NEW HOS-BASED PARAMETER ESTIMATION METHODS FOR
SPEECH RECOGNITION IN NOISY ENVIRONMENTS
Asuncion Mo eno, Se gio To ola, .losep Vidal, Jose A. R. Fonollosa
Dp . Signal Theo y and Communica ions
Uni e si a Poli ecnica de Ca alunya
Ba celona, Spain
ABSTRACT
In his pape he p oblem o ecogm lOn in noisy
en i onmen s is add essed. O en, a ecogni ion sys em is
used in a noisy en i onmen and he e is no possibili y o
aining i wi J� noisy samples. Classical speech analysis
echniques a e based on second-o de s a is ics and hei
pe o mance d ama ically dec eases when noise is p esen
in he signal unde analysis. In his pape new me hods
based on Highe -O de S a is ics (HOS) a e applied in a
ecogni ion sys em and compa ed agains he
au oco ela ion me hod. Cumulan -based me hods show
be e pe o mance han au oco ela ion-based me hods o
10w SNR.
1.INTRODUCTION
In he las ew yea s, he e has been an inc easing in e es
in he applica ion o Highe -O de S a is ics in signal
p ocessing. Signal analysis sys ems based on cumulan s
and hei Fou ie T ans o m a e a powe ul ool since hey
ha e e y use ul p ope ies. De ec ion o non linea i y,
iden i ica ion o non minimum phase sys ems and
immuni y o whi e o colo ed Gaussian noise a e some
examples [1]. Voiced-un oiced classi ica ion, phoneme
segmen a ion o pi ch es ima ion a e p oblems ha ha e
been add ess :d using HOS. Resul s show ha speech
signal can be cha ac e ized no only by i s au oco ela ion
bu also by i s hi d- and ou h-o de cumulan s.
Immuni y o noise is an impo an ea u e in any signal
p ocessing sys em. Classical speech analysis echniques
a e based on second-o de sL:'l is ics and hei pe o mance
d ama ically dec ease when noise is p esen in he signal
unde analysis. Cumulan s o o de g ea e han wo a e
ze o o whi : and colo ed Gaussian noise. Analysis o
noisy speech signals based on HOS pe mi s o sepa a e he
speech om noise and in his pape his p ope y is used.
We add ess he p oblem o ecogni ion in noisy
en i onmen s. O en, a ecogni ion sys em is used in a
noisy en i onmen and he e is no possibili y o aining i
wi h noisy samples. Paliwal [4] made use o HOS in a
This pape has been suppo ed by Spanish Go e nmen
g an TIC 92-0800-C05/04
429
ecogni ion sys em and he showed ha esul s emain
cons an unde a g ea a iabili y o SNR.
Howe e , in high SNR condi ions, he pe o mance o he
au oco ela ion meiliod was clea ly be e . This ac can be
ela ed o he ollowing poin s:
*Cumulan s-based no mal equa ions can gi e a non
minimum-phase il e . The es ima ion o he AR
pa ame e s can a ise a non s able solu ion in some ames
o speech signals.
"'The a iance o he es ima ion is g ea e ilian in he case
o au oco ela ion.
To a oid hose wo p oblems we ha e de eloped new
meiliods o es ima e he AR pa ame e s ha gi e s able
solu ions and ha e less a iance. These me hods use a
linea combina ion o cumulan slices (WS) [2] o an
unique slice CDS) [3]. In iliis pape iliose me hods a e
compa ed in a speech ecogni ion ask agains o de h ee
and ou Yule-Walke based equa ions and au oco ela ion
me hod.
2. HOS BASED PARAMETER ESTIMATION
METHODS
Conside he speech Signal gene a ed by a causal and s able
AR (p) model wi h added noise :
p
yen) = :£ a(i) yen-i) + yen) (1)
i=l
zen) -= yen) + wen)
The inpu p ocess (n) is a ze o mean non-Gaussian U.d.
sequence, wiili k-o de cumulan Yk,y�. The addi i e
noise wen) is independen o (n), ze o mean, and ei he
Gaussian wi h unknown powe spec um o non-Gaussian
wi h 'Yk,w=O. The il e H(z) is exponen ially s able. The
abo e condi ions imposed o e wen) gua an ee Ck,w-=O and
Ck,z=Ck,y.
In his sec ion we conside h ee algo i hms named Yule
Walke , W -slice and I-D slice.
0-7803-2431-5/95 $4.00 © 1995 IEEE
Highe -o de Yule- Walke algo i hm.
F om he welll nown equa ion [5]:
p
L a(l) Ck,y(m-l , leo, 0, .. 0)= 'Yk, hk-2( -m) h( -m+ko)
1=0
We ob ain
p
(2)
2. a(l) Ck,y(m-l , leo, 0, ... 0) =0 i m>o, o m>ko (3)
1=0
To sol e hose equa ions is necessa y o conca ena e p+ 1
slices, ko=-p ... , 0 in (3) because a single slice does no
gua an ee a ull ank sys em o equa ions [5].
Equa ions (3) mus be sol ed using cumulan s es ima es
and he solu ion using LS o TLS may yield a uns able
solu ion. In his pape we es wo al e na i es o imp o e
he s abili y: o inc ease be numbe o slices (ko=-p-M, ... O
and M>O), o o inc ease he numbe o equa ions pe slice.
In he la e case we conside h ee al e na i es: (S 1) p
equa ions pe slice (minimum), m=O .... p; (S2) he nega i e
slices (ko<O) ha e p+ 1 equa ions, m=O, ... ,p; (S3) he
nega i e slices ha e p-ko equa ions, m=l+ko, ... p.
W-slice algo i hm.
The w-slice [2] algo i hm is based on he ollowing
weigh ed sum o cumulan slices:
N
Cw(i) = W2C2,y(i) + 2. w3G) C3,y(i,j) +
j=-L
N N
2. L w4G,k) C4,y(i,j,k) + ...
j=-lk=-L
and i is de eloped in h ee s eps:
a). Choose w2, W3(j), W4(j,k), such ha :
i=-P, ... ,-l
(4)
(5)
being P � p , N � 0 and L� p+M, whe e M is he o e
de ennina ion.
b). Es ima e he II s P enus o he impulse esponse om
he weigh ed cumulan Cw(i).
h(i) = Cw(i) i=l, ... P.
430-
c). Sol e he il e coe icien s om he ollowing
equa ion:
p
2. a(l) b(i-I) = O. i= 1...P
1=0 (6)
S ep a) is sol ed by LS. To sol e (6) is p e e able o use
LS o TLS han backsus i u ion o minimize he a iance o
he solu ion and ob ain s able solu ions (P==p+M, M>O).
We ha e also conside ed he au oco ela ion me hod:
p
I a(k) Rhh(k-I) = 0, k==1...P
k=O
ha assu es a s able solu ion
i-D slice algo i hm
(7)
This algo i hm ob ains he AR coe icien s om a single
cumulan slice Ck,y(m, ko .... ,0). An one-dimensional slice
does no gua an ee a ull ank sys em o equa ions and o
his eason is no easonable o sol e (3) di ec ly; he
solu ion may no be unique o s able. Ins ead, we conside
he (de e minis ic) au oco ela ion o he cumulan s o o m
a Toepli z ma ix wi h a s able solu ion.
I we mul iply (2) by he one-dimensional slice Ck,y(m-l',
ko, O ...
,0) and we sum in an in e al wi h m>0, we ob ain:
p
L a(I) L Ck,y(m-l;ko, 0 ... ,0) Ck,y(m-l';ko, 0 ... ,0)= 0
1=0 m>0
This equa ion can be exp esed as:
p
LaO) 4>e 0,1', ko, 0, ... ,0) == 0
1=0
whe e
4>C<I,l', ko, 0, ••• , 0) =
(8)
= L Ck,y(m-l; ko, 0 ...
,0) Ck,y(m-l'; ko, 0 ...
,0) (9)
m>O
Ins ead o 4>e{l,l', ko, 0, ... , 0) we use he ollowing
ap oxima ion:
<!le(l,!', ko. 0, ... ,0) = Re(l-!" ko, 0, ... ,0)
whe e Re(i) is he au oco ela ion o he causal pa o he
one slice cumulan . Subs i uing his ap oxima ion in (8)
gi es:
""dl3 20dB lOdB OdB
M sillu�
amp
sinus
amp
sinus [(unp sinus
amp
YW3 ()
99.2
99.6 95.6 96.2 67.6 76.8 2l.2 25.6
16 99.4 99.2 94.R 96.4
65.4
72.4 28.2 24.8
YW4 0 99.2 100
95.4
96.6 68.2 69.0 19.4 24.4
16
99.4
99.0 96.4 96.4 64.8
79.4
19.0 25.4
Table I. Recogni ion a es in % o d( e en SNR. ob ained by me hods Yule
Walke o de 3 (YW3) and Yule-Walke o de 4 (YW4). M: is he a e 
de e mina ion .. andwindows: sinus and amp a e es ed.
00 dB 20 dB IOdB OdB
Me hod sinus
amp
sinus
nunp
sinus
amp
sinus
amp
YW3 LS 98.6 98.6 96.4 95.6 80.0 82.6 40.0 44.8
COR 99.4 99.0 94.6 96.0 8l.2 82.8 29.8 48.8
YW4 LS 98.R 97.8 95.6 93.6 82.2 83.8 39.8 46.0
COR 98.4 99.4 96.6 97.2 85.0 88.0 40.0 51.6
Table II. Recogni ion a es (%) o/l ained o d( e en SNR .. using W-Slice o de 3 (WS3)
and W-Slice o de 4 (l¥.S'4). Equa ions a e sol ed by me hods LS and Co ela ion.
and wo windows: sinus and amp a e es ed.
p
L a(l) Re(l-II') = 0
1=0
I' = l. .. p (10)
And can be sol ed o ming a Toepli z ma ix wi h a s able
solu ion.
The coe icien s ob ained using (10) di e om he ue
AR coe icien s bu he esul s con i m hei use ulness in
speech ecogni ion asks. In o de o imp o e he
ap oxima ion and o educe he a iance o he compu ed
au oco ela ion we used a la ge numbe o samples o he
cumulan s (»p) and ko=O.
3 • ...RESULTS
The expe imen chosen o comp u'e he di e en me hod� is
ile ecogni ion o he en digi s. The da abase is composed
by 10 speake s. Each speake Ul e s 10 epe i ions o each
digi . Speech signal is banopa);); il e ed be ween 100 ano
3400 Hz and samplcd al 8 KilL. 10 IIMM Sla es m'e used
wi h an o de 8 LPC analysis in ames o 37.5 ms (300
samples) delayed 150 smnples. Ceps lnn. del a eeps um
43'
and del a ene gy a e used in he ecogni ion sys em. Fi e
u e ances o all he speake s a e used o aining and i e
o es ing. Noisy signals a e ob ained adding whi e
Gaussian noise a SNR = O. 10. 20 dB. Resul s a e
compa ed agains ecogni ion a es ob ained wi h lle
au oco ela ion me hod (YW2) using a unp window
wi llou p eempha"is.
Highe -o de Yule- Walke algo i hm.
Table I shows he esul s ob ained using he Highe -o de
Yule- Walke algo iUlm. YW3 and YW4 co espond o lle
hi d- and ou h- o de cases espec i ely. When equa ions
a e solwd by LS. he numbe o equa ions a e chosen
ollowing S2 ( he nega i e slices (ko<O) ha e p+ 1
equa ions. m=O
.....
p). The esul s a e abula ed o
di e en alues o he o e de e mina ion ac o M and lle
window.
The numbe o slices chosen o sol e (3) is no c i ical in
high SNR condi ions. The esul s a e simila o he h ee
es ed condi ions named S1. S2 and S3. Wi h a SNR o
IOdB, S2 is he bes choice. lLS p oduces a g ea numbe
SNR AC YW3 YW4 WS3 WS4 DS3 DS4
Clean 99.6 99.6
99
99
99.4
98.4 98.0
20 dB 98.6 96.2 96.4 96
97.2
97.2 97.4
10 dB 82.2 76.8 79.4 82.8 88 90.4 92.4
o dB 34.4 25.6 25.4 48.8 51.6 61.6
64.4
Table III Summa y o ecogni ion a es a di e en SNR.
o uns able ames and he bes esul s a e ob ained wi h
LS. Th ee me hods we e compa ed o p ocessing uns able
ames: El: o elimina e uns able ames, E2: o in e
poles inside he uni ci cle, E3 o wo k equally wi h s able
and uns able ames. The bes choice wi h LS was E3.
Resul s o amp o sinus windows a e simila a high
SNR. When SNR=10 dB amp window shows a sligh ly
be e pe o mance.
Wi h he YW3 me hod, o e de e mina ion M=16
dec eases he ecogni ion a es a 10 dB. Howe e o he
YW4 me hod, o e de e mina ion imp o es he esul s.
W-slice algo i hm.
Table II shows he esul s ob ained using he W -slice
algo i hm. WS3 and WS4 co espond o he hi d- and
ou h- o de cases espec i ely. The ahle compa es he
esul s ob ained wi h he LS and he co ela ion me hod.
TI...S gi es poo e esul s. I LS o TI...S a e chosen, he
solu ion IIlay be uns able. I LS is used he p ocedu e
named E2, (in e ing he poles inside he uni ci cle) gi es
he bes esul s which a e shown in able II. Sol ing (6)
using he co ela ion o he cumulan s gi es always a s able
solu ion.
In noisy condi ions, amp window imp o es he
ecogni ion a e, specially wi h he co ela ion me hod.
J-D slice algo i hm
This is he simples me hod, i is always s able and gi es
he bes esul s, using eilhe hi d- o ou h- o de
cumulan s. The window e ec is no impo an and in
Table III only he esul s ob ained wi h a amp window a e
p esen ed.
Table III compa es he bes esul s ob ained wi h each o
he conside ed pa ame e iza ion me hods. A SNR=10dB,
ID3 and ID4 clea ly imp o e all hc HOS-based me hods
and he con en ional au oco ela ion me hod (YW2)
432
4. CONCLUSIONS-
The p esen ed esul s show ha he me hods based on a
weigh ing o hi d- o ou h-o de cumulan s a e mo e
obus han hose based on he cumulan Yule-Walke
equa ions as SNR is dec eased. Compa ed wi h he
au oco ela ion me hod, DS is be e o low SNR.
Mo eo e , DS is he simples me hod based on cumulan s
since only one slice has o be calcula ed.
The imp o emen in he ecogni ion a e is due only o he
changes in he pa ame e es ima ion s age. O he s ages in
he ecognize may also be modi ied o inc ease he
ecogni ion a e in noise, bu ou goal was only o compa e
di e en pa ame e iza ion me hods in he same s anda d
HMM-based ecogni ion ask.
5. REFERENCES
[1] J. M, Mendel "Tu o ial on Highe O de S a is ics
(Spec a) in Signal P ocessing and Sys em Theo y:
heo e ical esul s and some applica ions" P oc IEEE,
ol 79, no 3 Ma ch 1991
[2] J. Vidal , J. A. R. Fonollosa, Causal AR Modeling
Using a Linea Combina ion o Cumulan Slices,
Else ie Science: Signal P ocessing 36, Aug.1992
[3] A. Mo eno, J. A. R. Fonollosa, J. Vidal "HOS analysis
o speech. A ocode Applica ion". 3 d Eu opean
Con e ence on Speech Communica ion. EURO
SPEECH 93. pp 519-522 Be lin. Ge many.
[4] K.K. Paliwal, M.M. Sondhi. "Recogni ion o noisy
Speech Using Cumulan -Based Linea P edic ion
Analysis". ln Con on Acous ics, Speech and Signal
P ocessing. ICASSP'91. pp 429-432
[5] A. Swami, J.M.Mendei. AR Iden i iabili y Using
CUlllulan Slices. P oceedings o he Wo kshop in
Highc O de Spec al Analysis. CO pp 13-18 June
1989
[6] L.R.Rabine . A Tu o ial on Hidden Ma ko Models
and Selec ed Applica ions in Speech Recogni ion.
IEEE P oceedings Vol 77. N02. Feb . 1989