Counter-example to Nyquist's Sampling Theorem?

[chingkui]

The Sampling Theorem states that when you have a signal x(t) bandlimited to B Hz, then if you sample the signal at frequency f_s higher than or equal to 2B, then you can use the sample to reconstruct the original signal x(t) uniquely.
Yet, if you consider the simple sinusoidal signal x(t)=sin(2*pi*B*t), which obviously only has one frequency of B Hz, and sample with a frequency f_s=2B at time t=0,1/(2B),2/(2B),3/(2B),4/(2B),...,n/(2B),... then, all you get is a sequence x_sample(0)=x_sample(1)=...=x_sample(n)=sin(n*pi)=0, and you would naturally conclude that x(t)=0. Wouldn't it be a very simple counter-example to the sampling theorem?

 

[willib]

The Sampling Theorem states that when you have a signal x(t) bandlimited to B Hz, then if you sample the signal at frequency f_s higher than or equal to 2B, then you can use the sample to reconstruct the original signal x(t) uniquely.
Yet, if you consider the simple sinusoidal signal x(t)=sin(2*pi*B*t), which obviously only has one frequency of B Hz, and sample with a frequency f_s=2B at time t=0,1/(2B),2/(2B),3/(2B),4/(2B),...,n/(2B),... then, all you get is a sequence x_sample(0)=x_sample(1)=...=x_sample(n)=sin(n*pi)=0, and you would naturally conclude that x(t)=0. Wouldn't it be a very simple counter-example to the sampling theorem?

How do you figure this??
then, all you get is a sequence x_sample(0)=x_sample(1)=...=x_sample(n)=sin(n*pi)=0, and you would naturally conclude that x(t)=0.
the sampling theorem states that you must sample at least twice the highest expected frequency, which you are sampling..
what you get from sampling is a digital number..

 

[faust9]

How do you figure this??
then, all you get is a sequence x_sample(0)=x_sample(1)=...=x_sample(n)=sin(n*pi)=0, and you would naturally conclude that x(t)=0.
the sampling theorem states that you must sample at least twice the highest expected frequency, which you are sampling..
what you get from sampling is a digital number..

What chingkui is getting at is correct. Nyquist need not apply. Sampling of recurring wave patterns can be done at less than the wave frequence and still give enough useful information to reconstruct the original wave.

Here's a little device which utilizes this principle: http://www.bitscope.com/

I wouldn't use the bitscope to T/S a glitch problem (I.E. a messed up pulse at a frequence 1/2 higer than the bitscope frequence) because there's no guarantee that the bitscope will pick up the glitch--especially if said glitch occurred somewhat randomely. But, sampling of recurring or semirecurring wave patterns are easily and accuretly accomplished with a sample rate less than the input frequency.

 

[chingkui]

What I mean is that if you know the signal is bandlimited to B Hz, and you sample at frequency f_s=2B, if someone send you a sine wave x(t)=sin(2*pi*B*t), and you sample at t=0,1/(2B),2/(2B),3/(2B),4/(2B),...,n/(2B),... then you will be very unlucky and getting all zeros, which you probably will think there is no signal at all and you can have no way to reconstruct the original sinusoidal signal... thus violating the Sampling Theorem which garantee you can reconstruct the original signal.

 

[faust9]

What I mean is that if you know the signal is bandlimited to B Hz, and you sample at frequency f_s=2B, if someone send you a sine wave x(t)=sin(2*pi*B*t), and you sample at t=0,1/(2B),2/(2B),3/(2B),4/(2B),...,n/(2B),... then you will be very unlucky and getting all zeros, which you probably will think there is no signal at all and you can have no way to reconstruct the original sinusoidal signal... thus violating the Sampling Theorem which garantee you can reconstruct the original signal.

That is incorrect. I thought you were eluding to the fact that Nyquist need not apply in all situations. Here, plot sin(x) and sin(2x) you'll see these two functions cross at multiple points--enough to reconstruct a wave.

 

[rbj]

The Sampling Theorem states that when you have a signal x(t) bandlimited to B Hz, then if you sample the signal at frequency f_s higher than or equal to 2B, then you can use the sample to reconstruct the original signal x(t) uniquely.

no, the sampling theorem says that you must sample at a rate (f_s) strictly greater than 2B. f_s = 2B is not good enough (there are an infinite number of sinusoids at frequency B that alias to the same set of samples.

Yet, if you consider the simple sinusoidal signal x(t)=sin(2*pi*B*t),

yeah, that's a bummer. f_s must be bigger than 2B.

r b-j