How does Nyquist's Frequency affect aliasing in discrete signals?

[JimmyJimmy]

Hi everyone!

Struggling to get my head about Nyquist's Frequency.. Have I got this right?

The Nyquist Frequency is 1/2 of the sampling rate of a discrete signal, and is the highest frequency that will allow a signal to be fully reconstructed without aliasing, at a given sampling rate. The standard frame-rate for a certain video camera is 30 frames per second. Therefore the camera is sampling at 30Hz. This means that the Nyquist Frequency is 15 Hz. Any vibrations applied to the camera about 15Hz will therefore not be visible in the footage.

Or is that completely wrong? Is it 60Hz rather than 15Hz?


Any help would be much appreciated!


Thanks,
JJ.

 

[DrGreg]

The Nyquist Frequency is 1/2 of the sampling rate of a discrete signal, and is the highest frequency that will allow a signal to be fully reconstructed without aliasing, at a given sampling rate. The standard frame-rate for a certain video camera is 30 frames per second. Therefore the camera is sampling at 30Hz. This means that the Nyquist Frequency is 15 Hz. Any vibrations applied to the camera about 15Hz will therefore not be visible in the footage.

Not quite right.

It means that a vibration at 17 Hz would be indistinguishable from a vibration at 13 Hz.
Or a vibration at 19 Hz would be indistinguishable from a vibration at 11 Hz.
Or a vibration at 29 Hz would be indistinguishable from a vibration at 1 Hz.

To avoid aliasing you would need to ensure that there were no vibrations above 15 Hz.

 

[JimmyJimmy]

Not quite right.

It means that a vibration at 17 Hz would be indistinguishable from a vibration at 13 Hz.
Or a vibration at 19 Hz would be indistinguishable from a vibration at 11 Hz.
Or a vibration at 29 Hz would be indistinguishable from a vibration at 1 Hz.

Thanks for the reply.

Does this mean that if I isolate the vibrations below 15Hz, no other vibrations will be apparent?

 

[DrGreg]

Does this mean that if I isolate the vibrations below 15Hz, no other vibrations will be apparent?

No. In that case a real vibration of 29 Hz would look, on the video, like a vibration of 1 Hz.

Have you ever seen a film or video of a rotating wheel where the wheel gives the illusion of rotating backwards? (Wagon-wheel effect) A rotation of 29 spokes-per-second clockwise would look like a rotation of 1 spoke-per-second anticlockwise.

 

[FactChecker]

Aside from the problem of aliasing from higher frequencies there is another consideration. The accuracy of the amplitude estimate of the lower frequencies can be very bad if the data used is not over infinite time. Given an infinite data sample, the amplitude of anything below 15 Hz can be determined precisely (assuming no aliasing from higher frequencies). But shorter sample times give much larger amplitude errors. The Nyquist frequency is often misinterpreted as being adequate to reconstruct signals even though the amplitude calculations are based on data taken over a short time. There are formulas for the amplitude error limits if the data sample is not infinite.

 

[sophiecentaur]

Thanks for the reply.

Does this mean that if I isolate the vibrations below 15Hz, no other vibrations will be apparent?

Vibrations above 15Hz will appear as vibrations below 15Hz (aliases). Once they are present in the sampled signal, there is no way of getting rid of them; they are a part of the signal. You need to filter them out first (that's what the Nyquist filter is for) and, of course, because all filters are imperfect, you need to aim at a cut-off just below 15Hz. How far will depend on how good your filter is.
Factchecker is right about the time for the sampling. The contribution of noise from the sampling circuit is affected by the time taken for the sample to be made. It is common to use a sample time equal to the interval between samples (box-car) and this then requires an appropriate equalisation to adjust for the resulting effect on the frequency response.

 

[JimmyJimmy]

OK, thanks.

Going right back to basics - I've been told that there is a relationship between the frame rate which the camera is recording at, and an upper boundary of vibration frequencies, above which they will not be visible, explained(?) by Nyquist. Is this correct?

Sorry for the ignorance - this is definitely not my area of expertise!

 

[sophiecentaur]

OK, thanks.

Going right back to basics - I've been told that there is a relationship between the frame rate which the camera is recording at, and an upper boundary of vibration frequencies, above which they will not be visible, explained(?) by Nyquist. Is this correct?

Sorry for the ignorance - this is definitely not my area of expertise!

The length of the sampling interval will impose an upper limit to the frequency that can be reproduced (including aliases). This link shows the frequency response resulting from boxcar sampling. Naturally, you should be thinking in terms of Nyquist filtering in any case. The detailed spectrum of the signal you are sampling does have an effect on the basics of Nyquist's criterion and it is actually possible to sub sample if there are suitable 'holes' in the spectrum of the signal. (Analogue TV signals are an example where sub-Nyquist sampling actually works quite well.)

 

[FactChecker]

OK, thanks.

Going right back to basics - I've been told that there is a relationship between the frame rate which the camera is recording at, and an upper boundary of vibration frequencies, above which they will not be visible, explained(?) by Nyquist. Is this correct?

Sorry for the ignorance - this is definitely not my area of expertise!

I would restate it. Change they will not be visible to they will be mistaken for lower frequencies. Above the Nyquist frequency, the points that you sample are exactly matched by lower frequencies. You will never sample at a point where they are different so you will mistake them for the lower frequency.

 

[sophiecentaur]

I would restate it. Change they will not be visible to they will be mistaken for lower frequencies. Above the Nyquist frequency, the points that you sample are exactly matched by lower frequencies. You will never sample at a point where they are different so you will mistake them for the lower frequency.

But you cannot ignore the low pass effect of boxcar sampling. That will reduce the aliasing effect.