6.451 Principles of Digital Communication II
MIT, Spring 2005
Wednesday, March 2, 2005
Handout #10
Problem Set 4 Solutions
Problem 4.1
Show that if C is a binary linear block code, then in every coordinate position either all
codeword components are 0 or
6.451 Principles of Digital Communication II
MIT, Spring 2005
Wednesday, March 2, 2005
Handout #11
Due: Wednesday, March 9, 2005
Problem Set 5
Problem 5.1 (Euclidean division algorithm).
(a) For the set F[x] of polynomials over any eld F, show that the di
6.451 Principles of Digital Communication II
MIT, Spring 2005
Wednesday, March 9, 2005
Handout #12
Problem Set 5 Solutions
Problem 5.1 (Euclidean division algorithm).
(a) For the set F[x] of polynomials over any eld F, show that the distributive law holds
6.451 Principles of Digital Communication II
MIT, Spring 2005
Wednesday, March 30, 2005
Handout #15
Due: Wednesday, April 6, 2005
Problem Set 6
Problem 6.1 (rational realizations).
(a) Generalize Figure 2 of Chapter 9 to realize any causal rational impuls
Wednesday, April 6, 2005
Handout #16
6.451 Principles of Digital Communication II
MIT, Spring 2005
Problem Set 6 Solutions
Problem 6.1 (rational realizations).
(a) Generalize Figure 2 of Chapter 9 to realize any causal rational impulse response
g (D) = n(
6.451 Principles of Digital Communication II
MIT, Spring 2005
Wednesday, April 6, 2005
Handout #17
Due: Wednesday, April 13, 2005
Problem Set 7
Problem 7.1 (State space sizes in trellises for RM codes)
Recall the |u|u + v | construction of a Reed-Muller c
6.451 Principles of Digital Communication II MIT, Spring 2005 Problem Set 7 Solutions Problem 7.1 (State space sizes in trellises for RM codes)
Wednesday, April 13, 2005 Handout #18
Recall the |u|u + v | construction of a Reed-Muller code RM(r, m) with le
6.451 Principles of Digital Communication II
MIT, Spring 2005
Wednesday, April 20, 2005
Handout #19
Due: Wednesday, April 27, 2005
Problem Set 8
Problem 8.1 (Realizations of repetition and SPC codes)
Show that a reduced Hadamard transform realization of a
6.451 Principles of Digital Communication II MIT, Spring 2005
Wednesday, April 27, 2005 Handout #20R
Problem Set 8 Solutions (revised)
Problem 8.1 (Realizations of repetition and SPC codes) Show that a reduced Hadamard transform realization of a repetiti
6.451 Principles of Digital Communication II
MIT, Spring 2005
Wednesday, April 27, 2005
Handout #21
Due: Wednesday, May 4, 2005
Problem Set 9
Problem 8.3 (revised) (BCJR (sum-product) decoding of SPC codes)
As shown in Problem 6.4 or Figure 1 of Chapter 1
6.451 Principles of Digital Communication II MIT, Spring 2005 Problem Set 9 Solutions
Wednesday, May 4, 2005
Handout #22
Problem 8.3 (revised) (BCJR (sum-product) decoding of SPC codes) As shown in Problem 6.4 or Figure 1 of Chapter 10, any (n, n 1, 2) s
6.451 Principles of Digital Communication II
MIT, Spring 2005
Wednesday, May 4, 2005
Handout #23
Due: never
Problem Set 10
Problem 10.1 (Mod-2 lattices and trellis codes)
(a) Let C be an (n, k , d) binary linear block code. Show that
C = cfw_x Zn | x c mo
6.451 Principles of Digital Communication II
MIT, Spring 2005
Wednesday, May 11, 2005
Handout #24
Problem Set 10 Solutions
Problem 10.1 (Mod-2 lattices and trellis codes)
(a) Let C be an (n, k, d) binary linear block code. Show that
C = cfw_x Zn | x c mod
6.451 Principles of Digital Communication II
MIT, Spring 2005
Wednesday, February 23, 2005
Handout #9
Due: Wednesday, March 2, 2005
Problem Set 4
Problem 4.1
Show that if C is a binary linear block code, then in every coordinate position either
all codewo
6.451 Principles of Digital Communication II MIT, Spring 2005
Wednesday, February 23, 2005
Handout #8
Problem Set 3 Solutions
Problem 3.1 (Invariance of coding gain) (a) Show that in the power-limited regime the nominal coding gain c (A) of (5.9), the U
6.451 Principles of Digital Communication II
MIT, Spring 2005
Wednesday, February 16, 2005
Handout #7
Due: Wednesday, February 23, 2005
Problem Set 3
Problem 3.1 (Invariance of coding gain)
(a) Show that in the power-limited regime the nominal coding gain
Final Exam Solutions
Problem F.1 (60 p oints)
In this problem we consider a convolutional code C over the quaternary eld F4 . The ele
ments of F4 may be denoted as cfw_00, 01, 10, 11 (additive representation) or as cfw_0, 1, , 2
(multiplicative represent
Final Exam
You have 3 hours (9:00-12:00) to complete the test.
This is a closed-book test, except that ve 8.5 11 sheets of notes are allowed.
Calculators are allowed (provided that erasable memory is cleared).
There are three problems on the quiz.
Final solutions
Problem F.1 (70 p oints)
In this problem we will consider coded modulation schemes based on a one-to-one mapping
t : F3 A from the nite eld F3 to a 3-simplex signal set A in R2 with energy E (A)
per symbol. The symbols from A will be trans
Midterm Quiz
You have 110 minutes to complete the quiz.
This is a closed-book quiz, except that three 8.5 11 sheets of notes are allowed.
Calculators are allowed (provided that erasable memory is cleared), but will probably
not be useful.
There ar
Midterm
You have 110 minutes (9:05-10:55 am) to complete the test.
This is a closed-book test, except that three 8.5 11 sheets of notes are allowed.
Calculators are allowed (provided that erasable memory is cleared).
There are two problems on the
Midterm solutions
Problem M.1 (60 p oints)
Your boss wants you to do a feasibility study for a digital communication system with the
following characteristics.
You are allowed to use the frequency band B between 953 and 954 MHz. The allowed
signal power i
6.451 Principles of Digital Communication II
MIT, Spring 2005
Wednesday, March 16, 2005
Handout #13
Midterm
You have 110 minutes (9:05-10:55 am) to complete the test.
This is a closed-book test, except that three 8.5 11 sheets of notes are allowed.
C
6.451 Principles of Digital Communication II
MIT, Spring 2005
Monday, March 28, 2005
Handout #14
Midterm solutions
Problem M.1 (70 p oints)
In this problem, we will study a class of codes called product codes.
Suppose that C1 and C2 are two binary linear
6.451 Principles of Digital Communication II
MIT, Spring 2005
Wednesday, February 2, 2005
Handout #3
Due: Wednesday, February 9, 2005
Problem Set 1
These exercises use the decibel (dB) scale, dened by:
ratio or multiplicative factor of 10 log10 dB.
The fo
6.451 Principles of Digital Communication II
MIT, Spring 2005
Wednesday, Feb. 9, 2005
Handout #4
Problem Set 1 Solutions
Problem 1.1 (Compound interest and dB)
How long does it take to double your money at an interest rate of P %? The bankers Rule
of 72 e
6.451 Principles of Digital Communication II
MIT, Spring 2005
Wednesday, February 9, 2005
Handout #5
Due: Wednesday, February 16, 2005
Problem Set 2
Problem 2.1 (Cartesian-product constellations)
(a) Show that if A = AK , then the parameters N , log2 M ,
6.451 Principles of Digital Communication II
MIT, Spring 2005
Wednesday, Feb. 16, 2005
Handout #6
Problem Set 2 Solutions
Problem 2.1 (Cartesian-product constellations)
(a) Show that if A = AK , then the parameters N , log2 M , E (A ) and Kmin (A ) of A a