# Direction Of Arrival Doa Biology Essay

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Source direction of arrival estimation is one of the challenging problems in many applications. Such applications can be seen in wireless communications, radar, radio astronomy, sonar and navigation. The resolution of a source direction of arrival estimation can be enhanced by an array antenna system with innovative signal processing. Super resolution techniques take the advantage of array antenna structures to better process the incoming waves. These techniques also have the capability to identify the direction of multiple targets. This paper investigates performance of the DOA estimation algorithms namely; MUSIC and ESPRIT on the uniform linear array (ULA) in the presence of white noise. The performance of these DOA algorithms for a set of input parameters such as number of snapshots, number of array elements, signal-to-noise ratio are investigated. The simulation results showed that the resolution of the DOA techniques MUSIC and ESPRIT improves as number of snapshots, number of array elements, signal-to-noise ratio and separation angle between the two sources are increased.

Index Terms- Direction of Arrival (DOA); Multiple Signal Classification (MUSIC) algorithm; Estimation of Signal Parameters via Rotational Invariance Technique (ESPRIT); Array Signal Processing, ULA.

I. INTRODUCTION

In the last decades, accurate determination of a signal direction of arrival (DOA) has received considerable attention in radar system and communication of military and commercial applications. Wireless communications, radar, radio astronomy, sonar, navigation, tracking of various targets are a few examples of many possible applications. For example, in commercial applications it is necessary to identify the direction of an emergency cell phone call in order to dispatch a rescue team to the proper location. One example of defense applications it is to identify the direction of a possible treads [1]. In wireless mobile communication, the main objective of direction-of-arrival (DOA) estimation is to use the data received at the base-station sensor array to estimate the directions of the signals from the desired mobile users as well as the directions of interference signals. The results of DOA estimation are then used to adjust the weights of the adaptive beamformer, so that the radiated power is maximized towards

the desired users, and radiation nulls are placed in the directions of interference signals [2].

One of the most promising techniques for increasing the capacity in the third generation cellular is the adaptive array smart antenna. The smart antenna technology is based on antenna arrays where the radiation pattern is altered by adjusting the amplitude and relative phase on the different elements. If several transmitters are operating simultaneously, each source creates many multipath components at the receiver and hence receive array must be able to estimate the angles of arrival in order to decipher which emitters are present and what are their angular.

There are several methods to estimates the number of incidents plane waves on the antenna arrays and their angle of incidence. The various DOA estimation algorithms are Bartlett, Capon, Min-norm, MUSIC, and ESPRIT. But MUSIC and ESPRIT algorithms are high resolution and accurate methods which are widely used in the design of smart antennas. The popularity of MUSIC is more due to its accuracy and robust method. ESPRIT has two major problems related to its implementation and performance. First, the method requires a procedure to map the sub-band spatial frequency back to the full band. This manipulation is necessary because of the widening of the spatial frequency spacing. Second, the reduction of computational load in the singular value decomposition (SVD) is achieved at the expense of compromising the output SNR. In this paper, we concentrate the discussion on the application of estimating the DOA of multiple signals. The focuses in this study are on MUSIC and ESPRIT algorithm. Computer simulation programs using MATLAB were developed to evaluate the direction finding performance of an array processor [3].

This paper is organized as follows. Section II, describes receiving signal model. Section III, presents a description of the MUSIC and ESPRIT algorithm. Simulation results and discussion of MUSIC and ESPRIT are given in Section IV. Finally Section V presents the conclusion of the work.

II. SIGNAL MODEL

The following signal model is applicable for both MUSIC

and ESPRIT algorithms. The data model assumes that the

signal impinging upon an array of sensors to be narrow-band and emitted from a point source in the far field. Consider a number of plane waves from M narrow-band sources impinging from different angles ï±i , i = 1, 2â€¦, M, impinging

where x(t) and w(t) are assumed to be uncorrelated and w(t) is modeled as temporally white and zero-mean complex Gaussian process. Eq. 5 can be written in matrix form of size NÃ-K as:

into a uniform linear array (ULA) of N equi-spaced sensors, as shown in Fig. 1. At a particular instant of time t, t=1, 2â€¦ K,

X ï€½ A ïƒ- S ï€« W

(6)

where K is the total number of snapshots taken, the array output will consist of the signal plus noise components.

The signal vector x(t) can be defined as [4]:

M

where S=[s(1) â€¦ s(K)] is an MÃ-K matrix of source

waveforms and W=[w(1) â€¦w(K)] is an NÃ-K matrix of sensor noise. The spatial correlation matrix R of the observed signal vector x(t) can be defined as:

x(t ) ï€½ ïƒ¥ a(ï± m ) ïƒ- S m (t )

(1)

ï› ï€¨ ï€©ï€ ï€¨ ï€©H ï

mï€½1

where s(t) is an MÃ-1 vector of source waveforms, and for a

Rxx ï€½ E x t

Hï› ï€¨ ï€©ï€ ï€¨ ï€©ï€ ï€¨ ï€©H

ïƒ- x t

nH ï€¨ ï€©ï

(7)

ï› ï€¨ ï€©ï€ ï€¨ ï€©H ï

particular source at direction ï± from the array boresight; a(ï±) is an NÃ-1 vector referred to as the array response to that source or

Rxx ï€½ E A ï±

ïƒ- S t

ïƒ- S t

ïƒ- A ï±

ï€« E w t

ïƒ- w t

array steering vector for that direction. It is given by:

Rxx ï€½ Aï€¨ï± ï€©ïƒ- RSS ïƒ- A

ï€¨ï± ï€© ï€« RWW

(8)

a(ï± ) ï€½ ï›1

eï€ jï¦

...

eï€ j ( N ï€1)ï¦ ïT

(2)

ï€½ Aï€¨ï± ï€©ïƒ- RSS

ïƒ- AH ï€¨ï± ï€© ï€« ï³ 2 I

where T is the transposition operator, and ï¦ represents the electrical phase shift from element to element along the array. This can be defined by:

where Ïƒn2 is the variance of noise and I is the identity matrix.

For this signal model, the correlation matrix Rxx will have

MM signal eigenvalues, and N-M noise eigenvalues. Let Es be

ï¦ ï€½ (2ï° / ï¬ )d cosï±

(3)

the matrix constructed of the corresponding M signal eigenvectors ES=[e1 e2 â€¦ eM], and En be the matrix containing

where d is the element spacing and Î» is the wavelength of

the received signal. The signal vector x(t) of size NÃ-1 can be

the remaining N-M noise eigenvectors En=[eM+1 eM+2 â€¦ eN].

written as:

ï€½ ïƒ¥ ï¬ H ï€½

H ï€« ï³ 2 H

(9)

x(t ) ï€½ A(ï± ) ïƒ- S (t )

(4)

Rxx

m ï€½1

k ek ek

ES ES

n ES ES

where A (ï± ) ï€½ ï›a ï€¨ï± 1 ï€©... a ï€¨ï± M ï€©ï

is an NÃ-M matrix of

In real array measurements, the covariance matrices are

unknown and they can be estimated from a finite amount of measurement called snapshots. Therefore the natural estimate

steering vectors and S ï€¨t ï€© ï€½ ï›S 1 ï€¨t ï€©... S M ï€¨t ï€©ï is an NÃ-M matrix

of source vector. The model described in Eq.(4) can never explain the observed data; this is may be due to noise and

modeling errors. Therefore to account for these effects, an

of the correlation matrix or the sample covariance matrix is given by

KË† ïƒ¥ ï€¨ ï€© ï€¨ ï€©H

additive noise term w(t) is included. Hence the array output consists of the signal plus noise components, and it can be defined as:

RXX ï€½

Ë†

X

k ï€½1

K1

K

K ï€1

ïƒ- X K

H

xï€¨t ï€© ï€½ Aï€¨ï± ï€©ïƒ- S ï€¨t ï€© ï€« wï€¨t ï€©

Or

(5)

RXX ï€½ï€ ïƒ¥

k ï€½ 0

X k X k

(10)

ï±ï€ d sin ï±

ï±

1 2 3 M

d d

w0 w 1 w2 wM

ïƒ¥

where K is the number of samples or observation vectors, and X is the KÃ-M complex envelops matrix of M measured

ï¬ï€ signals.

III. DOA ESTIMATION ALGORITHMS

The DOA algorithms are classified as quadratic type and subspace type. The Barltett and Capon (Minimum Variance Distortionless Response) are quadratic type algorithms. The both methods are highly dependent on physical size of array aperture, which results in poor resolution and accuracy [4].

Fig. 1 A plane wave incident on a uniform linear array of N-equi-spaced sensors [4]

Subspace based DOA estimation method is based on the

eigen decomposition [5]. The subspace based DOA estimation algorithms MUSIC and ESPRIT provide high resolution, they are more accurate and not limited to physical size of

array aperture [2,5]. Capon and MUSIC algorithm performances is analyzed based on number of snapshots, number of users, user space distribution, number of array elements, and signal to noise ratio.

In the design of adaptive array smart antenna for mobile communication the performance of DOA estimation algorithm depends on many parameters such as number of mobile users and their space distribution, the number of array elements and their spacing, the number of signal samples and SNR [1].

A. MUSIC ALGORITHM

B. ESPRIT ALGORITHM

ESPRIT stands for Estimation of Signal Parameters via Rotational Invariance Techniques which is another subspace based DOA estimation algorithm. It does not involve an exhaustive search through all possible steering vectors to estimate DOA and dramatically reduces the computational and storage requirements compared to MUSIC. The goal of the ESPRIT technique is to exploit the rotational invariance in the signal subspace which is created by two arrays with a translational invariance structure [11]. ESPRIT assumes that

there are D < M narrowband sources centered at the centre

MUSIC is an acronym which stands for Multiple Signal

classification. It is high resolution subspace DOA technique

frequency

f 0 as shown in Fig. 2. ESPRIT further assumes

which gives the estimation of number of signals arrived, hence their direction of arrival. The algorithm is based on exploiting the eigenstructure of input covariance matrix. The incident signals are somewhat correlated creating non diagonal signal

multiple identical arrays called doublets and these arrays are

R

XXdisplaced translationally but not rotationally.

The steps of the ESPRIT algorithm are summarized as follows:

correlation matrix. The algorithm is used to describe experimental and theoretical techniques involved in determining the parameters of multiple wave fronts arriving at

an antenna array from measurements made on the signal

Step 1: Obtain an estimate measurements X.

Ë† XX of R XX from the

received at the array elements [7]. MUSIC deals with the decomposition of covariance matrix into two orthogonal matrices, i.e., signal-subspace and noise-subspace. Estimation of DOA is performed from one of these subspaces, assuming

Step 2: Perform eigen decomposition on RË† .

XXRË† ï€½ VïŒV H

(14)

that noise in each channel is highly uncorrelated. This makes the covariance matrix diagonal. The steps of the algorithm are summarized as follows:

where ïŒ=diag{ï¬0, ï¬1, â€¦ ï¬M-1} and E=diag{q0, q1, â€¦ qM-1} are the eigen values and eigenvectors, respectively.

Step 3: Using the multiplicity, K, of the smallest eigen-value

Step 1: Collect input samples X k ,

estimate the input covariance matrix

k ï€½ 0 ... N ï€ 1 and

ï¬min , estimate the number of signals LË† , as

LË† ï€½ M ï€ K .

K ï€1

Step

4: Obtain the signal subspace estimate

ï›

kRË† ï€½ 1 ïƒ¥ X

ïƒ- X H

(11)

Ë† Ë† ... Ë† ï

XX k k

k ï€½ 0

V S ï€½ V 0

V L ï€ 1

and decompose it into sub-array matrices,

0Ë† ïƒ©VË† ïƒ¹

XXStep 2: Perform eigen decomposition on RË†

VS ï€½ ïƒª Ë†

(15)

V

1

ïƒºïƒ« 1 ïƒ»

XXRË† E ï€½ EïŒ

(12)

Step5: Compute the eigen decomposition ï€¨ï¬ , ... ï¬

2 LË† ï€©

where

ïŒ ï€½ d iag ï» ï¬0 , ï¬1 , ... ï¬M ï€1 ï½ are the eigen values

ïƒ©VË† H ïƒ¹

and

E ï€½ d iag ï» q 0 , q 1 , ...q M ï€1 ï½

are the corresponding

V HV V ï€½ ïƒª Ë†

ïƒºï›V

V ïï€½ VïŒV

0Ë† H Ë†

0 Ë† H Ë† H

H (16)

01 01 1

XXeigenvectors of RË† .

ïƒ« 1 ïƒ»

Step 3: Estimate the number of signals LË† from the multiplicity

And partition V into LË† ï‚´ LË†

sub-matrices,

K, of the smallest eigenvalue ï¬ min as equation LË† ï€½ M ï€ K

ïƒ©V11

V ï€½ ïƒª

V12 ïƒ¹

ïƒº

V ,(17)

Step 4: Compute the MUSIC spectrum by the following Eq.

ïƒ«V21

V22 ïƒ»

PË† ï€¨ï± ï€©

AH ï€¨ï± ï€©Aï€¨ï± ï€©

Step 6: Calculate the eigen values of ï¹

K 12 22ï€ 1

ï€½ ï€V12 22

MUSIC ï€½ AH ï€¨ï± ï€©E E H Aï€¨ï± ï€©

(13)

ï†Ë† = eigen values of ï€¨ï€ V V ï€©, k ï€½ 0 , ... LË† ï€ 1

(18)

n n

Equation (18) indicates that if we are able to estimate eigen-

PStep 5: Find the LË† largest peaks of

estimates of the Direction -Of- Arrival.

Ë†

MUSIC

ï€¨ï± ï€© to obtain

value of which are diagonal elements of

DOA.

ï€© we can estimate

ï†

ï€ 1k

Fig.2. Principle of the ESPRIT algorithm

Step7: Estimate the Angle-Of-Arrival as

Case.3: MUSIC spectrum for varying number of snapshots

Figure 5 shows MUSIC spectrum as a function of the number of snapshots with K=100, K=500 and K=1000 and keeping all parameter constant. We can notice that by the increasing the number of snapshots peaks in the MUSIC spectrum become further sharper for higher number of snapshots e.g. with K=1000.

Case.4: MUSIC spectrum for varying the number of array sensor

The effect of increasing the number of array sensor on the performance of the MUSIC algorithm can be shown in Fig.6.

kï€1 ïƒ© argï€¨ï†Ë† ï€©ïƒ¹

ï±k ï€½ cos ïƒªï€ ïƒº

(19)

Small reduction in beamwidths and the noise level is observed. The spacing between the elements of the sensor array must be

ïƒ« ï¢d ïƒ»

increased resulting in a better resolution of the estimated peaks, as shown in Fig.7 for which d=0.5Î». Small peaks appeared in

As seen from the above discussion, ESPRIT eliminates the

search procedure inherent in most DOA estimation methods; ESPRIT produces the DOA estimates directly in terms of the eigen values.

IV. SIMULATION RESULTS AND PERFORMANCE EVALUATION

The MUSIC and ESPRIT techniques for DOA estimations are simulated using MATLAB tool. The performance of the algorithms has been analyzed by considering the effect of changing a number of parameters related to the signal environment as well as the antenna array. A uniform linear array with M elements has been considered in our simulation experiments and all inputs were made fixed when the effects of changing a parameter value was investigated. In these simulations, it is considered a linear array antenna formed by

10 elements that are evenly spaced with the distance of ï¬ / 2 . The noise is considered to be additive, having the 0.1 variance value. The simulation has been run for two signals coming

the response at angles of -500 and 650 in case of d=0.75Î» this is

due to grating lobes.

B. SIMULATION RESULTS OF ESPRIT ALGORITHM

Case.1: The effect of varying the SNR

The effect of varying the SNR, with higher values of SNR, the performance of ESPRIT algorithm will be better than lower values of SNR as shown in Fig. 8.

Case.2: The effect of varying number of elements:

ESPIRT algorithm has been tested for different number of array elements. The spectrum of the algorithm signals as shown in Fig.9 provides almost similar DOA estimates for different array elements with lesser variance than MUSIC algorithm.

Case.3: The effect of varying the horizontal angular separation between users:

1

2from different angles ï± ï€½ ï€250

and ï± ï€½ 350

for different

value of snapshots, SNR, and array elements. These two signals are considered to have equal amplitudes.

A. SIMULATION RESULTS OF MUSIC ALGORITHM

SNR=5dB SNR=15dB

SNR=25dB

Case.1: MUSIC spectrum for varying the values of SNR

The effect of changing the SNR with three different values (5, 15, and 25) dB is shown in Fig.3. It is clear that as SNR value increases, the ratio between the output peaks and the noise level at the output of the array increases proportionally.

Case.2: MUSIC spectrum for varying the horizontal angle separation

Figure 4 shows the effect of varying horizontal angle separation on MUSIC algorithm. Sharper peaks increases as the angle separation between signals increases, since MUSIC is a high resolution technique; it is capable to resolve the signal from two users.

ESPIRT algorithm has been tested for different angular

separation. From Fig.10 it is observed that for this algorithm DOA detection decreases as angular separation between arriving signals increases.

MUSIC Algorithm

80

70

60

Pseudo spectrum [dB]50

40

30

20

10

0

-10

-50 -40 -30 -20 -10 0 10 20 30 40 50

DOA [degrees]

Fig. 3. The effect of varying the SNR on MUSIC algorithm

MUSIC Algorithm

70

60

60

d=0.25 lambda

50 d=0.5 lambda d=0.75 lambda

MUSIC Algorithm

Pseudo spectrum [dB]40

40

30

30

20

20

10

Pseudo spectrum [dB]10

0 0

5 and 10 degrees

5 and 15 degrees

5 and 20 degrees

-10

-10 -5 0 5 10 15 20 25 30 35 40

DOA [degrees]

-10

-80 -60 -40 -20 0 20 40 60 80

DOA [degrees]

Fig. 4. The effect of varying the angular separation on the MUSIC algorithm

Fig.7. The effect of element spacing d on the MUSIC algorithm

K=100

K=500

K=1000

70

60

Pseudo spectrum [dB]50

40

30

20

10

0

-10

MUSIC Algorithm

Case.4: Effect of varying number of snapshots:

ESPIRT algorithm has been tested for different number of snapshots as shown in Fig.11. It can be seen that the spectrum and DOA estimates for this algorithm are almost close to the actual DOA even at lower number of snapshots. Therefore, it outperforms MUSIC technique.

Case 5: Element Spacing of the Sensor Array:

Figure 12 shows the ESPRIT for an element spacing of d=0.25Î», d=0.5Î» and d=0.75Î», respectively. When the elements of the sensor array are placed too close to each other, mutual coupling effects dominate resulting in inaccuracies in the estimated angles of arrival, as shown in Fig.12 for which

d=0.25Î». To overcome this problem, the spacing between the

-50 -40 -30 -20 -10 0 10 20 30 40 50

DOA [degrees]

Fig.5. The effect of varying the number of snapshot on the MUSIC algorithm

elements of the sensor array must be increased resulting in a better resolution of the estimated peaks, as shown in Fig.12 for which d=0.5Î».

MUSIC Algorithm

M=6 elements M=10 elements M=14elements

70

60

50

ESPRIT Algorithm

80

70 SNR=5dB SNR=15dB

Pseudo spectrum [dB]60 SNR=25dB

Pseudo spectrum [dB]40 50

30

40

20

30

10

20

0

10

-10

-20

-50 -40 -30 -20 -10 0 10 20 30 40 50

DOA [degrees]

Fig.6. The effect of varying the number of sensors in MUSIC algorithm

0

-50 -40 -30 -20 -10 0 10 20 30 40 50

DOA [degrees]

Fig.8. The effect of varying the SNR on ESPRIT algorithm.

ESPRIT Algorithm

70

ESPRIT Algorithm

60

M=6 elements

60 M=10 elements

M=14 elements

Pseudo spectrum [dB]50

Pseudo spectrum [dB]50 d=0.25 lambda d=0.5 lambda d=0.75 lambda

40

40

30

30

20

20

10

10

0

-50 -40 -30 -20 -10 0 10 20 30 40 50

DOA [degrees]

Fig.9. The effect of varying number of elements on ESPRIT algorithm

0

-80 -60 -40 -20 0 20 40 60 80

DOA [degrees]

Fig.12. The effect of element spacing d on the ESPRIT

ESPRIT Algorithm

70

60

Pseudo spectrum [dB]50

40

30

20

10

0

5 and 10 degrees

5 and 20 degrees

5 and 15 degrees

V. CONCLUSION

This paper presents results of direction of arrival estimation using MUSIC and ESPRIT algorithms. The two methods have greater resolution and accuracy than the other considered classical methods like Bartlett and CAPON. Extensive computer simulations were performed to demonstrate the effect of various parameters on the performance of the MUSIC and ESPRIT algorithms and their ability to resolve incoming signals accurately and efficiently. From the simulated results, it is observed that both investigated algorithms provide an accurate estimation of the DOA with improved resolution power than other classical DOA techniques. The simulation results show that their performance improves with more elements in the array, with large snapshots of signals and greater angular separation between the signals. These improvements are seen in form of the sharper peaks and a

-10 -5 0 5 10 15 20 25 30 35 40

DOA [degrees]

Fig.10. The effect of varying the horizontal angular separation between two users on the ESPRIT algorithm , the users located at 10 , 15 , and

20 degrees

ESPRIT Algorithm

70

60 K=100

K=500

K=1000

Pseudo spectrum [dB]50

40

30

20

10

0

-50 -40 -30 -20 -10 0 10 20 30 40 50

DOA [degrees]

Fig.11. The effect of varying the number of snapshot on the algorithm

smaller error in angle detection. The results obtained from

these two algorithms add new possibility of user separation through SDMA and can be widely used in the design of smart antenna system. The results also improve and accelerate the design of wireless networks.

VI. REFERENCES

[1] Godara L.C., "Application of Antenna Arrays to Mobile Communications Part-II: Beamforming and Direction-of- Arrival Consideration", In proceedings of IEEE, 85 (8), pp.

1195 - 1245, 2003

[2] J. Liberti and T. Rappaport, Smart Antennas for Wireless

Communications, Prentice Hall, 1999

[3] Shauerman, Ainur K. Shauerman, Alexander A.," Spectral- based algorithms of direction-of-arrival estimation for adaptive digital antenna.

[4] K. Al-Midfa, ''Investigation of Direction-of-Arrival

Algorithms''. Ph.D. Thesis, University of Bristol, UK, 2003.

[5] Tsoulos, G: "Smart Antennas for Mobile Communication Systems: Benefits and Challenges", IEE Electron. Comm. Eng. Journal, ll(2): 84-94, Apr. 1999

[6] Capon, J. "High-resolution frequency-wave number spectrum analysis," IEEE Proc, 57, 1408-1418, 1969

[7] R.O. Schmidt, "Multiple Emitter Location and Signal Parameter Estimation," IEEE Transactions on Antennas and Propagation, vol. AP- 34, issue 3, pp. 276-280, Mar. 1986.

[8] ROY, R. and Kailath, T., "ESPRIT - Estimation of Signal Parameters Via Rotational Invariance Techniques," IEEE Trans. On Acoustic, Speech, and Signal Processing, Vol.37, pp. 984-995, July 1986.

[9] John. F .Hanna," Application of Smart Antenna In Wireless communication," ,Ain Shams University ,July 2003.

[10] R.O.Schmidt, "Multiple Emitter Location and Signal Parameter Estimation," IEEE Transactions on Antennas and Propagation, vol. AP- 34, issue 3, pp. 276-280, Mar. 1986.

[11] Radich ,B.M. and Buckley ,K .M. "Effect of Source Number Under Estimation on MUSIC Algorithm, "IEEE Tran Signal Processing, vol.42,pp.2333-236,1994

[12] ROY,R. and Kailath , T. ,"ESPRIT -Estimation of Signal Parameters Via Rotational Invariance Techniques," IEEE Trans. On Acoustic, Speech, and Signal Processing, Vol.37 pp.984-995, July 1986.