Exercises 10 - 2011-01-20
Channel Coding
Winter Term 2010/11
Prof. Dr.-Ing. M. Bossert
Dipl.-Ing. Ing. ENST A. Zeh
TAIT, University of Ulm
Exercise 10.1
You are given a convolutional code Cwith the following generator matrix:
G(D) = (1 + D2, D +D2).
a) Draw the shift register of the encoder.
b) Draw the state diagram of the encoder.
c) Is the generator matrix G(D)catastrophic? Why?
d) Determine the free distance dfr ee.
Hint: Use the equivalent noncatastrophic generator matrix Geq(D) = (1 + D, D)and draw
the corresponding shift register and the state diagram.
e) Assume that the word r= (10 11 01 00 11) has been received after transmission of a
codeword of the terminated code Cover a binary symmetric channel (BSC) with p < 0.5.
Decode rusing the Viterbi algorithm.
Hint: Use the trellis corresponding to the encoder given in (a).
Is the desicion of the Viterbi decoder unique? Why?
Exercise 10.2
Given the generator matrix GA(D) = (1 + D2+D3,1 + D+D3,1 + D+D2+D3)of the
convolutional code C.
a) Determine the rate RAand the memory mAof GA(D).
b) Draw the convolutional encoder for the generator matrix GA(D).
c) Determine the submatrices G0,G1,G2, and G3, where GA(D)is given by GA(D) =
G0+G1D+G2D2+G3D3. Write the generator matrix GA(D)in the time domain (at
least the first four rows!).
d) Give the rate Rr
Aof Gr
A, when Gr
Ais generated by terminating the generator matrix in the
time domain GAafter 30 information bits. What is the maximum and the minimum possible
rate obtained by termination?
e) Give the generator matrix GB(D)of rate RB=1
2, where GB(D)is generated by puncturing
the code Cusing the pattern [110], i.e., the third code bit in every block of three code bits
is erased. Determine the memory mBof GB(D). What is the relation between GB(D)and
GA(D)?
Channel Coding 1
