This full-text paper was peer-reviewed and accepted to be presented at the IEEE ICCSP 2015 conference.

MUSIC and Improved MUSIC algorithm to Estimate Direction of Arrival

Pooja Gupta, S.P. Kar

Abstract- The DOA estimation in array signal processing is one

of the important and emerging research area. The effectiveness of

this direction of arrival estimation greatly determines the

performance of smart antennas. It works on the digitized output

from each sensor array. The estimation results for coherent

signals, broadband signals and multiple signals are of greater

consideration. Various information of the parameters relating to a

particular wave can be obtained by analyzing the incoming wave

that is received by a sensor or an array of sensors. The accuracy in the estimation of direction of arrival is very crucial in array signal processing. DOA estimation has vital applications in radar,

sonar, seismology, earthquake, astronomy, biomedicine and

communication. This paper presents an overview of direction of

arrival estimation using the MUSIC algorithm. This algorithm is

a peak search method to estimate the arrival angle. The paper

emphasizes on the factors that can affect the results of the

estimation process and describes the concept of estimating the

arrival angle. The MATLAB simulations shown here highlight

the factors that can improve accuracy. The factors include the

array element spacing, number of array elements, number of

snapshots and the signal incidence angle difference. Attempts are

made to bring about better resolution and contrast. As the

conventional MUSIC algorithm is found to have less efficiency for

coherent signals, better simulations using an improved MUSIC

algorithm in provided in this paper .

Index Terms- DOA (Direction of Arrival), ESPRIT

(Estimation of Signal Parameters via Rotational Invariance

Technique, MUSIC(Multiple Signal Classification), MUSIC

(Multiple Signal Classification), ULA (Uniform Linear Array).

I. INTRODUCTION

In signal processing a set of constant parameters upon which the received signals depends are continuously monitored. DOA estimation carried out using a single fixed antenna has limited resolution [1], as the physical size of the operating antenna is inversely proportional to the antenna main lobe beam width. It is not practically feasible to increase the size of a single antenna to obtain sharper beam width. An array of antenna sensors provides better performances in parameter estimation and signal reception. So we have to use an array of antennas to improve accuracy and resolution [2].

Pooja Gupta, Student, KIlT University, Bhubaneswar, Odisha (e-mail: poojainsky@gmail.com). S. P. Kar, Ass!. Professor ,KIlT University.

978-1-4 799-8081-9/15/$3l.00 © 2015 IEEE

Signal processing aims to process the signals that are received by the sensor array and then strengthen the useful signals by eliminating the noise signals and interference. Array signal processing (ASP) has vital applications in biomedicine, sonar, astronomy, seismic event prediction, wireless communication system [3], radar etc. Various algorithms like ESPRIT, MUSIC, WSF, MVDR, ML techniques [4] and others can be used for the estimation process.

The entire spatial spectrum is composed of target, observation and estimation stages [5]. It assumes that the signals are distributed in space is in all the directions. So the spatial spectrum of the signal can be exploited to obtain the Direction of Arrival. ESPRIT (Estimation of Signal Parameter via Rotational Invariance Technique) [6] and MUSIC (Multiple Signal Classification) [7] , [8] are the two widely used spectral estimation techniques which work on the principle of decomposition of Eigen values. These subspace based approaches depend on the covariance matrices of the signals. ESPRIT can be applied to only array structures with some peculiar geometry [9]. Therefore the MUSIC algorithm is the most classic and accepted parameter estimation technique that can be used for both uniform and non-uniform linear arrays. The conventional MUSIC estimation algorithm works on ULA where the array elements are placed in such a way that they satisfy the Nyquist sampling criteria. The design of non- uniform array [lO] is quite tedious and it requires various tools. It can compute the number of signals that are being incident on the sensor array, the strength of these signals and the direction i.e. the angle from which the signal are being incident. In this DOA estimation firstly the bearing space is sampled uniformly to get many discrete angles. Then we assume that the source and noise signals arrive from every small bearing angles and the estimation algorithm computes the angle of signal with stronger power. The simulations shown in the following sections makes it clear that the efficiency and resolution of the obtained spectrum using MUSIC algorithm can be improved by varying various parameters like the spacing between the array elements, number of array elements, number of snapshots and the signal incidence angle difference. An improved MUSIC algorithm which is efficient while detection of coherent signals is also discussed .

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This full-text paper was peer-reviewed and accepted to be presented at the IEEE ICCSP 2015 conference.

II. THE MATHEMATICAL MODEL

We assume that there are F narrowband source signals with the same center frequency fc which are impinging on an array of D sensor elements. These D elements are linearly spaced with equal distances between consecutive elements. Now we restrict the number of sensor to be greater than the number of signals being incident i.e. (D>F) for a better estimation result and suppose that the F signals are incident on the sensor with azimuth angles 9k, where k lies between 1 and F. The distances between consecutive elements of the array is maintained to be equal to the half of received signal wavelength [11]. This distance 'd' between the consecutive elements of the antenna array can be represented as a space matrix (dl d2 .... dD_1j

Direction of Normal

Oirection of Radiated source

Fig I. Structure of an Antenna Array

The figure above represents the structure of an antenna array where the distance is maintained to be dl=d2 =dD_1 .The array of antennas and the incoming signals lie in the same plane.

If ¢ (t) is taken as the phase of the source signal and aamp(t) is the amplitude for the same The representation of the source signal in its complex form is given as :

S ,ffl- hI i (f!: ( t) + �{t)) sOllree \ t/ - aamp\ l; .f} (1)

Considering the first element of the antenna array to be the reference, the signal that will be sensed by the pth array because of the kth signal source [12] is given as

-jz rrlip !J:' - 1 )s',., /I!::

Sk (t) .f} .l. where 1'S k 'S F (2)

Now calculating the total sensing at the pth array due to all the incoming signals is:

-j 2Tr!lp!J:' - 1) s". (} t Jp(t) = LK= 1 f} .l. S k (t) + tip (t)

and 1 'S p 'S.D. (3)

The (p_l)'h and the plh elements of the array are separated with a distance of dp , The k'h signal incident on the array is taken as Sk (t), np is used to represent the noise signal that is sensed by the plh array.

Now if the matrix representation of the received signal is given by J(t), the received signal information at the D array elements can be represented as :

J (t) = ASsol/relt) +N(t) (4)

The steering vector matrix is represented as 'A' and the total noise received by all the array elements is given by N(t). Now the received signal by the array elements can be represented in a matrix form as shown below:

J(t) = ( Jlt) J2(t) .... .Jdt)f (5)

Similarly the matrix form representation of the 'F' signal source being incident on the array of elements can be arranged as:

(6)

And the matrix form representation of the steering vector matrix [11] 'A' is:

(7)

This matrix forms the signal subspace. And

(8)

where k varies from 1 to F, representing the number of source signals being incident.

III. THE MUSIC ALGORITHM

Schmidt with his colleagues proposed the Multiple Signal Classification (MUSIC) algorithm in 1979[13]. The basic approach of this algorithm is the Eigen value decomposition of the received signal covariance matrix. As this algorithm takes uncorrelated noise into account, the generated covariance matrix is diagonal in nature. Here the signal and the noise subspaces are computed using the matrix algebra and are found to be orthogonal to each other. Therefore this algorithm exploits the orthogonality property to isolate the signal and noise subspaces.

Various estimation algorithms can be used to compute the angle of arrival, but this paper focuses on the most accepted and widely used MUSIC algorithm. The data covariance matrix forms the base of MUSIC algorithm. To find the direction of arrival we need to search through the entire steering vector matrix and then bring out those steering vectors that are exactly orthogonal.

The covariance matrix ' RJ ' for the received data 'J' is the expectance of the matrix with its hermitian equivalent.

RJ =E[JfJ (9)

Substituting the value of J from equation (5) we get:

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This full-text paper was peer-reviewed and accepted to be presented at the IEEE ICCSP 2015 conference.

RJ = E[(AS+N) (AS+N)H) = AE[SsH} AH + E[NNH} = ARJ AH + RN (10)

where RN is the noise correlation matrix and can be expressed as:

(11)

where I represents an unit matrix for the antenna array elements D*D. In a practical scenario, the signals are also associated with the noise, so now the computed correlation matrix along with noise can be represented as

where Rs represents the covariance matrix for the signal Ssource(t) and A is the steering vector matrix.

(12)

When this correlation matrix is decomposed it results in 'D' number of eigen values out of which the larger F eigen values corresponds to the signal sources and the remaining smaller D-F eigen values are related to the noise subspace. If Qs and QN denote the basis for signal and noise subspaces respectively, the resultant decomposed correlation matrix can be represented as :

(13)

As the MUSIC algorithm exploits the orthogonality relationship between the signal and noise subspaces, the following relation holds true:

(14)

The direction of arrival angle can be represented in terms of incident signal sources and the noise subspaces as shown in the following equation:

(15)

The above equation can be represented in terms of its reciprocal to obtain peaks in a spectral estimation plot:

(16)

The above equation results in high peaks when the direction of arrival of the signal source is exactly equal to that of 9. The F higher peaks are of greater power [13] and corresponds to the estimated arrival angle. If the element spacing between the antenna array is maintained to be half the received signal wavelength .The SNR and number of snapshot is kept to belOdB and 300 respectively. The signals considered are non-coherent and narrow banded. The number of antenna elements (D) is taken to be 10 and the noise here is considered to be additive white Gaussian noise.

The simulation for signal sources corresponding to arrival angles 20° and 70° is shown below:

·10 ...... ' ...... ' ....... : ....... � ...... ' ...... ...... , ....... , .. ... , .... .

·20 ...... , ...... : ....... � ....... � ...... , ...... ······ ,······r ... � .... . E : : : : i -30 -----.------- �-------l--- - - --l- -----. ------- --------------:--- ---�-----i 4 0 ······(···· ;·······i·······f······ :······ ······(·····r··· ·· r·····

, , , ·50 ...... ; .... . ; ....... : .. .... f .... ·

T .. )·, : \ .... · 1 .. · .... ' .... � .....

. : : : : , : ·60 ...... ,.... " 1 ' .. , ...... , .....

i .7.�0 Lo-. ..J.80�-' .6,L-O -4..J.O�-'.2,L-O --!0--'2,L-O --:'40:---'6� 0--!.80:----:C' 00 angle in degrees

fig 2, Spatial spectrum for MUSIC algorithm

The two independent spectrum peaks in the above graph correspond to the two signal sources and their arrival angles.

IV. FACTORS INFLUENCING THE EFFICIENCY OF MUSIC

ALGORITHM

The application environment [14] and the incoming signals are not the only parameters to influence the operation of MUSIC algorithm. The following parameters can vary the estimation efficiency.

A. The Spacing between Array Elements ('d'):

When we consider an array with 10 elements and vary the

spacing as IJ6, IJ2 and A respectively, we find the following

results keeping all the remaining parameters to be same as in

the above simulation.

no

20

30

�o

50

50 ��===c==�==����--�--=-�7-� ·100 ·80 -60 -40 -20 20 40 60 80 100 Fig 3. Efficiency result on varying the spacing between array elements.

Spectral functions for element spacing of IJ6,IJ2 and A is shown using solid, dashed and dash-dotted lines. It can be understood that we obtain better peaks when the distance increases from IJ6 to IJ2. But the spectrum losses efficiency when the element spacing is increased beyond IJ2. So the maximum allowable spacing in order to have the best efficiency is IJ2.

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This full-text paper was peer-reviewed and accepted to be presented at the IEEE ICCSP 20 IS conference.

B. Increasing Number 0/ Antenna Array

If the number of array elements is increased to 10, 20 and 30 respectively, we obtain the following results when all the other parameters are maintained to be same as in the first simulation:

·1 0 ...... , ... ... , ....... , ....... , ...... , ...... .. .... , ....... , .. ... , .... . , , , , . , . , , , , , . , . , , , , , . , . , , . , , . , . , , , , , . , . , -2:0 ······ ; ··- ··· ;···-··-!··-··-·,··-··+-··-· ·····-i·-··-+ . ... , -.. -.

, , , , . , . , , . , , . , . , E : : : : : : : : f -30 -··-·- j·-- ·-r·-·--j··-·--· j-··-··- t·-·--· "-"-'j'-'--'-:-'- .-. :-.

-- .

u " ' " ' " U5 -4 0 ------�------�-------:-------:-------;------ ------�-------:_-- --�-----

� : : : : : : : '

: .50 -.--.L ----L-.--L-----L-.--L-.-- ,--.--L----l !-, l-.--.

__ ! __ !=_!. __ �=_! __ /c_-=i_��.' \'t--&0 -.--.

-; -

-- - : ' , , -- ,.- ' -

-, -

-.--. ; -.

-- . , , , . , , . , . , , , , . , , . , . , , , -�� 0",,0-_-='80,,---_-='=60 ---4""0---:_2':- 0 --'----='20"--- 4-": 0---:&""0---:8':-0 --,-',00

angle in degree

Fig 4, Efficiency results on varying the number of antenna array.

The solid, dashed and dash-dotted lines give the simulation

result for 10, 20 and 30 antenna elements respectively. We

find better results on increasing the number of array elements.

C. Increasing the Number a/Snapshots We obtain the following result when the number of snapshot is increased to 10, 20 and 30 respectively. The solid, dashed and dashed-dotted lines give the simulations for 10, 20 and 30 number of snapshots respectively.

-10 ---.. - :-----. : .. ----., .... ---� .... --:------ -.-... :.----.. �.-, , . . , " I , , " " , , . . , " , , . . , " I , , " " , , . . , " , , . . , " ·20 ---.. - . -----. 0------.,.------" ..

----.

-..

--- ---..

-"----

..

-"

--

i :: 'iIII;L'A: , , , , , (..I ' '/' \ \;

-50

---�c'-�-�---!-------�-�--'--��/' ;\\---'---�y----+�---��0"= 0-_-,J8

'':-0 -·-�"='6

'0�-'-'- :....:�::J��--::"":::--:-"=' 2

'0=-'-::"":"'--::::J

'c:..- .=--' -='2

'0=--

' =-:J40:-=--==-

'-='6

'0=------='8�=------:-:',00

angle in degree

Fig 5. Enhancing efficiency by increasing the number of snapshots

We find that increasing the number of snapshots can increase the efficiency of the estimation algoritlun [lS].More the number of snapshots, more narrower are the peaks.

D. Increasing the Signal Incidence Angle Difference. We find the following result when the angle difference

between the incoming signals is increased to 10°, ISO and

20°.The solid, dashed and dashed-dotted lines give the result

for angle differences of S, IS and 20 degrees respectively. o.--,--,--,-�-,--.--,-_,_c_,-�

: : : : : : :: -10 ---·-- ' ----·· . --·---·!---··--r·---·- ' -··--- -

--.-- , .. ---.-,-. --. ,---.-

! �:�[IJj]jlljA[ -�� o""0-."""ao"---.6"' 0---:-4.L0 -."""20O-----'-0---:2.L0 --,-'40o------,L60,----:a"'"0 --,-J, 00

angle in degree

Fig 6: Enhancing efficiency by increasing the incidence angle difference

It can be seen that the estimation results are best when the difference between the incident angles is highest i.e. the when source signals are separated by an angle of 20°.

V. MUSIC AND IMPROVED MUSIC ALGORITHM

FOR DETECTION OF COHERENT SIGNALS

MUSIC algoritlun has advantages over other estimation

algoritluns because of the sharp needle spectrum peaks which

can efficiently estimate the independent source signals with

high precisions unlike the other estimation processes which are

limited with low precisions. It has many practical applications

as it provides unbiased estimation results. The MUSIC

algoritlun to estimate the direction has even proved to have

better performance in a multiple signal environment. MUSIC

algoritlun has better resolution, higher precision and accuracy

with multiple signals. But this algoritlun achieves high

resolution in DOA estimation [16] only when the signals being

incident on the sensor array are non-coherent. It losses

efficiency when the signals are coherent. Keeping all the

parameters same as those used for the conventional MUSIC in

all the previous simulations and considering the coherent

signals to be incident on the sensor arrays, we obtain the

following result.

. . . . . . . . �O_5 -----�------i-------�------�------�----- � ------(-----� -----� -----

: : : : : : : : E

-1 - ... - :---.. . :----... ,-----.,.----- : .--- , " ' - --i- - " ' - ,--..... ----.. f -1.5

-----�---

---�-------:---

----:---

--

�---- --:- -----t---- -�------+------'-' � -2 '" -2.5

-3:� 0"" 0---;;\80:;---;-6="'0---:-4"'"0 -.-;;\ 20:;------!;-0 --;;2"'"0 --,';400---;5 ="'0---;;8"'" 0 ----;-!" 00 angle in degree

Fig 7: MUISC algorithm for coherent signals

As the peaks obtain are not sharp and narrow, they fail to estimate the arrival angle for coherent signals. So we need to move towards an improved MUSIC algoritlun to meet the estimation requirements for coherent signals.

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This full-text paper was peer-reviewed and accepted to be presented at the IEEE ICCSP 2015 conference.

To improve the results for MUSIC algorithm [17], we simply introduce an identity transition matrix 'T' so that the new received signal matrix X is given as:

X= TJ* where J* is the complex conjugate of the original received signal matrix.

(17)

(18)

Now the matrices Rx and RJ can be summed up to obtain a reconstructed matrix 'R'.As the matrix are summed up they

will have the same noise subspaces.

R=RJ + Rx

R= ARsAH + T{ARsAHJ *T +2';1 (19)

(20)

The noise subspace obtained after decomposition of RJ is filtered out and the new noise subspace obtained by the characteristic decomposition of the resultant matrix R is used for the spatial spectrum construction and to obtain peaks.

The same simulation of coherent source signals using the improved MUSIC algorithm is shown below

·10 · · · · · ·:·· · ·· · 1· · ·· · ··:· ·· · ·· · :· · · · · ·:·· · ·· · · · · · · ·1 · ·· · ·· · :· ·

. , , " " m . . . . . . . i -20 ------t------i- - - - - --!- - - ----�------t------ ------i--- - ---�--� : , ' " " . 2 : : : : : : : g -30 - - - - - - f - - - - - - � -------:-- - - - - -� - - - - - - t - - - - - - - - - - - - �--- - - - -� --

2 : . . . . . . E 2 � -40 ...... ; .. . . . . , . . .. . .. , . . . . . . . , . . .... ; .. . . . . : . . . . . . , . . . . . . . , . .

� :

·50 ...... i .. . . · . ;. . .. . . + .. . . + . . . . ·i .. . . · ; . . . . ·; . . . . .. · f . . . . . . � .... , , , , . , , , , , , . , , , , , , . , , , , , , . , , " , ,

.60 L,---,l,----,L_ ---L._-,L-_..l..----,L---,l ,----,L_ ---L._...l ·100 ·80 ·60 -40 ·20 20 40 60 80 100

angle a/degree

Fig 7: MUISC algorithm for coherent signals

It can be seen that using the improved algorithm for direction of arrival estimation results in narrower peaks for coherent signals. Hence detection of coherent signals can be achieved satisfactorily by the using the improved MUSIC algorithm.

VI. CONCLUSION

The MUSIC uses the eigen values and eigen vectors of the signal and noises to estimate the direction of arrival of the incoming signals. It becomes easier to separate the signals from noise as the eigen vectors for signal and noise subspace are orthogonal to one another. It works efficiently when the signals that are being incident on the array of sensors are non­coherent. Efficiency of this estimation algorithm can be improved by increasing the inter element spacing, increasing the number of antenna sensors, number of snapshots and improving the incidence angle difference between the incoming signals. For coherent signals the conventional

MUSIC algorithm fails to obtain narrow and sharp peaks. An Improved version of the MUSIC algorithm as discussed in this paper can be implemented for coherent signals as well. This improved algorithm achieves sharp peaks and makes the estimation process much accurate.

ACKNOWLEDGMENT

It gives me great pleasure in acknowledging the support

and help of my guide Sambit Prasad Kar, KIlT University,

Odisha, India. I would love to thank my co-author Miss Ipsa

Pradhan and I am also thankful to Prof D.D. Seth for his

constant encouragement and support.

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