Channel Coding I Exercises – WS 2012/2013 – - Universität Bremen

4 CONVOLUTION CODES November 7, 2012 11
4 Convolution codes
4.1 Fundamental principles
Exercise 4.1
Convolution code
Given is a convolution code with the code rateR = 1/3, memorym = 2 and the generator polynomials
g 1 (D) = 1+D +D 2 and g 2 (D) = 1+D 2 and g 3 (D) = 1+D +D 2 .
a) Determine the output sequence for the input sequence u = (011010).
b) Sketch the corresponding Trellis chart in the case of the under a) given input sequence.
c) Sketch the state diagram of the encoder.
d) Determine the corresponding free distance d f .
4.2 Characterization of convolution codes
Exercise 4.2
Catastrophic codes
Given is the convolution code with the generator polynomial g(D) = (1+D 2 ,1+D). Show that this
is a catastrophic code and explain the consequences.
4.3 Distance properties of convolution codes
Exercise 4.3
Distance properties of convolution codes
a) Given is the non-recursive convolution code with the generatorsg 0 (D) = 1+D+D 3 andg 1 (D) =
1+D+D 2 +D 3 . Determine by means of the MATLAB-routine iowef conv the IOWEF and
present a w,d (in the script also called T w,d ) for even and odd input weights in separate diagrams
(maximal distance d max = 20). How large is the free distance of the code and what’s striking with
reference to the weight distribution?
b) Calculate the coefficients a d and c d .
c) Estimate the word- and bit error rates with the help of the Union Bound for the AWGN-channel
in the range 0 dB ≤ E b /N 0 ≤ 6 dB. Draw the determined error rates for several maximally
considered distances d.
d) The convolution code shall now be terminated and considered as block code of the length n = 50.
Determine the IOWEF with the function iowef block conv and calculate the bit error rate
with the help of the routine pb unionbound awgn (now, the block length n = 50 and the
information length k = 22 is to be specified). How can the difference to the results of item c) be
explained?
Exercise 4.4
Comparison of NSC- and RSC-codes