Mathematical Foundation For Information Technology

a) Consider a binary symmetric communication channel, whose input source is the alphabet X = {0, 1} with probabilities {0.5, 0.5}; whose output alphabet is Y = {0, 1}; and whose channel matrix is where x is the probability of transmission error.

1. Find the entropy of the source, H(X).
2. Find the probability distribution of the outputs, p(Y) and the entropy of this output distribution, H(Y).

b) State and prove Shannon’s Source Coding Theorem.

a) Define the Kolmogorov algorithmic complexity K of a string of data. What relationship is to be expected between the Kolmogorov complexity K and the Shannon entropy H for a given set of data?

b) A Hamming Code allows reliable transmission of data over a noisy channel with guaranteed error correction as long as no more than one bit in any block of 7 is corrupted. What is the maximum possible rate of information transmission, in units of (data bits reliably received) per (number of bits transmitted), when using such an error correcting code?

a) What is the entropy H, in bits, of the following source alphabet whose letters have the probabilities shown below.

b) Why are fixed length codes inefficient for alphabets whose letters are not equiprobable? Discuss this in relation to Morse Code.

a) Describe fuzzy relation. Explain the operation of fuzzy sets with a suitable example.

b) Explain fuzzy Cartesian and composition with a suitable example and the concept of fuzzy set with suitable examples.

a) i) Given a signal x[n] = δ[n] + 0.9 δ[n – 6]. Find the Discrete Time Fourier Transform for 8 points.

ii) Find the Z transform of �� (n – m).

b) Find the period of the signal x(t) = 10 sin 12 π t + 4 cos18 πt .

a) Calculate similarity between two time series using discrete wavelet transform coefficients.

b) What is the relation between lifting scheme, biorthogonal wavelet and translation invariance?

a) Two fair coins are flipped. As a result of this, tails and heads runs occurred where a tail run is a consecutive occurrence of at least one head. Determine the probability function of number of tail runs.

b) The length of life of an instrument produced by a machine has a normal distribution with a mean of 9.4 months and a standard deviation of 3.2 months. What is the probability that an instrument produced by this machine will last between 6 and 11.6 months?

a) The annual salaries of workers in a large manufacturing factory are normally distributed with a mean of Rs. 48,000 and a standard deviation of Rs. 1500. Find the probability of workers who earn between Rs. 35,000 and Rs. 52,000.

b) A meeting has 12 employees. Given that 8 of the employees is a woman, find the probability that all the employees are women.