gold-music: a variation on music to accurately determine peaks of the spectrum

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100 IEEE TRANS ANTENNA & PROPAGATION

gold-MUSIC: A Variation on MUSIC to AccuratelyDetermine Peaks of the Spectrum

Kaluri V. Rangarao E-mail: [email protected] Venkatanarasimhan E-mail: [email protected] of Electronics and Communication Engineering

Jawaharlal Nehru Technological UniversityHYDERABAD

INDIA

Abstract— This paper presents a new way of accuratelydetermining peaks of the MUSIC [1] spectrum, here consid-ered from the point of view of estimating the directions ofarrival (DOAs) of narrow-band signals. It can be used, withany smart antenna geometry and for any purpose whereMUSIC is applicable.

The MUSIC algorithm for DOA estimation evaluates theMUSIC spectrum for various angles and chooses the max-ima or peaks as the angles of arrival. The values obtaineddepend on the interval at which the spectrum is evaluated.The coarser the interval, the less accurate are the resultsin case of MUSIC. To improve accuracy and not depend onthe interval, Root-MUSIC [2], which involves finding theroots of a polynomial, is available. However, Root-MUSICis applicable, in its original form, only to uniform lineararrays (ULA).

The gold-MUSIC algorithm proposed in this paper is atwo-stage process. The first stage evaluates the objec-tive function at coarse intervals and determines peaks fol-lowed by an iterative approach based on gold-section uni-variate(GSU) minimization [3] to find accurate values ofthe peaks. If the number of peaks found so far is equal tothe number of estimated peaks, the algorithm stops withthis first stage. The second stage is an iterative step for fineresolution using finer intervals around the peaks found so farfor finding peaks that were missing in previous iterations.

This paper, also presents a method, based on a partition-ing algorithm for estimating the number of emitters.

The performance of gold-MUSIC is described, includingits advantages and comparison of time complexities for MU-SIC, Root-MUSIC and gold-MUSIC. The proposed algo-rithm gives good results even when the number of snap-shots is small. This gives it an additional computationaladvantage. It does not compromise on the resolving powerof MUSIC.

Index Terms: gold-MUSIC, Root-MUSIC, Directions ofArrival (DOAs), Gold Section Univariate Minimization,Smart Antenna, Sensor Arrays, Partitioning Algorithm,Resolution Technique.

I. Introduction

Directions of arrival for narrow-band signals have beencomputed using high and super-resolution algorithms suchas those discussed in [1], [2], [4], [5] and [6] over the last fewdecades. In this paper, we present a variation on MUSIC.In this method, which we call gold-MUSIC, the peaks ofthe MUSIC spectrum are evaluated quite accurately usinggold-section univariate(GSU) minimization.

GSU minimization [3], which is an iterative method has

been similarly and successfully used by the authors in de-lay estimation [7] and DOA estimation using spatial tunedfilter [8]. The spatial tuned filter approach [8] is based ona filter described in the book [9].

Stage I of gold-MUSIC finds the peaks in two passes.In the first pass, the spectrum is evaluated as in MUSICfor −90◦ to 90◦ using coarse intervals. This provides theintervals with left and right points within which accuratepeaks are located. The second pass, then estimates veryaccurate measures of the peak DOAs using gold-sectionunivariate minimization. If all the emitters are not foundin Stage I, Stage II finds them by repeating the two passesusing fine sweep angles around the peaks found so far.

II. MUSIC Spectrum

This section revises the concepts of MUSIC spectrum asused in [1]. Let M be the number of sensors in a smartantenna array. Let there be K snapshots at each sensor.For any k ∈ {1, . . . , K}, the kth snapshot at each sen-sor is for the same instant in time. Let Y represent theK × M matrix of snapshots. This matrix has complexvalues in general, representing in-phase and quadraturecomponents. The covariance matrix is given by:

S = YHY (1)

where H denotes Hermitian. The eigenanalysis of S yieldsthe eigenvalues L and the corresponding eigenvectors E.It has been shown that L will contain only positive realvalues as S is positive definite.

If there are D < M independent signals, M − D of theeigenvalues will ideally be 0 under no noise condition, but,will be close to 0 depending on the signal-to-noise ratio(SNR). After sorting the eigenvalues in L in ascendingorder, the M × (M − D) matrix EN , the matrix of theM − D eigenvectors corresponding to the M − D lowesteigenvalues is found. EN contains the noise eigenvectors.E can be written as:

E = {EN |ES} where ES are the signal eigenvectors.

The array manifold for a direction of arrival θ, is theM × 1 column vector a(θ). It depends on the geometry.



KALURI: GOLD-MUSIC: A VARIATION ON MUSIC TO ACCURATELY DETERMINE PEAKS OF THE SPECTRUM 101

For example, for a ULA, it is given by:

at(θ) = [1, e−jω, e−j2ω, . . . , e−j(M−1)ω ] (2)= [1, z−1, z−2, . . . , z−(M−1)] (3)

z = ejω and ω is given by:

ω = 2πd

λsin θ (4)

where d is the spacing between the ULA elements and λis the wavelength corresponding to the center frequencyof the narrow-band signal. The filtered vector VMUSIC

which is analogous to the output of the spatial tuned filteras described in [8] is here given by:

VMUSIC(θ) = at(θ)EN (5)

which is a 1 × (M − D) row vector. Then, the energy inthe filtered output is:

EMUSIC(θ) = VMUSIC(θ)VHMUSIC (θ) (6)

= at(θ)ENENHa∗(θ) (7)

For MUSIC, this energy is minimized, i.e., almost close tozero, when the angle θ corresponds to a DOA. Thus, theobjective function for maximization is:

JN (θ) =1

EMUSIC(θ)(8)

It is interesting to note that the evaluation of JN (θ) is lin-ear in θ while non-linear in ω unlike conventional frequencyresponse. The spectrum can be thought of as the inverseof the magnitude squared of the frequency response of thefilter given by Eq 5.

III. MUSIC, Root-MUSIC and gold-MUSIC

We begin with the signal snapshots Y, which is a K ×M matrix. Covariance matrix S is computed via Eq 1.The next step is eigenanalysis, getting the eigenvalues Land the corresponding eigenvectors E. In this paper, wedescribe a partitioning method in section IV on the setof the normalized eigenvalues, which gives us an estimateof D. We separate the eigenvectors into M − D noiseeigenvectors EN and the remaining D signal eigenvectorsES.

MUSIC, Root-MUSIC and gold-MUSIC differ after thisstep as depicted in Fig 1. In MUSIC, the spectrum JN (θ)(Eq 8) is evaluated for different angles and the peaks arechosen. Numerically, the estimate is limited by the evalu-ation interval. In Root-MUSIC, the complex polynomialsassociated with EN (Eq 5) are solved and the D most com-mon roots are treated as the DOA angles. gold-MUSIC isexplained in section V.

IV. Estimating the Number of Signals

We estimate the number of signals, D, as the numberof significant eigenvalues in L, i.e., the cluster of eigenval-ues which carries maximum mean energy. The size of thecluster provides an estimate of D.

Root-MUSIC

Solve Polynomialin EN. Find DCommon Roots

gold-MUSIC

Coarse JN (θ)SpectrumGSU Minimization

MUSIC

Evaluate JN (θ)SpectrumFind Peaks.

Partition Eigenvectors{EN |ES}

Estimate Emitters D

Eigenvalues LEigenvectors EEigenanalysis of S

Covariance S = YHY

Signal Snapshots Y

�

�

��

�

� ��

Fig. 1. Steps involved in MUSIC, Root-MUSIC and gold-MUSIC

A. Partitioning Algorithm

The partitioning method we propose has a number ofiterations starting from k = 1. After iteration k, the resultis a collection of clusters {p1, p2, ..., pk} which have beenformed and a remainder set Pk, which has to be brokendown into further clusters if possible. Each cluster pk isa set consisting of normalized eigenvalues of covariancematrix S. The algorithm is started with P0 in Eq 9 as theset of normalized eigenvalues for k = 1.

∀x ∈ Pk−1 pk = {x ≥ σk−1} and Pk = {x < σk−1} (9)

where σk−1 =

√∑(x − µ)2

n − 1∀x ∈ Pk−1

in which n = size[Pk−1] and µ = E(Pk−1)

The choice of the dynamic threshold, σk(second moment)is heuristic [10]. The recursive set of operations as given inEq 9 is performed till either set pk or Pk becomes a NULLset, in which k is an integer. In case pk is NULL, the setpk is replaced by Pk.

B. Numerical Example of the Method

A numerical example of how this algorithm works isdemonstrated below. Only for this purpose, narrow-bandsignals from three emitters arriving from 20◦, −35◦ and25◦ were chosen. The SNR was about −2 dB. The numberof snapshots was K = 50 and the number of sensors in thereceiving ULA was M = 20.

{P0} Initial Set of Normalized Eigenvalues0.018377 0.020537 0.024351 0.030716 0.0408480.045091 0.059093 0.060620 0.068971 0.0702510.088147 0.096373 0.104326 0.118996 0.1323630.151811 0.162140 0.233112 0.499889 1.000000Threshold σ0 = 0.226902




k = 1 size[P0] = 20 size[p1] = 3 and size[P1] = 17{p1} First Partition E[p1] = 0.577667 size = 30.233112 0.499889 1.000000{P1} First Remaining Part0.018377 0.020537 0.024351 0.030716 0.0408480.045091 0.059093 0.060620 0.068971 0.0702510.088147 0.096373 0.104326 0.118996 0.1323630.151811 0.162140Threshold σ1 = 0.045736

k = 2 size[P1] = 17 size[p2] = 11 and size[P2] = 6{p2} Second Partition E[p2] = 0.10118999 size = 110.059093 0.060620 0.068971 0.070251 0.0881470.096373 0.104326 0.118996 0.132363 0.1518110.162140{P2} Second Remaining Part0.018377 0.020537 0.024351 0.030716 0.0408480.045091Threshold σ2= 0.009065

k = 3 size[P2] = 6 size[p3] = 6 and size[P3] = NULL{p3} Third Partition E[p3] = 0.02998 size = 60.018377 0.020537 0.024351 0.030716 0.0408480.045091

Now, having formed the clusters, we choose D as the sizeof the cluster with maximum E[p]. In this case D = 3corresponding to size of p1.

2 4 6 8 10 12 14 16 180

0.1

0.2

0.3

0.4

0.5

0.6

0.7CLUSTER DIAGRAM Snap Shots = 15 at SNR = −2.1042

Estimated Emitters = 3

Str

engt

h

Clusters using Algorithm Eq 9

Fig. 2. Partitioning Algorithm

For a better visual understanding clusters are depictedin Fig 2. Ideally( SNR ≥ +10 dB ) there must be only twoclusters one with full strength (1.0) and second with (0.0).The reason for the pop-up of two other non-zero clustersis due to the presence of noise ( SNR about −2 dB ).

The estimation of number of emitters has been testedexhaustively over different angles in the [−90◦, 90◦] rangeand for various SNR values as found in Fig 6 in sectionVI.

V. DOA Estimation Using gold-MUSIC

The objective function JN (θ) as given by Eq (8) peaksat each DOA. The sharpness of JN (θ) is dependent onthree factors:

1. K the number of snap-shots2. M the number of ULA elements3. SNR Signal to Noise Ratio

The resolving power of the MUSIC algorithm is also re-lated to the above parameters. To combat poor SNR con-ditions parameters K and M are chosen to fit the reality.

A. gold-MUSIC Algorithm

The gold-MUISC algorithm has two stages:

1. Finding a number of emitters using a coarse scan in[−90◦, 90◦]. Let these be the initial D1 ≤ D DOAs.

2. Determining the remaining D−D1 angles using finerscanning.

The second part is not required if all the initial emittersfound are equal to the total number of estimated DOAs.

A.1 Stage I : Algorithm for Initial Angles

By evaluating at suitable points, intervals are foundwithin which the peaks, i.e., the DOAs are present. Then,by using Gold Section Univariate (GSU) minimizationtechnique peaks are located very accurately.

For demonstrating the gold-MUSIC algorithm, wechoose three emitters with relative strengths 1.0, 1.3 and1.2, arriving from 30◦, −10◦ and 26◦ respectively. In thefirst pass, peaks or energy clusters are found by evaluat-ing JN (θ) at suitable intervals in [−90◦, 90◦]. Fig 3(a)(toppart) shows how 2 peaks (of the red curve) have been foundin a MUSIC spectrum evaluated between −90◦ and 90◦ atevery 10◦ interval. The corresponding gold-MUSIC peaksfound are also marked in the same figure by blue stemswith a bubble.

GSU minimization method [3] finds the extreme val-ues of a univariate function accurately. The algorithmworks by successively narrowing the range of values in-side which the extremum is known to exist. To locate thepeaks of JN (θ), we need to consider −JN (θ) as our uni-modal function. We require 4 points which include thelimits of the interval in which the maximum is to be ob-tained. Let [θa, θb] represent the coarse interval such thatθa < θb for a possible peak. This interval is obtained forall possible peaks by performing a coarse sweep in the in-terval [−90◦, 90◦]. We choose intermediate points θc andθd such that θc = θa + g(θb − θa) and θd = θc + g(θb − θc)where g = 3−√

52 , called the Golden Ratio . The objec-

tive function, −JN (θ), is evaluated at θa ,θb, θc and θd. If−JN (θc) ≤ −JN (θd), θb takes the value of θa and θa takesthe value of θd in the next iteration. Otherwise, θa takesthe value of θc and θc takes the value of θd in the next it-eration. Typically a fixed number of iterations is used andthe minimum value is quickly obtained quite accuratelyresulting in DOA value. Using a simple logic this processis repeated till all peaks,i.e., DOA values are found.

In Fig 3(a), for each of the 2 peaks, θa and θb are chosen2 × 10 = 20 degrees apart, keeping the local maximumin between them. In the same figure, θa and θb pointto these positions for one of the peaks. For this figure,noise was present in the signal with SNR about −2 dB.The accurate angles are found corresponding about −10.43(corresponding to −10◦) and 28.67◦ (corresponding to 26◦




−100 −80 −60 −40 −20 0 20 40 60 80 1000

0.5

1

1.5

Nor

mal

ized

JN

Coarse: ULA 32 SNR −2.0593dB Samples/Cycle 5 Step 10 Snap Shots 15

gold−MUSIC peaks θa θ

b

20 22 24 26 28 30 32 34 36 380

0.5

1

1.5

DOA in degrees

Nor

mal

ized

JN

Fine: ULA 32 Snap Shots 400 Sweep 20 to 38 in Step 0.9

gold−MUSIC peaks

20 22 24 26 28 30 32 34 36 380

0.5

1

1.5

Nor

mal

ized

JN

Fine: ULA 32 Snap Shots 100 Sweep 20 to 38 in Step 0.9

gold−MUSIC peaks

−10.4315328.672252

26.23872329.723267

26.98689329.669132

UNABLE toRESOLVEblue color

Fig. 3. gold-MUSIC Spectrum

or 30◦). Even though the actual and estimated numberof emitters is 3, we find only two. This is due to thehigher sampling interval and lower snap-shots, which mayselect only one peak whereas several may be present. Closeresolution is obtained by the technique mentioned in sub-section A.2 below. If the number of peaks found is equalto the number of emitters, there is no need for furtherresolution.

A.2 Stage II: Resolution Technique

If two emitters are too close for the chosen sweep, i.e.,sampling angle, the above procedure may give less numberof peaks as it has been shown for the two angles around30◦. This is a single value of 28.67◦ (corresponding toeither one of 26◦ and 30◦). However, as in this case, therecould be two (or more) peaks such as θ1 and θ2 and onlyone of them, say θ1 will show up by gold-MUSIC usingcoarse scanning. However, here, we know apriori by usingthe partitioning algorithm of section IV that there can bemore (in this case 3) peaks.

To resolve these peaks, we repeat the gold-MUSIC al-gorithm near each of the 2 peaks, but, this time using afine sweep angle around on either side of the respectivepeak and an increased number of snapshots. We obtainthe close resolution as shown in Fig 3(b) for K = 100, giv-ing values about 26.24◦ (corresponding to original angle of26◦) and 29.72◦ (30◦).

For K = 400, where 27◦ was used instead of 26◦ asan original DOA, Fig 3(c) shows how we get a superiorresolution of about 26.99◦ (27◦)and 29.67◦ (30◦) due to thehigher number of snapshots. However, using K = 100 and27◦ instead of 26◦, the fine scan fails to resolve between 27◦

and 30◦. This is shown by the blue line in Fig 3(b). Thisvariation of the results with K shows how increasing thenumber of snapshots (from 100 to 400 in this case) givesmore resolution power with accurate results. However, italso shows that increasing K from 100 to 400 has improvedresolving power only from 4◦ to 3◦. Also, as Fig 4 shows,

increasing the number of snapshots improves resolution.The finer sweep angle used in both of these is about 0.9

degrees. This way gold-MUSIC does not compromise onthe resolution but is limited by the choice of the numberof sensors M , the number of snapshots K and the SNRconditions as in Root-MUSIC.

If the number of angles found so far is still less than D,the algorithm of section A.2 is repeated using still finersweep angles till all the D emitters are found.

B. Snapshots and Coarseness

As the number of snapshots, K is increased, peaks be-come sharper. This effect is depicted in Fig 4. Highercoarse scanning will tend to miss sharp and close peaks.To counter this it is preferable to use less number of snap-shots which impacts the resolving power. This is overcomeby using the fine sweep of stage-II as explained in sub-section A.2 above. Improved resolving ability can be seenin the Fig 4(b), with a higher number of snapshots (in thiscase 500).

−100 −80 −60 −40 −20 0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

Nor

mal

ized

JN

Sharpness Variations ULA 32 SNR −2.1232dB

20 25 30 35 40 45 500

0.2

0.4

0.6

0.8

1 ZOOM from 21 to 47 degrees

DOA in degrees

Nor

mal

ized

JN

Snapshots 15100500

Fig. 4. Sharpness Variations with Snapshots

A unique dual computational advantage is thus achievedby the gold-MUSIC method by lowering K and the numberof angles to evaluate the spectrum at the same time.

VI. Performance

gold-MUSIC has some advantages which are mentionedin this section. As mentioned in the sub-sections below,gold-MUSIC has quick convergence, so that objective func-tion needs to be evaluated overall less number of times.There is trade-off between sharpness of peaks and the num-ber of snapshots which affects the way the scanning inter-val of the first pass and the number of snapshots are tobe chosen. The time-complexity of gold-MUSIC is derivedand compared with MUSIC and Root-MUSIC algorithms.The accuracy of gold-MUSIC is shown for various SNRconditions in Fig 6.




0 5 10 15 20 25 30 35−15

−10

−5

0ULA 32 SNR −2.1042dB Samples/Cycle 5

1st D

OA

val

ue

0 5 10 15 20 25 30 3529

29.5

30

30.5

2nd D

OA

val

ue

0 5 10 15 20 25 30 3526.5

27

27.5

28

3rd D

OA

val

ue

Number of iterations

Fig. 5. Convergence of GSU Minimization

A. Convergence of gold-MUSIC

As can be seen from Fig 5, it does not take many iter-ations for gold-section to converge. The first part of thefigure is for −10 degrees and the remaining two below itcorrespond to finer angles around 26 and 30 degrees. Thefixed number of iterations used was 32. However, Fig 5shows the quick convergence for all the three angles, sothat about 10 iterations would have sufficed.

B. Comparison of Complexity

Till the isolation of the noise eigenvectors EN, Root-MUSIC and gold-MUSIC follow the same steps. Thetime-complexity is that of computing covariance and doingeigendecomposition. The detailed analysis of these stepsgive

∆ = O(KM2 + M3)

.After that, conventional Root-MUSIC solves M − D

polynomials of degree M −1 and finds D common roots ineach of the polynomials. The time complexity for polyno-mial rooting is O(Mn) where n >> 2 is a relatively largeinteger for known algorithms. So, the overall complexityof these steps is ∆ + O((M − D)Mn).

In gold-MUSIC, the objective function is evaluated onceper coarse spectrum point and a fixed number of timesper peak. The time complexity for objective functionevaluation is that of multiplying a 1 × M matrix withan M × (M − D) followed by multiplying 1 × (M − D)matrix with its Hermitian. That is O(M2) ignoring D.Considering evaluating the objective funtion as a unit ofcomputation (of complexity O(M2)), 180

CAunits are spent

when the spectrum is evaluated with coarse angle of CA

degrees. If all the D1 initial emitters are scanned finerwith a finer coarse angle FA degrees, the total numberof units added for this is D1 × CA

FA+ D2NF , where D2 is

the number of new peaks for which the objective func-tion is evaluated NF times in the gold-MUSIC for de-

termining the peaks. Hence the overall complexity is:O(KM2 + M3 + M2 × [ 180CA

+ D1NF + D2CA

FA+ D2NF ]).

Assuming NF , CA and FA are fixed values and lettingD = D1 + D2 as the total number of emitters, the overallasymptotic time-complexity is O(KM2 +M3 +DM2).LetCA degrees be the coarse scanning angle of the first passand let NF be the number of iterations used. We havedemonstrated gold-MUSIC with less number of snapshots.K = 15 was used in the 3-emitter example. This improvesthe computational speed of gold-MUSIC.

The following table summarizes the time-complexitiesof MUSIC, Root-MUSIC and gold-MUSIC:

Algorithm Complexity

MUSIC ∆ + O(DM2 × 180CA

)

Root-MUSIC ∆ + O((M − D)Mn)

gold-MUSIC ∆ + O(DM2)

If the number of emitters that show up initially are small,gold-MUSIC also has an advantage in less number of eval-uations of the objective function during fine resolution cor-responding to the less number of initial angles.

C. Accuracy

For evaluating the algorithm, consider θ as true DOAand the corresponding estimate by gold-MUSIC as θg. Letus define error Eg for a given ULA size, number of snap-shots K and SNR conditions as:

Eg(θ) = θg − θ ∀θ ∈ {−90◦ < θ < +90◦} (10)

The standard deviation of the error Eg(θ) is:

σg =

√∑θ Eg(θ)2

Nθ − 1(11)

where Nθ is the number of angles for which θg is estimated.

−80 −60 −40 −20 0 20 40 60 80−100

−50

0

50

100ULA 10 SNR −1.0056dB at 5 Samples/Cycle Step 15 Snap Shots 15

DOA in Degrees

Est

imat

ed D

OA

True DOA Estimated DOA σ

g 0.96357

−3 −2 −1 0 1 2 3 4 50

0.5

1

1.5

2

SNR in dB

σg

mean of σg (100) = 0.3850 .. mean of σ

g (15) = 0.9871

Snap Shots 100 Snap Shots 15

Fig. 6. Performance of Gold-MUSIC




The Fig 6a shows θg against the actual DOA angles fora single emitter case with DOA values ranging from −90◦

to 90◦ for a fixed value of SNR.Fig 6(b) depicts σg for K = 15 and K = 100 for various

SNR values in dB. Each value of σg for a given SNR isgenerated using Eq 10 and Eq 11 in which Nθ was chosenas 180. SNR values are varied from −3 dB to 5 dB insteps of 0.1 dB. As can be seen from the Fig 6(b), themean values are µ(σg)|K=100 = 0.3850o and µ(σg)|K=15 =0.9871o

The bias in DOA estimation is present and dependsupon the values of the number of snapshots K, numberof sensors M , the number of sources D and the SNR asdiscussed in [11].

Discussion of the asymptotic accuracy of Root-MUSICcan be found in [12]. Asymptotic efficiency of the MUSICalgorithm is given in [13].

[14] gives a detailed derivation of the Cramer-Raobounds for MUSIC and explains its relationship to theMaximum Likelihood method. The performance analysisis based on the minimization of the objective function. Itis clear from the paper that the asymptotic bound reachedby different forms of the MUSIC algorithm are the sameand depend on the objective function. Thus the Cramer-Rao bound is reached by gold-MUSIC also, under the con-ditions mentioned in that paper.

VII. Conclusions

We have presented a new variation on MUSIC calledgold-MUSIC. This method is useful for getting accurateresults for different kinds of array geometries where MU-SIC can be applied. gold-MUSIC makes use of gold-sectionunivariate minimization, which is an iterative method. Afixed number of iterations gives good results. Results pre-sented in this paper show that it is a very effective methodto get quite accurate results under different SNR condi-tions. We get resolution measured by the standard devi-ation σg better than 1◦ on the average and always within2◦ where SNR is varied from −3 dB to 5 dB for K = 15and better than 0.5◦ for K = 100. We have analyzedthe time complexity of gold-MUSIC and compared it withRoot-MUSIC.

We have presented a novel method for estimating thenumber of emitters, based on a partitioning approachwhich forms clusters of normalized eigenvalues. Thismethod works well under a wide range of SNR conditions.

The fact that gold-MUSIC works well with less numberof snapshots gives it a computational advantage.

VIII. Acknowledgements

The authors wish to thank Defence Electronics ResearchLaboratories, (DLRL), Defence Research and Develop-ment Organization, (DRDO), Ministry of Defence, Gov-ernment of India, Hyderabad, India for sponsoring the ini-tial parts of this research work. Special thanks are toDr. M. Laxminarayana and Dr. Bhoopathi, of DLRL.We thank Dr. L. Pratap Reddy and Dr. D. Srinivasa

Rao of the Electronics and Communication EngineeringDepartment, Jawaharlal Nehru Technological University,Hyderabad (JNTU H) for helping us by providing supportat the Department.

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Kaluri V. Ranga Rao:

Earned his BSEE from Andhra University in 1974 and MSCS from

the Indian Institute of Science, Bangalore in 1977. He received an

MSSE in 1991 from the US Naval Post Graduate School and a Ph.D.

from the Indian Institute of Technology (Madras) in 1994. He is a

senior member of the IEEE. He is the author of a text on DSP appli-

cations, published by John Wiley(UK). Currently he is a Professor

Emeritus at the Dept of ECE, JNTU-H, Hyderabad, and Director

at Voxta Communications, India.




Before that, he was the Chief Scientist of Tanla Solutions, VP

Engineering at Sarayu Softech, Chief Engineer at General Elec-

tric (Hyderabad), Vice President Satyam Computer Services, CEO

of GMR Vasavi Infotech and Project Director,Senior Scientist in

DRDO. Principal Investigator for DRDO project. He has been a

Visiting Professor at IITD, Visiting Scholar at Oklahoma State Uni-

versity, Visiting Professor at IIIT-H, and Visiting Scholar at CMU.

Prof Rangarao published 25 IEEE transaction and conference pa-

pers, and holds several US and Indian patents.

Shridhar Venkatanarasimhan: Shridhar received BSCS from

the Indian Institute of Technology (Mumbai) in 1991 and MSCS

from Rutgers University in 1993. Currently he is a Research Scholar

at JNTU-H.

gold-music: a variation on music to accurately determine peaks of the spectrum

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