1.
2.
7.
x(t)
LTI
y(t)
h(t)
(a) Suppose x(t) is Wide Sense Stationary (WSS). Show that y(t) is also WSS.
(b) Please find the output power spectrum density in terms of
the input power spectrum density and transfer function and prove it.
Solution:
8.Textboo

The Digital Communications Framework
s1 (t ), s2 (t ), . sM (t )
Transmitter
s (t )
sk (t )
Receiver
n(t )
( Gaussian White Noise )
The diagram above shows a very general framework for a digital communications
system. This can include coded, as well as

1
Numerically Efficient Direct-Optimization
Filter Design
Juan Fang and I-Tai Lu
Department of ECE, NYU Wireless,
Polytechnic Institute of New York University
AbstractNovel direct-optimization approaches for
designing prototype filters are developed. Trad

The Q Function
In this course certain partial areas under the gaussian probability density function will be
used extremely often. For that reason this brief note on the Q function will be useful.
The Q function is defined, for positive values of x , as
Q(

The Signal Space Version of the Optimum Receiver
Suppose the M signals to be used for transmission are s1 (t ) , s2 (t ), . , sM (t ) . Assume that
we have found a basis f1 (t ) , f 2 (t ), . , f N (t ) for the signal space in which N M . The
number N of

Orthonormal Basis Functions and the Signal Vector Space
Two functions, f (t ) and g (t ) , are said to be orthogonal if
f (t ) g (t ) dt 0
(3.1)
where the integral is over the domain of the functions. A typical domain might be the
interval [0, T ] , in w

Channel Capacity
I-Tai Lu
NYU Poly
1
Channel Capacity
Maximum data rate for arbitrarily low
probability of error
Theoretical upper bound on system
throughput
2
Part I
SISO Channel Capacity
3
Information and Entropy
Less uncertainty implies less informat

Lecture 5. Digital
Communications Model
5.2. Block and Convolutional
Codes
I-Tai Lu
Department of ECE
Polytechnic Institute of NYU
1
Lecture 5.
Digital Communications Model
5.1. Information Processing
5.2. Block and Convolutional Codes
5.3. TCM, Turbo and

Signaling Analysis & Design
for Modulation
Basedon
Class Notes
&
Chapter 2 of
EL6013 Digital Communications
Digest from Textbook:
Fundamentals of Digital Communication
by Upamanyu Madhow
0.BriefOverviewon
Modulation/Demodulation
SeeNote1a
Modulation
Modul

BLOCK CODING
Consider a communication scheme that uses simple BPSK to transmit bits sequentially.
If the information sequence is 1 0 1 1 0 0, The transmitted signal would look like that
shown below
s(t )
Tb
2Tb
3Tb
t
Where the pulse shape can be arbitrary

Convolutional Coding
An example of a binary convolutional encoder is shown below
1
input
output
2
3
The encoder operates as follows. The shift register is initially filled with zeros before the
input information bit stream begins entering the encoder. The

Channel Capacity and Random Coding
Time, Bandwidth and Dimensionality
We shall see that the dimensionality, N , of the signal space is a very important
parameter. Recall that N is the number of orthonormal basis functions used to represent
the signals und

Demodulation
EL6013 Digital Communications
Digest from Textbook:
Fundamentals of Digital Communication
by Upamanyu Madhow
3. Demodulation
3.1 Gaussian basics
This is because the statistics of a Gaussian random process are characterized by its
first and se

Synchronization and noncoherent
communication
EL6013 Digital Communications
Digest from Textbook:
Fundamentals of Digital Communication
by Upamanyu Madhow
4. Synchronization and noncoherent
communication
Synchronization and noncoherent communication
In Ch

Channel equalization
EL6013 Digital Communications
Digest from Textbook:
Fundamentals of Digital Communication
by Upamanyu Madhow
5. Channel equalization
5.1 The channel model
5.2 Receiver front end
5.3 Eye diagrams
5.4 Maximum likelihood sequence estimat

Channel coding
EL6013 Digital Communications
Digest from Textbook:
Fundamentals of Digital Communication
by Upamanyu Madhow
7. Channel coding
The key idea of channel coding is to introduce redundancy in the
transmitted signal so as to enable recovery from

EL601 3
Name
Poly ID
A set of M equally likely signals is defined by s m (t ) a m s (t ) , where s (t ) is a known
deterministic signal and the a m are amplitudes that are used to distinguish the different signals
(and, hence, carry information). The allo

3/22014 Nyq uist ISI criterion - Vlﬁkipedia, the free encyclopedia
Nyquist ISI criterion
From Wikipedia, the free encyclopedia
In communications, the Nyquist ISI
criterion describes the conditions which,
when satisﬁed by a communication channel
(including

1. Prove the following two equations:
a)
b)
Sol.
2.
Complex baseband equivalent of passband filtering: find the relation between the
following passband filter and baseband filter in both time and frequency domains.
Sol:
3.
Problem 2.5 of the text book
Sol

Problem
Problem 2.25 of the text book
Problem 2.27 of the text book
Problem 2.29
Problem 2-24 of the text book
Prove the Sampling Theorem
Consider a signal x(t) with the frequency domain response X(f) sketched as below:
X(f)
1
-fM
fM
f
n
and with the s

HW31
1.Sumoftworandomvariables
Twoindependentuniformrandomvariables:XandY.ThepdfofXis1when0 x 1
andthepdfofYis1when0.5 y 0.5.FindthepdfofX+Y.
(hint:
To know the pdf of of Z=X+Y is very important in communications since it
represents the results of many pr

HW 1-1. Barker Spectrum
Derive analytically the spectrum of the Barker code (11 chips)
<Diagram of Barker code for 11chips>
1s
T
Sol:
HW 1-2 . Convolution
1
a.) Define the spectrum ( )
0
or f 0.5
or f 0.5
Prove that the inverse Fourier Transform of ( )

The Complex Baseband Representation of Bandpass Signals
Consider the two signals below.
x(t ) 2 R (t ) cos 0t (t )
(1)
s (t ) R(t ) e j (t )
(2)
x(t ) is a real, bandpass signal with center frequency 0 , and s (t ) is a complex
baseband signal. Note that