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Relation between Gaussian derivatives and Gausian-windowed Cosine function

I am doing some research on UWB radars which transmit a pulse given by: $$ g(t) = p(t) cos(2\pi f_{c}t) = A e^{(-\frac{t^{2}}{2\tau ^{2}})} cos(2\pi f_{c}t) $$ In some other papers, the transmitted pulse
Charis Hadjipanayi's user avatar
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Expression for Frequency-shifted Gaussian pulse

p(t)cos(2\pi f_{c}t) = V_{TX}exp(-\frac{t^{2}}{2\tau^2})cos(2\pi f_{c}t) $$ where $f_{c}$ is the center or carrier frequency, $V_{TX}$ is the peak amplitude and $\tau$ determines the Bandwidth of the pulse
Charis Hadjipanayi's user avatar