Mason George University Electrical and Computer Engineering Department ECE 754 Optimum Array Processing I
Problem Set 5
Spring 2009 Issued: Friday, March 20, 2009 Assigned reading 3/18/09 3/25/09 Due: Wednesday, March 25, 2009
Sections 6.2-6.5 in Optimum Array Processing Sections 6.6-6.7 in Optimum Array Processing
Please include a hard-copy of any Matlab code you develop. This week youll need to develop code to compute optimum beamformer weights and the associated array gains. ECE-754 Problem 5-1 Suppose we assume the following model for the signal and noise incident on a receiving array. The signal consists of a single planewave with a random complex amplitude. The replica vector associated 2 with the signal is denoted vs . The complex amplitude of the signal (s ) is zero-mean with variance s . The noise is random with zero mean and covariance matrix Sn . Under these conditions the received signal and its associated covariance are: x = s vs + n,
2 H Sx = s vs vs + Sn .
The MVDR and MPDR beamformers are designed for this signal using the steering vector vm . Show that as long as vm = vs , the MVDR and MPDR weight vectors are identical. Problem 6.2.8 in Optimum Array Processing Problem 6.3.1 in Optimum Array Processing ECE-754 Problem 5-2 Consider a standard 10-element linear array. In this problem you will implement a conventional beamformer and an MVDR beamformer for this array, and apply it to the simulated data stored in the le prob5 2.mat on the course website. The le contains data for multiple scenarios. In each scenario there is a single planewave interferer located at the u = u1 (u is the directional cosine) plus spatially white noise. The signal is always 3 dB below the white noise. The interference-to-noise ratio (INR) and the interferer location u1 varies with scenario. The source signal is located at broadside (u = 0), so thats the direction the beamformer should be steered. The source and interference signals are complex exponentials in the time domain. To generate this data set, Ive assumed that the source and the interferer are out of phase by a factor of /2. (This will help you see how much of each signal and/or interferer is making it through your spatial lter.) The le contains 100 snapshots of data. You are to process each data snapshot to create a 100-point time series (yconv and ymvdr ) for each of your beamformers. The data is complex, so when you plot your results, you should just plot the real part. Youll do this for each of the scenarios and compare the results. The following variables are included in the data le: INR: This contains vector the INR (in dB) for the cases of interest. The three INRs are -20 dB, 0 dB, and +20 dB.
u int: This vector contains the interferer locations for the cases of interest. The interferers are located at u1 = 0.02, 0.0433, 0.09, 0.18, 0.3, 0.5. Note: these are the same cases considered in examples in the textbook. Snmatrix: This multidimensional matrix contains the 10x10 noise covariance matrix for each of the 18 cases (3 INRs times 6 interferer locations). Snmatrix is a 10x10x6x3 matrix. The 3rd dimension is associated with the interferer location and the 4th dimension is associated with the INR level, e.g., Snmatrix(:,:,1,1) contains the covariance matrix for the case where u1 = 0.02 and INR=-20 dB. Xdata: This multidimensional matrix contains the data snapshots for the 18 cases. Xdata is a 10x100x6x3 matrix. As with Snmatrix, the 3rd dimension is associated with the interferer location and the 4th dimension is associated with the INR level, e.g., Xdata(:,:,1,1) contains the 10x100 matrix of data snapshots for the case where u1 = 0.02 and IN R = 20 dB. vm: Steering vector for the desired signal direction. sigs: Desired signal time series. (Actual signal used to generate the snapshots.) sig1: Interference signal time series. (Actual signal used to generate the snapshots.) rcvr: Receiver parameters (locations along z-axis, information about wavelength). You shouldnt need this, but Ive included it for reference. You should design the conventional weight vector (uniform weighting), as well as the MVDR weight vector for ...