Suppose We Have a Source Consisting of Two Symbols $X$ And $Y$

Essay

Suppose we have a source consisting of two symbols, $X$ and $Y$ , with probabilities $P_{X}=2 / 3$ and $P_{Y}=1 / 3$ .
(a) Using Arithmetic Coding, what are the resulting bitstrings, for input $XXX$ and $YXX$ ?
(b) A major advantage of adaptive algorithms is that no a priori knowledge of symbol probabilities is required.
(1) Propose an Adaptive Arithmetic Coding algorithm and briefly describe how it would work. For simplicity, let's assume both encoder and decoder know that exactly $k=3$ symbols will be sent each time.
(2) Assume the sequence of messages is initially $3 XXX \mathrm{~s}$ , followed by $11 YYY \mathrm{~s}-$ the sequence is $X X X X X X X X X Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y$ $Y Y Y$ .
Show how your Adaptive Arithmetic Coding algorithm works, for this sequence, for encoding: (i) the first $XXX$ , (ii) the second $XXX$ , and (iii) the last $YYY$ .