Question

 Consider the sinusoidal signal X(t) = sin(t + Θ), where Θ ∼ Uniform([−π, π]).

Let Y (t) = d/dtX(t). 

(a) Find the first-order PDF of the process Y (t). 

(b) Find E[Y (t)]. 

(c) Find the autocorrelation function of Y . 

(d) Find the power spectral density of Y . 

(e) Is Y ergodic with respect to the mean?

Answer 1

(a) First-order PDF of the process Y(t): To find the first-order PDF of Y(t), we need to determine the probability distribution of its derivative, which is the second derivative of X(t).

The derivative of X(t) with respect to t is given by:

Y(t) = d/dt X(t) = d/dt sin(t + Θ)

Differentiating the sine function, we have:

Y(t) = cos(t + Θ)

Since Θ follows a uniform distribution on the interval [-π, π], its probability density function (PDF) is constant within that interval.

Therefore, the first-order PDF of Y(t) is also constant within the range [-1, 1] since the cosine function oscillates between -1 and 1.

(b) Expected value of Y(t) (E[Y(t)]): To find the expected value of Y(t), we need to calculate the average of Y(t) over its entire range.

E[Y(t)] = ∫ Y(t) * f_Y(t) dt

Since the PDF of Y(t) is constant within the range [-1, 1], the expected value can be calculated as:

E[Y(t)] = ∫ Y(t) * c dt

Where c is the constant value of the PDF of Y(t).

(c) Autocorrelation function of Y: The autocorrelation function of Y(t) measures the correlation between Y(t) and Y(s), where s and t are time points.

The autocorrelation function can be defined as:

R_Y(t, s) = E[Y(t) * Y(s)]

To calculate this, we need to consider the joint probability distribution of Y(t) and Y(s). Since Y(t) and Y(s) are derived from the same process, their joint distribution depends on the phase difference between t and s.

(d) Power spectral density of Y: The power spectral density (PSD) of Y(t) is a measure of the distribution of power with respect to frequency in the random process.

To find the PSD, we need to take the Fourier Transform of the autocorrelation function obtained in part (c). The Fourier Transform will give us the PSD as a function of frequency.

(e) Ergodicity of Y with respect to the mean: To determine if Y is ergodic with respect to the mean, we need to analyze whether the time average and ensemble average of Y converge to the same value.

In this case, since Y(t) = cos(t + Θ) and Θ is a random variable with a uniform distribution, the mean of Y(t) is zero, i.e., E[Y(t)] = 0.

If the time average of Y(t) over a long period converges to zero, and the ensemble average (expected value) is also zero, then Y is ergodic with respect to the mean.

To determine the time and ensemble averages, you may need to provide a specific integration range or further details about the duration and properties of the signal Y(t) that you want to analyze.