Synchronous detection of DSB-SC; phase and frequency error

Synchronous (coherent) detection of DSB-SC

A DSB-SC wave $s(t) = A_c m(t)\cos\omega_ct$ has no carrier, so its envelope is $|m(t)|$, not $m(t)$. It is detected by multiplying with a locally generated carrier of the same frequency and phase, then low-pass filtering.

 s(t) --->( X )--->[ LPF, BW = W ]---> v_o(t)
            ^
            |
   [ Local oscillator cos(wc t) ]

With an ideal local carrier $\cos\omega_ct$:

$$v(t) = A_cm(t)\cos^2\omega_ct = \frac{A_c}{2}m(t) + \frac{A_c}{2}m(t)\cos 2\omega_ct$$

The LPF removes the $2\omega_c$ term, so $v_o(t) = \frac{A_c}{2}m(t)$, an exact copy of the message.

Effect of phase error

Let the local carrier be $\cos(\omega_ct + \phi)$:

$$\begin{aligned} v(t) &= A_cm(t)\cos\omega_ct\cos(\omega_ct+\phi) \\ &= \frac{A_c}{2}m(t)[\cos\phi + \cos(2\omega_ct + \phi)] \\ v_o(t) &= \frac{A_c}{2}m(t)\cos\phi \end{aligned}$$

• The output is the message scaled by $\cos\phi$: an attenuation, not a distortion, if $\phi$ is constant.
• At $\phi = \pm 90^\circ$ the output is zero (quadrature null effect).
• If $\phi$ drifts randomly with time, the gain varies and the output fades.

Effect of frequency error

Let the local carrier be $\cos(\omega_c + \Delta\omega)t$:

$$\begin{aligned} v(t) &= \frac{A_c}{2}m(t)[\cos\Delta\omega t + \cos(2\omega_c + \Delta\omega)t] \\ v_o(t) &= \frac{A_c}{2}m(t)\cos(\Delta\omega\,t) \end{aligned}$$

• The message is multiplied by a slowly varying cosine at $\Delta f$. The output rises and falls periodically and passes through zero twice every $1/\Delta f$ seconds: a "beating" effect.
• In spectrum terms, $m(t)$ is shifted to $\pm\Delta f$, which badly distorts speech and music.

So DSB-SC needs exact carrier synchronization in both frequency and phase, using a pilot carrier or a Costas loop / squaring loop.