Reference no: EM132893190

Problem 1 Classify the following signals as energy signals or power signals and find the normalized energy or power of each:

(a) g(t) = 5 cos(2Π x 10⁵t), -∞ < t < ∞

(b) g(t) = {A cos(2Πf_ot), -T₀/2 ≤ t ≤ T₀/2, T₀ = 1/f₀
      g(t) = {  0,                elsewhere

(c) g(t) = { 5e^-3t,     t> 0, a > 0

               { 0,        elsewhere

Problem 2 Prove Parseval's Theorem for the following signal
g(t) = 20 cos(10t) + 30 cos (20t),

by calculating:

(a) the average power using time averaging,
(b) the average power using the coefficients of the exponential Fourier series.

Problem 3 The two-sided PSDof g(t) is given by S_g(f) = 10^-7 f².
(a) Find the average power in g(t) over the frequency band from 0 to 11 kHz
(b) Repeat (a) for the band from 5 to 6 kHz

Problem 4 Given a random process X(t) = A cos(2Πf_ot + Φ), where ,f₀ is a constant and A and (Φ) are independent random variables uniformly distributed in the ranges (-1,1) and (0, 2Π), respectively.
(a) Determine its mean, E{x(t)}
(b) Determine its autocorrelation function, R_X(t₁, t₂)
(c) Is the process WSS? Why?
(c1) Is the process ergodic in mean? Why?
(e) If the process is WSS, what is its power P_X?

Problem 5 A signal x(t) = 5∏(10⁴t) is applied at, the inplit of an ideal low pass filter with t musfer function H(f) = ∏(f/2x10⁴). Sketch X (f); the transfer function, H(f); and the output Y(f).

Problem 6 In this problem, we prove that distortionless transmission reguires a linear phase of the transfer function in addition to its constant magnitude. Assume that the magnitude and phase of the transfer function H(f) is given by

|H(f)| = 1,   |f| < B

|H(f)| = 0,  otherwise

Show that the output y(t) of a signal input x(t) that is band limited to B Hz is given by

y(t) = x(t -to) + k/2 [x(t -t0 -T) - x(t - to + T)]

Problem 7 The autocorrelation function of a random process X(t) is given by

Rx(r) = N0/2 δ(T).

Suppose that X (t) is the input to aim ideal bandpass filter with bandwidth B and center frequency fc. Determine the total power of the signal at the output of the filter.

Problem 8 For the signal

x(t) = 2a/t²+a²

Determine the essential bandwidth B Hz of x(t) such that the energy contained in the spectral components of x(t) of frequencies below B Hz is 99% of the signal energy E_x.