Electrical engineering - energy of the following discrete sequence and samll questions

Question

Can anyone help me with the working for the following problems or just one of them. Coould you show the working so i know how the answer is obtained, thanks.

Calculate the energy of the discrete sequence, x[n] = {0,−1, 0, 1, 2}:

Calculate the power in the signal x(t) = 3 sin(5t)

Calculate the output, y[n], when the sequence x[n] = {1, 0, 1, 1}, is input to a linear time-invariant (LTI) system that has an impulse response, h[n] = {1, 2, 1}:

If an 8 kHz sinewave is sampled at 12 kHz what is lowest (positive) frequency present in the sampled signal?

Regarding Fourier representations which of the following is FALSE:

(a) if the time domain is periodic then the frequency domain is discrete and vice versa,

(b) linear discrete-time convolution can be efficiently calculated using the periodic convolution property of the Discrete Fourier Transform,

(c) the frequency response of a stable, continuous-time LTI system is given by the Fourier Transform of its impulse response,

(d) the continuous- and discrete-time Fourier series are not applicable to LTI systems analysis because of Gibbs’ phenomenon,

(e) if the time domain is continuous then the frequency domain is non-periodic and vice versa.

Answer 1

The energy of a signal, in this case is the sum of the squares of its components. So your answer would be: E = 0^2 + (-1)^2 + 0^2 + 1^2 + 2^2 = 4.

The power, or normalized average power, is defined as the Lim(T->Inf) Integral(-T/2,T/2, (3 sin(5t))^2 dt).

You should be able to compute that in Mathematica, Maple, or HP or TI graphing calculator.

The easiest way to determine y[n] from x[n] and h[n] is through convolution. The equation is (pardon the notation):

Sum(k=-Inf, k=Inf, x[k] h[n-k])

Here you 'reverse' one of the sequences, start with a k that's at the edge of the given sequence and compute the sum of their multiples. This can be tricky at first, but you should get:

{ (1*1+0*0+0*1+0*1), (2*1+1*0+0*1+0*1), (1*1+2*0+1*1+0*1), (0*1+1*0+2*1+1*1), (0*1+0*0+1*1+2*1), (0*1+0*0+0*1+1*1) }

which gives you:

{1, 2, 2, 3, 3, 1}. Notice that the x[n] sequence is intact and the h[n] sequence is "dragged" through it. h[n] would be reversed in order, but it's a palindrome, so it looks the same.

Nyquist theorem states that a sampled signal can only be reconstructed to a maximum frequency of half its sampling rate. A sinusoid sampled at 12kHz can only accurately describe a maximum 6 kHz signal. Since harmonics exist, the lowest positive frequency will be 4 kHz.

Regarding Fourier representations: (d) is obviously false, because Gibbs phenomenon is accurately modeled by differential equations and any book on Fourier analysis shows how it applies directly to analysis of continuous- and discrete-time systems.

(a) is misleading, because it shouldn't be "discrete" but should be an impulse. The frequency domain is still continuous, but contains two impulses (at the frequency of the periodic signal and the negative frequency as well).