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Why compressing a train of impulses in time domain produce a wide-spaced train of impulses in frequency domain and vice versa? i want to know the intuition behind that.

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  • $\begingroup$ Do you mean impulses? $\endgroup$
    – MBaz
    May 5 at 23:38
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    $\begingroup$ yes sorry for confusion $\endgroup$ May 6 at 0:04
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    $\begingroup$ In my opinion, you never really understand impulses, you only get used to them. $\endgroup$
    – MBaz
    May 6 at 2:13
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    $\begingroup$ I like that @MBaz - I think it would make a great poster $\endgroup$ May 6 at 21:01
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The intuition is simply the update rate in the time domain. If I repeat an impulse in time once per second, it should be quite intuitive that a 1 Hz tone will exist in frequency. When a waveform isn't exactly sinusoidal (such as a distorted sinewave), it will have higher frequencies that will only exist at harmonics of that waveform. Since the waveform repeats over a given time $T$, the only other frequency components that can exist must also repeat over that same time interval $T$, which are the harmonics. In general, whenever you repeat any waveform exactly the same in the time domain, as we see with the Fourier Series Expansion, the only non-zero frequencies that can possibly exist will be at DC, the repetition rate (fundamental frequency) and all higher harmonics. (Some of these may themselves be zero in the general case, but any non-zero frequencies can only exist at this location). If I said your name exactly the same way once per second, the only frequency content of my voice track will be at 1 Hz spacings.

This is the same with a train of impulses in time, just in this case we are repeating an impulse.

Further, you can take the Fourier Transform of any base signal (such as the impulse, which is a constant for all frequencies), and then take the property above showing where the only non-zero values can be, and we see that we end up "sampling" in frequency the Fourier Transform. This also works with any base signal where a Fourier Transform exists, and explains why a square wave has odd harmonics (which are simply samples of the underlying Sinc function as the Fourier Transform of a pulse).

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