Converting a continuous, aperiodic spectrum to a discrete spectrum can be achieved without energy loss if the frequency samples are spaced appropriately, specifically less than 1/(2T) apart, allowing for lossless reconstruction. This process involves sampling the spectrum by multiplying it with a Dirac Comb function, leading to a convolution of the original signal and the Fourier Transform of the comb function. The discussion emphasizes that this method aligns with Shannon's theorem, but applied in the frequency domain rather than the time domain. Additionally, there is interest in understanding how to implement a practical Nyquist low-pass filter in this context. Overall, the conversion maintains signal integrity under the right conditions.
