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

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

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

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

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

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

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 (

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

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

Adversary
Lower Bounds in Streaming
Piotr Indyk
MIT
Streaming Algorithms
Norm estimation: (1+)-approximation of
|x|p, xRm, under a sequence of n
updates
O(log(n+m)/2) bits for p(0,2]
(excluding randomness)
Heavy hitters/sparse approximations
Question:

Streaming Algorithms, etc.
MIT
Piotr Indyk
Data Streams
A data stream is a sequence of data that is too
large to be stored in available memory
(disk, memory, cache, etc.)
Examples:
Network traffic
Database transactions
Sensor networks
Satellite data fee

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

L2 Norm Estimation
MIT
Piotr Indyk
Lecture 2
L2 Norm Estimation
Vector x:
1 2 m
A stream is a sequence of updates (i,a)
xi=xi+a
Want to estimate |x|2 up to 1
Last week, we have seen how to do that for |x|0 :
Space: (1/ + log m)O(1)
Technique:
Linear sk

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)

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

Heavy Hitters
Piotr Indyk
MIT
Lecture 4
Last Few Lectures
Recap (last few lectures)
Update a vector x
Maintain a linear sketch
Can compute Lp norm of x
(in zillion different ways)
Questions:
Can we do anything else ?
Can we do something about linea

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

Estimating Lp Norms
Piotr Indyk
MIT
Lecture 3
Recap/Today
Two algorithms for estimating L2 norm of a stream
A stream of updates (i,1) interpreted as
xi=xi+1
(fractional and negative updates also OK)
Algorithms maintain a linear sketch Rx, where R is a k

Chapter 1
Analysis of Discrete-Time Linear
Time-Invariant Systems
1.1 Signals
1.1.1 Denitions and Notation
A signal is a function: signal and function are synonymous. The two notions are
the same, and we will be using them interchangeably. The historical

18
CHAPTER 1. ANALYSIS OF DISCRETE-TIME LINEAR TIME-INVARIANT SYSTEMS
T
T
Domain of f ,
D, a set of T
T numbers
T
T
signal f
T
T
T
Domain of S, system S
T
D, a set of
T
T signals
T
T
T
Range of f ,T
R, a set of T
T numbers
T
T
(a)
T
T
Range of S, T

MEL ZG520: Digital Signal Processing
Assignment 1
Due date : 7th May 2016
1. The following sequence represent one period of a sinusoidal sequence of the form x[n] =
Acos(0 n + ):
cfw_0
2 2
2
0
2
2
2
Determine the values of the parameters A, 0 and .
2. F