comparision mpvsft

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GITAL SIGNAL P ROCESSING 6, 108 125 (1996)RTICLE NO. 0011

Comparison between the Matrix Pencil Methodand the Fourier Transform Technique forHigh-Resolution Spectral Estimation

ose Enrique Fernandez del R Bo and Tapan K. Sa rkar *Department of Electrical and Computer Engineering, 121 Link Hall,

yracuse University, Syracuse, New York 13244-1240

where j is 0 1 , K is t he num ber of frequency com-Ferna ndez del R Bo, J . E., and Sarkar, T. K., Comparison ponents , an d Am i s t he complex a mpl itude a t f re-

etween the Ma trix P encil Method a nd t he Fourier Tra ns- quency f m .orm Technique for High-Resolution Spectral Estimation, The t ime function is sampled at N equispacedigi tal Signal Processing 6 (1996) , 108 125. points, D t a part . Hence (2.1) reduces t oThe objective of this paper is to compare the perfor-

ma nce of the Mat rix P encil Method, part icularly the Totalorward Backward Matrix Pencil Method, and the Fou- g ( i D t )

K

m 1

A m e j 2p f m i D t ;

ier Transform Technique for high-resolution spectral esti-mation. Performance of each of the techniques, in terms i 0, . . . , N 0 1. (2.2 )

f bias and variance, in the presence of noise is st udiednd the results are compared to those of the Cramer Rao

The signa l in (2.2) may be conta mina ted by noiseBound. 1996 Academic Press, Inc.to pr oduce z ( i D t ). The a dditive w hite n oise w (i D t )is assumed to be Ga ussian with zero mean and vari-

. INTRODUCTION an ce 2s 2 , a nd it is included in our model via

In th is work, the Tota l Forward Ba ckward Ma- z ( i D t ) g ( i D t ) / w (i D t );r ix Pencil Method (TFBMPM) is ut i l ized for the

i 0, . . . , N 0 1. (2.3 )igh-resolution est imator and i ts results are com-ared with those of the Fourier Transform Tech-

In order t o simplify t he nota tion, Eq. (2.3) will beique, w hich is a stra ightforwa rd implementa tion ofrewri t ten a sh e Fourier Tra nsform. The root m ean squ a red error

or both of th e methods is a lso considered in ma kingcomparison in performance. z i g i / w i ; i 0, . . . , N 0 1. (2.4 )

Simulation results a re presented t o i llustra te t heerformance of each of the techniques. The frequency estimation problem consists of esti-mat ing K frequency components from a known set. SIGNAL MODELof noise contaminated observations, z i , i 0, . . . ,N 0 1.Consider a time domain signal of the form In this paper, the frequency est imat ion problemwill be solved by using a n extension of t he Ma trix

g ( t ) K

m 1

Am e j 2p f m t , (2.1 ) Pencil Method (MPM) [1] cal led Total Forward

B a ckwa rd Mat rix P encil Method a nd compa red with* Fax: (315) 443-4441. E-mail: [email protected]. th e results obta ined from the Fourier Techniques.

051-2004/96 $18.00opyright 1996 by Academic P ress, In c.ll rights of reproduction in any form reserved.

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Z 1 f b 2 ( N 0 L ) 1 L z1 z2 zL 0 1 zL

z *L 0 1 z *L 0 2 z *1 z *0 , ( 3.2 )

where * denotes complex conjugate, L is called thepencil para meter, and the tr a nspose of z j ( j 0, . . . ,L ) is d ened a s

zT j [z j , z j / 1 , . . . , z N 0 L / j 0 1 ]; j 0, . . . , L . (3.3 )

The new Z 0 f b a nd Z 1 f b are better conditioned [2,Appendix B ] th a n Z 0 a nd Z 1 , which are formed forthe ordinary MPM; that i s , Z 0 f b a nd Z 1 f b are lesssensit ive than Z 0 a nd Z 1 to smal l changes in theelement values.

With (3.1) a nd (3.2) one ca n build the Mat rix Pen-cil, Z 1 f b 0 j Z 0 f b (j is a complex scalar), and follow th e m ethod proposed in [1, S ection II ] to estima tethe frequency components, but , for noisy da ta , t hebest st ra tegy is to perform a Singula r Value Decom-position (S VD ) [3 ] on t he a ll da t a ma t rix [4 ]. Thismatrix is given by

IG.1. Real and imag inar y par ts of a n undam ped cisoid formedy two frequency components of equal power. Z f b 2 ( N 0 L ) 1 (L / 1 )

z 0 z1 zL 0 1 zL

z *L z *L 0 1 z *1 z *0 . ( 3.4 )

In F ig. 1, a possible noiseless da ta record (r eal a ndmagina ry pa rt of the signa l ) is shown. The function I t i s easy t o see tha t Z f b contains both Z 0 f b a ndepresented was generated using Eq. (2.2) with the Z 1 f b :ara meters given in Ta ble 1.

This function w ill be utilized in ma king a compa ri- Z f b 2( N 0 L ) 1 (L / 1) [Z 0 f b 2( N 0 L ) 1 L , cL / 1 ] (3.5 )on betw een t he Mat rix Pencil Method and th e Fou-ier Tra nsform Techniq ue. Z f b 2( N 0 L ) 1 (L / 1) [c 1 , Z 1 f b 2 ( N 0 L ) 1 L ]; (3.6 )

. TOTAL FORWARDBACKWARD MATRIX here c 1 a nd c L / 1 represent , respect ively, th e rst a ndENCIL METHOD (L / 1)t h columns of Z f b .

On th e other ha nd, the SVD of Z f b is

The estimat ion of frequencies in th e presence ofZ f b 2 ( N 0 L ) 1 (L / 1)oise is considered by the TFBMPM. When the com-

lex exponent ials in ( 2.2) (so-called cisoids ) ar e un - U 2 ( N

0L )

12 ( N

0L ) S 2 ( N

0L )

1(L

/1 )

V H (L

/1 )

1(L

/1 ) , (3.7)amped 1 (w hich is th e ca se in th is work ), to improve

he est imation accuracy we consider the matrices0 f b a nd Z 1 f b a s dened by TABLE 1

Input D at a Considered in Fig. 1Z 0 f b 2( N 0 L ) 1 L

z 0 z1 zL 0 1 zL 0 1

z *L z *L 0 1 z *2 z *1 (3.1)

64 samples (N 64)Sampling period 0.25 ms (D t 1/4000 s)

2 frequency components (K 2)1 Note tha t the Ma trix P encil Method can solve a more generalA 1 1e j 2.7( p /180 )roblem [1], the pole estimation, p m , for damped cisoids (p m A 2 1 e j 00 s

m / j v

m ) D t , s m 0, m 1, . . . , K ) and t he undam ped cisoids ar e f 1 580 Hzpa rticular ca se of the dam ped exponentials (in tha t it is enough

f 2 200 Hzo set s m to zero for all m ) .

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wh ere th e superscript H denotes complex conjugat e an d r ight mult iplying (3.19) by Z O /0 f b , t he result ingeigenproblem can be expressed asra nspose of a ma trix and U , S , and V a re given by

q H (Z O 1 f b Z O /

0 f b 0 j I ) 0 H , (3.20)S d iag {s 1 , s 2 , . . . , s p };

p min {2(N 0 L ) , L / 1} (3.8 )where Z O /0 f b is th e Moore P enrose pseudoinverse [3 ]of Z 0 f b and i t can be wri t ten ass 1 s 2 rrr s p 0 (3.9 )

U [u 1 , u 2 , . . . , u 2 ( N 0 L ) ];Z O

/

0 f b (V O H 0 )

/

S O 0 1

U O /

. (3.21)Z H f b u i s i v i , i 1, . . . , p (3.10)

Substi tut ing (3.17) and (3.21) into (3.20), theV [v 1 , v2 , . . . , vL / 1 ]; equiva lent genera lized eigen-problem becomesZ f b v i s i u i , i 1, . . . , p (3.11)

q H (V O H 1 0 j V O H 0 ) 0 H . (3.22)U H U I , V H V I . (3.12 )

It can be shown tha t (3.22) is equivalent t oi a re the singular values of Z f b a nd the vectors u i nd v i a re, respectively, the i th left singula r vector

q H (V O H 1 V O 0 0 j V O H 0 V O 0 ) 0 H , (3.23)nd the i th right singular vector.The problem ca n be comput a tiona lly improved by

w hich is a genera lized eigenproblem of dimension K pplying th e singula r va lue ltering, w hich consist s1 K .f [1] using the K lar gest singula r va lues of Z f b , i.e.,

Using the values of the generalized eigenvalues,j , of (3.23), the frequency component s ca n be esti-Z O f b 2 ( N 0 L ) 1 (L / 1 ) U O 2 (N 0 L ) 1 K S

O

K 1 K V O H K 1 (L / 1 ) , (3.13)mated.

In t he follow ing, the a lgor i thm a pplied to estima tewhere the frequencies is summarized as:

Step 1: Construct the matrix Z f b , (3.4) , with theS

O d iag {s 1 , s 2 , . . . , s K } (3.14 ) corrupted sam ples, w here z T j ( j 0, . . . , L ) is d e-

ned a s in (3.3), a nd L ha s to sat isfyas the K la rgest s ingular values of S and the col-mn s of U a nd V a re formed by extra cting t he singu- K L N 0 K . (3.24)ar vectors corresponding t o those K singular va lues.

Eq. (3.13) can be rewrit t en a s Step 2: Realize the SVD of Z f b , (3.7) , an d, fromits singula r va lues, estimat e K (n umber of frequ ency

Z O f b U O SO V O H U O S O [ t 1 , t 2 , . . . , t L / 1 ] components ). This problem is equiva lent to solving

th e eigenproblem Z H f b Z f b ; i .e., i t can be proved that [U O SO t 1U O SO t 2 rrr U O SO t L U O SO t L / 1 ] . (3.15) the singular values of Z f b , s i , a re the nonnegativesquare roots of hi , w here hi are the eigenvalues ofComparing (3.5), (3.6), a nd (3.15), t he equa tions th e eigenproblem

Z O 0 f b U O S O

V O H 0 (3.16) (Z H f b Z f b 0 hi I )r i 0 . (3.25)Z O 1 f b U O S

O

V O H 1 (3.17)Step 3: Extract V 0 a nd V 1 from V , (3.18), w here

V is the K -truncation of V ((3.7) to (3.14)) .an be esta blished, w here V 0 a nd V 1 are obtainedStep 4: Es t ima te the K frequencies using the K rom V , deleting, respectively, its (L / 1) t h a nd rst

genera lized eigenvalues, j m , of (3.23) , such tha tolumns, i.e.,th ose eigenva lues ca n be expressed a s

V O [V O 0 , vL / 1 ], V O [v 1 , V O 1 ]. (3.18 )j m Rea l (j m ) / j I ma g (j m );

B y considering the ma trix pencil m 1, . . . , K , (3.26)

where Real(j m ) and Imag (j m ) are, respectively, theZ O 1 f b 0 j Z O 0 f b (3.19)

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eal and imaginary par ts of j m , but th ose eigenval- i 0, . . . , N 0 1, (4.2.1 )es are related to the frequencies as

ha s been follow ed, wh erej m e j 2p f m D t ; m 1, . . . , K . (3.27 )

A m A m e j um ; m 1, . . . , K (4.2.2)And, from (3.26) and (3.27),

v m 2p f m ; m 1, . . . , K . (4.2.3 )

For th e noisy da ta problem it is enough t o considerf m 12pD t t a n

0 1I m a g (j

m )

Real (j m ) ; (2.4), which, in vectorial notation, can be denoteda s

m 1, . . . , K . (3.28 )

z g / w , (4.2.4 )

. LIMITS OF TFBMPM FOR FREQUENCIESwhereESTIMATION

z T [z 0 , z 1 , . . . , z N 0 1 ] (4.2.5 ).1. T he Fr equency Esti m ati on Problem g T [g 0 , g 1 , . . . , g N 0 1 ] (4.2.6 )

The frequency estimation problem consists of [5, w T [w 0 , w 1 , . . . , w N 0 1 ] (4.2.7 )Chapter 6] determining the frequency componentsf a signal, which obeys the mathematical model of

and those vectors could be briey described as fol-ection 2, from a set of noisy samples.lows:Any est imate of the frequency parameter evalu-

g is formed by t he noise free sam ples, (4.2.1). Thisted from a set of sam ples involves a ra ndom processvector may be seen l ike a determinist ic unknownnd, thus, it is necessary t o consider the estimat e asmagnitude. The deterministic model for g is usedra ndom var iable. Consequently, it is not correct t owhen K (n umber of frequency components ) and th epeak of a part icular value of a n est imat e, but i t isnum ber of sna pshots (in th is work just one sna pshotecessary to know its s ta t is t ical distr ibution if theor picture is considered) are small [9].ccuracy of the estimate is analyzed.

w represents the complex white Gaussian noise,An efcient estima te ha s t o be a s nea r a s possible with the chara cterist icso the true value of the parameter to be est imated

6, Chapter 32]. This idea of concentration or dis-zero mean: E [w ] 0 (4.2.8)ersion a bout the t rue va lue may be measured us-

ng several s ta t is t ical ma gnitudes (va riance, mean uncorrelated, with variance 2s 2 :qua red error, etc.).One of the rst w orks concerned w ith th e a pplica- R w 2s 2 I N 1 N , (4.2.9)

ion of the E stima tion Theory by Fisher a nd C ra mero the problem of est imating signal parameters is where E [r ] mea ns expected va lue, R w is the correla -ha t of Slepian [7]; lat er, in [8] , the sta t is t ical t he- t ion matr ix of the noise , and I N 1 N i s the ident i tyry is applied to the estimation of the Direction of matrix.

Arriva l of a plane w a ve impinging on a linea r pha sed z is the vector cont a ining th e observed da ta . Ob-r ray. viously, from it s denit ion, (4.2.4) , it is a ra nd omIn th is work, the limits of TFB MP M for frequency vector.

st imation wil l be pointed out and the variance of In order to dene t he C RB it is rst necessa ryh is m et h od w i ll be com pa r ed w i t h t h a t of t h e to introduce the joint probability density function

Cra mer Ra o Bound (CRB ) [6, Cha pter 32]. (jpdf). The jpdf of a complex Gaussian random vec-tor of N components, x , is dened [5, p. 478] a s

.2. T he Cram er Rao B oun d In this section, the notation

f x (x ) 1

p N det (R x ) e 0 ( x 0 E [ x ])

H R 0 1x ( x 0 E [ x ]) , (4.2.10)

g i

K

m 1

Am

e j um e j v m i D t ;w here det (r ) means determina nt of a mat rix, H de-

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otes complex conjuga te t ra nspose, a nd0 1 in dica t es a r e a lm ost u nbia s ed3 in t h e r eg ion w h er e t h eTFBMPM works.he inverse of a matrix.

Therefore, t he jpdf of w ca n be evalua ted by using For unbiased est imat es, the CRB sta tes tha t i f a P

4.2.8) (4.2.10): is an unbiased est imat e of a , t he va riance of eachelement, a P l (l 1, . . . , 3K ), of a

P can be no smallerth a n t he corresponding dia gona l term in t he inverse

f w (w ) 1

(2ps 2 )N e 0 1 /2 s

2 N 0 1i 0 w i

2 . (4.2.11) of the Fisher Information Matrix

v a r (a P

l ) [F 0 1

]l l , (4.2.17 )The jpdf of z can be obtained from (4.2.11) bya king int o a ccount the relationship [10, p. 61] be-ween z a nd w , which is given by (4.2.4), where a P l is the est imate of the para meter a l (l 1,

. . . , 3K ) , [F 0 1 ]l l i s the l th diagonal element of t heinverse of F , a n d F 3 K 1 3 K is t he Fisher Information

f z a (za ) 1

(2ps 2 ) N e 0 1 /2 s

2 N 0 1i 0 z i 0 g i

2 , (4.2.12) Matrix.The ( m , n )th element of F is dened a s

where a denotes tha t the jpdf i s condi t ioned ton unknown vector parameter, a , a n d g i is given [F ]m n E

ln f z a (za ) a m

r l n f z a (za )

a n ;

n (4.2.1).

Fr om (4.2.12) one ca n d educe tha t z is a G aussianandom vector with m , n 1, . . . , 3K . (4.2.18 )

E [z ] g (4.2.13) The last equa tion, using (4.2.12), can be rew rit ten[1] asR z 2s 2 I N 1 N . (4.2.14)

Also, a i s the vector formed by the parameters [F ]m n 12s 2

N 0 1

i 0

2 Real g i a m

r g *i a n

;o be est ima ted. In th is w ork t he complex a mpl i-udes of the s ignals , Am , 2 and the va r i ab le v m in4 .2 .1 ) wi l l be chosen as unknown pa ramete r s . m , n 1, . . . , 3K , (4.2.19 )

N ot e t h a t Am i s g iven by (4 .2 .2) and, therefore ,ach Am corresponds t o tw o para meters , A m a nd where Real [r ] denotes t he real part .m . On the oth er han d, v m is rela ted to the frequen- It can be proved [11] th a t F 0 1 may be decomposedies t hrough (4.2.3) . a sConsequent ly, t he vector a can be writ ten a s

F 0 13 K 1 3 K s 2 S 3 K 1 3 K P 0 13 K 1 3 K S 3 K 1 3 K , (4.2.20)a T [a 1 , a 2 , a 3 , . . . , a 3 K 0 2 , a 3 K 0 1 , a 3 K ] , (4.2.15)

wherewhere

S 3 K 1 3 K a 3 m 0 2 v m 2p f m ; a 3 m 0 1 A m ; d iag {[S 1 ]3 1 3 , [S 2 ]3 1 3 , . . . , [S K ]3 1 3 } (4.2.21)a 3 m um ; m 1, . . . , K . (4.2.16)

[S m ]3 1 3 d iag {A m 0 1 , 1, Am

0 1 };The CRB provides th e goodness of a ny est ima te of

m 1, . . . , K (4.2.22)random parameter. The es t imates of th is workave been computed via the TFBMPM, and i t wil le pointed out, through simulat ion results, tha t t hey

P 3 K 1 3 K [P 11 ]3 1 3 [P 1 K ]3 1 3

[P K 1 ]3 1 3 [P K K ]3 1 3 (4.2.23)

2 In order to estimat e the complex a mplitudes, Am , us ing theesults obtained from the TFBMPM for the frequency compo-ents, one ma y solve a least -squar es problem z E a , where zre the corrupted samples, a conta ins the complex am plitudes 3 An estimat e a

P of the vector parameter a is unbiased if E [a P ]

a .m , and E is the m at rix which applied to a gives g .

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P m n

(D t ) 2 N 0 1

i 0

i 2 cos D ( i , m , n ) 0 D t N 0 1

i 0

i sin D ( i , m , n ) D t N 0 1

i 0

i cos D ( i , m , n )

D t N 0 1

i 0

i sin D ( i , m , n ) N 0 1

i 0

cos D ( i , m , n ) N 0 1

i 0

sin D ( i , m , n )

D t N 0 1

i 0

i cos D ( i , m , n ) 0 N 0 1

i 0

sin D ( i , m , n ) N 0 1

i 0

cos D ( i , m , n )

(4.2.24)

r 2 i (i 0, . . . , N 0 1), a re obta ined to constr uct theD ( i , m , n ) i (v m 0 v n )D t / um 0 un ;complex sequence

i 0, . . . , N 0 1; m , n 1, . . . , K . (4.2.25)

x i r 1 i / j r 2 i ; i 0, . . . , N 0 1. (4.3.1.1).3 . Sim ul at ion Resul t s 4.3 .1 . In put Data . In this section several graphs

Ta king into account th a t t he var iance of the com-re presented and discussed in order to facilitate aplex noise, w i , w a s d ened a s 2s 2 , i t is easy t o de-et t er understa nding of t he TFB MPM and i ts est i-

duce the relat ionshipmation limits.The methodology followed to obtain the differentlots has been to generate a set of N complex sam-

w i 2s 2 x i ; i 0, . . . , N 0 1. (4.3.1.2)les, using ((4.2.1) to (4.2.4)) and then to apply theTFBMPM as proposed in the algorithm of Section 3.

The S NR, for ea ch frequ ency component, h a s beenThis algorithm was iterated several times when thedened a sar iance of the frequency estima te w a s numerica lly

omputed.The input da ta ma y be described a s follow s:

SNRm 10 log10A m 2

2s 2 ;

(1 ) Observation interval 8 samples have been considered (N 8 ).

m 1, . . . , K . (4.3.1.3 )Th e s a m pl in g p er iod w a s n or m a l iz edD t 1 s ) .

(4 ) T FBM PM remar ks (see Sect i on 3 ) (2 ) Description of the signal The rst st ep in t he TFB MP M consist s of choosing2 freq uency component s ha ve been chosen

a value for t he pencil para meter, L , in order t o formK 2 ) .t he Z f b mat rix.A1 A 2 1: Tw o components of equa l

The best choice for L is [2]ower.u1 , u2 : A deterministic model has been as-

umed for t he pha ses of the frequency component s. N 3

L 2N

3 , (4.3.1.4 )The difference u1 0 u2 is taken from values in [0,

80 ). TFBMPM performance depending on u1 0 u2s show n in the n ext section.

but , at the same t ime, L ha s t o sat isfy (3.24).f 1 0.200 Hz.To numerically compute the variance of the fre-f 2 : The second frequency var ies betw een

quencies the algori thm proposed in Section 3 has.270 a nd 0.290 Hz a nd, therefore, th e va lue of D f been i terated 500 t imes ( tr ials) . For each tr ial , atudied i s in the in terval [0.070 Hz, 0.090 Hz ] ,different vector w wa s ra ndomly ta ken.where D f f 2 0 f 1 .

(3 ) S t at i st i ca l consider a t i ons for t he noi se ( see 4 .3 .2 . Per form ance of t he TFBM PM as a f unct i on of u1 0 u2 . The a ccura cy in th e frequencies estima-ection 4.2)

The noise wa s generated by using ISML [12] FOR- t ion, using the TFB MPM, depends strongly on t hedifference of phases between the components of theTRAN subroutine GG NML. This subroutine is a

Ga ussian (0, 1) pseudo-ran dom number generator. s igna l . I t has been proved [2] tha t the inverse of theva rian ce of t he frequencies estima tes,With GGNML two sets of N real numbers, r 1 i a nd

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IG. 2. Inverse of the var iance of the rst frequency estima te, a s a function of the difference of pha ses of the t wo frequency componentsnd the difference of frequencies. SNR 17 dB and the pencil par ameter for the TFB MPM is L 5.

have been explained in Section 4.3.1. SNR is 17 dB10 log10

1v a r (f O m )

; m 1, . . . , K , (4.3.2.1) a nd L 5. In Fig. 3 the same input da ta are t aken,and the CRB fo r the va r i ance o f f 1 is shown. Toobtain this 3D plot , the method in S ection 4.2 haseaches a maximum ifbeen follow ed, determin ing the CRB for th e va ria nceof v P 1 a nd a pplying t he relat ionship in (4.2.3) to ca l-v m 0 v n )( N 0 1 )D t / 2 (um 0 un )

culate t he CRB for f

1 . ( 2l )p (4.3.2.2) Comparing Fig. 2 to Fig. 3 one can deduce thatthe C RB is reached by the est imate obtained using

nd a minimum if TFBMPM when f 2 0 f 1 is close to 0.090 Hz or, in theentire interval [0.070 Hz, 0.090 Hz], when u1 0 u2

v m 0 v n )( N 0 1 )D t / 2 (um 0 un ) is far from the worst case.4.3.3. Esti mat in g the nu mber of fr equency compo- l p . (4.3.2.3)

nents f r om the singular va lues of Z f b . As w a s ex-plained in S ection 3, to estimat e th e number of fre-In both Eq s. (4.3.2.2) a nd (4.3.2.3), m , n , a n d l qu ency components K t he eigenva lues of Z H f b Z f b willave t o sat isfybe used. This idea will be followed in this section for

both the ideal sampling (neglecting the noise) andor a ll m x n ; m , n 1, . . . , K ; th e corrupted sam ples.l integer. (4.3.2.4) Figures 4 to 11 show the normalized magnitude,

in dB , of t he eigenvalues, j n (n 1, . . . , L / 1), ofWe will call, respectively, best case and worst caseZ H f b Z f b . This norma lized m a gnitude is given byo (4.3.2.2) a nd (4.3.2.3). The mean ing is simple; w hen

4.3.2.2) is given, (4.3.2.1) reaches a maximum andhus the va riance ta kes its minimum va lue. In other

10 log10j n

j m a x; n 1, . . . , L / 1, (4.3.3.1)words, the distribution of the estimates reaches its

maximum of concentration around the true value ofhe vector parameter being estimated. The explana-ion for the w orst ca se is a na logous. w here L i s the penci l parameter and j m a x i s t h e

largest eigenvalue.In F ig. 2 th a t dependence is show n. The input da ta

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IG. 3. Inverse of the CR B of the rst frequency estima te, a s a function of the difference of phas es of the tw o frequency componentsnd the difference of frequencies. SNR 17 dB.

The input d a ta for SNR, L , f 2 0 f 1 , and u1 0 u2 a r e n u mber of s ig na l s is es t im a t ed fr om t he K largesteigenva lues of Z H f b Z f b ). This ga p is much grea ter foriven in Table 2.

Compar ing the noiseless case (Figs. 4 to 7) to the the noiseless samples than for the samples in noise,a s w a s expected. In fa ct, t he n oise is t he culprit oforrupted samples (Figs. 8 to 11) one can see that

he main difference is the gap betw een the second the gap reduction.To enha nce this ga p, for t he noisy da ta ca se, digi-igenvalue and the th i rd one (note tha t two f re-

uency components ar e being considered an d the ta l ltering techniques in the original set of samples,z i , can be a pplied [13].

FIG.5. Normalized magn itude of the eigenva lues of Z H f b Z f b . TheIG. 4. Normalized magn itude of the eigenvalues of Z H f b Z f b . In-ut data : N 8, K 2, A 1 A 2 1, u1 0 u2 88.2 ( w or st s a m e i npu t d a t a a s i n F ig . 4, b ut u1 0 u2 113.4 (worst case)

a nd f 2 0.290 Hz.ase), f 2 0.270 Hz, f 1 0.200 Hz, SNR (noiseless) , L 3.

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IG.6. Norma lized ma gnitude of the eigenvalues of Z H f b Z f b . The FIG.8. Normalized magn itude of the eigenva lues of Z H f b Z f b . Theame input dat a as in Fig. 4, but u1 0 u2 178.2 (best case ) and

same input dat a as in Fig. 4, but SNR

20 dB. 6.

ance of f 1 is referred to the CRB , which mean s tha t4 .3 .4 . TFB M PM for f r equenci es est ima t i on i n t he (SNR ) (f 2 0 f 1 ) plane represents the CRB . B othr esen ce of n oise. In t his section the number of fre-gures demonstra te tha t the TFB MPM works be-uency components, K , is a ssumed to be know n a ndyond a certain threshold of SNR.qual t o 2.

Consequently, t he threshold is an indica tor of th eFigures 12 and 13 show the TFBMP M perfor-estimation limits. For example, for the worst case,mance as a function of SNR and f 2 0 f 1 . Figure 12 and for f 2 0 f 1 0.070 Hz, the threshold is betweenas been obtained for the worst case of u1 0 u2 a c- 17 a nd 19 dB, a s is shown in Fig. 12; therefore th isording to (4.3.2.3), w hile Fig. 13 corresponds to t heis the SNR low er limit in order for t he TFB MP M to

est case est imat ion, (4.3.2.2) . Note tha t the vari- provide rea sona ble results.

IG.7. Norma lized ma gnitude of the eigenvalues of Z H f b Z f b . Theame input da ta as in Fig. 4 , but u1 0 u2 23.4 (best case), f 2 FIG.9. Normalized magn itude of the eigenva lues of Z H f b Z f b . The

same input data as in Fig. 5, but SNR 20 dB. 0.290 Hz, and L 6.

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TABLE 2

Input Data Considered for Figs. 4 to 11

Figure S NR (dB ) L f 2 f 1 (Hz) u1 u2 ( )

4 (noiseless) 3 0.070 88.2 (w orst ca se)5 (noiseless) 3 0.090 113.4 (w orst ca se)6 (noiseless) 6 0.070 178.2 (best ca se)7 (noiseless) 6 0.090 23.4 (best ca se)8 20 3 0.070 88.2 (w orst ca se)9 20 3 0.090 113.4 (w orst ca se)

10 20 6 0.070 178.2 (best ca se)11 20 6 0.090 23.4 (best ca se)

For the best est imat e, and f 2 0 f 1 0.070 Hz, th e 5. THE FOURIER TRANSFORM ESTIMATORower l imit is between 5 and 6 dB, as is shown inig. 13.Figures 14 an d 15 ha ve been extracted from the 5.1. T he Per iodogram

ata used in Figs. 2 and 3 and thus correspond to a The Fourier Transform Estimator (FTE ) for fre-NR of 17 dB . Also 0.070 Hz is t he designa ted va lue quency components estimat ion considered in this

or f 2 0 f 1 in Fig. 14 and 0.090 Hz is the value inw ork is ba sed on t he classic periodogram. The esti-ig. 15. mates of the frequencies, f m (m 1, . . . , K ) , w illI n F i g . 1 4 t h e C R B i s r e a c h e d f o r a l l u1 0 u2 be the values of the var iable f (frequency) which

xcept in the interval (70 , 105 ) , a pproximat e ly, maximize ( local ma xima ) the periodogram, (f ) .where the TFBMPM is not performing wel l . The The periodogra m is an estima te of th e pow er densityeason can be found in Fig . 12 , obta ined for the spectr um a nd can be dened [14] as

w orst case of u1 0 u2 , wh ere one ca n see th a t for f 2f 1 0.070 H z, a S NR of 17 dB is below t he th resh -

( f ) 1N D t

Z ( f )2 , (5.1.1 )ld a nd, by d enit ion, t he est ima tor cea ses func-ioning. Never th eless, the CR B is a lw ay s reachedn Fig . 15 because 17 dB is above the thresholdor a l l u1 0 u2 ( for the w ors t case e s t ima t ion the where Z ( f ) is the Discrete-Time Fourier Transformh reshold for f 2 0 f 1 0.090 Hz is betw een 13 an d (D TFT) of th e noise sa mples,4 dB, a s i s show n in F ig . 12) .

IG. 10. Norma lized ma gnitude of the eigenvalues of Z H f b Z f b . FI G. 11. Norma lized ma gnitude of the eigenvalues of Z H f b Z f b .The same input dat a as in Fig. 7, but SNR 20 dB.he same input dat a as in Fig. 6, but SNR 20 dB.

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12/18

FIG. 13. Varia nce of f O 1 compared t o the C RB for t he best case estimat ion. The peaks show t he threshold of the TFB MPM.

N D t ) 0 1 is 0.125 H z an d th e maximum D f s tud ied In F igs. 17 and 18 the windows a re shown in bothtim e an d frequency doma ins. The number of sampless 0.090 H z a nd , in conseq uence, th e FTE d oes not

w or k u nd er t hose con dit ion s. ha s been t aken a s 12 a nd t he sa mpling per iod is 0.25ms. The ma in difference a mong th ese w indows is th eThree w indows h a ve been considered in t his w ork:

Rect a ngula r w indow reduct ion in t he side lobes. The S t a nda rd w indow a chieves the la rgest r eduction of the bias, but it doesso at th e expense of broadening t he ma in lobe, whichh i

1, 0 i N 0 1

0, otherwise;(5.2.4)

results in a loss of spectral resolution.

The window in the t ime domain i s a ppl ied byweighting the input samples, z i , w i th the w indow Sta ndar d w indow [11]coefcien t s, h i , by modifying Eq. (5.1.2) in the fol-lowing way:

i

13

3

k 0

a k cos2p i k

N , 0 i N 0 10, ot herw ise;

Z ( f ) D t N 0 1

i 0

z i h i e 0 j 2p fi D t , 0

12D t

f 12D t

.

(5.2.5) (5.2.7)

wi th a 0 1, a 1 0 1.43596, a 2 0.497536, a 3 Eq. (5.2.7) is s imply the DTFT of the weighted0.061576.

samples, z i h i , a n d it w i ll be u sed , join t ly w i t hKaiser window [17, p. 232](5.1.1), to estima te th e frequency components.

5.3. Compar ison betw een t he FT E and the

i

I 0[b r 1 0 ( (i 0 N /2)/ N /2 ) 2 ]I 0[b ]

,T F B M P M

0 i N 0 1 The frequency component est imation using the0, ot herw ise; Four ie r Tra ns form ha s been wide ly s tud ied by

Rife and Boorstyn in [11]. Figure 19 provides the(5.2.6)com p a r i son b et w e en v a r i ou s w i n d ow s a n d t h eTFBMPM.ere I 0[r ] is t he modied B essel fu nct ion of t he rst

ind and order zero and b is a pa r a m et er, a n d in Th e in pu t d a t a for F ig . 19 a r e g iv en by F ig . 1,a nd the SNR, w hich is dened in (4.3.1.3), var ieshis work it has been chosen according to Table 3.

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reduction of the bias but a t t he expense of increasingthe va riance of t he est ima te.

The bias shown in Fig. 20 wa s comput ed a ccordingt o

bias(f O 1 ) E [f O 1 ] 0 f 1 , (5.3.3 )

a nd one ca n see tha t for SNR below 10 dB the FTE

with the Standard window offers less bias than theTFB MP M. Nevertheless, the rmse obta ined with theTFB MPM is less tha n the one computed using t heSt a nda rd w indow a s seen in F ig. 19. This is beca usethe S ta ndard w indow reduces the b ias but a t t hesame t ime increases t he var iance . On the otherhand, the use of the Rectangular window makes aFTE biased even for high SNR.

In Fig . 21 the behavior of the es t imator as thenumber of samples increases is shown. The inputdata are the same as in Fig. 19, but a Du of w orstcase was taken for each N , a nd SNR 0 dB. TheFTE uses th e Kaiser window for t his simulation an d

IG. 14. Compar ison between the inverse of the va riance and i t can be seen tha t for long data record the FTEhe CR B for the rst frequency estima te. f 2 0 f 1 0.070 Hz andreaches the CRB .NR 17 dB . The TFB MP M produces ina ccura te results in u1 0

2 ( 70 , 105 ) because the SNR is below the threshold. Figure 22 shows a compar at ive study of the rmseas a function of the difference of frequencies f 1 0 f 2for t w o components of equal power w hen th e SNR is

etween 0 and 40 dB. The values corresponding to 20 dB. As in Fig. 19 the sampling period is 0.25he C RB (dark squares in Fig. 19) have been com- ms but the number of sam ples ha s been dra sticallyuted by the square root of the corresponding diago-al term in the inverse of the Fisher Information

Mat rix (4.2.20) a nd th e pencil par a meter, L , for t heTFB MP M ha s been ta ken as 22. The sta tistical ma g-

itude rep resen ted in F ig . 19 i s the root meanqua red error (rm se ), dened a s

rmse( f O 1 ) E [(f O 1 0 f 1 ) 2 ] , (5.3.1 )

where E [r ] mea ns expected va lue, f 1 is the par ame-er being est imat ed, and f 1 is the true value of thea rameter. The rmse i s r el a t ed to the va r iancehr ough t he bias, i .e.,

rmse 2 ( f O 1 ) bia s2 ( f O 1 ) / va r (f O 1 ) , ( 5.3.2 )

nd, evident ly, for un biased estimat ors t he rmse be-omes th e squa re root of the va ria nce. The rmse w a somputed u sing 200 tr ials for ea ch a lgorithm .

From Fig . 19 one can see tha t the TFBMPM iser forming bet t e r than the FTE in a l l t he SNRa nge. On t he oth er han d, an d in spite of th e smallerias presented by the Standard window (see Fig .

FIG. 15. Compar ison between the inverse of the var iance and0), the Ka iser w indow provides bett er results t ha n the C RB for the rst D OA estima te. f 2 0 f 1 0.090 Hz and SNRhe St a nda rd w indow for SNR below 30 dB . The rea- 17 dB. The TFB MPM reaches t he CRB for a ll u1 0 u2 becau sethe S NR is a bove the thr eshold.on for t his is tha t t he Sta ndar d w indow achieves a

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IG. 16. Norma lized periodogram of the unda mped cisoid of Fig. 1. A Rectangu lar w indow wa s used to weight t he samples in the timeomain.

educed f rom 64 to 12 samples . The penci l parameter The las t s imula t ion included in th is paper i sshown in F igs. 24 to 28. While in t he previous sim u-or the TFBMPM is L 7, f 2 i s 200 Hz, and Du

worst case ) is assumed according to (5.3.2.3) . Two lat ions the two frequency components ha d the samepow er, in Figs. 24 to 28, th e rst frequency compo-ma in conclusions can be dra w n from F ig. 22; on thene hand the FTE does not work for D f below 460 nent has 10 t imes more power tha n the second one,

Hz ( (N D t ) 0 1 is 333 H z ) while TFB MPM sti l l per-orms w ell up to 180 Hz a nd, on the other ha nd, th e

TFB MP M performs better t ha n t he FTE even w henTE works, i.e., for D f greater tha n 460 Hz.In F ig . 23 the accuracy of the es t ima tor s de-

ending on the number of samples, N , is shown . ANR of 15 dB for tw o frequency components of equ a lower a t , respectively, 1300 a nd 1000 H z wa s consid-

red. Also a D t of 0.25 ms and a Du of w orst caseor each N were taken. Similar conclusions to thenes for Fig. 22 can be derived.

TABLE 3

b Values for the Kaiser Window

Figure b Value

17, 18, 19, 20, 21, 22 6FIG. 17. The three windows used in this w ork for t he Fourier23 5.5 Tra nsform Estima tor (FTE ). The graph shows 12 samples for

24, 25, 26, 27 5 each of them in the t ime domain.

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FIG. 20. Bias for the es t imate of Fig . 19. In spi te of the fa ctIG. 18. Compar at ive spectrum of the windows. The Discrete tha t the St anda rd window offers less bias t han t he TFB MPM forime Fourier Transform (DTFT) was used to obtain H ( f ) . SNR below 10 dB, i ts rmse performance is worse because the

Standard window increases the variance.

e., A12 10 A 22 , which supposes that SNR1ter used in t he TFB MP M is 6, f 2 400 Hz, SNR2 dB ) 10 dB / SNR2 (dB ). On t he other ha nd, 0.2510 dB, a nd Du of w orst case for ea ch D f is chosen.ms for the sampling period and 12 samples charac-From F igs. 24 an d 25 one can see the better perfor-erize th e observat ion interva l. The pencil para me-ma nce of t he TFB MP M for both f 1 a nd f 2 est imatesa nd a lso a larger spectra l resolution for this estima-tor. At this point it is important to indicate that the

IG.19. A rst compa rison betw een the TFB MP M and t he FTE.everal SNR were considered for the signal of Fig. 1. A bettererforma nce of the TFB MP M is observed in the entire SNR ra nge FIG. 21. The signal of Fig. 1 wa s conta minat ed with a SNR

0 dB. For long data records the FTE reaches the C RB.nder study.

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IG. 22. The signal was built with two components of equal FIG. 24. rmse of the rst est ima te, f O 1 , for a signa l composed byower and SNR 20 dB. The observat ion interval is character- tw o frequency components. The rst component ha s 10 timeszed by 12 samples and D t 0.25 ms. B etter performa nce and more power tha n the second one. SNR2 10 dB.igher spectral resolution are observed for the TFBMPM.

27, where the main lobe, centered in 1000 Hz (f 1 ),riterion a pplied t o consider w hether a n estimat e is is ma sking t he lobe corresponding to t he second fre-alid, w hen t he FTE is used, ha s consisted of being quency component, f 2 , a t 400 Hz. The Recta ngularble to distinguish the two frequency components. window was not considered in this s imulation be-

This idea is reected in Fig. 26, where the Kaiser cause, for some frequencies, t he sma ller frequencyw indow is used for the F TE, f 1 is 1400 Hz, f 2 is 400 component, f 2 , wa s hidden for side lobes, as is shownHz, and Du 45 . The opposite case is shown in Fig. in F ig. 28. The t w o frequency component s of th e sig-

nal for that example are f 1 1860 Hz a nd f 2 400Hz, and Du 177.3 .

IG. 23. A dual behavior to the one of Fig. 21 is derived. A SNRf 15 dB wa s chosen for tw o components of equa l power an d 300 FI G. 25. The same input da ta as in Fig . 23 but the es t imate

evaluated is f O 2 .z apa r t .

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FIG. 28. For some difference of frequencies, the second mainIG. 26. The tw o ma in lobes cent ered, respectively, at 1400 a ndlobe is h idden by s ide lobes w hen the Rectangular window is00 H z, can be dist inguish ed from ea ch other. The rst ma in lobeapplied and, consequently, the FTE will not work.as 10 times more power than the second one.

. CONCLUSIONS th e expense of spectr a l resolut ion. The Recta ngula r,Sta nda rd, and Ka iser windows have been chosen a sth e represent a tives for numerical simulat ion. It ha s

The objective of this pa per ha s been to present t he been shown that when TFBMPM works beyond aTFBMPM and the Fourier Transform Technique for certain threshold of SNR, it provides better varianceh e es t im a t ion of u n da m ped cis oid s in w h it e estimat es tha n the Fourier techniques, a lthough the

Gaussian noise. The accuracy of TFBMPM has been bias may be large. However, the root mean squaredrought out in the presence of noise and its variance error is less for the TFBMPM than for the Fourierompared to tha t of the Cra mer Rao B ound. Techniques w ith var ious w indow s.

It has been shown that applying windowing in theourier Transform provides unbiased estimates at

REFERENCES

1. Hua , Y., and Sa rkar, T. K. Mat rix pencil method for estimat -ing pa ra meters of exponentially da mped/undamped sinu-soids in noise. I EE E Tr ans. Acoust. Speech Si gnal Process.38, No. 5 (May 1990), 814 824.

2. Hua, Y. On techniques for estimating parameters of exponen- t ially damped / undamped sinusoids in noise. P h.D. disser ta-tion, Syr a cuse University, New York, Aug. 1988.

3. Golub, G. H., and Van Loan, C. F. M atr i x Computat i ons . TheJ ohn Hopkins Univ. Press, Baltimore, 1985.

4 . Hua, Y. , and Sarkar, T. K. Matr ix penci l method and i tsperforma nce. In Pr oc. ICA SSP-88, N ew Y ork, 1988, pp. 2476 2479.

5. J ohnson, D. H., an d Dudgeon, D. E. Array Signal Processing: Concepts an d Techn iqu es. Prentice Hall , Englewood Cliffs,NJ , 1994.

6 . Cramer, H . M athemat i ca l M ethods of Sta t i st ics. PrincetonUn iv. P ress, P rinceton, NJ , 1966.

7. Slepian, D . Estima tion of signal para meters in the presenceof noise. I RE Trans . I n form . Theory PGIT-3 (March 1954),68 89.

IG. 27. The rs t main lobe, a t 1000 Hz, i s masking the second 8. Brennan, L. E. Angular accuracy of phased array radar. I RE Trans. Antennas Propagation AP-9 (May 1961), 268 275.main lobe centered at 400 Hz.

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9. Hua, Y. , and Sarkar, T. K. A. Note on the Cramer Rao bound NY. He came t o Syracuse wi th a fel lowship from t he Marcel inBot Bn Foundat ion (Santa nder, S pain) . H e is current ly w orkingfor 2-d direction nding based on 2-d a rra y. I E E E Tr a n s .

Signal Process. 39, N o. 5 ( Ma y 1991 ) , 1215 1218. t ow a r d h is P h . D . d eg ree i n t h e U n iv er si ty of C a n t a b ria , s t ud yin gdifferent topics relat ed to applied electr omagnet ics. His r esearch0. Shanmugan, K. S. , and Breipohl , A. M. Random Signals : interests include signa l processing a nd electroma gnetic compat i-Detect ion, Es t im at ion and Data Analys is. Wiley, New York,bility.1988.

TAP AN KU MAR S ARK AR w as born in Ca lcutt a , India , on Au-1. Rife, D. C., an d Boorstyn, R. R. Multiple tone para meter esti-gust 2, 1948. He received the B. Tech. degree from the Indianma tion from discrete-time observat ions. Bell System Tech. J .Institute of Technology, Kharagpur, India, in 1969, the M.Sc.E.55, No. 9 (Nov. 1976), 1389 1410.degree from the University of New Brunswick, Fredericton, Can-2. IMSL, INC. IMSL Library. Problem-Solving Software Sys- ada , in 1971, and the M.S. an d P h.D. degrees from Syra cuse Uni-

tems for M athemat i ca l and Sta t i s t ical F ORTRAN Program - versity, Syra cuse, NY in 1975. Fr om 1975 to 1976 he wa s withming . Nov. 1984. the TACO Division of the G enera l Inst ruments Corpora tion. H e

3. Sarkar, T. K., Hu, F., Hua, Y., and Wicks, M. A real-t ime was with the Rochester Institute of Technology, Rochester, NY,signal processing technique for approximating a function by from 1976 to 1985. He was a Research Fel low at the Gordona sum of complex exponentials uti l izing the ma trix-pencil McKay Laboratory, H arvard Univers i ty, Ca mbridge, MA, f romapproach. Digital Signal Process. 4, (1994) , 127 140. 1977 to 1978. H e founded OHRN Ent erprises in 1985, which ha s

4. Ka y, S. M., a nd Mar ple, S. L., J r. Spectrum a na lysis A mod- been engaged in signal processing resear ch and development,ern perspective. Proc . IEEE 69, No. 11 (Nov. 1981), 1380 with several governmental and industr ia l organizat ions . He is1419. also a professor in the Departm ent of Electrical and Computer

Engineering, Syra cuse U niversity, Syr acuse, NY. His current re-5. Ma rple, S. L., J r. Di gi ta l Spect ra l Analys is wi th Appl i ca t ions.search interests deal with adaptive polarization processing andP rentice Ha ll, Englewood Cliffs, NJ , 1987.numerical solutions of operator equations arising in electromag-6. Ha rr is , F. J . On the use of w indows for harmonic ana lysisnet ics and s ignal process ing wi th applicat ion t o ra dar sys temwith the discrete Fourier transform. P roc . IEEE 66, No. 1design. He obta ined one of the best solution a wa rds in May

(J an 1978), 51 83. 1977 at the Rome Air Development Center (RADC ) Spectral Esti-7. Kuo, F. F., and Ka iser, J . F. System Analysis by Digital Com- ma tion Workshop. He ha s au thored or coau thored more tha n 154

puter. Wiley, N ew York, 1966. journal ar ticles and conference papers a nd ha s w ritten chaptersin eight books. Dr . S arka r is a registered professional engineerin the State of New York. He received the Best Paper Award oft he I EE E Tr ansact ions on El ect romagnet ic Compat ib i l i ty in 1979.He wa s an Associa te Editor for fea ture a rticles of the IE EE An ten -

J OSE ENRI QUE FE RNANDEZ DEL RIO wa s born in Sant on a , nas and Propagation Society N ewsletter, the Technical Programa nta bria, S pain, on December 28, 1965. He gra duat ed in 1992 Cha irman for the 1988 IEE E Antennas and P ropaga tion Societys the valedictorian of his class with a B.S. degree in Physics Interna tional Symposium and URSI Radio Science Meeting, andlectronics from the University of Ca nta bria, Sa nta nder, Spain. a n Associat e Edit or of the IEEE Transactions of Electromagnetic

n 1994 he received the M.S. degree in Electrical Engineering, Compat ib i l i ty. He w as an Associat e Editor of the Journal of Elec- l so f rom the Univers i ty of Cantabr ia . For two years he was a t romagnet ic Waves and Appl i ca t ions and on the editorial board

member of a research team of the University of Cant abria , where of the I n t erna t i onal Jou rna l of M ic rowave and M i l l im eter Wave e worked on POWERCAD, a project which is part of ESPRIT, Computer Aid ed Engin eer i ng. He has been a ppointed U.S. Re-ne of the resear ch progra ms sponsored by the E uropean Commu- search Counci l Representa t ive to many URSI G eneral Assem-ity. H is t ask consisted in modeling the inductive coupling a nd blies. He is the Cha irma n of the Int ercommission Working G rouphe radiated noise in switched mode power Supplies. From 1994 of Interna tional URSI on Time Domain Metrology. Dr. Sarka r iso 1995 he was a visiting scholar in the Department of Electrical a member of Sigma Xi and Interna tional Union of Radio Science

Commissions A and B .nd Computer Engineer ing a t Syracuse Univers i ty, Syracuse ,

comparision mpvsft

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