6.450 Principles of Digital Communication
MIT, Fall 2009
Monday Dec 14, 2009
Final Exam
You have 180 minutes to complete the quiz.
This is an open-book quiz. You may use your book and six pages of notes. Calculators
are allowed, but probably wont be u

6.450 Introduction to Digital Communication
MIT, Fall 2002
December 9, 2002
Lecture 24: Coding, IS-95, and CDMA
1
Coding and Decoding
We have spent quite a while discussing modulation and demodulation. Modulation can
be viewed as the entire process of map

6.450 Introduction to Digital Communication
MIT, Fall 2002
November 20, 2002
Lecture 19: The irrelevance theorem and orthogonal signal sets
1
Review
We are looking at channels with real valued waveforms that can be modeled as Y (t) =
X(t)+N (t) where the

6.450 Principles of Digital Communications
MIT, Fall 2002
Wednesday, Sept. 4
Handout #3
Lecture 1: Introduction to Digital Communication
1
Introduction and Objectives
The digital communication industry is an enormous and rapidly growing industry, roughly

6.450 Introduction to Digital Communication
MIT, Fall 2002
October 2, 2002
Lecture 8: Analog Sources: waveforms sequences
1
Analog sources
We have now studied coding both for discrete sources and analog-valued discrete-time
sources. We have emphasized sev

6.450 Principles of Digital Communications
MIT, Fall 2002
September 11
Handout #6
Lecture 3: Coding for Discrete Sources (cont.)
Variable-length source codes were introduced in the previous lecture. Their study is based
on the intuitive idea that the bit

6.450 Principles of Digital Communications
MIT, Fall 2002
September 9
Handout #5
Lecture 2: Coding for Discrete Sources
Literature: Todays lecture and the three subsequent lectures deal with coding for discrete sources. Good reference books on this topic

6.450 Introduction to Digital Communication
MIT, Fall 2002
September 18, 2002
Lecture 4: Coding for Sequences of Source Symbols
1
Review
We have been considering a discrete memoryless source model whose output is a sequence
of iid chance variables. We hav

6.450 Principles of Digital Communications
MIT, Fall 2002
September 18
Handout #12
Lecture 5: Sources with Memory and the Lempel-Ziv Algorithm
1
Markov Sources
In previous lectures, we developed the basic coding results for discrete memoryless sources.
Ma

6.450 Introduction to Digital Communication
MIT, Fall 2002
September 25, 2002
Handout #13
Lecture 6: Quantization
1
Review
In previous lectures, we discussed coding for discrete sources. As described in lecture
2, however, many sources are analog (such as

6.450 Introduction to Digital Communication
MIT, Fall 2002
October 23, 2002
Lecture 13: QAM and Noise
1
Implementation of QAM
Last time we described QAM and the principles of its implementation. The incoming
bits are encoded into complex valued symbols in

6.450 Introduction to Digital Communication
MIT, Fall 2002
September 30, 2002
Lecture 7: High-rate entropy-coded quantization
1
Introduction
We will now take a somewhat deeper look at the quantization of analog-valued, discretetime sources. Surprisingly,

6.450 Introduction to Digital Communication
MIT, Fall 2002
October 20, 2002
Lecture 12: QAM
1
Review
In the previous lecture, we discussed pulse amplitude modulation (PAM) as a very simple
mode of digital communication. The input data sequence arrives at

6.450 Introduction to Digital Communication
MIT, Fall 2002
October 30, 2002
Lecture 14: Noise and Gaussian random processes
1
Review
The previous lecture started to introduce channel noise into the problem of communicating
over a channel. We are assuming

6.450 Introduction to Digital Communication
MIT, Fall 2002
November 27, 2002
Lecture 21: Input/output models for wireless
1
Input/Output Models of Wireless Channels
Suppose a transmitting antenna sends a sinusoid, cos(2f t), which is received at a receivi

6.450 Introduction to Digital Communication
MIT, Fall 2002
December 2, 2002
Lecture 22: Stochastic wireless models
1
Statistical Channel Models
We defined Doppler spread and multipath spread in the previous lecture as quantities
associated with a given ce

6.450 Introduction to Digital Communication
MIT, Fall 2002
December 4, 2002
Lecture 23: Channel measurement and Rake receivers
1
Channel measurement
The lesson learned from binary detection in Rayleigh fading is that there is substantial
probability that

6.450: Principles of Digital Communication 1
Digital Communication: Enormous and normally
rapidly growing industry, roughly comparable in size
to the computer industry.
Objective: Study those asp ects of communication
systems unique to those systems. Litt

DISCRETE MEMORYLESS SOURCE
(DMS) Review
The source output is an unending sequence,
X1, X2, X3, . . . , of random letters, each from
a nite alphabet X .
Each source output X1, X2, . . . is selected
from X using a common probability mea
sure with pmf pX

ENTROPY OF X , |X | = M , Pr(X =j ) = pj
H (X ) =
pj log pj = E[ log pX (X )]
j
log pX (X ) is a rv, called the log pmf.
H(X ) 0; Equality if X deterministic.
H(X ) log M ; Equality if X equiprobable.
If X and Y are indep endent random symb ols,
then the

MARKOV CHAINS
A nite state Markov chain is a sequence S0, S1, . . .
of discrete cvs from a nite alphab et S where
q0(s) is a pmf on S0 and for n 1,
Q(s|s ) = Pr(Sn=s|Sn1=s )
= Pr(Sn=s|Sn1=s , Sn2 = sn2 . . . , S0=s0)
for all choices of sn2 . . . , s0, We

The Lemp el-Ziv algorithm matches the longest
string of yet unenco ded symb ols with strings
starting in the window.
Window size w is large p ower of 2, mayb e 217.
log w bits to enco de u, 2 log n + 1 bits for n
w = window
P
n=3
Match
b c d a c b a b a c

Measure and complements
We listed the rational numb ers in [T /2, T /2]
as a1, a2, . . .
cfw_
k
ai =
k
([ai, ai]) = 0
i=1
i=1
The complement of k=1 ai is k=1 ai where ai
i
i
is all t [T /2.T /2] except ai.
Thus k=1 ai is a union of k+1 intervals, lling
i

Functions not limited in time
We can segment an arbitrary L2 function into
segments of width T . The mth segment is
um(t) = u(t)rect(t/T m). We then have
u(t) = l.i.m.m0
m0
m=m0
um(t)
This works b ecause u(t) is L2. The energy in
um(t) must go to 0 as m .

Fourier
series
f
2ik 2f
W rect
u (f ) = k u k e
2W
f
1 W u(f )e2ik 2W d
u k = 2W W
f
T/F dual
t
uk e2ikt/T rect( T )
k=
1 T /2
uk = T T /2 u(t)e2ikt/T dt
DTFT
u(t) =
Fourier
transform
Sampling
u(t) = 2W uk sinc(2W t k)
k=
uk = 21 u( 2k )
W
W
1
The

6.450 Principles of Digital Communication
MIT, Fall 2009
Due Mon, Sep 21, at beginning of class
Problem Set 1
Problem 1- Problem 2.14 from Gallagers book.
Problem 2- In class, we proved Kraft inequality by mapping each codeword to a rational
number in the

6.450 Principles of Digital Communication
MIT, Fall 2009
Wednesday October 21, 2009
Quiz 1
You have 90 minutes to complete the quiz.
This is an open-book quiz. You may use your book and three pages of note. Calcu
lators are allowed, but probably wont

6.450 Introduction to Digital Communication
MIT, Fall 2002
November 6, 2002
Lecture 16: Spectral Density, Orthonormal Expansions
1
Review of linear functionals and filters
Let g 1 , . . . , g j0 be a given set of real L2 waveforms (such as a finite orthon

6.450 Introduction to Digital Communication
MIT, Fall 2002
November 25, 2002
Lecture 20: Wireless Communication Systems
1
Introduction
During the next three weeks we provide a brief treatment of wireless digital communication
systems. As the name suggests