Continuous and discrete spectra

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Discussion Overview

The discussion revolves around the conversion of a continuous, aperiodic spectrum to a discrete spectrum in signal processing. Participants explore the implications of this conversion, particularly regarding energy loss and reconstruction fidelity, as well as the relationship to established sampling theorems.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant inquires about the possibility of converting a continuous spectrum to a discrete spectrum and whether this process results in energy loss.
  • Another participant explains that sampling the spectrum involves multiplying it with a Dirac Comb function, leading to a convolution in the time domain, and suggests that lossless reconstruction is possible under certain conditions.
  • A later reply confirms the initial inquiry, asking for clarification on whether any losses occur during this process.
  • One participant asserts that there are no losses in this conversion, likening it to Shannon's theorem applied in the frequency domain.
  • Another participant reiterates the explanation of sampling and introduces the concept of the Nyquist criterion, expressing a desire to understand how to specify a practical Nyquist low-pass filter in this context.

Areas of Agreement / Disagreement

Participants express differing levels of understanding regarding the implications of the conversion process, with some asserting no losses occur while others seek further clarification. The discussion does not reach a consensus on all aspects, particularly regarding practical applications and specifications.

Contextual Notes

Some assumptions about the conditions for lossless reconstruction are not fully explored, such as the limitations of the original signal's duration and the specific requirements for frequency sampling. The relationship between the Nyquist criterion and the proposed conversion process remains partially unresolved.

Domenico94
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Is there any way to convert a continuous, aperiodic spectrum, to a discrete spectrum, in a signal? If so, would part of he energy of this signal be lost, I am this process of conversion, or would it be " distributed" amomg the various frequencies?
 
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Not sure I fully understand your question. Sampling the spectrum is equivalent to multiplying it with a Dirac Comb function and then the signal you get from transforming back to the time domain would be a convolution of the original signal and the FT of the comb function. If the original signal is only present for a limited time 0 - T , this would allow lossless reconstruction when the frequency samples are less than 1/(2T) apart
 
BvU said:
Not sure I fully understand your question. Sampling the spectrum is equivalent to multiplying it with a Dirac Comb function and then the signal you get from transforming back to the time domain would be a convolution of the original signal and the FT of the comb function. If the original signal is only present for a limited time 0 - T , this would allow lossless reconstruction when the frequency samples are less than 1/(2T) apart
Yes. Thanks..It s exactly what I was asking for. No losses than? Of any kind?
 
No losses. It is the equivalent of Shannon's theorem but now from the frequncy domain to the time domain instead of the other way around.
 
BvU said:
Not sure I fully understand your question. Sampling the spectrum is equivalent to multiplying it with a Dirac Comb function and then the signal you get from transforming back to the time domain would be a convolution of the original signal and the FT of the comb function. If the original signal is only present for a limited time 0 - T , this would allow lossless reconstruction when the frequency samples are less than 1/(2T) apart
This is the equivalent of the Nyquist criterion for temporal sampling. I'm trying to get my head around how to specify the equivalent to a practical Nyquist LP filter in this process.
 

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