Question E8C03:
Why should the phase of a PSK signal be changed at the zero crossing of the RF signal?  

Choices:  
A: To minimize bandwidth  
B: To simplify modulation  
C: To improve carrier suppression  
D: All these choices are correct    

Correct Answer is  A: To minimize bandwidth  

To minimize bandwidth:  
Changing the phase of a PSK signal at the carrier’s zero crossings keeps the RF waveform continuous in amplitude at the instant of the phase change, which reduces spectral splatter and occupied bandwidth.

1) Why the correct answer is correct (A)
- Model the RF as:
  $$s(t) = A \cos\big(2\pi f_c t + \phi(t)\big)$$
  Suppose the phase jumps by $\Delta\phi$ at $t=t_0$. Just before and after the jump:
  $$s(t_0^-) = A\cos\theta,\quad s(t_0^+) = A\cos(\theta+\Delta\phi)$$
  where $\theta = 2\pi f_c t_0 + \phi(t_0^-)$.  
  If $s(t_0^-)\neq s(t_0^+)$, the waveform has an amplitude discontinuity, which produces high-frequency components (wide sidelobes), i.e., increased bandwidth.
- At a zero crossing, $\cos\theta=0$, so $s(t_0^-)=0$ and $s(t_0^+)=0$ for any $\Delta\phi$. The waveform remains continuous in amplitude, greatly reducing spectral splatter and thus minimizing occupied bandwidth.

2) Why the other choices are incorrect
- B: To simplify modulation — Incorrect. Timing phase changes to carrier zero crossings actually requires more precise timing; it does not simplify the modulator.
- C: To improve carrier suppression — Incorrect. Carrier suppression is set by the balance and architecture (e.g., I/Q modulator, balanced mixer), not by when the phase is changed.
- D: All these choices are correct — Incorrect because only bandwidth is affected by zero-crossing phase changes.

3) Useful formulas and rules of thumb
- RF model with phase modulation:
  $$s(t) = A \cos\big(2\pi f_c t + \phi(t)\big)$$
- Amplitude discontinuity from a phase step:
  $$\Delta s = A\big[\cos(\theta+\Delta\phi) - \cos\theta\big]$$
  If $\cos\theta=0$ (zero crossing), then $\Delta s=0$ and the amplitude is continuous.
- Rule of thumb: Time-domain discontinuities (sharp steps) create wide spectra. For a transition with rise time $t_r$, a rough bandwidth estimate is:
  $$\text{BW} \approx \frac{0.35}{t_r}$$
  Avoiding an amplitude step (by switching at zero crossings) effectively lengthens the “edge” and reduces occupied bandwidth. Continuous-phase methods (e.g., OQPSK, MSK, raised-cosine shaping) further reduce splatter.