Is discrete or continuous a matter of frequency?
For example, would saying, this signal is in "discrete frequency" or "continuous frequency" make sense?
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$\begingroup$ Maybe this post (dsp.stackexchange.com/questions/58859/…) will help. There is no answer to it but it provides some equations that relate various frequency "quantities". $\endgroup$– ZaellixAMar 22, 2022 at 11:17
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2 Answers
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- A signal that's periodic in time is discrete in frequency.
- A signal that's aperiodic in time is continuous in frequency.
- A signal that's periodic in frequency is discrete in time.
- A signal that's aperiodic in frequency is continuous in time.
These are basic mathematical properties and hence there 4 different types of Fourier Transforms depending which case you are looking at.
Is discrete or continuous a matter of frequency?
Both time and frequency can be discrete or continuous. Any combination is possible, so there are signals that are discrete in frequency but continuous in time.
For example, would saying, this signal is in "discrete frequency" or "continuous frequency" make sense?
Yes it does. It's equivalent to saying the signal is periodic or aperiodic in time.
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Yes signals can be discrete or continuous in the frequency domain. Discrete in the frequency domain means that the signal only consists on discrete frequencies (up to an infinite amount). Any signal that repeats in time with no variation from each repetition interval in time will have the property of having discrete non-zero values in the frequency domain.
A good explanation of this is the Fourier Series Expansion which shows how an analytic signal over any finite duration of time $T$ can be represented by an infinite number of sinusoids, each only existing at the frequencies that are an integer multiple of $1/T$. Since each of these sinusoids repeats exactly the same for time durations beyond $T$, if we considered the sinusoids as waveforms extending to infinity, summing those sinusoids would then result in an exact repetition of the result we achieved over the interval $T$ (periodic). We also see then why other sinusoids at any other frequency can't exist- any non-zero value for other frequency terms would start at a different position in each new interval of $NT$ (where $N$ is any integer) and thus would destroy the periodicity we otherwise had.
Similarly, and given by the reciprocity property of the Fourier Transform, a signal that is discrete in time will be periodic in the frequency domain.
I give further details and examples of this at these links:
Sine Wave Aliasing during IFFT
And see Richard Lyon's great philosophical question here:
answered Mar 22, 2022 at 8:25
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$\begingroup$ I think, what I try to ask is if in signal processing, "discrete" would generally mean "discrete frequency" AND "continuous" would generally mean "continuous frequency", while "generally" here means about 99% of the cases. $\endgroup$– yaraklisMar 22, 2022 at 8:41
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$\begingroup$ No not at all. Discrete in time and discrete in frequency are two different things and we must clarify. We can have any combination as I detail in the links: A signal can be simultaneously continuous in one domain (time or frequency) and discrete in the other, or discrete in both domains, or continuous in both domains. Discrete in both domains is NOT 99% of the cases in signal processing. Consider frequency responses of filters for one example. $\endgroup$ Mar 22, 2022 at 8:43
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$\begingroup$ So I don't think it's about if signals can be discrete or continuous in the frequency domain (only) but if they are, in essence, in and of themselves, signals of "discrete frequency" (instead just "discrete signals") or signals of "continuous frequency" (instead just "continuous signals". $\endgroup$– yaraklisMar 22, 2022 at 8:44
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$\begingroup$ So in general we face discrete time due to the plethora of "digital electronics" and "digital processing". We sample our analog world to create discrete time waveforms. This alone results in a periodic frequency waveform. The next question to ask is if we can treat the sampled time domain waveform as periodic-- then this would add the additional "feature" that the frequency domain is also discrete. Note that the DFT, which is over a finite time interval has the property of being equivalent mathematically to a periodic time domain waveform over that interval for the same reasons I gave with FSE $\endgroup$ Mar 22, 2022 at 8:49