Least-Squares Design
Example dataFor our particular example, we have the following parameters:
Matlab data
N = 10; % discretization parameter n = 16; % number of antennas lambda = 8; % wavelength Phi = pi/6; % sidelobe parameter Least-squares approachWe examine how least-squares can be used to strike a trade-off between the sideloble level attenuation and
thermal noise power. The initial problem reads
The basic idea is to penalize the constraints, as follows: we choose a ‘‘trade-off parameter’’ CVX implementationThe above problem can be solved via linear algebra (SVD) methods. Here, we do not even bother to invoke SVD — CVX will do. A CVX implementation of this problem is given below. Note how we use a new vector variable to handle the objective function. Note also that CVX understands complex variables. CVX implementation
Angles = linspace(Phi,pi,N); % angles in the stop band a = 2*pi*sqrt(-1)/lambda; % intermediate parameter cvx_begin variable z(n,1) complex; variable r(N,1) complex; minimize( z'*z+mu*r'*r ) subject to for i = 1:N, exp(a*cos(Angles(i))*(1:n))*z == r(i); end real( exp(a*(1:n))*z) >= 1; cvx_end Note that the penalty approach adopted here will not enforce the desired constraints. We can only hope that, for
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