Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

9.11 MEAN-SQUARE-ERROR ESTIMATION

Consider two random variables X and Y that are correlated with joint pdf fX,Y(x, y). Assume that X is measurable whereas we do not have direct access to Y. A simple example is

(9.144)

where V is not measurable and is independent of Y. Such a model is useful for many problems where an estimate of Y is needed but additive noise V corrupts the samples. From Chapter 4, we know that when Y and V are independent (which is often the case in many problems), the pdf of X is obtained from the following convolution:

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In order to estimate Y from X, the joint pdf fX,Y(x, y) is required. In the notation of Section 9.10, the unknown “parameter” is , and there is only one sample X such that N = 1. For the simple model above, the joint pdf is

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Using the techniques in Chapter 4:

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where in the last expression we have used the fact that Y and V are independent. Thus

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and

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Returning to the general problem of estimating Y from