Machine learning research has largely been devoted to binary and multiclass problems relating to data mining, text categorization, and pattern/facial recognition. Recently, popular machine learning algorithms, including support vector machines (SVM), have successfully been applied to wireless communication problems. The paper presents a multiclass least squares SVM (LS-SVM) architecture for direction of arrival (DOA) estimation as applied to a CDMA cellular system. Simulation results show a high degree of accuracy, as related to the DOA classes, and prove that the LS-SVM DDAG (decision directed acyclic graph) system has a wide range of performance capabilities. The multilabel capability for multiple DOAs is discussed. Multilabel classification is possible with the LS-SVM DDAG algorithm presented

Abdallah, Chaouki T.

Christodoulou, Christos G.

Rohwer, Judd A.

English

University of New Mexico

University of New Mexico
UNM Digital Repository
Electrical & Computer Engineering Faculty
Publications Engineering Publications
6-22-2003
Least squares support vector machines for direction
of arrival estimation
Chaouki T. Abdallah
Judd A. Rohwer
Christos G. Christodoulou
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Recommended Citation
Abdallah, Chaouki T.; Judd A. Rohwer; and Christos G. Christodoulou. "Least squares support vector machines for direction of arrival
estimation." IEEE Antennas and Propagation Society International Symposium (2003): 57-60. doi:10.1109/APS.2003.1217400.
Least Squares Support Vector Machines for Direction of Arrival 
Estimation 
Judd A. Rohwer'. Chaouki T. Abdallah*. and 'Chtisros G. ChristodoulouZ 
ISandia National Labanitotics, Albuquerque. NM. jamhwe@sandia.gov 
2Depamenf of Electical and Computer Engineenng. University of New Mexico, 
Albuquerque, NM. (chaoulu,christosJ @eece.unm.edu 
1 Inlmduelio" 
Machine laming research has largely been devoted to binary and multiclas problems relating to 
dara mining. curl ealegotizarian. and patledfacial recognition. Recently. popular machine leaming 
algotilms, including ~ u p p n  vector machines (SVM), have successfully been applied to wireless 
cammulucarian problems. notably spread specmm receiver design [I]. channel equalization [ZJ. 
This papr  presents a m u l f i c l a ~ ~  implementation of SVMs fordlrection of a n i v i  (DOA) e\limalian. 
2 Loas1 Squares Suppad Vector Maehins 
In  a bina ~lassifi~ation syilem the input sequence and a set of mining labels are represented as 
(xk,yk},=, , whereyk = (-1,l)~presentrtheclassification"label"oftheinputvectorx~. l f the 
two clases are linearly separable in the input space then the hyperplane separaring the d a s s a  i s  
defined as w x + b  = 0, w is a vector of weigh8 and b is a b i s  twm. I f  the input space i s  projected 
to a higher dimensional feature space then the hyperplane becomes wr [x ]  +b = 0, the nonlinear 
function r [.) maps the input space to the feature space. 
Suykens. et.al.. 131 introduced a l e s t  squares SVM (LS~SVM) which i s  based an the Vapnik SVM 
ElaSSifiCT. 
(1) 
2 
1 y[x)=s ign ~ e ~ y X ( x , x t ) + b  
The U - S V M  classifier i s  generated fmm the optimization problem: 
ik:l 
Miiclasrifications. due 10 overlapping distributions, are accounted far wirh l e  slack variables &, 
+ i s  a regu1ariwlion parameter that governs the complexity of the SVM. and the margin between 
the hyperplane and the data p i n t s  in the feature space i s  maximize when w i s  minimized and 
the relationship P (xkfr P (a) = r (x.xk) i s  due to Mercer's Theorem I41. The Lagrangian of 
equation (2) i s  defined as 
x 
ZLS (w.b,4, a) = .CLS (w,b.Ql - cm ( ~ b  [w'r (ztl +b] - 1 + 4,) (4)  
where 01 are Lagrangian multipliers, also known a\ "support V ~ C ~ O R " ,  that can eitherbe positive or 
negative. 
One-vswne muI11cIass cla~~ificalion IS based on binary L.-SVMr. For P dirtincl classes there am 
hywmlanes that separate the classes with maximum marein. The hvcemlanes with maxi- 
*=I 
0-7803-7846-6/031$17.00 02001 IEEE 57 
.'. i ~ '  I 
(DDAG)'is aipecific technique for one-wane multiclass classification 151. The technique uses a 
uee smcture to compare the test data to each of @he hyperplanes. Thmugh a series O f  elimination 
steps the best labcl i s  assigned to the input data. The LSSVM algorithm for DOA eslimtion i s  
based on the DDAC mhitecture with each node coilraining a binary U - S V M  classifier of the i t h  
and j'" classes, sec Figure I .  
MI -1 
Figure I :  DAGSVM for Four Classes 
3 SVMs and DOA Estimation 
The pra~ess of DOA estimation i s  to monitor the outputs of D vnlennn elements and predict the 
angle of arrival of L signals, L < D. The output vector for each incident signal from !he antenna 
elements is B (0 , )  = [ 1 e-Ji' . . . e - ~ ~ O - ' ) k f l  I T ,  and the vector of incident DOAs is E = 
[ Si, . . . , SL 1. With h training p i ~ e s s  the leaming algorithms generate DOA esrimnes based 
FartheLS-SVM b~sedapp_muchloDOAestimation rheoutput_of rhereceiverisusedtocvlculateIhe 
samplecavanancematrirR,oflhei"putdatasig"alx, (k).R, = ~~=,.,+, x, ( k ) x : ( k ) .  
The dimension of the observation matrix is D x M, P4 i s  ideal sample sire (window length). and the 
dimension Of the sample covariance mvvix 15 D x D. The principal eigenvectors, "1,. . . , vg, are 
calculated via cigcn decompositi%n (ED) or subspace tracking techniques. Each eigenvector is used 
on the responses f" the antenna elements, B (00 
tocalculate acovanance mavin. R',.,, . . . , 
The LSSVM DOA estimation algorithm includes preprocssing. mining, and testing steps 
hpWXSsing far SVM TG3inlng 
I )  Generate the D x N mining Y ~ C L O ~ S  for the P SVM das~es. D i s  the number of rnlenna 
elnnents. N is the number of input data samples. 2) Generate the P sample covarknce 
matrices. e),with M samples from the D x N daw vector 3) Calculate the signal eigenveelor. 
S. f" each of the P sample covariance matrices. 4) Calculate Ibc D x I pmjeclion vectors. 
C . S .  for each of the P classes. 5 )  Store the projection vectors for the !mining phase and the 
eigenvectors for the testing phase. . LSSVMTrrining 
I )  With the P projection ve~tors train the 
rithm. 2) Stare the LS-SVM variables. nk and B from equalion (1) 
nodes with the one-vs-one LS-SVM algo- 
. hprWeSSing for SVM TSLing 
I) Acquire D x N input signal from antenna m a y .  2) Genemle the smple covariance matrix 
with M samples from the P Y N data vector. 3) Calculate the eigenvectors for the signal 
subspace and the noise subspace. 4) Generate the Covariance marnccs for each eigenvector 
58 
. L.-SVM Testing for Ihc t ' " / j ' "  DDAG Nodi. 
I )  Calculate two D x 1 projection vectors with the desired eigenvector covariance malrix and 
the it" and 3L'' eigenvectors from the training phase. 2) Test bath projection vectocs against 
the Ls-SVM hypcrplanc for the i"L/3L" node. I )  Calculate the mean value of the IWO IS- 
SVM ou'put vectors (labels). Select the man  value that i s  closcsl Io il decision boundaiy, 0 Or 
1. Compare this value 10 the label definirion at the DDAG node. then select [he pmper output 
label. 4) Repeat process for the next DDAG node in the eval~ation path or declare the final 
DOA label. . IS-SVMDDAGErrorConfrol 
I )  Review the MSE ~ a l ~ ~ l a i i o n ~  for the DDAG ev i l l~~ l ion  path 2) Apply error c~ntrol  and 
validation measures 10 classify the label as eiiher an nccurale DOA erlimvte or a3 NOISE. 
3.1 Simulation Results 
Simulvrions a i  (he IS-SVM ODAG DOA cstimrkn algorithm are based On 0 complex syhtcm 
model that includes amplitude and phase distributions representative af the COmmuniCalion channel. 
The received signal at the receiver i s  modeled as 
D L  
.,(I) =CCH(8t)oasd(f  -.~,)."s(.:,(I-rii))tni(I) ( 5 )  
d=1 / = I  
Thls model inclvdea DrntennuvrrByelementrwilh steeringrnvyvectnrH(0,) andaddiiiveGaussian 
noise n,, ( 1 ) .  In addition. [he model assumes L independent. resolvable signal paths. The multipath 
variable " ,U  i s  defined as oiii = p cJ(2nl, 1'-'"fi)+4,1. The rmplitude of the received algnal P,,,. 
includes the transmitted power &and the attenuation due tu the link gain and shadowing q'. 
This variable i s  modeled as a fixed. Rayleigh. Ricean. or log~nurmvl distributed random variable. 
The Doppler shift for a c h  resolvable path i s  defined by f, = e: zjC i s the velocity of the mobile 
in  2, i s  the cmier frequency. and c IS the speed of propagation. A uniformly distributed carrier 
phaseshift.~~..andalimedelvyforeachsignal path,rn,vrealsoincludcdinrhemultip~th variable. 
ulit The C D M 4  w a d i n g  code, m (I - T,u), pmvides the processing gain PI the E O T T ~ ~ ~ O T O U ~ Q U I .  
The an~enna rray includes eight elements and the Wining and test signals ure the complex outputs 
from the antenna array. The LS-SVM system includes four DOA classes and six DDAG nodes. 
F~gure2showsresultsfaruIendegreerangeperclvrs. TocampletelytcsrtheIS-SVM DDAG'sca- 
pabiliiics the  simulation^ were automated 10 test a wide range of DOAs. As can been seen from Fig- 
ure 2 the LSSVM DDAG W A  esrimalion algorithm i s  extremely accurate. NO misclassificariona 
were logged. Additional simulations show that the IS-SVM DDAG system ~ccurately classifies the 
DOAs for three to ten classes and DOA ranges from one degree IO twenty degrees. 
3.2 Multilabel Capability for Multiple DOAs 
In DOA estimation far cellular systems there can be multiple DOAs for a given signal. This resulls 
from multipath induced by the envimnmenl. The machine leaning System must be able 
IO discriminate between a Small number of independent DOA. that include signal componenw with 
similvr!imcdelayr. With ihisconstrainiIhemachine levrning vlgorithmmuet beamulticlvss system 
and able to process mulciple labels. The machine laming algoethm mut  generate mdt ic la~s labels. 
y, t x. where x t [-go. 901 1s a set of real numbers that represent an appropriate range of expeclcd 
DOA values. and multiple labels y,. i = I . .  . L for L dominant signal paths. If anlenm sectoring is 
used i n  the cellular system the muluclass labels are from the set x t [SJ where S. i s  field of view 
for the i"' sector. Multilabel classification is possible with the IS-SVM DDAG algorithm presented 
in Section 3. The LS-SVM algorithm for DOA estimation assigns DOA labels 10 each eigenvector 
in the signal subspace. By repealing the DDAG cycle for each eigenvector the multiclass algorithm 
has the capability of assigning multiple labels to the input ~ignul. 
59 
Figure 2: U - S V M  for DOA cstimution. four C I ~ S S ~ E  with ten degree separation between each. 
4 Conelusion 
In this ppcr we presented P miilticla~s IS-SVM anhilecture far W A  estimation as applied lo a 
CDMA cellular system. Simulation results show 2 high degree af accuracy. as relaled to the W A  
E l ~ s e S  and prove that the L S S V M  DDAG system has a wide nngc of perfomance capabililiw. 
The bmad range of our research in machine leaming bued DOA estimation includes multilabcl 
and ~UIUCIPPS clwsification, c1;ssificarian accuracy, error mnml m d  validation. kemel seh~tion, 
estimation of signal subspace dimension. and overall performance characreti-ion of the L.-SVM 
DDAG DOA estimation algorithm 
Referenas 
I I 1  S .  Chen, A.K. Samingan. and L. Hanm. "Support Vecior Machine Multiu~er Receiver for DS- 
CDMA Signals in Multipath Channels", IEEE Transactians On Neural Networks. vol. 12. no. 
3.pp.6111-611.May 2WI. 
121 Daniel J. Sebald and James A. Bucklew, "Support Vector Machine Techniques for Nonlinear 
Equaliwtion". E E E  Trans;elions On Signal Recessing, vol. 48, no. 11. pp. 3217.3226, No- 
vember 2wO. 
131 J.A.K. Suykcns. L. Lukrs. 1,. Van Dooren. B. DeMoor, and J. Vandcwallc, "Least Squares Sup- 
port Vector Machine Classifiers: a Large Scale Algorithm". ECCTD'99 Europcan Conf. on 
Circuit Theory and Design. pp. 839-842. August 1999. 
141 J.A.K. Suykenr and J. Vandcwa1le;'Least Squares Support VectorMachim Clasifien", Neural 
Rocerring Letlers. August 1998. 
151 John C. Platl. Nello Chrisiianini. and John Shuwe-Taylor. "Large Margin DAGS for M u l l i d a ~ ~  
Clunifica!ion". in Advoricer i n  Neural blformoriori Pmcerring Sysrrm. vol. 12. pp. 547-553. 
Cambridge. MA, MIT P m i .  2oW. 


Least squares support vector machines for direction of arrival estimation

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Least squares support vector machines for direction of arrival estimation

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