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Design what?

207.207.243.89

If you mean a damped 20khz tone that has no out of band components, that's easy, and it's old.

Use a guassian impulse with a 10 millisecond sigma. That will mean that its 1-sigma bandwindth is 100 Hz. That means that at 1khz, it's down more dB than you care to talk about.

Now that's not an e^-at damping, but please, just PLEASE go to Morrison or any other Fourier Analysis book, it will show you that there are (literally) an infinite class of what you ask for.

I'm not sure what you're asking for here.

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