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)
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
Low-Density Parity-Check Codes. Detailed solutions to problems
8.1) a) there are at least 7 cycles of length 4; the1s involved in these cycles are seen in
matrix H below:
0
1
0
1
H=
0
1
0
0
1 0 1 0 1
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
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
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
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 no
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
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
0