Attention, please show the calculations in details, comments have to be provided.
You have information source with M symbols.
1. Calculate the amount of the information in bits for one M-ary symbol, considering that the probabilities of the symbols are the same.
2. Calculate the amount of the information in bits for M symbols.
3. Calculate the entropy of the source, considering that the probabilities of the symbols are the same. Check and explain the result.
4. Calculate the redundancy of the source, considering that the probabilities of the symbols are the same. Check and explain the results.
You have probability distribution for the source with the M symbols.
5. Calculate the amount of the information in bits for M symbols, considering given probabilities.
6. Calculate the entropy of the source, for given probability distribution. Make conclusions.
7. Calculate the source redundancy , for given probability distribution. Compare with the value You got in the 4. Explain.
8. Perform economy coding with the Haffman code.
9. Decode bit sequence with the number B. Explain.
10. Calculate the source redundancy after coding, check, compare with the values got in 4 and 7, explain.
3 Dinckan Alper M10 Source8 B3
Nr.3. 11011111 10110010 11111010 01001111 10111100
Nr 8
1 0.2
2 0.2
3 0.11
4 0.01
5 0.01
6 0.3
7 0.07
8 0.04
9 0.02
10 0.04
Sure! Let's go through each question step-by-step and explain the calculations and process involved.
1. Calculate the amount of information in bits for one M-ary symbol, considering that the probabilities of the symbols are the same.
To calculate the amount of information in bits for one symbol, we use the formula:
I = log2(M)
Where M is the number of symbols in the source. Since the probabilities of the symbols are the same, each symbol has an equal chance of occurring.
2. Calculate the amount of information in bits for M symbols.
To calculate the amount of information in bits for M symbols, we multiply the amount of information for one symbol by M:
Total Information = I * M
3. Calculate the entropy of the source, considering that the probabilities of the symbols are the same. Check and explain the result.
To calculate the entropy of the source, we use the formula:
H = -∑(P * log2(P))
Where P is the probability of each symbol. Since the probabilities of the symbols are the same, we can substitute P with 1/M.
4. Calculate the redundancy of the source, considering that the probabilities of the symbols are the same. Check and explain the results.
The redundancy of the source is calculated using the formula:
Redundancy = Total Information - Entropy
Since the probabilities of the symbols are the same, the redundancy will be zero. This means that there is no redundant information, and the source is as efficient as possible.
5. Calculate the amount of information in bits for M symbols, considering given probabilities.
To calculate the amount of information in bits for M symbols with given probabilities, we multiply each probability by the amount of information for that symbol and sum them up:
Total Information = ∑(P * log2(P))
6. Calculate the entropy of the source, for given probability distribution. Make conclusions.
To calculate the entropy of the source with a given probability distribution, we use the same formula as in question 3:
H = -∑(P * log2(P))
The entropy measures the average amount of information required to represent each symbol from the source. A lower entropy value indicates a more predictable source, while a higher entropy value indicates a more unpredictable source.
7. Calculate the source redundancy, for given probability distribution. Compare with the value you got in question 4. Explain.
The source redundancy can be calculated by subtracting the entropy from the total information:
Redundancy = Total Information - Entropy
Compare this value with the redundancy calculated in question 4. If the given probability distribution has unequal probabilities for the symbols, the redundancy will be non-zero. This means that there is redundant information in the source that can be compressed or optimized.
8. Perform economy coding with the Huffman code.
Huffman coding is a variable-length prefix coding algorithm that assigns shorter codes to more frequently occurring symbols and longer codes to less frequently occurring symbols.
To perform Huffman coding:
1. Order the symbols based on their probabilities in decreasing order.
2. Combine the two least probable symbols into a merged symbol with a probability equal to the sum of their probabilities.
3. Repeat step 2 until all symbols are combined into one merged symbol tree.
4. Assign a binary code to each symbol, with a shorter code for more probable symbols and a longer code for less probable symbols.
9. Decode the bit sequence with the number B. Explain.
Decoding the bit sequence with the Huffman code involves reversing the process of encoding. We start at the root of the Huffman tree and follow the path based on the bits in the sequence until we reach a leaf node, which represents a symbol.
10. Calculate the source redundancy after coding, check, compare with the values obtained in questions 4 and 7. Explain.
After coding with Huffman code, the source redundancy can be recalculated using the same formula:
Redundancy = Total Information - Entropy
Compare this value with the redundancies calculated in questions 4 and 7. The source redundancy after coding should be lower than the values obtained in questions 4 and 7. This indicates that Huffman coding has reduced the redundancy by efficiently compressing the source.