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# 3.1: Introduction

We turn to a problem that is dual to the overconstrained estimation problems considered so far. Let $$A$$ denote an array of $$m$$ vectors, $$A=\left[a_{1}|\cdots| a_{m}\right]$$, where the $$a_{i}$$ are vectors from any space on which an inner product is defined. The space is allowed to be infinite dimensional, e.g. the space $$\mathcal{L}^{2}$$ of square integrable functions mentioned in Chapter 2 . We are interested in the vector $$x$$, of minimum length, that satisfy the equation

$y=\prec A, x \succ \ \tag{1a}$

where we have used the Gram product notation introduced in Chapter 2.

Example 3.1

Let y[0] denote the output at time 0 of a noncausal FIR filter whose input is the sequence $$x[k]$$, with

$y[0]=\sum_{i=-N}^{N} h_{i} x[-i]\nonumber$

Describe the set of input values that yield $$y[0] = 0$$; repeat for $$y[0] = 7$$. The solution of minimum energy (or RMS value) is the one that minimizes $$\sum_{i=-N}^{N} x^{2}[i]$$.

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