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rs6 - EE263 S Lall 2011.05.16.02 Review Session 6 46075 1...

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EE263 S. Lall 2011.05.16.02 Review Session 6 1. Estimating a signal with interference. 46075 This problem concerns three proposed methods for estimating a signal, based on a mea- surement that is corrupted by a small noise and also by an interference, that need not be small. We have y = Ax + Bv + w, where A R m × n and B R m × p are known. Here y R m is the measurement (which is known), x R n is the signal that we want to estimate, v R p is the interference, and w is a noise. The noise is unknown, and can be assumed to be small. The interference is unknown, but cannot be assumed to be small. You can assume that the matrices A and B are skinny and full rank ( i.e. , m > n , m > p ), and that the ranges of A and B intersect only at 0. (If this last condition does not hold, then there is no hope of finding x , even when w = 0, since a nonzero interference can masquerade as a signal.) Each of the EE263 TAs proposes a method for estimating x . These methods, along with some informal justification from their proposers, are given below. Nikola proposes the ignore and estimate method. He describes it as follows: We don’t know the interference, so we might as well treat it as noise, and just ignore it during the estimation process. We can use the usual least-squares method, for the model y = Ax + z (with z a noise) to estimate x . (Here we have z = Bv + w , but that doesn’t matter.) Almir proposes the estimate and ignore method. He describes it as follows: We should simultaneously estimate both the signal x and the interference v , based on y , using a standard least-squares method to estimate [ x T v T ] T given y . Once we’ve estimated x and v , we simply ignore our estimate of v , and use our estimate of x . Miki proposes the estimate and cancel method. He describes it as follows: Almir’s method makes sense to me, but I can improve it. We should simulta- neously estimate both the signal x and the interference v , based on y , using a standard least-squares method, exactly as in Almir’s method. In Almir’s method, we then throw away ˆ v , our estimate of the interference, but I think we should use it. We can form the “pseudo-measurement” ˜ y = y B ˆ v , which is our measurement, with the effect of the estimated interference subtracted off. Then, we use standard least-squares to estimate x from ˜ y , from the simple model ˜ y = Ax + z . (This is exactly as in Nikola’s method, but here we have subtracted off or cancelled the effect of the estimated interference.) These descriptions are a little vague; part of the problem is to translate their descriptions into more precise algorithms. (a) Give an explicit formula for each of the three estimates. (That is, for each method give a formula for the estimate ˆ x in terms of A , B , y , and the dimensions n, m, p .) (b) Are the methods really different? Identify any pairs of the methods that coincide ( i.e. , always give exactly the same results). If they are all three the same, or all three different, say so.
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