2. Probability and Random Processes

Question 1

Variance of Uniform distribution

Answer 1

(b-a)^2/12

Question 2

PDF of Normal (Gaussian) Distribution

Answer 2

See notes

Question 3

Does two signals being uncorrelated imply that they are independent?

Answer 3

No, but two independent signals are also uncorrelated

Question 4

What is the Central Limit Theorem?

Answer 4

Given a sum of i.i.d random variables, the distribution of the sum tends to the Normal distribution

Question 5

What is a random process?

Answer 5

It assigns a function of time to a random variable, where given a fixed value of time, a random process is a random variable, and for a fixed outcome, it will vary over time

Question 6

What is the ensemble of signals?

Answer 6

The signals obtained by scanning of all possible outcomes of the underlying random experiment

Question 7

What is the auto-correlation function and its equation?

Answer 7

Measures the correlation between between two samples of signals (could be the same signal at different times or different signals at the same time.
Equation in notes.

Question 8

What is a strict-sense stationary process (SSS)?

Answer 8

A process where the joint density function of all the outcomes of the process, are equal to the joint density function of the outcomes of the same process, but shifted by a time c (for all t)

Question 9

What is a wide-sense stationary process (WSS)?

Answer 9

A random process where:
Its mean is time independent (constant for all t)
Auto-correlation function depends only on the difference in time tau

Question 10

If a process is WSS, is it also SSS?

Answer 10

Only if the process is Gaussian.

Note: any SSS process is also WSS

Question 11

What is an ergodic process?

Answer 11

One where the time average (the average of a single sample of the process over all time) is equal to the ensemble average (average of the set of random variables of the process for all time)

Question 12

How is the output y(t) related to the input u(t) and impulse response h(t)?

Answer 12

y(t) = h(t) * u(t)

Question 13

If the mean of the input process X(t) is finite for all t and the LTI system is stable, how can you obtain the output mean?
If X(t) is WSS, is there a another way?

Answer 13

Output mean given by the convolution of the input mean and impulse response.

When X(t) is WSS, output mean given by input mean x H(0), where H(0) is the zero-frequency (dc) response of the system.

Question 14

If the power of the WSS input process X(t) to an LTI system is finite for all t, and system is stable, what is the auto-correlation of the output process?

Answer 14

Ry(t) = h(tau)[h(-tau)Rx(tau)]

Question 15

True or false. If the input process to an LTI system is WSS, then the output is also WSS.

Answer 15

True

Question 16

True of false. If the input process to an LTI system is SSS, this doesn’t necessarily mean that the output is SSS.

Answer 16

False

Question 17

What is the Einstein-Wiener-Khintchine relation between the PSD and auto-correlation function?

Answer 17

The PSD of a WSS process is equal to the Fourier transform of the processes’ auto-correlation function.

Question 18

How can you obtain the average power, given the PSD?

Answer 18

Integrate the PSD over the frequency spectrum (since it is often bounded)

Question 19

True of false. Rx(tau) an even function if X(t) is WSS.

Answer 19

True

Question 20

What is the PSD of an output process Y(t) of an LTI system, if the input X(t) is WSS?

Answer 20

Sy(f)=|H(f)|^2 x Sx(f)

Output PSD is input PSD times impulse response (frequency domain) squared.

Question 21

If the input process X(t) is complex, and passes through a complex valued LTI system, what is the auto-correlation function of Y(t).

Answer 21

Ry(tau)=h(tau)[h(-tau) * Rx(tau) ]