Reference no: EM131210866
Problem 1: The Fourier series expansion of a real periodic signal is in the form of
f(t) = a0 + n=1∑∞an cos(nω0t) + n=1∑∞ bnsin(nωot)
(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 HL(z) to a high pass filter
HH(z) = HL(-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.
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