QPSK modulation and demodulation

Quadrature Phase Shift Keying (QPSK) is a 4-level PSK in which each symbol carries a pair of bits (a dibit) and the carrier takes one of four phases spaced $90^\circ$ apart.

$$s_i(t)=\sqrt{\frac{2E}{T}}\cos\left[2\pi f_ct+(2i-1)\frac{\pi}{4}\right],\quad i=1,\dots,4,\ 0\le t\le T$$

$T=2T_b$ is the symbol duration and $E=2E_b$ the symbol energy. Using $\cos(A+B)$:

$$s_i(t)=\sqrt{E}\cos\left[(2i-1)\tfrac{\pi}{4}\right]\phi_1(t)-\sqrt{E}\sin\left[(2i-1)\tfrac{\pi}{4}\right]\phi_2(t)$$

with $\phi_1(t)=\sqrt{2/T}\cos 2\pi f_ct$ and $\phi_2(t)=\sqrt{2/T}\sin 2\pi f_ct$. So QPSK is the sum of two BPSK signals on quadrature carriers.

Signal space (Gray coded)

             phi2
   01 *       |       * 11
              |
  ------------+------------ phi1
              |
   00 *       |       * 10
 points at (+-sqrt(E/2), +-sqrt(E/2))

QPSK modulator

 bits-->[polar NRZ encoder]-->[Demux S/P]
                               |        |
                         a1(t) odd  a2(t) even
                         (rate Rb/2) (rate Rb/2)
                               |        |
                     phi1 ->(X)      (X)<- phi2
                               |        |
                               +->(+)<--+
                                   |
                                 QPSK s(t)

1. The data is converted to polar NRZ (+1, -1).
2. The demultiplexer splits it into odd and even bits, each lasting $T=2T_b$.
3. Odd bits modulate the in-phase carrier and even bits the quadrature carrier (two BPSK signals).
4. The adder gives one of four phases; e.g. dibit 11 gives the point $(+\sqrt{E/2},+\sqrt{E/2})$, phase $7\pi/4$.

QPSK demodulator (coherent)

          +->(X)->[int 0..T]-x1->[x1>0?1:0]--+
          |   ^ phi1                          |
 x(t) ----+                                 [Mux P/S]-> bits
          |   v phi2                          |
          +->(X)->[int 0..T]-x2->[x2>0?1:0]--+

1. The received signal goes to two correlators fed with the recovered carriers $\phi_1$ and $\phi_2$.
2. Because $\phi_1\perp\phi_2$, each correlator sees only its own BPSK component: $x_1=\pm\sqrt{E/2}$, $x_2=\pm\sqrt{E/2}$ (plus noise).
3. Each output is compared with a zero threshold to give the odd and even bits.
4. The multiplexer recombines them into the original serial stream.

Properties

QPSK doubles bandwidth efficiency without any loss in bit-error performance, which is why it is used in satellite links, DVB-S and CDMA systems. Offset QPSK (Q branch delayed by $T_b$) avoids $180^\circ$ phase jumps.