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Is it possible for two identical waveforms to have different spectrograms ? I am rookie in this field so please pardon if something is wrong with this question.

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A waveform is a period of superimposition of all the frequencies involved, each of which can be deconstructed into sine waves. Therefore a waveform will always represent those frequencies, theoretically. If the waveform is cut or sampled too short, some lower frequencies might be excluded.

It is possible however to have different waveforms represent near-identical spectrums. This is because a waveform can be deconstructed into not only harmonics but the phase angles of harmonics, and spectrograms do not indicate phase angles. Interestingly, sometimes this can alter the timbre of the sound.

A great visual is this part of a video on Additive synthesis; it shows how a Saw wave is constructed with many harmonics from the beginning (0 degrees of phase), therefore using different phases will not result in a Saw wave, but still maintain the spectrum.

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If two waveforms are strictly identical, it means they represent the same signal, and the spectrograms of these two signals will also be identical.

So the answer is : no.

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Mathematically, converting a waveform (based on a signal changing with time elapsing) to a spectrum is called a fourier transform and the reverse operation (not surpringly) is called an inverse-fourier transform. So, you create a spectrum from the time elapsing signal and then you do in inverse and you theoretically get back to the same time elapsing waveform.

Every rise/fall or undulation in the timebased waveform has a precise meaning and value in the frequency spectrum - they are mathematically locked-in to each other.

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Without respect to the frequency axis scale, two waveforms may look identical. Only if the frequency is the same for both waveforms you will see the same spectrogram.

On the other hand, since a spectrogram shows the absolute amplitude as intensity you can get nearly identical spectrograms from different waveforms (though you should be able to spot more or less vague harmonic changes, i.e. changes elsewhere than the fundamental frequency)

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