Phase and frequency error in local oscillator (DSB and SSB)

In coherent (synchronous) demodulation, the receiver multiplies the received signal by a locally generated carrier and low-pass filters. A PLL (or Costas loop) is used to generate this carrier. If the local oscillator is not exactly matched, i.e. it has a frequency error $\Delta\omega$ and phase error $\phi$, the output is degraded.

Let the local carrier be $c(t)=2\cos[(\omega_c+\Delta\omega)t+\phi]$, and define $\theta(t)=\Delta\omega t+\phi$.

 s(t) -->(X)-->[ LPF ]--> e0(t)
          ^
          | 2cos[(wc+dw)t + phi]
     [ PLL / local osc ]

Effect on DSB-SC

Received $s(t)=m(t)\cos\omega_ct$:

$$\begin{aligned} s(t)c(t)&=2m(t)\cos\omega_ct\cos(\omega_ct+\theta)\\ &=m(t)\cos\theta+m(t)\cos(2\omega_ct+\theta)\\ e_0(t)&=m(t)\cos(\Delta\omega t+\phi)\quad(\text{after LPF}) \end{aligned}$$

• Phase error only ($\Delta\omega=0$): $e_0=m(t)\cos\phi$. The output is undistorted but attenuated by $\cos\phi$. If $\phi$ varies randomly, the output fades; at $\phi=\pm90°$ the output is zero (quadrature null effect).
• Frequency error only ($\phi=0$): $e_0=m(t)\cos(\Delta\omega t)$. The message is multiplied by a slowly varying sinusoid, so the output beats (rises and falls periodically at rate $\Delta f$), causing serious distortion even for small $\Delta f$.

So DSB-SC demodulation needs both frequency and phase to be correct; a Costas loop or squaring PLL is used.

Effect on SSB

Received USB: $s(t)=m(t)\cos\omega_ct-\hat m(t)\sin\omega_ct$. Multiplying and filtering:

$$\begin{aligned} e_0(t)&=m(t)\cos\theta+\hat m(t)\sin\theta\\ &=m(t)\cos(\Delta\omega t+\phi)+\hat m(t)\sin(\Delta\omega t+\phi) \end{aligned}$$

(For LSB the sign of the $\hat m$ term changes.)

• Phase error only: $e_0=m(t)\cos\phi+\hat m(t)\sin\phi$. Every frequency component of the message gets a constant phase shift $\phi$ (for a tone $A_m\cos\omega_mt$, $e_0=A_m\cos(\omega_mt-\phi)$). Amplitude is not reduced, but there is phase distortion. The human ear is insensitive to phase, so speech is acceptable, but data and video are affected.
• Frequency error only: for a tone, $e_0=A_m\cos(\omega_m-\Delta\omega)t$. Every component is shifted in frequency by $\Delta f$, so harmonic relationships are destroyed. Speech sounds unnatural ("Donald Duck" effect); a shift of a few Hz is tolerable for speech, but music becomes badly distorted.

Comparison

Remedy: a PLL locks the local oscillator to a transmitted pilot carrier (SSB, DSB) or recovers the carrier from the signal itself (Costas loop for DSB-SC), reducing both errors close to zero.