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I have a hard time teasing apart the conceptual difference between these three domains, and constantly mix them up in my head. I've been reading up on it, but I can't wrap my head around it.

  1. a time-domain graph shows how a signal changes over time: oscilliscope)

^ connected through Fourier transform v

  1. a frequency-domain graph shows how much of the signal lies within each given frequency band over a range of frequencies.

    • the 'spectrum' of frequency components is the frequency domain representation of the signal
  2. spectral domain - ??

  3. cepstral domain - ??

Any help or insight you could provide, at an undergraduate level of physics, would be so helpful to me.

Thanks a lot.

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Say you have some signal in 'real' (eg. time or space) domain.

When you perform a Fourier transform of it, you project it into the Fourier spectral domain. More specifically, as you convolve your signal with imaginary exponentials (ie. infinite sines), you end up with the signal Fourier spectrum that shows which infinite sines (~references) your signal contains. When you do an Inverse Fourier transform of your spectrum, you go back into the real domain (and hopefully your original signal if your IFT was done properly considering both amplitude and phase).

Another transform can be the Wavelet transform which results from convolving your signal with wavelets instead of sines -hence obtaining a Wavelet spectrum instead of a Fourier spectrum.

The Cepstral domain results from taking the Inverse Fourier Transform of the LOGARITHM of your Fourier spectrum (instead of your 'plain' Fourier spectrum). Hence it is on the 'real' side of the representations.

Now concerning the 'Frequency domain'... the term 'frequency' is more general than it seems, as frequency designates the number of occurrences in a second. The 'Frequency domain' and 'Spectral domain' are distinct, because they imply different references. For example a 1Hz pulse signal and 1Hz sine signal have the same frequencies (1 Hertz). But their Fourier spectral representations give VERY different results (1 line for the sine wave, infinite number of lines for the pulse signal).

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when a signal is analyzed in time domain they are called spectrum or you can say that signal is in spectral domain. but when the signal is analyzed in frequency domain and amplitude of such signal is taken to analyzed the signal then they are said to be in cepstral domain.

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