QPSK fundamental

Question

I am trying to understand the QPSK fundamental and when I read that the phase shifts are 45°, 135°, 225°, and 315° in QPSK, I can not figure it out how.

In other word, I do not understand how the wave form creates 45°, 135°, 225°, and 315° angles. It seems to me they are 45°, 315° (Since falls in 4th quadrant), again 315° (same, since falls in 4th quadrant), and 45° (since falls in 4th quadrant).

Could you please help me to understand how to wave forms are forming 45°, 135°, 225°, and 315° angles?

Answer 1

The individual waveforms are just one period of the signal

$$x_k(t)= \begin{cases} \sin(2\pi t/T+\phi_k),\qquad & 0\le t<T \\ 0,\qquad & \text{otherwise} \\ \end{cases}\tag{1}$$

where the phase angles $\phi_k$ are the ones given in your question:

$$\phi_k=(2k+1)\frac{\pi}{4},\qquad k\in\{0,1,2,3\}\tag{2}$$

The QPSK signal shown in the figure is then given by

$$s(t)=\sum_{k=0}^{3}x_k(t-kT)\tag{3}$$