Final answer:
The entropy of the source is approximately 2.8966. The Huffman code for the source is x1: 001, x2: 10, x3: 110, x4: 11, x5: 00, x6: 111, x7: 01. The average codeword length is approximately 2.13 and the efficiency is approximately 1.36.
Step-by-step explanation:
The entropy of a discrete memoryless source can be calculated using the formula: H(X) = -Σp(x)log2 p(x), where p(x) is the probability of each symbol in the source's alphabet and log2 represents the logarithm base 2. In this case, we have the following probabilities: p(x1) = 0.02, p(x2) = 0.11, p(x3) = 0.07, p(x4) = 0.21, p(x5) = 0.15, p(x6) = 0.19, p(x7) = 0.25. Calculating the entropy using these probabilities:
H(X) = -(0.02log2(0.02) + 0.11log2(0.11) + 0.07log2(0.07) + 0.21log2(0.21) + 0.15log2(0.15) + 0.19log2(0.19) + 0.25log2(0.25))
After performing the calculations, the entropy of the source is approximately 2.8966.
To design a Huffman code for this source, we need to arrange the symbols in decreasing order of probability. Starting with the least probable symbol, we combine the two least probable symbols into a single node and keep repeating this process until we form a tree. Assigning 0 to the left branch and 1 to the right branch, we can obtain the Huffman code for each symbol:
x1: 001, x2: 10, x3: 110, x4: 11, x5: 00, x6: 111, x7: 01
The average codeword length can be found by multiplying each symbol's probability by its codeword length, and summing these products:
Average codeword length = (0.02 * 3 + 0.11 * 2 + 0.07 * 3 + 0.21 * 2 + 0.15 * 2 + 0.19 * 3 + 0.25 * 2) ≈ 2.13
The efficiency can be calculated by dividing the entropy by the average codeword length:
Efficiency = Entropy / Average codeword length = 2.8966 / 2.13 ≈ 1.36