Latest revision as of 07:08, 9 November 2010

The Sampling Theorem, in your own words

One question on an ECE301 midterm was the following:

State the Sampling Theorem in your own words.

As the results were much worse than expected, students were given an opportunity to earn bonus points by posting their answers online so we could all read/discuss them.

What did you answer ? How many points did you get?

I Wrote

The sampling theorem states that for a signal x(t) to be uniquely reconstructed, its X(jw) = 0 when |w| > wm, and the sampling frequency, ws, must be greater than 2wm

I got a 7/10 on this because I did not say what it is being reconstructed from. Also I used w because I did not know how to type omega in this file.

I Wrote

My Definition:

A signal can be recovered from sampling if

• The Signal is bandlimited and the Sample Frequency ($ \omega_s $) is greater than $ 2\omega_{max} $ (maximum frequency)

                    $ \omega_{s}>2\omega_{max} $  

Recieved 9/10 Points because it is not clear if I meant $ 2\omega_{max} $ or $ \omega_{max} $ is the maximum frequency

I Wrote

My definition of the sampling theorem:

In order to sample a signal that can be recovered back into the original sample, the sampling frequency, $ \omega_{s} $ , must be more than twice the highest frequency of the signal, $ \omega_{m} $.

I got $ \frac{7}{10} $ on it because I forgot to say that the signal must be band limited.

I Wrote

The sampling theorem states that a set of samples of a signal can be reconstructed into the original signal iff the original system is band limited and the sampling frequency is greater than twice the maximum frequency for non-zero values of the original function

I lost 1 point for saying "iff" since it is not an if and only if I lost 2 points for "for non-zero values of the original function" Not too sure why but I'm sure something about the statement must be ambiguous.

I Wrote

If there exists an w_m such that X(w) = 0 for |w| > w_m (band limited), then a signal sampled at a frequency w_s of > 2*w_m will be capable of being reconstructed. This frequency of 2*w_m is called the Nyquist rate.

I Wrote

If there's a signal, it can be recovered in the case of ws > 2wm" That's what I wrote. I got 3 points because I think I didn't mention signal should be band limited, what wm is. Also, I needed to mention what signal can be recovered from.

I Wrote

"If a signal x(t) is band limited X(j$\omega$)=0 for $\omega$>wm, then it can be recovered from its samples."

I Wrote

Here is the definition I put on the test (got a 7/10 - I forgot to mention it must be band-limited):

"In order to accurately reproduce* a signal, you must sample it at more than twice its maximum frequency. (This ensure accurate reproduction.) Otherwise, aliasing may occur."

• Mimi commented that "reproduce" is the wrong word. I'm guessing she wanted "reconstruct", so what's the difference between the two? They seem to mean pretty much the same thing to me, in this context.

I Wrote

The only way a signal can be perfectly reconstructed from its sampled version is if the original signal is band limited. It is sufficient (but not always necessary) for the signal to be sampled higher than its Nyquist rate (two times its $ \omega_m $ in order to be perfectly reconstructed. $ \omega_{sample}>2\omega_{max} $ sufficient for perfect reconstruction

I got a score of $ \frac{8}{10} $ because I did not define what $ \omega_m $ is.

I Wrote

My definition of the sampling theorem:

The sampling theorem states that any band-limited signal can ALWAYS be reconstructed from its samples without aliasing if the sampling frequency is greater than twice the highest frequency in the signal being sampled. Twice the highest frequency is not always required, but it guarantees a reconstruction that is free of aliasing.

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