Continuous and discrete spectra

[Domenico94]

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?

 

[BvU]

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

 

[Domenico94]

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?

 

[BvU]

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.

 

[sophiecentaur]

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.