I. Introduction
Information, Signals and Systems
Signal processing concerns primarily with signals and systems that operate
on signals to extract useful information. In this course our concept of a signal
will be very broad, encompassing virtually any dat

The z -transform
We introduced the z -transform before as
h[k ]z k
H (z ) =
k=
where z is a complex number. When H (z ) exists (the sum converges), it can be
interpreted as the response of an LSI system with impulse response h[n] to
the input of z n . The

Fourier Representations
Throughout the course we have been alluding to various Fourier representations.
We rst recall the appropriate transforms:
Fourier Series (CTFS) x(t): continuous-time, nite/periodic on [, ]
1
X [k ] =
2
x(t)ejkt dt
1
x(t) =
2
X [k

Inner Product Spaces
Where normed vector spaces incorporate the concept of length into a vector
space, inner product spaces incorporate the concept of angle.
Denition 1. Let V be a vector space over K . An inner product is a function
, : V V K such that f

Vector Spaces
Metric spaces impose no requirements on the structure of the set M . We
will now consider more structured M , beginning by generalizing the familiar
concept of a vector.
Denition 1. Let K be a eld of scalars, i.e., K = R or C. Let V be a set

I I. Signal Representations in Vector Spaces
We will view signals as elements of certain mathematical spaces. The spaces
have a common structure, so it will be useful to think of them in the abstract.
Metric Spaces
Denition 1. A set is a (possibly innite)

Linear Operators
Denition 1. Def: A transformation (mapping) L : X Y from a vector space
X to a vector space Y (with the same scalar eld K ) is a linear transformation
if:
1. L(x) = L(x) x X , K
2. L(x1 + x2 ) = L(x1 ) + L(x2 ) x1 , x2 X .
We call such tr

Orthobasis Expansions
N
Suppose that the cfw_vj j =1 are a nite-dimensional orthobasis. In this case we
have
N
x=
x, vj vj .
j =1
But what if x span(cfw_vj ) = V already? Then we simply have
N
x=
x, vj vj
j =1
for all x V . This is often called the reprod

I II. Representation and Analysis of Systems
Linear systems
In this course we will focus much of our attention on linear systems. When our
input and output signals are vectors, then the system is a linear operator.
Suppose that L : X Y is a linear operato

Hilbert Spaces in Signal Processing
What makes Hilbert spaces so useful in signal processing? In modern signal
processing, we often represent a signal as a point in high-dimensional space.
Hilbert spaces are spaces in which our geometry intuition from R3

Approximation in
p
norms
So far, our approximation problem has been posed in an inner product space,
and we have thus measured our approximation error using norms that are induced by an inner product such as the L2 / 2 norms (or weighted L2 / 2 norms).
So

Poles and zeros
Suppose that X (z ) is a rational function, i.e.,
X (z ) =
P (z )
Q(z )
where P (z ) and Q(z ) are both polynomials in z . The roots of P (z ) and Q(z )
are very important.
Denition 1. A zero of X (z ) is a value of z for which X (z ) = 0

Discrete-time systems
We begin with the simplest of discrete-time systems, where X = CN and Y =
CM . In this case a linear operator is just an M N matrix. We can generalize
this concept by letting M and N go to , in which case we can think of a linear
ope

Stability, causality, and the z -transform
In going from
N
m
ak y [n k ] =
k=0
bk x [ n k ]
k=0
to
H (z ) =
Y (z )
X (z )
we did not specify an ROC. If we factor H (z ), we can plot the poles and zeros
in the z -plane as below.
Im[z ]
Re[z ]
Several ROCs

The DTFT as an eigenbasis
We saw Parseval/Plancherel in the context of orthonormal basis expansions.
This begs the question, do F and F 1 just take signals and compute their
representation in another basis?
Lets look at F 1 : L2 [, ] 2 (Z) rst:
1
F 1 (X (