A modulating signal $m(t)= \sin(2 \pi \nu_m t)$ is transmitted via a carrier of frequency $\nu_c$ using upper sideband (USB) modulation. Show that the USB modulated signal can be demodulated using a reconstructed carrier signal and a low pass filter.
Attempt:
USB (SSB signals in general) can be demodulated by multiplying them with a reconstructed carrier $\cos(2\pi \nu_c t).$ So I first tried to write an expression for the transmitted USB modulated signal:
$$f(t)=A \Big[ m(t) \cos(2 \pi \nu_c t) - \hat{m}(t) \sin(2\pi \nu_c t) \Big]$$
$$f(t)=A \Big[ \sin(2 \pi \nu_m t) \cos(2 \pi \nu_c t) - \hat{m}(t) \sin(2\pi \nu_c t) \Big] \tag{1}$$
Where $A$ is some amplitude, and $\nu_c >> \nu_m$. So to proceed further, I think we need to calculate the Hilbert transform $\hat{m}(t).$ It is possible to do this by finding the imaginary part of the analytic signal:
$$m_a (t) = m(t) + j \hat{m} (t)= 2 \int^\infty_0 M(\nu) \exp(j 2 \pi \nu t) d\nu \tag{2}$$
But the Fourier transform of $m(t)$ is tedious, so I am wondering if there is a simpler way of doing this?