# Demodulation distortion in single-sideband and double-sideband

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.

• how do you define "worse distortion"? – Marcus Müller May 25 '17 at 13:13
• I believe in SSB, the only observable distortion would be the frequency shift (and for speech, for instance, it is still understandable as long as the detuning is relatively small). From what I've read in DSB, there would also be a beating, where the resulting demodulated signal fades in and out at a rate equal to the detuning. So, the distortion is 'worse' in DSB. How can I show this mathematically using the method above? – Merin May 25 '17 at 13:56
• so, your application is audio transport, and specifically speech? You see, how "bad" some distortion is depends on what you need to do with the distorted signal. Thus, a mathematical model would need to follow from the characteristics of the intended receiver, and in the case of speech, from the characteristics of human hearing. I'm not 100% sure where the fading aspect would come into play here; I think you might be assuming a specific hardware receiver that has exposes that behaviour, and that might or might not align with the demodulation equations you use! – Marcus Müller May 25 '17 at 16:57