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I'll give this a shot. The Fourier Transform of a Gaussian is also a Gaussian. The standard deviations in each domain are related as $\sigma_t \cdot \sigma_F = \frac{1}{2\pi}$ The time standard deviation, $\sigma_t$ has units of time and the frequency domain standard deviation $\sigma_F$ has units of Hz. We can define the "bandwidth" of a gaussion ...

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The fact that it's modulated with a sinusoid doesn't change the FWHM bandwidth of your pulse – the $e^{jx}$ function has $\left\lvert e^{jx}\right\rvert\equiv 1$ at every point. That doesn't change the amplitude, so the FWHM of a sinusoid-modulated gaussian is just the same as of the unmodulated gaussian.

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No such thing as a single frequency of the noise. That's exactly why it's called white; it has power in all frequency ranges, but not at a single frequency. Finally, is there a frequency-domain representation of Gaussian white noise? Yes, a constant power spectral density for all frequencies. That's like white light (which contains also a continuum of all ...

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