Non-linear adaptive signal processor
Abstract
This paper is a first attempt to give formalism to non-linear system design and in which context, related with similar linear processing techniques, they are located. A summary on the relation-ship of linear objectives and classical adaptive algorithms, in non-linear design problems, introduces the paper; giving the potential of random search techniques in order to open the different problems in non-linear objectives that could be handled with them. After, the similarity between probability distribution functions and power spectral density in linear processing is shown. This is supported by a nice example of non-linear system design. Finally, some prospective work is reported in the problem of adaptive companding design.
Full text
NON-LINEAR ADAPTIVE SIGNAL
PKN!I?SSOR
M.A.Lagunas,F.Vall e du,M.E.San ama ia
P ocesado de Seiial en Camunicaciones
E.T.S.I. Telecmunicacion
Agio.
30002
08034 Ba celona
SPAIN
ABSTRACT
This pape is a i s a emp o gi e
o malism o non-linea sys em design and in which
con ex , ela ed wi h simila linea p ocessing
echniques, hey a e loca ed.
A sma y on he ela ion-ship o linea
objec i es and classical adap i e algo i hms, in
non-linea design p oblems, in oduces he pape ;
gi ing he po en ial o andom sea ch echniques
in o de o open he di e en p oblems in
non-linea objec i es ha could
be
handled wi h
hem.
A e , he simila i y be ween p obabili y
dis ibu ion unc ions and pe spec al densi y
in linea p ocessing
is
shown. This
is
suppo ed
by a nice example o non-linea sys em design.
Finally, some p ospec i e wo k
is
epo ed
in he p oblem o adap i e companding design.
INTRowcTIcbl
The i s applica ion epo ed in
non-linea il e ing is ha one whe e a
memo yless non-linea sys em o o de Q
is
cascaded wi h a non-linea channel o be
equalized. This case is depic ed in ig. 1.
L
I
Fig.
1.
Non-linea adap i e equalize .
Because in his si ua ion he objec i e
(
i.e. he quad a ic mean-squa e e o be ween he
aining sequence and he ou pu ) becomes linea
in he weig hs o he non-linea equalize , he
minimum
is
sol ed in a linea way like in a Wiene
il e in he linea case. Thus, w i ing down he
o mulas o he case unde s udy, he objec i e
will
be
(1).
This wo k
is
suppo ed by CAICYT g an numbe
21096/84
p
=
E (
(Z)-X)"
(1)
Whe e
(z)
is a polynomic de ice wi hou memo y
as
shown in
(2).
I is easy o p o e ha he
non-mm y cons ain in he analysis
is
no
a
limi a ion o he main ideas summa yzed he ein.
Taking de i a i es o
(I)
wi h espec
he weig hs a(q) and s ing hem o ze o,
equa ions (3) esul s; whe e
/u
deno es he
n-o de men o he andom "a ia!%?
z
,
and
0
he espec al alue o he c oss-p oduc
09
z
and x.
E
=?
E
[zpx]
=
P 4
gP
(3.a)
No e ha he expec ed alue
0
can be
ew i en as
(4)
,
whe e he e ec
o
he
non-linea sys em o be compensa ed becomes mo e
ele an .
INLCp(x) x p(x) dx
=
gP
(4)
A he same ime he design equa ions means ha
(5)
holds.
I (z)
zp
p( ) dz
=
In
summa y,
he design equa ions check ou , in and
in eg al o m, he simila i y be ween (NLC(x))
and x. In o he wo ds, is a es o up o wha
deg ee (.) is close o he in e se o NLC(.).
Also,
i is in e s ing o ema k ha his
ela ionship
is
jus o con ol ha he
p obabili y dis ibu ion ucn ion pd (x) and he
pd (y) a e he same. This las poin
is
impo an
because a mo e in e es ing design objec i e would
be o educe he di e ences be ween bo h pd 's
ins ead
o
minimizing he
MSE
Wi h espec o he use
o
non-linea
adap i e equalize s,
i
is
wo hwhile
o
no e ha
he high dynamic ange, needed in he upda e
equa ion o he weig hs and he non-linea
cha ac e o he objec i e, o ces
o
conside
andom sea ch algo i hms
as
a
good
candida e o
he adap i e algo i hm
o
be used in
he
non-linea p ocesso .
In his sense,
i
is
impo an o ema k
ha bo h
ED
and andom sea ch algo i hms
do
no
use he snapsho da a in he weig hs upda e. In
(7)
i
is
deno ed he upda e o a gene al
adap i e algo i hm whe e
is
he objec iye
g adien wi h espec
he
weig h ec o
A
=
a(l) a(2),...la(Q)
.
A
his pin , could
be
compu ed in
wo
ways:
one
as
a g adien o he objec i e in
(1
)
o
as
a
pe u ba ion measu e. In he g adien o m
i
is
compu ed di ec ly m he objec i e ob ainig
(8).
being
Tnus, he dynamic ange
o
ec o
2
in he
adap i e loop ep esen s
a
disad an age in he
ha dwa e implemen a ion
o
he sys em.
Unde
a
pe u ba ion algo i 'm e en using
g adien like algo i hms
a
pe u ba ion
Lj
is
se
o e e y weig h and he g adien d /da(q)
is
compu ed as
(10).
P
Once he global g adien ec o
is
compu ed,
equa ion
(7)
is
used
o
upda e he weig hs.
men using andom sea ch algo i hms linea
o
no he philosophy
i
is
se
a andm
pe u ba ion and, once he esul ing objec i e
is
ob ained, he
e o
e olu ion dic a es
o
keep he
pe u ba ion
as
pe manen in he andom sea ch
case:
o
used in
a
guided
o _
unguided o m in a
g adien basis upda e
131
,
147
-
This sho desc ip ion
o
adap i e
algo i hms p o es he po en ial
o
andom sea ch
echniques in non-linea p ocessing.
A
he
same
ime
and when
i
is
decided o
go
o andom
sea ch, he quad a ic mean squa e
e o
is
no
longe a cons ain
as
he only ealis ic choice
o
he design objec i e. In ac , gene al
e sions
as
(11)
o he objec i e could be used
oge he wi h a andom sea ch adap i e algo i hm.
Wi h
p
and q in ege s,
i
is
easy o
conclude ha i
q
is
high,
low
dynamic ange in
he ou pu signal
(z)
will
ecei e sligh
a en ion in he upda e loop.
Fo
low
q
he
same
will
be ue
€o
high dynamic ange ou pu s. Wi h
espec pa ame e p he
same
easoning could be
made bu
i
will
be abou he esidual signal
ins ead o he non-linea equalize dynamic ange,
As
he eade can
see,
s a ing m
a
non-linea
sys em design, he andom sea ch algo i hms
allow
he designe
o
cope
wi h
in e es ing ea u es
o
he esul ing adap a ion loop by handling non
MSE
c i e ia.
In Fig.
2
i
can be iewed he
NLC
used in
a
es
wi h a non-gaussian inpu and
he
co esponding non-linea euqalize
o
o de
7
oge he wi h he lea ning cu e o he adap i e
algo i hm.
Pig.
2.
(a)
NLC
sys em;
(5)
non-linea equalize ;
(c)
lea ning cu e o he adap i e algo i hm.
PDF
CONTROL
AND
NON-LINERR
SYSTEMS
This sec ion
is
de o ed
o
show
ha
he
pd
o
a
andom a iable in non-linea sys ems
plays
almos
he
same
ole
ha he powe densi y
unc ion in he linea case.
Fi s a all, no e ha he pd sha es wi h
he pme spec um he
sam
basic ea u es; in
o he wo ds, p(x)
is
always posi i e and desc ibes
in he ampli ude damin he e olu ion
o
a andom
a iable in a dis ibu ion o m.
Because he p elimina y cha ac e o his
wo k and he limi a ion in
i s
ex en only
a
ew
poin s
o
he
unde going esea ch
will
be
epo ed. Seems
o
be
ha , keeping in mind he
simila i ies
o
p(x) wi h
he
powe spec um, he
i s
a enp,
in wo king ou o e p(x),
will
be
o
ep oduce he whi ening p ocessing, which p o ed
o
be e y use ul1 in a be e unde s anding
o
he
s uc u e
o
a
powe
densi y. In
sma y,
he
p oblem o be sol ed
is
gi en
a
p(x) ind he
non-linea sys em which p oduces om
x
an
114
uni o mly dis ibu ed andom a iable
u.
This
si ua ion
is
depic ed in Fig.
3.
Fig.
3.
The whi ening p ocessing in
a
non-linea
scheme.
g(.)
is
a
non-linea sys em wi hou
memo y.
The ela ionship be ween p(x), p(u) and he
non-linea ans e unc ion g(x)
IS
shown in
(12).
me e
g(x)
is
he de i a i e o p(x) wi h
espec
x.
Also
i
is
assumed ha g(.)
is
a
mono one
inc easing unc ion. Thus i p(u1
is
desi ed
o
be
cons an he desi ed ans e unc ion
is
ob ained om he in eg al o he gi en p(x)
as
i
is
shown in
(13).
o
.x
(13.a)
(13.b)
Now,
le
us
assume
ha
we
a e
in e es ed
in
a
pa ame ic e sion o his ans e ucn ion.
This
will
be he
case
when he sys em
is
ob ained
a
he ansmi e loca ion and he in e se sys em
(i.e.
he sys em which p o ides p(x) om an
uni om p.d. .
)
is
needed
a
he ecei e
loca ion. Thus, in ob aining
a
pa ame ic e sion
o g(x)
we
ace he p oblem o
a
pa ame ic
e sion o p(x) i sel .
Assuming ha p(x)
is
he unc ion
we
a e
looking o , looks
clea
ha
some
cons ain s in
i s
design mus be
se
in o de
o
gua an ee
he
simila i y wi h he ac ual pd .
A
his
ime
seems
$0
be
ha
he
bes way
is
o
o ce ha p(x) and
p( x) bo h ha e he
same
momen s,
le s
say, up
o
an
o de
0.1.
These
cons ain s
a e
shown in
(14).
[$(x)+xq dx
=Yq;
q=O,Q-1
being
=
I
p(x) x"
dk
(14. a)
(14.b)
In selec ing
he
objec i e many choices can
be
made, bu he only one which gua an ees he
op imali y
o
a
polynomial s uc u e
is
(15).
Sol ing his a ia ional p oblem, he solu ion
(16)
a ises o
D(x).
whe e
A9(u=O.O-1)
a e
he
Lag ange pa ame e s o e e y cons ain (14.a).
The
se
o equa ions which p o ide hese
pa ame e s
is
shown
in
(17),
and he esul ing g(x)
is
de i ed a e using he
in eg a ion p ocedu e (18).
The
iden i ica ion o
he non-linea sys em weig hs
is
ob ious.
Tl
q
This sol es he p oblem o how
o
modi y a
gi en p obabili y densi y unc ion o ob ain
a
la pd
and, o cou se
,
he way ou
o
ob ain
he ans e esponse which p oduces
a
gi en pd
when he inpu
is
uni o m.
To no e he simila i y wi h linea
p ocessing p oblems
is
in e es ing
o
cmpu e how
la p(u)
is
ac ually. The esul ing p(u)
is
shown
in
(19)
and,
as
he eade can
See,
i
can be said
ha he p ocedu e
is
op imum when a
"MA"
model
is
adequa e o he p(x) unde p ocessing.
This new concep e eals ha he polynomial
cha ac e o he pd
o
handle
will
p e eal in
non-linea p ocessing
as
pu e
AR
model
spec a
does in he linea
case.
Mo eo e , in
some
sense
he way
o
ob ain he ce icien s could be
enhanced in
a
simila way
ha
linea 4p edic ion
heo y was s a ed. To do his,
le
us suppose ha
we
a e
dealing wi h a non-linea model which
p edic s om powe s o
a
T.V.
he andm a iable
i sel .
X
+
U
Fig. 4. Non-linea "p edic ion".
Minimizing he a iance
o
second o de
momen
o
he ou pu u
will
a ise
o
a
di e en
non-linea sys em han be o e. men
i
is
desi ed
o
ha e he
same
pd a he ou pu o he
non-linea sys em,i
is
he
case
whe e bo h
sys ems
will
be he
same.
F om
he abo e he ex ension o linea
p edic ion heo y
o
he non-linea
case
is
s aig o wa d. In ac ,
he
ha d decision
o
he
115
designe
a ises
when he s uc u e o he op imal
es ima e
E
x/da a
is
selec ed. Once his
s uc u e
is
gi en, and assuming
i
is
a linea
weig hed
sum
o
pwe s
g ea e
o
equal han one,
o
pas
samples, he design can be ca ied ou in
he adap i e o m using bo h g adien
o
andom
sea ch p ocedu es.
THE
ADAPTIVE
COMPANDING
PROBLEM
"he e
is
o he applica ions o non-linea
p ocessing whe e he objec i e
is
o ced
o
be
non-linea also. This
is
he
case
o
he
companding p oblem whe e he adap i e non-linea
p ocesso
is
loca ed be o e he non-linea de ice
o
be compensa ed.
This
si ua ion
is
shown in Fig.
5.
The i s decision
is
o
selec
which
is
he
ksidual
o
be minimized in he adap i e p ocesso
design.
Fig.
5.
ie
p oblem o adap i e ccanpanding.
The i s choice could be
o
minimize
he
di e ence be ween
z
and y
(i.e.
e
=y-z he
e o
due
o
he non-linea de ice). This esidual
will
p cmo e ha he compande a emps
o
educe he
dynamic ange o z o he linea ange o he
ans e esponse o NLC(z).
A
mo e
in e es ing
way
o
desc ibe his e ec
is
o
say ha he pd
o he inpu andom a iable
will
be concen a ed,
by he non-linea p ocesso (x), a ound he
linea ange,
o
NLC(x); o
a
leas , a ound he
minimum dis o ion ange o he non-linea de ice.
An in e es ing
case
is
when (.)
is
designed
as
an adap i e compande o an one bi
uuan ize
1'
31
,
Fig.
6.
Adap i e compande o one-bi
quan ize .
When minimizing he esidual y-z,
seems
o
be
clea
ha
(x)
will
concen a e he pd o x,
he posi i e a gumen s side, a ound
z
equal
o
.
The o mula ion o he objec i e
o
be minimized
is
as
(20).
In oducing
he
polynomic
cha ac e
o
(x) inslde
(20)
and se ing de i a i es espec
o
he weig h
o
ze o,
he
op imum ec o
is
ob ained.
The esul ing equa ion ha e he o m o (22)
(21)
As
an example, using
an
o de
wo
equalize
o compaye , he esul ing ans e yc ion
is
(4x-l0/3x
)
and he quan ize
e o
is
b
/g,
which
p o ides he same
e o
o
a
(5=0.866
ha
a
op imum quan ize o he
sane
p obabili y densi y
inpu (pd o x). No e ha using he compande
he signal
o
noise
a io
in he cmunica ion
channel inc eases due o he di e ence o he
quan ize s eep
0.866
in ou
case
and he op imum
0.5
in
he
classical
app oach.
The
dis ibu ion o
he inpu andom a iable was uni m be ween
-1
and
1.
I
is
wo hwhile
o
men ion ha , in
some
cases,
unde such app oach
i
is
needed
o
a oid
he i ial solu ion o he
ze o
ou pu ou o he
compande . %cause, in gene al, NLC(z) p o ides
ze o ou pu wi h a ze o inpu ;
he
p e ious
men ioned solu ion p oduces
a
minimum
e o
bu do
no ull il ou objec i e in designing he
canpande . In
his
cases, a dynamic ange
cons ain
like
(x )=xmx should
be
se
in he
adap i e p ocess. gxsu mna y,
i
could be said
ha non-linea objec i es would equie e
addi ional cons ain s in he adap i e andom
sea ch algo i hm in o de
o
a oid undesi ed
solu ions
o
local minima.
This
las
cmen
is
ela ed wi h he
second choice
we
can
se
o
he p oblem depic ed
in Figu e
5.
This second choice consis s in
selec ing
as
esidual he global di e ence
be ween he ou pu y and he inpu x.
The
solu ion
needs also
o
be cons ained by dynamic ange
o
esponse ange cons ain s. Taking
he
same
example,
o
he one bi quan ize ,
i
is
well
know
ha he minimum squa e
e o
be ween y and x
is
ob ained whene e (23) holds.
Unde his scheme,
(23)
is
a
cons ain in
he design o (x), and he objec i e
is
o
selec
(.) such ha
i s
in e se g( (x))=x minimices he
noise e ec s
a
he ecei e .
REFERENCES
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Va iables, and
S ochas ic
P ocesess",Mc.G aw-Hill,l965.
P.
Eykho , "Sys em iden i ica ion".
J.
Wiley
&
Sons,
(1979).
L.R.
Rabine
R.W.
Scha e , "Digi al
P ocessing o Speech Signals". P en ice Hall.
Monzingo,
Mille ,
"In oduc ion
o
Adap i e
A ays".
J.
Wiley
&
Sons.
4.3.
116