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Presents leasts squares estimation in subspaces of Hilbert spaces, with applications.

Projections with orthonormal bases

Having an orthonormal basis for the subspace of interest significantly simplifies the projection operator.

Lemma 1 Let x X , a Hilbert space, and let S be a subspace of X . If { b 1 , b 2 , ... } is an orthonormal basis for S , then the closest point s 0 S to x is given by s 0 = i x , b i b i .

We begin by noting that

i x , b i b i = i x - s 0 + s 0 , b i b i = i x - s 0 , b i b i + i s 0 , b i b i .

Now, since s 0 is the projection of x onto S , we must have that x - s 0 S , and so for each basis element b i we must have x - s 0 , b i = 0 . Additionally, since s 0 S and { b 1 , b 2 , ... } is an orthonormal basis for S , we must have that i s 0 , b i b i = s 0 . Thus, we obtain

i x , b i b i = s 0 ,

proving the lemma.

Application: communications receiver

Consider the case of a communications receiver that records a continuous-time signal r ( t ) = s ( t ) + n ( t ) over 0 t 1 , where s ( t ) is one of m codeword signals { s 1 ( t ) , ... , s m ( t ) } , and n ( t ) is additive white Gaussian noise. The receiver must make the best possible decision on the observed codeword given the reading r ( t ) ; this usually involves removing as much of the noise as possible from r ( t ) .

We analyze this problem in the context of the Hilbert space L 2 [ 0 , 1 ] . To remove as much of the noise as possible, we define the subspace S = span ( { s 1 ( t ) , ... , s m ( t ) } ) . Anything that is not contained in this subspace is guaranteed to be part of the noise n ( t ) . Now, to obtain the projection into S , we need to find an orthonormal basis { e 1 ( t ) , ... , e n ( t ) } for S , which can be done for example by applying the Gram-Schmidt procedure on the vectors { s 1 ( t ) , ... , s m ( t ) } . The projection is then obtained according to the lemma as

r S ( t ) = i = 1 n r ( t ) , e i ( t ) e i ( t ) ,

where r ( t ) , e i ( t ) = 0 1 r ( t ) e i ( t ) d t .

After the projection is obtained, an optimal receiver proceeds by finding the value of k that minimizes the distance

d 2 ( r S ( t ) , s k ( t ) ) = 0 1 | r S ( t ) - s k ( t ) | 2 d t = 0 1 r S ( t ) 2 d t + 0 1 s k ( t ) 2 d t - 2 0 1 r S ( t ) s k ( t ) 2 d t ;

note here that the first term does not depend on k , so it suffices to find the value of k that minimizes the “cost”

c k : = 0 1 s k ( t ) 2 d t - 2 0 1 r S ( t ) s k ( t ) 2 d t = s k ( t ) , s k ( t ) - 2 r S ( t ) , s k ( t ) = s k ( t ) , s k ( t ) - 2 i = 1 n r ( t ) , e i ( t ) e i ( t ) , s k ( t ) = s k ( t ) , s k ( t ) - 2 i = 1 n r ( t ) , e i ( t ) e i ( t ) , s k ( t ) .

In practice, the codeword signals are designed so that their norms s k ( t ) 2 = s k ( t ) , s k ( t ) are all equal. This design choice reduces the problem above to finding the value of k that maximizes the score

c k ' : = i = 1 n r ( t ) , e i ( t ) e i ( t ) , s k ( t ) .

Thus, the receiver can be designed according to the diagram in [link] .

Diagram of a communications receiver designed in accordance with the projection theorem.

Least squares approximation in hilbert spaces

Let y 1 , ... , u n be elements of a Hilbert space X and define the closed, finite-dimensional subspace of X given by S = span ( y 1 , ... , y n ) . We wish to find the best approximation of x in terms of the vectors y i , that is, the linear combination i = 1 n a i y i with the smallest error e = x - i = 1 n a i y i . To measure the size of the error, we use the induced norm e = x - i = 1 n a i y i .

To solve this problem, we rely on the projection theorem: we are indeed looking for the closest point to x in S = span ( y 1 , ... , y n ) . The projection theorem tells us that the closest point s 0 = i = 1 n a i y i must give x - s 0 S , i.e., e S , which implies in turn that x - i = 1 n a i y i , y j = 0 for all j = 1 , ... , n . The requirement can be rewritten as x , y j = i = 1 n a i y i , y j = i = 1 n a i y i , y j for each j = 1 , ... , n . These requirements can be collected and written in matrix form as

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Source:  OpenStax, Signal theory. OpenStax CNX. Oct 18, 2013 Download for free at http://legacy.cnx.org/content/col11542/1.3
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