[SOLVED] Nyquist Rule in signal sampling

[kent486]

"Explain the use and limitations of the Nyquist rule in signal sampling"

 

[barry]

Basically, the bandwidth of the signal being sampled must be less than 1/2 the frequency of the sampling clock, or you'll get aliasing (images of the signal appearing in the wrong place).

 

[biff44]

Here is a sine wave that is sampled at below the nyquist rate (i.e. sampled at every "x"). The output of the sampler looks like a much lower frequency than it really is.

 

[barry]

That's not a valid representation. In between samples you can't assume any value. If you took enough samples and did a Fourier analysis, you'd see the fundamental frequency is really there. (And I'm assuming that's supposed to be a sine wave you drew. )

 

[digi001]

Check this out

The Sampling Theorem

 

[permute]

There are two common definitions. The first is the easy to remember "you must sample at 2x (or faster) than the highest frequency in the measured signal". This is actually false in general, but many courses and even interview questions are focused on the very common but specific case of baseband sampling.

The more accurate is that "the uniform sampling rate has a lower bound of 2x the information bandwidth of the signal". For example, if you have a 4khz sine wave, the average sampling rate would be 0 -- using only a handful of samples the signal will be exactly known. If the bandwidth were 4kHz-5kHz, then you could sample at 2kHz. This is because no frequency between 4 and 5khz would map to the same digital frequency. had the bandwidth been between 4.5kHz and 5.5kHz, then the (uniform) sampling rate would need to be higher to prevent destructive aliasing.

The second definition can come up in some specific applications, though it is common with modern communication systems.

and yes, you CAN assume values between signals. You just have to have a correctly defined anti-aliasing filter (or in the original formulation, an assumption of a bandlimited signal). In such a case, the analog signal can be reconstructed from the samples. The example in the picture shows a signal which has a bandpass filter as the anti-aliasing filter. The image also shows what happens when a signal is then reconstructed using a different anti-aliasing filter.

 

[sivamani]

Please refer the attached pdf.I think it will be helpful to you.

 

[barry]

and yes, you CAN assume values between signals.

No, you CANNOT assume the values between samples, that's the nature of sampling. For example, suppose you have a sine wave where your sampling rate is such that you sample it a 0 degrees and then at 100 degrees. The value measured at 100 degrees would be identical to sampling a slower sine wave at 0 and 80 degrees. The values between samples for the two sine waves is quite different.

 

[permute]

barry, the entire point of the sampling theorem is that you CAN determine the value between samples. The requirements for this are given by the nyquist sampling rate (no aliasing), and bandlimiting (known mapping of digital to analog frequency).

you have a degenerate example which also seems to rely on some assumptions on different amplitudes. In any case if there are only two non-zero samples then the signal clearly isn't a sine wave of any sort. To reconstruct the waveform exactly, all samples would be used for every instant that is interpolated (bandlimiting is the same as a noncausal filter of infinite duration). Fewer samples are required if there is additional known information about the signal, or if some degradation is allowed.

 

[biff44]

That's not a valid representation. In between samples you can't assume any value. If you took enough samples and did a Fourier analysis, you'd see the fundamental frequency is really there. (And I'm assuming that's supposed to be a sine wave you drew. )

when someone says "here is a sine wave"...it is pretty safe to assume it is a sine wave. And that is a valid representation of undersampling.