Solving systems of symmetric equations
In the previous section, we saw a few examples of how to set up
systems of linear equations for signal processing problems of interest.
In this section, we will discuss how to solve, and analyze the solution
to, syst

Notes by J. Romberg January 8, 2012 11:58
Notes by J. Romberg January 8, 2012 11:58
Discretizing linear inverse problems
In many real-world applications, the signal or image we are measuring is a function of a continuous variable (or variables for images)

Basis representation fundamentals
Having a basis representation for our signals of interest allows us
to do two very nice things:
Take the signal apart, writing it as a discrete linear combination of atoms:
x(t) =
X
(t)
for some fixed set of basis signa

`1 minimization
We will now focus on underdetermined systems of equations:
# samples
resolution/
bandwidth
=
data
acquisition
system
unknown
signal/image
Suppose we observe y = x0, and given y we attempt to estimate
x0 by applying the pseudo-inverse of to

c J. Fessler, March 8, 2004, 11:40 (FULL version)
updown.1
Properties of up-sampling and down-sampling
Impulse train function
Define
sM [n] =
X
k=
By the DTFS (or by direct evaluation):
[n kM ] =
1,
0,
n an integer multiple of M
otherwise.
M 1
1 X 2 kn
e

Notes on General Frame Operators in Infinite
Dimensions
1.
a) A mapping L : H G from a Hilbert space H into a
Hilbert space G is called a linear operator if for all
, C and f, g H
L[f + g] = L[f ] + L[g].
b) The operator norm of L is defined as
kLk := sup

The cosine-I transform
The cosine-I transform is an alternative to Fourier series; it is an
expansion in an orthobasis for functions on [0, 1] (or any interval on
the real line) where the basis functions look like sinusoids. There
are two main differences

Notes by J. Romberg January 8, 2012 11:58
Notes by J. Romberg January 8, 2012 11:58
Notes by J. Romberg January 8, 2012 11:58
Notes by J. Romberg January 8, 2012 11:58
Notes by J. Romberg January 8, 2012 11:58
Notes by J. Romberg January 8, 2012 11:58
Not

Homework #1 Odd Numbered Answers
1. From Wikipedia, there are many very readable accounts of the life and technical achievements of J.C.
Maxwell, and H.R. Hertz, G. Marconi, and N. Tesla. Their backgrounds, personalities, professional recognitions, and fi

Notes by J. Romberg January 8, 2012 16:47
Notes by J. Romberg January 8, 2012 16:47
Notes by J. Romberg January 8, 2012 16:47
Notes by J. Romberg January 8, 2012 16:47
Notes by J. Romberg January 8, 2012 16:47
Notes by J. Romberg January 8, 2012 16:47
Not