Find the hilbert-transform pair

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Reference no: EM131210866

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

f(t) = a0n=1an 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

2340_Fig.png

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

304_Fig1.png

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

Reference no: EM131210866

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