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 $YYY$.

Question

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 $YYY$.

Quizplus · Accepted Answer

The Answer is (a) Using Arithmetic Coding, the resulting bitstrings for input

XXX

and

YXX

are as follows:
- For input

XXX

, the resulting bitstring would be

0.100

.
- For input

YXX

, the resulting bitstring would be

0.010

.

(b)
(1) An Adaptive Arithmetic Coding algorithm would work by dynamically updating the probabilities of symbols as new symbols are encountered. The algorithm would maintain a probability model that is updated based on the frequency of occurrence of symbols. When encoding, the algorithm would use the updated probability model to determine the intervals for each symbol and generate the corresponding bitstring. For simplicity, let's assume both encoder and decoder know that exactly

k=3

symbols will be sent each time.

(2) For the given sequence of messages:
- Initially, the sequence is

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 Y Y Y Y Y Y Y Y Y Y Y

.

(i) Encoding the first

XXX

:
- The initial probability model would assign

P_{X}=2 / 3

and

P_{Y}=1 / 3

.
- The algorithm would use the probability model to encode the first

XXX

and generate the resulting bitstring.

(ii) Encoding the second

XXX

:
- After encountering the first

XXX

, the algorithm would update the probability model based on the frequency of occurrence of symbols.
- The updated probability model would be used to encode the second

XXX

and generate the resulting bitstring.

(iii) Encoding the last

YYY

:
- After encountering the sequence of

11 YYY \mathrm{~s}

, the algorithm would again update the probability model based on the frequency of occurrence of symbols.
- The updated probability model would be used to encode the last

YYY

and generate the resulting bitstring.

Overall, the Adaptive Arithmetic Coding algorithm would adapt to the changing symbol probabilities and generate the corresponding bitstrings for the given sequence of messages.

Suppose We Have a Source Consisting of Two Symbols XXX And YYY

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