Information Theory Assignment | Get Paper Help
This work requires knowledge of Information Theory, NOT a paper writing but Questions with CALCULATIONS. Just follow the instruction of t he PDF to answer all the. questionsHomework Information Theory
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1.) Convolutional codes
A rate-1/3 convolutional code is given in octal representation as (4,7,7).
a) Show the corresponding shift-register circuit.
b) Draw the state diagram.
c) Determine the free (minimum) Hamming distance.
d) Draw the trellis starting from the initial state 00 and decode the received sequence
101,111,010,000 with Viterbi’s algorithm. Make clear, which intermediate
steps are necessary and make a final decision after all four segments have been
received. What is the most probable information sequence?
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2.) Reed-Solomon code
We know that an RS code can be defined by means of an IDFT over some number
field. The following pictures show the area(s) which are set to zero in DFT domain.
Which ones have the same minimum Hamming distance. Mark them by crosses.
How big is this minimum distance?
63
10 10
20
= 0
20
10 10
How many symbols can be corrected in case of erasures or errors?
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3.) Galois-field arithmetic
Let us work in a Galois field GF(2m) with a primitive polynomial a5+a+1.
a) m = ……….
b) a40 = ……….
c) (1+a)2(1−a)3 = ……….
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4.) Rate region
The following drawing shows an interference situation where two source signals
add up and both receivers receive the same combined information apart from the
independent noise contributions. The noise shall be considered to be zero mean,
unit variance AWGN (Additive White Gaussian Noise), i.e., we normalize the
signal and noise such that the noise is normalized to unit variance.
X1
X2 Y2
Y1
N
N
Determine the rate region limited by I(X1;Y1|X2), I(X2;Y2|X1), I([X1,X2]; [Y1,Y2]).
Remember the capacity formula for a vector channel with i.i.d. normalized noise
given by
I(X;Y) = H(Y)−H(Z) =
1
2
log2 |I+HRXHH|
Think about how to formulate the channel H for the three limits of the rate region.
Since we normalize the noise variance to one, the mutual informations shall be
written in terms of the signal variance, which we call S for each transmit signal
X1/2.
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5.) Some mixed questions
a) How can you produce equally distributed random numbers derived from an m
sequence shift register.
b) Looking at the block diagrams of the two Slepian-Wolf coding approaches, the
syndrome and the parity approach, let us try to allocate N, K, M, i.e., codeword
length, number of information bits, and number of parity bits, respectively, of
some code to the input and output of the encoder operations.
c) What is the relation between single parity-check code and repetition code?
d) Does allowing for a distortion increase of decrease the required encoding rate
for a source? Explain shortly!
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6.) Some mixed Crypto questions
a) Let us consider a physically degraded Gaussian channel, where Alice (transmit
power S) experiences an attenuation of 6 dB to Bob and the noise power N1 and
Eve sees another attenuation “behind” Bob of 6 dB and an additional noise power
N2. The original signal power at Alice shall be S.
Determine the Secrecy Capacity.
b) How would you in a few words describe the difference between secrecy and
authentication?
c)Which property of a Galois field can easily be used to generate the public/private
key pair?
d) You have to place encryption, source and channel coding in the right sequence.
Choose that suitable sequence and explain shortly.
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7.) Rate distortion theory
We assume a somewhat unusual distortionmodel in reminiscence of the Z-channel.
Let the probabilities for the source data X be given as u and (1−u).
1
q = 1− p
p
1
ˆX
X
0
1
0 P(X = 0) = u
P(X = 1) = 1−u
P( ˆX = 0) = b
P( ˆX = 1) = a
Determine the rate expressed by I(ˆX ;X) for the given distortion model.
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