Advanced Digital Communications
Suhas Diggavi
Ecole Polytechnique Fdrale de Lausanne (EPFL)
ee
School of Computer and Communication Sciences
Laboratory of Information and Communication Systems (LICOS)
23rd October 2005
2
Contents
I
Review of Signal Proces
ECE 562
Fall 2011
HOMEWORK ASSIGNMENT 1 SOLUTIONS
1. Use your knowledge of Gaussian and jointly Gaussian pdfs to get the answers to the
following directly (without resorting to integration).
(a) Find the variance of the random variable that has density
(x
Turbo codes. Detailed solutions to problems
Concatenated codes:
The output codeword of a block code C b ( 6 ,3 ) generated by the generator
7.1)
matrix G is then input to a convolutional encoder like that seen in the following figure,
operating in pseudo-
Convolutional Codes. Detailed solutions to problems
6.1)
The following figure corresponds to the convolutional encoder with generator
polynomials g ( 1 ) ( D ) = D + D 2 , g ( 2 ) ( D ) = 1 + D and g ( 3 ) ( D ) = 1 + D + D 2 .
c (1 )
c( 2 )
m
c(3 )
The c
Reed-Solomon Codes. Detailed Solutions to problems
Problems
5.1)
The generator polynomial of a cyclic code over GF ( 4 ) is g ( X ) = X + 1 , with code length
n = 3 . The field elements are generated by the polynomial 2 + + 1 = 0 .
Since 2 = + 1 , the fie
BCH Codes. Detailed solutions to problems
Problems
4.1)
The polynomial is irreducible if it is not divisible by polynomials over GF( 2 ) of degree
less than 5.
Polynomials of degree 1:
X , X + 1 do not divide p( X ) = 1 + X 2 + X 5
because this polynomial
Cyclic Codes. Detailed solutions to problems
The polynomial 1 + X + X 3 + X 4 is a generator polynomial of a binary linear
3.1)
cyclic block code with code length n 7 if this polynomial is a factor of X n +1 , with
n = 5 ,6 ,7 . The number of redundant bi
Block Codes. Detailed solutions to problems
2.1)
P = [ 1] ,
1
[
G = [ 1 1] ,
1
H = I n k
P
T
]
1 0 1
=
,
0 1 1
1 0
H = 0 1
1 1
T
Code table
c
m
0
000
1
111
Syndrome table
r
S
000 00
001 11
010 01
011 10
100 10
101 01
110 11
111 00
There are 8 error pat