Answer the following questions for the continuous-time periodic signals x(t), y(t), and z(t):
x(t):
T - T.
T + To
To 0T
y(t):
T To
T + 7
(a) Express y(t) and z(t) in terms of x(t) for an appropriate value of T/T.
(b) Using the result of (a), determine the Fourier series Y[k] for the signal y(t) and sketch the value for the range -5 ≤ k ≤ +5. [Hint: Set 3π/T = C.]
(c) Using y(t) = dz(t)/dt, write the relation of Fourier series Y[k] and Z[k] and calculate Z[k].
(d) Calculate the power of the signals y(t) and z(t).
(e) The partial sum P of the power is defined as follows:
Px = |X[0]| + {|X[k]|^2 + |X[-k]|^2}
For signals y(t) and z(t), find the least J for which P is greater than 99% power. Answer using the following table.
k Y[k] Pv/2 Z[k] Pz*3
1 0.826993 0.683918 0.39486 0.935488
2 0.413497 0.854897 0.098715 0.993956
3 4 0.206748 0.897642 0.001542 0.99397
5 0.165399 0.924999 0.015794 0.995467
6 7 0.118142 0.938957 0.000164 0.995467
8 0.103374 0.949643 0.00617 0.995695
10 0.082699 0.956482 3.95E-05 0.995695
40 0.020675 0.98874 1.54E-07 0.995809
41 0.020171 0.989147 0.000235 0.99581
42 43 0.019232 0.989517 1.15E-07 0.99581
44 0.018795 0.98987 0.000204 0.99581
45 46 0.017978 0.990194 8.82E-08 0.99581
47 0.017596 0.990503 0.000179 0.99581
48 49 0.016877 0.990788 6.85E-08 0.99581
50 0.01654 0.991062 0.000158 0.99581