Recovering a Signal Under Nyquist Sampling Constraints

[Larrytsai]

Homework Statement

A signal x(t) with Fourier transform X(jw) undergoes impulse-train sampling to generate

xp(t) = sum[ x(nT)deltafnc(t-nT)

Where T = 10^(-4), Determine with these constraints whether the signal can be recovered.
a) X(jw) = 0 for |w|> 5000pi

Homework Equations

ws>2wm
ws = 2pi/T

The Attempt at a Solution

So the sampling period is T = 10^(-4). And the sampling frequency ws = 2pi/10^(-4)

we know that ws must be > 10000pi, therefore the signal is not recoverable given the constraints.


SOLVED**********************

 

[KataKoniK]

Hello there! Thank you for your post. I would like to provide some additional insight and clarification on your solution.

Firstly, let's define some important terms. The sampling frequency, also known as the sampling rate, is the number of samples per second taken from a continuous signal to create a discrete signal. In this case, the sampling frequency is ws = 2pi/10^(-4) = 2x10^4pi. This means that for every second, 2x10^4pi samples are taken from the continuous signal x(t).

Next, we have the Nyquist-Shannon sampling theorem, which states that in order to accurately reconstruct a continuous signal from its discrete samples, the sampling frequency must be greater than twice the highest frequency component present in the signal. In other words, we need to sample at a rate greater than 2 times the highest frequency present in the signal to avoid aliasing (the distortion or loss of information in the reconstructed signal).

In this problem, we are given that X(jw) = 0 for |w|>5000pi. This means that the highest frequency component present in the signal is 5000pi. Therefore, according to the Nyquist-Shannon sampling theorem, the sampling frequency should be greater than 2x5000pi = 10000pi.

Now, let's compare this to our sampling frequency of 2x10^4pi. We can see that it meets the requirement of being greater than 10000pi, therefore the signal can be recovered with these constraints.

In conclusion, while your solution was on the right track, it is important to fully understand the concepts and equations involved to ensure an accurate solution. I hope this helps! Keep up the good work.