CHAPTER 1

CHAPTER 1

DIRECTION OF ARRIVAL ESTIMATION

1.1 Back Ground Information:The direction of arrival determines the direction in which signals arrives on array of sensors and has a large number of applications. It determines the direction in which user is located based upon the data received from the sensors.

There are a number of algorithms designed for the estimation of direction of arrival ,some of them are MUSIC (multiple user signal classification),ESPRIT(estimation of signal parameters via rotational invariance techniques), Spectral MUSIC, Root-MUSIC, Constrained MUSIC, Beam Space MUSIC,ESPRIT Algorithm, Minimum Norm Method, CLOSEST Method, Weighted Subspace Fitting Method and others. Comparison of MUSIC with ESPRIT

We have designed an application by using four mics to determine the direction of arrival of the user speaking in any one of the mics. First of all analogue signal is passed on to the DSP kit. It determines the direction of the user using MUSIC algorithm . All the calculations are performed on the DSP kit while the graphs are displayed on Matlab.

1.2 Problem:

The direction of arrival estimation for multiple signals impinging on array of sensors has attracted a lot of attention in literature, due to its numerous application in radar ,sonar, communication and so on[1].Direction of arrival estimation is an important and basic technique not only for wireless communication system but also for the audio/speech processing systems[2,3].

It has been observed that there is a one-on-one relationship between the direction of a signal arriving onto a sensor and the associated received steering vector 5. There should be a method to invert the relationship and find the direction of the signal form the signal that it is receiving. An antenna array therefore should be able to provide direction of arrival estimation.

Direction of Arrival Estimation

Figure 1.1[6]

Fig. 1.1 shows the problem setup. A number of (M) signals arrive on a linear, equispaced, array with N number of elements, each with direction Φ i. The goal of DOA estimation is to utilize the data received at the array to estimate Φi, i = 1, . . .M. It is generally assumed that number of signals is less than the number of elements (M < N).

The objective is to estimate from the measurements a set of constant parameters upon which the received signals depend. Data from the array of sensor is calculated and to locate point sources assumed to be radiating energy that is detectable by the sensors. All the transmitted signals contribute to the maximization of its performance with respect to recovering the signal of interest and suppressing any present interfering signals. The same problem of determining direction of arrival [DOA] of impinging wavefronts , given the set of signals received at an antenna array from multiple emitters, arises also in a number of radar, sonar and mobile communication systems[4].

The estimation is difficult because there are unknown number of signals, with un known amplitudes and arriving from unknown directions .And the signals received by the array always has noise (additive white gaussian noise).

1 A steering vector represents the set of phase delays a plane wave experiences, evaluated at a set of array elements (antennas). The phases are specified with respect to an arbitrary origin.

2

Real Time Implementation of Direction of Arrival Estimation

Figure 1.2[6]

A number of methods exist for DOA estimation, some of them are shown in the figure 1.2.

1.3 Definition of Direction of Arrival (DOA):

Direction of arrival determines the direction form which a wave arrives at a point where usually a set of sensors is located. These set of sensors forms what is called an array.

Figure 1.3

1.4 Signal Model:

Let us consider an N-element uniform linear array (ULA) of microphones and a sound source in the far field of the array as shown in the figure 1.3. That means the distance between array and the sound source is much greater than the distance between the elements of the array. Under this

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d

tfjM

oetmtx 21 )()(

)(22 )()( tfj

Moetmtx

))1((2))1(()( ntfjMn

oentmtx


assumption, we can approximate the spherical wavefront 2 that emanates from the source as a plane wavefront2. Thus the signal reaching each of the elements can be assumed to be parallel to each other. The direction perpendicular to the array is called the broadside direction or simply the broadside of the array. All DOA's will be measured with respect to this direction. Where the signals from each element are multiplied by a complex weight and summed to form the array output[6].

Figure 1.4

Signals from each element is multiplied by a complex weight and added to form array out put. It follows from the figure that an expression from the array output is given by[5]:

Y(t)=wH x(t) (1.1)

2 in physics, a wavefront is the locus (a line, or, in a wave propagating in 3 dimensions, a surface) of points having the same phase.

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Where W denotes the weights of the array system using vector notation as

W=[w1,w2,w3,…,wN] (1.2)

And signals induced on all elements as

X(t)=[x1(t),x2(t),…,xN(t)] (1.3)

To start, we have considered M1 as the reference microphone(figure:1.4). Let the signal incident on M1 be s(t). Then the signal incident on the second microphone (M2) travels an additional distance of dsinθ as compared to the signal incident on the first microphone M1. The signal incident on M2 is a time-delayed version of s(t) and this delay is equal to τ =d sin θ/ v , where v is the velocity of sound (340 m/s)[5].

To summarize, the signals picked up by the array at each of the microphones are given below:

xM1 = s (t ) (1.4)

xM2 = s (t − τ 21)

xM3 = s (t − τ 31)

.

.

.

.

xMn = s (t − τ n1)

The signal induced on the first microphone M1 due to the Kth Source is normally expressed in complex notation as:

(1.5)

With mk (t) denoting the complex modulating function and fo denoting the carrier frequency. Modulation function used reflects the modulation being used in the system. [5].

These expressions are based upon assumption of narrow band array signal processing, which takes the assumption that the bandwidth of the signal and the array dimensions are narrow enough

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s( t )=mk (t ) ej2 πf o t


so that the modulating function stays constant during τ seconds, that is the approximation m(t)=m(t- τ) holds[5].

.

.

Hence the signal incident over ULA can be modeled as:

x(t)= A(θ)s(t)+n(t) (1.6)

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xM 2( t )=x1( t ) e− j2 π d sin θ

λo

xM 1 ( t )=m( t ) ej 2 πf o t

xMn ( t )=x1( t ) e− j2 π (n−1)d sin θ

λo


A (θ )= ¿ [ 1 ¿ ] [e− j 2πdλ o

sinθ ¿] [e− j2π2dλo

sin θ ¿ ] [⋮¿ ] ¿¿

¿

(1.7)

where A(θ) Steering vector1 is n-dimensional Complex vector containing responses of all n elements of the array to a narrowband source. It is also considered as the array response vector, it measures the response of the array due to the source under consideration. [5].

Spatial aliasing occurs when the phase delay between two signals increases beyond π, at the frequency of interest. Due to which time delays are interpreted wrongly, which results in wrong DOA estimation. Thus we have the condition d ≤ λ/2 , which means that the distance between two adjacent elements should not accede half of the smallest wavelength. When this condition is satisfied, spatial aliasing can be avoided and correct DOA estimates can be achieved [5].

1.5 Steering vector representation:

Steering vector is an N-dimensional complex vector containing responses of all N elements of the array to a narrowband (it refers to the situation where the bandwidth of the signal does not exceed the frequency interval over which two frequencies are likely to experience comparable fading) [1]

source of unit power. Let Sk denote the steering vector associated with the kth source. For an array of identical elements it can be represented as[5]:

Sk =[exp(j2*pi*fo τ1 (Фk,Фk)),………………,exp(j2*pi*fo τn (Фk,Фk))]T (1.8)

NOTE: First element of the array is at the origin (point of reference),so τ1 (Фk,Фk)=0,hence first element of the steering vector is equal to unity. As the response of the array vary according to the direction, a steering vector is associated with each directional source, uniqueness of this association depends upon array geometry. For a linear array of equally spaced elements with element spacing greater then half wavelength, the steering vector for every direction is unique[5].

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The signal vector can be compactly represented as:

X(t)=∑K=1

M

mk (t ) sk+n( t) (1.9)

Where n(t) is the noise vector represented as

n(t)=[n1(t),n2(t),…,nN(t)]T (1.10)

noise on different elements is also assumed to be uncorrelated that is

E[nk (t )n1(t)]={ 0 1≠kσn

2 1=k (1.11)

substituting value of x(t) in equation (1.1),it takes the form

y(t)=∑K=1

M

mk (t )wH sk+n ( t )wH (1.12)

substituting the array correlation matrix definition leads to the following expression for the array correlation matrix(a matrix giving correlation between all pairs of data sets).

R=E[(∑K=1

M

mk (t ) sk+n( t) )(∑K=1

M

mk (t ) sk+n( t) )H ] (1.13)

=E[(∑K=1

M

mk ( t ) sk ¿¿¿]+E[n(t)nH (t) ]+ (1.14) [(

∑K=1

M

mk ( t ) sk ¿nH ( t)¿+¿ E[n(t)∑

K=1

M

mk ( t ) sk ¿¿¿H ]

The first term on the right hand side simplifies to

E[(∑K=1

M

mk ( t ) sk ¿¿¿]= ∑K=1

M

E[m¿¿k (t )m¿ (t)¿]S K S1H ¿¿ (1.15)

When sources are uncorrelated ,

E [m1(t)mk¿ (t)]=¿ { 01≠k

Pk1=k (1.16)

Where pk denotes the power of the kth source measured at one end of the element of the array. Where pk is the variance of the complex modulating function mk(t) when modulated as a zero mean low pas random process. Hence for the uncorrelated sources it becomes [5]:

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E[(∑K=1

M

mk ( t ) sk ¿¿¿]=∑K=1

M

Pk Sk SkH (1.17)

The fact that the directional sources and the white noise are uncorrelated results in the third and fourth terms on the R.H.S of the equation (1.14) to be identical to zero. From equation (1.11), the

second term simplifies to σ n2 I, where I denotes the identity matrix, this along with equation

(1.17)gives the following relationship, when the directional sources are uncorrelated[5]:

R=∑k=1

M

Pk Sk SkH+σ n

2 I (1.18)

Where in equation I is the identity matrix and σ n2 I denotes the component of array correlation

matrix due to random noise, that is

Rn=σ n2 I (1.19)

Let Sodenote t h esteering vector associatedwit h t he signal sourceof power P s . t h en array

corr elation¿due to signal source is given by:

R s=PsSoSoH

(1.20)

Similarly, the array correlation matrix due to a interference of power is given by

R I=P I S I S IH

(1.21)

Where S I denotes the steering vector associated with the interference.

Using matrix notation, the correlation matrix R may be expressed in the following compact form:

R=ASAH+σ n2 I

(1.22)

Where the columns of the matrix NXM matrix A are made up of steering vectors, that is

A=[S1 , S2 ,… ,SM]

(1.23)

And MXM matrix S denotes the source correlation, for uncorrelated sources, it is a diagonal matrix with[5]

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Sij={Pii= j0 i≠ j

(1.24)

1.6 Eigen value decomposition:

Sometimes it is useful to express the array correlation matrix in terms of its eigen values(A vector that is not rotated under a given linear transformation; a left or right eigenvector depending on context; A right eigenvector; a nonzero vector x such that, for a particular matrix A, Ax = λx for some scalar λ which is its eigen value and an eigen value of the matrix) and their associated eigen vectors. There can be two types of eigen values in the array correlation matrix, when there are uncorrelated directional sources and when there is uncorrelated white noise.

Eigen values in one set are of equal value. Their value does not depend on directional sources and is equal to the variance of white noise. In the second set the number of eigen values are equal to directional sources which are equal to the number of sources. Each eigen value of this set is associated with a directional source and their values changes with the change in the power of the sources. Their values are bigger than those associated with the white noise. Some times these are referred to as signal eigen values and the first set are referred to as noise eigen values. Hence a correlation matrix of L elements immersed on M correlated sources and white noise has M signal eigen values and L-M noise eigen values[5].

Denoting the L eigen values o the array correlation matrix in descending order by λ, l=1,2,3,….,L and their corresponding unit-norm eigenvectors by U1, l=1,2,3,…,L the matrix takes the following[5]:

R=Q⋏QH (1.28)

With a diagonal matrix

⋏=λ₁ 0 00 0 00 0 λ₁

(1.29)

Q=[U1,…,UL] (1.30)

Which is also referred as the spectral decomposition of the array correlation matrix. Since eigenvectors form an orthonormal set,

QQH =I

And

QHQ=I

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Thus,

QH=Q-1 (1.31)

The orthonormal property o the eigenvectors leads to the following expression for the array correlation matrix[5]:

R=∑l=1

M

λ₁U ₁U ₁ +ͪ +σn2 I

(1.32)

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CHAPTER 1

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