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When we need to analyse band pass signals we use complex envelope and preenvelope representations, wherein a signal is given an equivalent low pass representation.

My issue is why it is done, though I agree that using a preenvelope or complex envelope remove the carrier amplitude and shift the signal around zero frequency, which eases analyses. But is this the only reason.

I have even searched literature but was not able to get a proper answer. Can somebody help me on this. I will grateful for a detailed mathematical answer for this question.

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