| # Implementation Errors in an Antenna Array Design Problem Return to the antenna array design problem described here. Let us specifically consider the problem of minimizing the sidelobe level subject to a normalization constraint, described here. This problem takes the form of an SOCP: ### Uncertainty model We assume that the antenna weights (contained in the complex vector ) are subject to implementation errors. It is natural to model these errors as multiplicative perturbations on the antenna weights, of the form where the relative errors are bounded: , . Here, is a measure of the maximum amount of implementation errors. Note that such uncertainty errors amount to uncertainties in the coefficients of the original SOCP. ### Impact on solution Assume that we have solved the above SOCP, and found a solution . Now let us perturb the solution according to the uncertainty model above. We generate random samples of the relative error vector , inside the box . When we change the antenna weights, the normalization requirement is not necessarily met. To compare diagrams meaningfully, we scale the sample diagrams to make equal to , and look at the resulting diagrams. When solving the problem with nominal data and an equidistant -point angular grid, we end up with the nice nominal solution depicted here. For this solution and no data perturbations, the optimal sidelobe level is as low as . Unfortunately, these nice results are nothing but a dream. In reality, with random actuation errors of level as low as , our supposedly optimal design becomes a complete disaster. With samples, the sidelobe level for the nominal design jumps, on the average, from its nominal value to , even attaining a maximum of ! In fact, in some of these random realizations of the diagram, most of the energy is sent in the opposite direction, as seen below.
| The dream: the optimal magnitude diagram. The attenuation is excellent in the stop band, so that the stop-band magnitude is not distinguishable from zero in this plot.
| The reality: a particular magnitude diagram obtained after perturbing the optimal weights of the above design with a relative error of . This design is a complete disaster: most of the energy is sent in the opposite direction from the target.
| The reality: another particular magnitude diagram obtained after perturbing the optimal weights of the above design with a relative error of . This design is also a complete disaster: the sidelobe level is has jumped from to almost , an increase by four orders of magnitude.
### What is going on? What happens is that the optimal antenna weights are quite high in magnitude, to the order -. Thus, even a small relative error in the weights can have consequences. The physics of our setup does not help: the distances between consecutive oscillators is equal to of the wavelength, and the spatial angle of interest — the one where we would like to send as much energy as possible — is comprised of directions whose angular distances from the axial direction are at most . This is a problem similar to what happens when trying to find a solution to a least-squares problem where the data matrix has low rank: the solution is highly sensitive to changes in the data.