Hilbert Spaces and Least Squares Methods for Signal Processing
Yoram Bresler
Samit Basu
January 13, 2013
Christophe Couvreur
2
Acknowledgement: The help of Kiryung Lee in drafting the gures and helping with the editing of the text
c
is gratefully acknowle
Chapter 9
The Hilbert Space of Random
Variables
9.1
Introduction
So far in this course, we have been working with Hilbert spaces like L2 (R), Cn or 2 (Z), all of which are
generalizations of vectors in two or three dimensions. Indeed 2 (Z) is a natural ex
Chapter 8
Review of Probability
In this chapter, we will briey review probability theory. The material given here is meant only to refresh
the readers memory, and introduce our notation.
8.1
introduction
Underlying the concept of randomness is a probabili
Chapter 7
Complete Inner Product Spaces
Hilbert Spaces
When a Banach space is also equipped with an inner product, it is called a Hilbert space. Equivalently, a
Hilbert space is any inner product space that is complete (completeness measured with respect
Chapter 6
Innite Dimensional Linear Vector
Spaces
6.1
Introduction
While nite dimensional vector spaces are very useful, especially for computational purposes, many important
problems cannot be phrased exclusively in terms of nite dimensional problems. He
Chapter 5
Finite Dimensional Linear Vector
Spaces
5.1
Introduction
In the previous chapter we introduced abstract linear vector spaces in order to generalize our work with
matrices. However, while we could construct the vector space, and supply it with st
Chapter 4
Inverse Problems in Linear Vector
Spaces
4.1
Why general linear vector spaces?
One of the most basic concepts that underlies the theory of inverse problems, as treated in this text, is that
of an abstract vector space. The reason for this is sim
Chapter 3
Temporal and Spatial Spectrum
Estimation
3.1
Harmonic Retrieval
The rst problem we will consider is called harmonic retrieval. In general, we have M samples of a signal
y (m) for m = 0, 1, . . . , M 1. We would like to model y (m) as a sum of we
Chapter 2
Singular Value Decomposition
2.1
Introduction
So far, we have looked at the problem of solving Ax = b for dierent classes of the matrix A, e.g., full
column rank, full row rank, etc. Now, however, we turn to an entirely dierent kind of problem.
Chapter 10
Least Squares and Random Processes
10.1
LTI Transformations of Zero Mean, WSS Processes
Our Hilbert space of random variables L2 (U ) allows us to study a number of interesting problems involving
lters that arise in signal processing problems.