Array Processing and MIMO Systems Multiple Signal Classification and Interference Rejection Techniques PGEE11124 Coursework Due March 4th 2021 Aims: To investigate the use of an array of sensors for detecting and estimating multiple signals and re- jecting in-band interferences. Equipment: PC and Matlab Introduction and Motivation By distributing a number of receiving elements (from now on called sensors) in a 3-dimensional Cartesian space, an array is formed; the region over which the sensors are distributed is called the aperture of the array. The general array signal processing problem is the obtaining of information about a signal environment from the waveforms received at the array elements, where the signal environment consists of a number of emitting sources plus noise. These sources, in the case of radar- based systems, are often targets which either reflect the transmitted signal (as in active radars) or emit their own signals (as in passive radars). Other typical examples of applications of arrays of sensors are: • radio telescope where the sensors are antennas and the emitting sources are radio sources; • sonar where the sensors are hydrophones and the emitting sources are ships (ship noise); • geophysics where the sensors are seismometers and the sources are earthquakes, etc. An important topic in the array signal processing problem is concerned with interference rejection. Since the emitting sources are distributed in space the array can perform both spatial and temporal filtering in order to optimize the reception of a signal from a desired source (desired signal). This can be achieved by using an array-pattern-network so as to place relative high gain in those desired signal directions and place nulls in the remaining unwanted interfering source directions. Other array signal processing problems of great interest are: • the detection problem; • the directions of arrival (DOA) estimation problem. With respect to detection problems the arrays estimates the number of emitting sources present in the array environment. On the other hand, Spatial Spectral Estimation techniques are concerned 1 with the DOA estimation problem. Classical spatial estimation techniques are based in the Fourier Transform (Conventional Beamformer). The main drawback of the Fourier-type methods is that they offer limited resolving capabilities. Thus, in the last decade the so called Superresolution methods have been introduced, their main objective being to improve the resolving capabilities by using a model for the signal better than that used by Fourier methods. These methods have given fresh impetus to the array signal processing problem by dealing with the question of resolution of the array in such a way that there is elimination of the effects of the Signal-to-Noise Ratio (SNR) on resolution, in contrast to the conventional methods where the resolution is limited by noise. This design experiment is concerned with a compact examination of the above concepts and provides a tutorial mechanism regarding array signal processing techniques with main objectives to: • estimate the number of the emitting sources; • provide complete information about the location of the emitting sources; • isolate a desired source and provide the complete cancellation of interferences at the output of the array; • estimate relative power cross-correlations, etc. Task: Consider a linear array of 7 sensors uniformly distributed along the x-axis in the 3 dimensional real space. i.e. type: array=[-3 0 0;-2 0 0;-1 0 0;0 0 0;1 0 0;2 0 0;3 0 0]; Consider that three narrowband transmitting sources are present in this environment. 1. form the pattern of the above array. type: pat2D(array) 2 if you know that the directions of the above three sources are 40◦, 45◦ and 90◦ i.e type: directions =[40 0; 45 0; 90 0] check the gain provided by the array. For each source there is a vector, called the Source Position Vector (SPV), which is a function of the position of the source and the geometry of the array. This means if you know the location of the sources and the array you may form the SPVs. Consider the matrix S with columns the Source Position Vectors Si, i = 1, 2, 3. This matrix can be seen by typing: S =SPV(array, directions) Comment on the form of the vectors Si. 3. Consider that you know that the above mentioned three sources are uncorrelated and of equal power (say 1, normalised). Form the covariance matrix of the 3 sources Rmm. 4. Assume that the noise present in the array is: additive isotropic noise, uncorrelated with the transmitting sources and of power σ2 = 40 dB below the power of the signals i.e. type: sigma2=0.0001 2 Form the covariance matrix of the signals at the input of the array Rxx. This is known as the data covariance matrix and is given by Rxx = S.Rmm.S H + σ2I i.e. type: Rxx=S*Rmm*S’+sigma2*eye(7,7) 5. Comment on the form of the matrix Rxx. Remember that your comments are valid only for the case of linear array. 6. Forget that you know that there are 3 sources present and in addition forget their directions and their powers i.e. type: directions = []; Rmm = []; S = []; sigma2 = []; Consider that the only information provided to you is the matrix Rxx at the input of the above array. Form the eigendecomposition of the matrix Rxx i.e. type: eig(Rxx) Is there any way to estimate • the number of sources which are present and • the power of noise simply by observing the eigenvalues of Rxx? 7. Write your conclusion from 6 as a formal STATEMENT (or THEOREM). 8. Let us consider that the source located at 90◦ is the ‘desired’ source. i.e. type: directions=[90,0]; Consider that we have knowledge of this direction and that in addition you know the covariance ma- trix Rxx. The remaining two sources (i.e. 40 ◦, 45◦) known as interferences are completely unknown. Let us estimate firstly the SPV Sd which corresponds to the desired source i.e. type: Sd=SPV(array, directions) Next let us weight the array elements by the following vector wopt = R −1 xxSd i.e. type: wopt=inv(Rxx)*Sd This is the optimum Wiener-Hopf solution. Estimate once again the array pattern of the array when it is weighted by wopt i.e. type: pat2d(array,wopt) 3 Can you distinguish the directions of the two interferences from that array pattern? 9. Repeat Questions 4-8 but with noise level 10dB below the level of the sources. 10. What conclusion can be drawn from Question 8 and 9? 11. Questions 8-10 deal with the suppression of interferences and sometimes (why sometimes?) the estimation of their directions. Next the case where the only information provided is the covariance matrix Rxx is considered. The aim now is to estimate the directions of the three sources. We are able to identify (detect) that there are three sources present simply by using your Theorem-Statement in Question 6. One way to achieve this aim is by using the so called MUltiple SIgnal Classification (MUSIC) algorithm [?]. Study and implement the MUSIC algorithm in order to estimate the location of the three sources. 12. Consider the cases where two sources ie. 40◦, 45◦ are fully correlated (or coherent). Repeat question 3 to 11. Compare your results from the uncorrelated and correlated cases. 13. Is it possible to overcome the problem resulting when the sources are coherent? The answer is YES [?] by applying the so called spatial Smoothing technique before using the MUSIC algorithm. Can you apply this technique? Write a small program in Matlab. 14. Consider that the covariance matrix Rxx is known to you and that, by using the MUSIC algo- rithm (or any other direction finding technique) the directions of the transmitted sources have been successfully estimated. Is it possible to receive one of the transmitted signals and, at the same time, to suppress completely the effects of the rest of the transmitted signals (i.e. to provide complete interference cancelation)? 15. Finally, consider the situation described in Question 1 to 4. However, this time, instead of using the theoretical covariance matrix Rxx = S.Rmm.S H + σ2I use an experimental (practical) data covariance matrix which results from only 250 snapshots. Apply your detection criterion (Question 7) in order to detect the three sources. Identify any problem and write it down in your report together with your conclusion. Then find papers associated with the following criteria AIC (Akaike Information Criterion) and MDL (Minimum Description Length). Apply these criteria to detect the three sources. What conclusion can be drawn from these criteria? References [1] R. O. Schmidt, “Multiple Emitter Location and Signal Paramenter Estimation”, IEEE Tran- scations on Antennas and Propagation, vol. AP-34, No.3, pp. 276-280, March 1986. [2] T. Shan, M. Wax and T. Kailath, “On Spatial Smoothing for Direction-of-arrival Estimation of Coherent Signal”, IEEE Transcations on Acoustic Speed and Signal Processing, vol. ASSP-33, No.4, pp. 806-911, August 1985. [3] H.L. Van Trees, Optimum Array Processing (Part IV of Detection, Estimation, and Modula- tion Theory), Wiley 2002 [4] D.H. Johnson and J.E. Dudgeon, Array Signal Processing: Concepts and Techniques, Prentice Hall, 1993 [5] S. U. Pillai, Array Signal Processing, Springer-Verlag, 1989. 4
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