Before delving into the true comprehensive definition of the sampling theorem, I will first discuss an INAPPROPRIATE definition of the sampling theorem.
For example, if one were to have wrote, "The sampling theorem specifies that to properly recreate a signal (without aliasing) you must sample at a frequency above the Nyquist rate. (Nyquist rate= $ 2 \omega_m\ $)" on the third midterm exam they would have merited only 5 points out of a possible ten. Since the proper definition should be easily understood be the common layman, one MUST specify EXACTLY what $ \omega_m\ $ is. It is also very important to note that the minute difference between should and must cost this person an entire point on their exam. In fact, for some unique signals, it is possible (though not probable) to actually be able to reconstruct a signal when sampling at frequencies less than the Nyquist rate. Finally, this definition is not correct because it does not specify from what the signal is being recreated. If the signal is for example not band-limited, it cannot be reconstructed at all.
Wow, this individual must be pretty thick-headed to have studied this material for so long and only achieve a failing grade (50%) on the most straight forward question on the exam. If he had only assumed that the audience he was writing the theorem for was completely uneducated (rather than the our esteemed professor) he may have been able to do a little better on this question.
If he was thinking clearly earlier, he may have written something like this:
- The sampling theorem specifies that if the original signal is continuous (which means that it is valid at all real values of time) and band-limited (which means that the Fourier transform of the signal is zero for frequencies above the absolute value of any real frequency) that if you sample it at a frequency greater than the Nyquist rate (which is equal to two times the absolute value of the frequency at which the Fourier transform of the signal is zero for values greater than said frequency) you are guaranteed to be able to successful reconstruct the signal without aliasing (which is what happens when you mess up!!). However, it is important (in the opinion of some) to note that when sampling at frequencies less than the Nyquist rate in rare cases you are able to properly reconstruct the signal.
Additional Definition --andrew.c.daniel.1, Thu, 06 Dec 2007 23:02:19
Sampling Theorem: The sampling theorem states that a band-limited signal is able to be completely recovered from it's samples as long as the frequency it is sampled at exceeds the Nyquist rate, $ 2 \omega_m\ $, where $ \omega_m\ $ is the frequency where the Fourier transform of the signal equals zero for all frequencies greater than $ \omega_m\ $.
Additional Definition --corey.e.zahner.1, Fri, 07 Dec 2007 13:36:43
Sampling Theorem: Given a signal $ x(t)\ $, it can be uniquely reconstructed from its transform $ X(j \omega)\ $, if $ \omega_s > 2 \omega_m\ $, where $ \omega_s\ $ is the sampling frequency and $ |\omega_m|\ $ is the point above which $ X(\omega) = 0\ $
This definition is worth 0/10 points and a frowny face. The word transform should be changed to sampling to get any points
... --matthew.a.harger.1, Sat, 08 Dec 2007 19:43:45
Don't Know what I'm doing, so my def goes here.
Sampling Theorem: The sampling theorem states that given a signal $ x(t)\ $, the signal is sampled with shifted deltas by some period T. [$ x(t)\ $ multiplied by $ \sum_{-\infty}^{\infty} \delta (t-nT) $. However, the sampling frequency must be at least twice the maximum signal frequency or aliasing occurs and it can't be guaranteed the original signal can be recovered.
4/10 -- Grading comments as follows:
- >> "No. It states that a signal can be recovered from its samples provided two
conditions are met."
- >> Should have added "Band Limited" to definition >> Must be "Greater Than" twice the maximum signal frequency >> Aliasing "May" occur
