3.1: Introduction

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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 \(\sum_{i=-N}^{N} x^{2}[i]\).