As seen in the previous problem, the Fourier series is one of a possible class of representations in terms of orthonormal functions. Consider the case of the Walsh functions which are a set of rectangular pulse signals that are orthonormal in a finite time interval [0, 1]. These functions are such that: (i) take only 1 and −1 values, (ii) φ_k(0) = 1 for all k, and (iii) are ordered according to the number of sign changes.

(a) Consider obtaining the functions {φ_k}_k=0,...,5. The Walsh functions are clearly normal since when squared they are unity for t ∈[0, 1]. Let φ₀(t) = 1 for t ∈[0, 1] and zero elsewhere. Obtain φ₁(t)with one change of sign and that is orthogonal to φ₀(t). Find then φ₂(t) which has two changes of sign and is orthogonal to both φ₀(t) and φ₁(t). Continue this process. Carefully plot the {φ_i(t)},i = 0,...,5. Use the function stairs to plot these Walsh functions.

(b) Consider the Walsh functions obtained above as sequences of 1s and −1s of length 8, carefully write these 6 sequences. Observe the symmetry of the sequences corresponding to {φ_i(t),i = 0, 1, 3, 5}, determine the circular shift needed to find the sequence corresponding to φ₂(t) from the sequence from φ₁(t), and φ₄(t) from φ₃(t). Write a MATLAB script that generates a matrix ­Ф with entries the sequences, find the product (1/8) ФФ^T, explain how this result connects with the orthonormality of the Walsh functions.

(c) We wish to approximate a ramp function x(t) = r(t), 0 ≤ t ≤ 1, using {φ_k}_k=0,...,5. This could be written

r = ­Фa

where r is a vector of x(nT) = r(nT) where T = 1/8, a are the coefficients of the expansion, and ­the Walsh matrix found above. Determine the vector aand use it to obtain an approximation of x(t). Plot x(t) and the approximation x̂(t) (use stairsfor this signal).