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I am looking for examples of signals that are band limited but also sparse. For instance, the spiking of the neurons can be modeled as banded in frequency domain and sparse in time domain. What other examples exists for such type of signals?

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  • $\begingroup$ How about the chirping of a bird or an orca whale sound? $\endgroup$ – Starhowl Aug 21 '18 at 17:51
  • $\begingroup$ Is it sparse in time domain? $\endgroup$ – Maxtron Aug 21 '18 at 19:20
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    $\begingroup$ Let's talk in the Continuous World. In order to be sparse it means the function must have a measurable closed set on which it vanishes. If I'm not mistaken this means it is not analytic function -> It has some discontinuity built into it (At least it has a an order of derivative which is not continuous). This suggests it is not band limited. $\endgroup$ – Royi Aug 21 '18 at 20:17
  • $\begingroup$ @Royi How would you explain spiking activity observed in the brain at a given frequency band? We have prior information that it belongs to a specific band. Further, as the activity appears as spikes, it is sparse in time-domain. $\endgroup$ – Maxtron Aug 21 '18 at 21:32
  • $\begingroup$ I'm not familiar with the specific signals you're describing. If you show something we'll be able to address it. What I gave you is intuition in continuous domain. What you describe sounds like periodic signal. Are you talking about that? $\endgroup$ – Royi Aug 21 '18 at 21:52
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If a signal is sparse in the time domain, it has infinite support in the frequency domain, thus is not band-limited in the mathematical sense. However it might be so close to band-limited that the out-of-band spectrum disappears under the quantization or other noise floor.

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  • $\begingroup$ Counter example: In the link, look at the ECG data. sleepdata.org/datasets/shhs/images/a/ar1a.jpg?inline=1 Notice, that the data (maybe the first/second difference of the signal, if not the original data) appears as sparse in time domain. Further, the ECG spectrum lies between 3-15 Hz, which makes it band limited. Much simpler example: Consider a square wave generated at particular frequency. The first difference of the square wave is sparse in time domain. $\endgroup$ – Maxtron Aug 23 '18 at 4:58
  • $\begingroup$ I did not understand your comment. Can you explain which of the above examples are misleading? I gave you the simplest counter example which shows a signal (first difference) can be both sparse and band limited. $\endgroup$ – Maxtron Aug 23 '18 at 5:03
  • $\begingroup$ But the square wave (in time domain) is generated at a specific frequency. $\endgroup$ – Maxtron Aug 23 '18 at 5:13
  • $\begingroup$ Don't forget all the odd harmonics of that specific frequency that are part of any square wave. Odd numbers can get arbitrarily large. $\endgroup$ – hotpaw2 Aug 23 '18 at 5:23

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