What is the maximum or Nyquist frequency of a Gaussian signal?

[Jiho]

Hello.

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I'm studying Fourier analysis. If we look at attached graph where Gaussian functions are transformed by Fourier analysis, we can find Gaussian functions in frequency domain have maximum value at 0 hertz.

So I confused what is the Nyquist frequency at Gaussian signal. I need to know Nyquist frequency for Fourier analysis, but alll of the Gaussian signal's critical frequcny is 0hertz.

 

[tech99]

The zero frequency component is because the pulse is unidirectional i.e. it is DC.
The max frequency is 5 GHz so we would need to sample at at least twice this = 10 GHz.

 

[marcusl]

The transform of a Gaussian is also Gaussian, and although the tails fall off very rapidly (faster than exponential), they never reach zero. Thus a Gaussian technically has no upper frequency limit.
If you sample a Gaussian, frequencies above Fs/2 will alias. If you pick Fs high enough, then the energy in the aliased segment is wholly negligible.

 

[Jiho]

The zero frequency component is because the pulse is unidirectional i.e. it is DC.
The max frequency is 5 GHz so we would need to sample at at least twice this = 10 GHz.

I can't understand why max freq is 5Ghz. Do you mean that value corresponding freq lower at about 5Ghz is close to 0?

 

[Jiho]

The transform of a Gaussian is also Gaussian, and although the tails fall off very rapidly (faster than exponential), they never reach zero. Thus a Gaussian technically has no upper frequency limit.
If you sample a Gaussian, frequencies above Fs/2 will alias. If you pick Fs high enough, then the energy in the aliased segment is wholly negligible.

How cna I choose Fs high enough when no upper frequency limit?? Is there any criterion in signal processing??

 

[FactChecker]

How cna I choose Fs high enough when no upper frequency limit?? Is there any criterion in signal processing??

The highest frequency that you wish to consider is application dependent. There is usually some physical limitation that allows one to ignore the extremely high frequencies.

 

[tech99]

I can't understand why max freq is 5Ghz. Do you mean that value corresponding freq lower at about 5Ghz is close to 0?

Your plot of amplitude versus frequency falls to zero (visually) at about 5 x 10^9 Hz, which is 5 GHz. I agree it goes on for ever, but we have to set some limit due to practical constraints like the lowest quantising level.

 

[sophiecentaur]

Is there any criterion in signal processing??

It will depend on the application. The Nyquist filter cut off (determined by the sampling rate) will determine the quantising noise / alias level in the baseband bandwidth.

 

[marcusl]

How cna I choose Fs high enough when no upper frequency limit?? Is there any criterion in signal processing??

Yes, in general you cut off the spectrum of the input signal with an anti-aliasing filter. If you want to set the cutoff frequency to capture all relevant data in a Gaussian signal in a real application, there will be noise present and you can determine where the spectrum goes below the noise. Above that frequency, no spectral components can be seen.

For your noiseless simulation, you can calculate where the energy in the spectrum becomes negligible compared to the total energy in the Gaussian. Look up the transform of a Gaussian (which will also be a Gaussian). The area under a Gaussian is generally normalized to one. You can then compute where energy in the tails is, say, 0.1% or 0.01%--this is where the error function erf reaches 0.999 or 0.9999. The high frequency components will alias, but their energy is so small that it can't be seen.