Problem 1: The Fourier series expansion of a real periodic signal is in the form of

f(t) = a₀ +  _n=1∑^∞a_n cos(nω₀t) + _n=1∑^∞ b_nsin(nω_ot)

(a) Find the Hilbert-transform pair.

(b) Find the average power of the Hilbert-transform pair.

(c) Evaluate the inner product of the Hilbert-transform pair.

Problem 2:

(a) Show that the z → -z^-1 frequency transformation coverts a low pass filter H_L(z) to a high pass filter

H_H(z) = H_L(-z^-1)

(b) The speech scrambler utilizes the modulation sequence [+1, -1, +1, -1, +1, -1, +1, -1,...] for the redistribution of frequency components. The z → -z^-1 is a low pass-high pass frequency transformation technique. Compare these two methods and identify the similarities and differences.

Problem 3: Below, two data sequences were used for the interpolation experiments.

1. Consider a simple single-mode 128-point sequence

x(n) = sin(πn/64)  for n = 0, 1, ... 127

The 32-point subsequence corresponding to a lower sampling rate is in the form of

x?(n) = sin(πn/16)    for n = 0, 1, ... 31

Use FFT method to interpolate the subsequence x?(n) to 128 points. Plot the resultant 128-point sequence: it shows below

2. Consider the 64-point sequence

x(n) = 2^(-n/16)  for n = 0, 1, ... 63

The 32-point subsequence corresponding to a lower sampling rate is in the form of

x?(n) = 2^(-n/8)  for n = 0, 1, ... 31

Use FFT method to interpolate the subsequence x?(n) to 64 points. Plot the resultant 64-point sequence: it shows below

Answer the question: Does the interpolation technique work equally well for both sequences? Explain why so.