chapter 2

CHAPTER 2

MUSIC ALGORITHM (MULTIPLE USER SIGNAL CLASSIFICATION

ALGORITHM)

2.1 Introduction:

There have been several solution to the problem including maximum likelihood (ML) of Capon (1969) and Burg’s maximum entropy method (ME) although successful and implemented these methods have fundamental limitations especially bias and sensitivity in parameter estimates largely because they use incorrect method ( i.e AR 1 instead of the specialized ARMA2) of the measurements. Pisarenko (1973) was one of the first in exploiting the structure of data model doing so in context of estimation of parameters of cissoids(a curve generated from two given curves) in additive noise using a covariance approach. Schmidt (1977) and Bienvenu (1979) were the first to correctly exploit the measurement model in the case of sensor arrays of arbitrary form ,Schmidt in particular ,accomplished this by first deriving a complete geometric solution in the absence of noise, then cleverly extending the geometric concepts to obtain a reasonable approximated solution in the presence of noise. The resulting algorithm was called MUSIC(Multiple Signal Classification)[7 ].

Other eigen structure methods include ESPIRIT (estimation of signal parameters via rotational invariance techniques)method, minimum norm methods and the weighted subspace fitting method.

1 In statistics and signal processing, an autoregressive (AR) model is a type of random process which is often used to model and predict various types of natural and social phenomena.

2 In statistics and signal processing, autoregressive moving average (ARMA) models, sometimes called Box-Jenkins models after the iterative Box-Jenkins methodology usually used to estimate them, are typically applied to auto correlated time series data. The ARMA model is a tool for understanding and, perhaps, predicting future values in this series. The model consists of two parts, an autoregressive (AR) part and a moving average (MA) part. The model is usually then referred to as the ARMA(p, q) model where p is the order of the autoregressive part and q is the order of the moving average part .

11

MUSIC (Multiple User Signal Classification) Algorithm

Figure: 2.1 antenna array system

Antenna resolution properties have been nvestigated by a large number of researchers to compare the DOA estimation and beam forming methods to different array geometries. In detailed performance evaluation based upon hundred of evaluation it has been declared by MIT’s Lincoln laboratory , has declare MUSIC as the most promising and leading algorithm for detailed study and hardware implementation amongst high resolution algorithms then available.

The popularity of MUSIC is due to its generality. For example it is applicable to arrays of any dimension and known configuration and response and can be used to estimate various parameters per source. MUSIC requires priori knowledge of second order spatial statistics of the background noise and interference field[4].

2.2 Implementation of MUSIC Algorithm:

The MUSIC algorithm was developed by Schmidt by noting that the desired signal array response is orthogonal to the noise subspace. The signal and noise subspace are first identified using eigen decomposition of the received signal covariance matrix. Following , the MUSIC spatial spectrum is computed from which the DOA’s are estimated. Inside the algorithm,we first define the general array to be the set:

A={a(ѳi): ѳi Є Ѳ (2.1)

For some region Ѳ of interest in the DOA space .The array manifold 3 is assumed unambiguous and known for all values of angle either analytically or through some calibration procedure. The objective is to apply appropriate methods to the received signals so as to extract region of ѳ out of

range of Ѳ[4].

If noise was absent in equation n(t)=a(ѳ) the observation could be confined entirely to M-dimensional subspace defined by the span of A(ѳ).For the no noise case ,it is simply a matter of finding unique elements of A that intersects with this subspace. A different approach is necessary in the case when noise is present since observation becomes “full rank” .The approach of MUSIC and other subspace based algorithms is to estimate dominant subspace of the

12

Real Time Implementation of Direction of Arrival Estimation

observations and then find the elements of A(ѳ) that are in some sense closest to this subspace[4].

The subspace estimation step is typically achieved by eigen decomposition of the auto covariance matrix of the received data R. For MUSIC to be applicable, the emitter covariance is required to be full-rank, i.e that M’=M. The eigen vectors can assumed to be forming an orthonormal basis 4

i.e QQH=QHQ. Once the subspaces are determined the DOA of the desired signals can be calculated over the spectrum region of interest using

PM(ѳ)= (a*(ѳ) a(ѳ))/( a*(ѳ)EN EN* a(ѳ)) [7] (2.2)

Number of signals that an array can detect depends upon the number of elements . It has been verified that N elements array can detect upto N-1 uncorrelated signals. [4].

2.3 Simulation Results:

To demonstrate the efficiency of the algorithm we choose a ULA(uniform linear array) and different parameter variations.

2.3.1-Variation of Number of Antenna Elements:

There are four sources located in far field of the array with ѳ1= -5°,ѳ2= 0°,ѳ3= 5° and ѳ4=10°. The spacing between two elements, d=λ/2. The number of snap shots (number of samples) is 1000. The result of the simulation is given

Figure 2.1 The number of elements is 8. Figure 2.2 The number of elements is 16

3 array manifold is a set array response vectors corresponding to all possible DOA

4In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module.

13


Figure 2.3 The number of elements is 32. Figure 2.4 The number of elements is 64.

2.3.2-Changing the Spacing between Elements:

There are four sources located in far field of the array with ѳ1= -5°,ѳ2= 0°,ѳ3= 5° and ѳ4=10°. The

antenna array has 8 elements. The number of snap shots (number of samples) is 1000.

Figure 2.5 d= λ/4. Figure 2.6 d= λ/5.

14


Figure 2.7 d= d= λ/6. Figure 2.8 d= λ/7.

Figure 2.9 d= λ/8. Figure 2.10 d= λ/9

Figure 2.11 d= λ/2. Figure 2.12 d= 0.8 λ.

15


Figure 2.13 d= 1 λ. Figure 2.14 d= 2 λ.

2.3.3-Changing Frequency:

There are four sources located in far field of the array with ѳ1= -5°,ѳ2= 0°,ѳ3= 5° and ѳ4=10°.The antenna array has 8 elements and the spacing between two elements ,d= 0.5 λ. The number of snap shots is 1000.

Figure 2.15 f= 1 GHz. Figure 2.16 f= 1MHz.

λ=v/f; λ=v/f

λ =(340)/(1000); λ=340/(1x10⁶)

λ=0.340m λ=3.4x10-4m

16


2.4 CONCLUSION:

From the simulation results discussed:

Greater the number of elements better the response of an antenna . Best performance at an antenna spacing of λ/2. If the spacing is increased beyond λ/2 alaising occurs. If the spacing is less than λ/2 the performance is not good. Greater frequency smaller the value of λ and hence better response.

17

chapter 2

Documents