec o is ully blind as i does no use he sp eading
codes [16], [7], [17], bu a he cons uc s hem.
Now, we include a discussion on some heo e ical aspec s o
he me hods abo e. We ocus on he nea - a p oblem, noise,
con e gence and complexi y.
The main poin in using he na u al g adien in MUD is
ha i is equi a ian [5], i.e., he con e gence has a uni o m
pe o mance. Le ’s de ine D=CS and igh mul iply (19)
by H o ob ain
D←− D+λK(Db)D(30)
Ma ix S=AH is only p esen a he ini ial alue D0=
C0AH. Thus, he con e gence does no depend on ma ix
AH. I can be concluded ha he me hods p oposed in his
pape a e nea - a esis ance, as con e gence is independen
o he use ’s ampli udes.
Algo i hms in BSS usually do no cope wi h he noisy case
whene e he numbe o sou ces and mix u es a e he same
m=n. By using he M-EASI algo i hm we comba he e ec
o noise in he sepa a ion p ocess o digi al communica ion.
On he o he hand i m>na signal subspace p ojec ion
allows noise educ ion. P e ious BSS app oaches o ully
blind MUD’s use a SVD decomposi ion, a MPLL,[17] ...
Howe e , his in ol es a highe compu a ional complexi y. As
he sp eading codes a e usually a ailable (a leas a he base
s a ion), i is s aigh o wa d o in oduce hem as a subspace
algo i hm a a null complexi y cos [4]. Besides, he s uc u e
o he MMSE de ec o has been used in (22) and (24) o cope
wi h noise.
Ano he impo an cha ac e is ic o na u al g adien BSS
echniques is ha o supe e iciency [42]. In his sense, p o-
ided E[ϕ(y)] = 0 (an usual case), he co a iance be ween
wo ou pu s dec eases o he o de o 1/ 2in ba ch es ima ion
and o he o de o λ2in on-line lea ning. Fu he mo e, λ= 1/
gi es, asymp o ically, he bes pe o mance, which is he same
as he op imal ba ch es ima o . Thus, wi h λ= 1/ we achie e
an on-line algo i hm wi h ba ch ea u es a e e y . On he
o he hand, i adap i e ea u es a e needed, we may use λ < 1
o achie e an ou pu co a iance dec easing as λ2.
On he ques ion o complexi y, he exac MMSE solu-
ion a e e y ime ein ol es compu ing he eigen alues o
he au oco ela ion ma ix. Thus, he compu a ional esou ces
needed a e signi ican . The me hods p oposed in his pape
allow compu ing, as desc ibed in he las sec ion de o ed o
supe e iciency, he ba ch solu ion as an on-line algo i hm a
a low compu a ional bu den.
B. SICA Based HyHOS
In he sec ion abo e, i is p oposed o subs i u e he ma ix
p oduc H∗HW ∗
W in (5) by a ma ix B ha makes he
ou pu s yias s a is ically independen as possible. The new
de ec o yields
CBH =BH∗=QW H∗
(31)
whe e Bis a whi ening– o a o compu ed by using he SICA
algo i hm [19]. No ice ha in he cen alized case he dimen-
sional educ ion is ca ied ou by he ma ched il e H∗, hus
J=Iin (6).
In [19] au ho s p o e ha unde whi eness cons ains some
ou h-o de con as s may be app oxima ed by a sinusoid.
Thus, he minimiza ion o he con as educes o compu ing i s
phase. The s a ing poin is he Minimum En opy (ME) ICA
con as gi en by Comon in [15] so ha we use he ’Jacobi
op imiza ion’ o cope wi h highe dimensions. This me hod,
called SICA (Sinusoidal ICA,) has a good pe o mance along
wi h a low compu a ional cos , ou pe o ming he ME by
Comon and he JADE me hods [43].
Applying SICA o ou p oblem, assuming inpu s iha e
been deco ela ed, his algo i hm compu es y=B minimiz-
ing he ollowing unc ion, deno ed as con as ,
φME [y]≈φ4[y] = 1
48 X
ijkl6=iiii
(Cumy
ijkl)2(32)
whe e Cumy
ijkl is he ou h o de cumulan o ou pu s
yi, yj, ykand yl.
Thus, his ICA algo i hm may be seen as a ou h o de
deco ela ion, and we go u he han he second o de based
MMSE o ind a new solu ion o he p oblem. The algo i hm
SICA is a block me hod ha compu es ma ix B om a whole
se o obse a ions, simila o he MMSE me hod.
As he sp eading codes a e usually a ailable (a leas a
he base s a ion), i is s aigh o wa d o in oduce hem as a
subspace algo i hm a a null complexi y cos [3]. Besides, he
s uc u e o he MMSE de ec o has been used o cope wi h
noise.
We will compa e he pe o mance o his algo i hm o ha o
he M-EASI algo i hm, an adap i e me hod o compu e ma ix
B ha y o minimize a simila ou h o de based con as
by using he na u al g adien [16].
V. EXPERIMENTAL RESULTS
The es en i onmen used he e is a synch onous CDMA
sys em in which he use s he e a e sp ead using Gold se-
quences wi h sp eading ac o 31. We will p esen he e wo
6
main esul s: i s , he ypical e alua ion o he pe o mance
in a digi al communica ions sys em, i.e., Bi E o Ra es om
10−3 o 10−6and he equi alen SNR ≥8dBs, a Low
Noise En i onmen (LNE); second, an e alua ion in a High
Noise En i onmen (HNE) whe e we es he noise esis ance
capabili ies o he HyHOS algo i hm wi h Bi E o Ra es
as low as 2×10−1≤BER <5×10−1and equi alen
Signal o Noise Ra io o −15 ≤SNR ≤0dBs. As he BSS
p oblem is posed in he absence o noise, one o he main
issues is o es he pe o mance o he esul ing algo i hms in
noisy en i onmen s. The compa ison o he a ious algo i hms
p esen ed he e mus be done wi h equal con e gence speed
(i.e. lea ning speed) on one hand, and equal inal a iance
on he o he , o ai h ully cha ac e ize hei pe o mance. The
me hods desc ibed in [44], [45] and [46] a e o be used o
analyze he inal a iance and he con e gence speed o each
algo i hm. Those me hods a e use ul and s aigh o wa d in he
case o linea algo i hms such as he MMSE bu , in he case
o he nonlinea unc ionals o he HyHOS algo i hms, his is
a much di icul ask and, as such, me i s a sepa a e analysis
which is cu en ly unde way. In he p esen wo k, and as
a means o compa e he algo i hms on an equal oo ing, he
aining leng h o each o he de ec o s is 2000 samples, he
con e gence speed (i.e. s ep-size) and he inal a iance a e
ixed h oghou he expe imen s. The numbe o Mon e Ca lo
simula ions a e aged is 100 in he HNE and 500 in he LNE,
which amoun s o a con idence ma gin o a leas 95% o any
BER es ima ion p esen ed in his wo k.
A. Low Noise En i onmen
In Low Noise En i onmen we p esen , o cla i y, he
con e gence pe o mance o he HyHOS I and he MMSE
algo i hms bo h wi h 15 and 20 use s. We see a clea di e ence
in he con e gence beha iou o he HyHOS I and he MMSE
algo i hm. Tha o he HyHOS II algo i hms closely esembles
ha o he HyHOS I. The a e age dec ease in SINR o he
HyHOS I algo i hm whi an inc ease o 5use s is hal o
he dec ease su e ed by he MMSE. In Figu e 1 we can see
ha 2000 samples is mo e han enough aining leng h o he
HyHOS I algo ihm whils o he MMSE, con e gence canno
be achie ed in a highly occupied channel.
In he e alua ion o he Bi E o Ra e (see Figu e 2), his
speed o con e gence o he di e en algo i hms is e lec ed in
he pe o mace o bo h algo i hms, HyHOS I and II and ha
o he MMSE. In his Low Noise En i onmen (LNE), he
sough ea u es in mul iuse de ec o is ha o a con olled
deg ada ion o Bi E o Ra e wi h an inc ease in he numbe
o use s. Bo h he SICA based HyHOS algo i hm (HyHOS I)
and he Na u al G adien based HyHOS (HyHOS II) show a
pe o mance close o ha o he MMSE (i no be e in he
HyHOS I case) wi h a low occupa ion o he channel (5use s)
bu he di e ence in beha iou a ises in a highly occupied
channel (25 use s) , whe e he BER o he MMSE deg ades
om 10−6a app oxima ely 14 dB o SNR o 10−5, while bo h
HyHOS algo i hms keep he BER o app oxima ely 10−6in
he same condi ions.
0 500 1000 1500 2000
Samples
8
9
10
11
12
13
14
15
SINR (dB)
HyHOS (15 use s)
HyHOS (20 use s)
MMSE (15 use s)
MMSE (20 use s)
Fig. 1. Signal o In e p e ence plus Noise Ra e, wi h SNR o he Use O
In e es (UOI) = 15 dB, o he HyHOS de ec o and he MMSE de ec o ,
bo h wi h 15 and 20 use s. The powe s o he in e e ing use s is dis ibu ed
homogeneously be ween 0and 30 dB abo e ha o he UOI.
8 10 12 14
SNR (dB)
1×10-6
1×10-4
1×10-2
BER
HyHOS I (5 use s)
HyHOS I (25 use s)
HyHOS II (5 use s)
MMSE (5 Use s)
MMSE (25 use s)
Fig. 2. Bi E o Ra e in Low Noise En i onmen (LNE), i.e. SNR o he UOI
≥8dB, o he HyHOS de ec o and he MMSE de ec o , bo h wi h 5and
25 use s. The powe s o he in e e ing use s is dis ibu ed homogeneously
be ween 0and 30 dB abo e ha o he UOI.
B. High Noise En i onmen
In he case o High Noise En i onmen , o HNE, he
con e gence speed (see Figu e 3) o bo h HyHOS algo i hms
is comple ely simila bu o highe a iance in he HyHOS
I algo i hm. Bo h in a low o high occupa ion channel, he
con e gence is ai ly simila . The MMSE shows a lack o
con e gence in a high-noise, low-in e e ence channel, qui
expec ed o he MMSE because he in e e ence noise s uc-
u e is comple ely masked by gaussian noise, a oiding his
algo i hm o “lock” on o he signal o he Use o In e es .
Qui e he con a y is he case o he HyHOS algo i hms.
Thei in insic nonlinea p ope ies show he e he limi a ions
o linea unc ionals in linea de ec o s.
The Bi E o Ra e o he HyHOS algo i hms show he
noise esis ance p ope o he MMSE wi hou showing i s
demeano s. The con e gence o a Ma ched Fil e solu ion in
he asymp o ic limi o σ→ ∞ is shown expe imen ally in
Figu e 4.
7
0 500 1000 1500 2000
Samples
-26
-24
-22
-20
-18
-16
-14
SINR (dB)
HyHOS I (5 use s)
HyHOS I (25 use s)
HyHOS II (5 use s)
HyHOS II (25 use s)
MMSE (5 use s)
MMSE (25 use s)
Ma ched Fil e
Fig. 3. Signal o In e p e ence plus Noise Ra e, wi h SNR o he UOI =−15
dB, o he HyHOS de ec o and he MMSE de ec o , bo h wi h 5and 25 use s.
The powe s o he in e e ing use s is dis ibu ed homogeneously be ween 0
and 30 dB abo e ha o he UOI.
-15 -10 -5 0
SNR (dB)
2.0×10
-1
2.5×10
-1
3.1×10
-1
4.0×10
-1
5.0×10
-1
BER
HyHOS I (5 use s)
HyHOS I (25 use s)
HyHOS II (5 use s)
HyHOS II (25 use s)
MMSE (5 use s)
MMSE (25 use s)
Ma ched Fil e
Fig. 4. Bi E o Ra e in High Noise En i onmen (HNE), i.e. SNR o he
UOI ≤0dB, o he HyHOS de ec o and he MMSE de ec o , bo h wi h 5
and 25 use s. The powe s o he in e e ing use s is dis ibu ed homogeneously
be ween 0and 30 dB abo e ha o he UOI.
VI. CONCLUSION
In his pape he au ho s p oposed bo h ICA and NG
as alid echniques o exploi he ma ix es uc u e o he
pa ame e s in ol ed in he MUD p oblem, an applica ion
o he mo e gene al na owband m-senso linea -a ay case.
He e, we p opose a no el Hyb id MUD by in oducing an
ICA ma ix in o he s uc u e o he blind cen alized MMSE
MUD. This way we pallia e one o he main p oblems o he
BSS/ICA algo i hms, hei sensi i eness o noise. The esul s
included he e show a good nea - a esis an pe o mance in
synch onous CDMA.
ACKNOWLEDGMENTS
w The au ho s would like o acknowledge D . Za zoso and
D . Ma ´
ınez-Ram´
on o hei aluable commen s.
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