Power spectral density of a stationary random process

Question

The stationary random process X(t) has a power spectral density denoted by Sx(f).

a. What is the psd of Y(t) = X(t) - X(t-T)?

b. What is the psd of Z(t) = X'(t) - X(t)?

What should the approach to this question be and the detailed solution?

Answer 1

You need to know how the power spectrum is transformed when the process is filtered by a linear time-invariant filter. Let $Y(t)$ be the output of a linear time-invariant filter with frequency response $H(f)$, and $X(t)$ is the input signal. Then the power spectral density of the output process $Y(t)$ is given by

$$S_Y(f)=S_X(f)|H(f)|^2$$

What remains is to find the frequency responses of the systems with outputs $Y(t)$ and $Z(t)$. They are given by

$$H_1(f)=1-e^{-i2\pi fT}\\ H_2(f)=i2\pi f - 1$$

Computing the magnitudes of these functions should be easy for you.