Eighth Assignment, Convolutional Codes: due Wed.
- Find the inverse of 1 + x + x^3
as a formal power series.
Explain how this question relates to convolutional codes.
- CT 8.3.2 (Don't make the state diagrams, just find an
infinite message and the corresponding finite codeword.)
- (See CT 8.2.11, 12, 13; 8.3.3, 6)
Consider the convolutional code, C
with generator polynomials:
g_1(x) = 1 + x + x^2
and g_2(x) = 1 + x^2 + x^3
- a. Draw the convolutional encoder for C ,
the state diagram and the table
showing the output for each state and choice of X_3.
- b. Encode
(i) 1+ x^3
(ii) 1+ x + x^3
(iii) 1+ x + x^2 + x^3+ ....
- c. The following are codewords. Decode them.
(i) ... 00, 01, 10, 01, 01, 10, 11, 10, 11
(ii) ... 00, 01, 11, 11, 00, 11, 01, 00, 10, 01, 11
- Show that the minimum distance of C is
d=6. Find codewords with weight 6
(There are 5, up to shift).
- We can correct e errors provided e
is less than half of d . What size windows should be used
to correct e=1 errors and to correct e=2
- For each case, e=1 and e=2 ,
in the basic decoding algorithm how many paths do you need to keep
- Decode ... 00, 00, 00, 11 with the 1-error correcting algorithm.
- Can you find an error pattern that will be correctly decoded
by the 1-error correcting algorithm but not by the 2-error correcting
- CT 8.3.6