I am trying to compare the demodulation for single sideband (SSB) and dual sideband modulation formats (DSB) when the reconstructed carrier is not perfect (i.e. shifted in both frequency and phase). I want to show mathematically in which case distortion would be more problematic.
For instance, if the carrier is $\cos (2 \pi \nu_c t)$, the shifted reconstructed carrier would be:
$$\cos (2 \pi (\nu_c + \Delta \nu)t + \varphi) \tag{1}$$
Attempt:
SSB: A signal transmitted with the single sideband (SSB) modulation is given by:
$$f(t)=A\ \Re\left\{m_a (t) e^{j2\pi \nu_c t}\right\} = A\left[m(t) \cos(2\pi \nu_c t)- \hat{m}(t) \sin (2\pi \nu_ct)\right] \tag{2}$$
where $m_a$ is the analytic signal and $\hat{m}(t)$ is the Hilbert transform. Demodulation is obtained by multiplying the transmitted signal with the reconstructed carrier given in $(1)$. With some manipulation this became:
\begin{align} A\left[\frac{m(t)}{2}\left[\cos [(2\omega_c + \Delta \omega)t+ \varphi]+ \cos(\Delta \omega t + \varphi)\right]\\ - \frac{\hat{m}(t)}{2}[\sin[(2\omega_c+\Delta \omega)t+\varphi]+\sin(\Delta\omega t +\varphi)]\right] \end{align}
Low pass filtering:
$$\therefore \frac{A}{2}\left[m(t)\cos(\Delta \omega t + \varphi)- \hat{m}(t)\sin(\Delta\omega t +\varphi)\right] \tag{i}$$
DSB: And the dual sideband modulation (DSB) signal is given by
$$f(t)=Am(t) \cos(2\pi \nu_c t) \tag{3}$$
Demodulation gives:
$$Am(t) \cos(2\pi \nu_c t) \cos (2 \pi (\nu_c + \Delta \nu)t + \varphi)=\frac{A m(t)}{2}\left[\cos((2\omega_c + \Delta \omega)t + \varphi)- \cos (\Delta \omega t + \varphi)\right]$$
Low-pass filtering:
$$\therefore -\frac{A m(t)}{2} \cos (\Delta \omega t + \varphi) \tag{ii}$$
So, if my approach is correct, how can we interpret equations (i) and (ii) to see which modulation format results in having more distortion?
Any explanation would be greatly appreciated.
