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

In summary, the conversation discusses the Nyquist frequency of a Gaussian signal and how it relates to Fourier analysis. It is mentioned that the maximum frequency for a Gaussian signal is 5 GHz, but technically it has no upper frequency limit. The importance of choosing a high enough sampling rate to avoid aliasing is also discussed, and it is mentioned that the cutoff frequency will depend on the application and can be determined through calculations or by setting a threshold for the noise level.
  • #1
Jiho
20
4
Hello.
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.
upload_2019-2-14_1-7-8.png

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.
 

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  • #2
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.
 
  • #3
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.
 
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  • #4
tech99 said:
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?
 
  • #5
marcusl said:
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??
 
  • #6
Jiho said:
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.
 
  • #7
Jiho said:
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.
 
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  • #8
Jiho said:
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.
 
  • #9
Jiho said:
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.
 
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1. What is the maximum frequency of a Gaussian signal?

The maximum frequency of a Gaussian signal is infinite, as it is a continuous function. However, in practical applications, the maximum frequency is limited by the bandwidth of the system.

2. What is the Nyquist frequency of a Gaussian signal?

The Nyquist frequency of a Gaussian signal is half of the sampling rate. It is the highest frequency that can be accurately represented in a digital signal without aliasing.

3. How is the maximum frequency related to the Nyquist frequency?

The maximum frequency is directly related to the Nyquist frequency, as it cannot exceed half of the sampling rate. This ensures that the signal is accurately represented without any aliasing.

4. Can a Gaussian signal have a maximum frequency higher than the Nyquist frequency?

No, a Gaussian signal cannot have a maximum frequency higher than the Nyquist frequency. This would result in aliasing and distort the signal, making it difficult to interpret and analyze.

5. How does the bandwidth of a system affect the maximum frequency of a Gaussian signal?

The bandwidth of a system limits the maximum frequency of a Gaussian signal that can be accurately represented. A wider bandwidth allows for higher maximum frequencies, while a narrower bandwidth limits the maximum frequency that can be accurately captured and analyzed.

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