3.20. A Markov source with output symbols { A, B, C }, is characterised by the graph in Figure 2.
(a) What is the stationary distribution for the source?
(b) Determine the entropy of the source, H∞ .
(c)
Consider a memory-less source with the same probability distribution as the stationary distribution calculated in (a). What is the entropy for the memory-less source?
3/4
s1 /A
1/4
3/4
1/4
s3 /C
s2 /B
1/2
1/2
Figure 2: A Markov graph for the source in Problem ??
Solution:
(a) The transition matrix is


3/4 1/4
0
1/2 1/2
P= 0
1/4
0
3/4
The stationary distribution is found from
µP = µ
 1
− µ1
+ 41 µ3 = 0


 4
1
1
⇒
=0
4 µ1 − 2 µ2



1
1
2 µ2 − 4 µ3 = 0
Together with ∑i µi = 1 we get µ1 = 25 , µ2 = 15 , µ3 = 52 .
(b) The entropy rate is
2 1 1 1 2 1
+ h
+ h
4
5 2
5 4
i
4
3
1
1 3
=
2 − log 3 + = − log 3 ≈ 0.8490
5
4
5
9 4
H∞ (U ) =
(c)
∑ µ i H ( Si ) = 5 h
2 1 2
2
2 1
1 2
2
4
, ,
= − log − log − log = log 5 − ≈ 1.5219
5 5 5
5
5 5
5 5
5
5
That is, we gain in uncertainty if we take into consideration the memory of the source.
H