(a) Huffman trees enable construction of uniquely decodable prefix codes with optimal codeword lengths. The five codewords shown here for the alphabet (A.B.C.D.E} form an instantaneous prefix code. (i) Give a probability distribution for the five letters that would result in such a tree. (ii) Calculate the entropy of that distribution. (iii) Compute the average codeword length for encoding this alphabet, and relate your results to the Source Coding Theorem. 00 B01 D110 E 111 [3 2 marks] (b) What does it mean for a function to be "self-Fourier"? Show that the Gaussian function is self-Fourier. Name two other functions of importance in information theory that are self-Fourier, and in both cases mention a topic or theorem exploiting this property. [6 marks] (c) (i) In the FFT algorithm, if a discrete data sequence consists of N sample values (nominally N is some power of 2), what complex number is the primitive Nth root of unity which, raised to various powers, generates all the complex numbers needed to perform a discrete Fourier transform? [2 marks] If all the Nth roots of unity are known, by what mechanism are sequences of them selected that are needed for the kth frequency component? [2 marks] (d) Define the Kolmogorov algorithmic complexity K of a string of data. What approximate relationship is expected between K and the Shannon entropy H for the same source? Give a reasonable estimate of the Kolmogorov complexity K of a fractal, and explain why it is reasonable. [4 marks]