QPSK modulator, demodulator and signal space

Quadriphase-shift keying (QPSK) is a digital modulation in which the carrier takes one of four phases, each carrying a pair of bits (a dibit). It sends 2 bits per symbol, so it needs half the bandwidth of BPSK for the same bit rate, with the same bit error probability.

Signal set

With symbol duration $T = 2T_b$ and symbol energy $E$:

$$s_i(t) = \sqrt{\frac{2E}{T}}\cos\left[2\pi f_ct + (2i-1)\frac{\pi}{4}\right], \quad 0 \le t \le T,\ i = 1, 2, 3, 4$$

Expanding the cosine:

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

with two orthonormal basis functions

$$\phi_1(t) = \sqrt{\frac{2}{T}}\cos(2\pi f_ct), \qquad \phi_2(t) = \sqrt{\frac{2}{T}}\sin(2\pi f_ct), \quad 0 \le t \le T$$

($f_c$ is a multiple of $1/T$, so $\int_0^T\phi_1\phi_2\,dt = 0$ and each has unit energy.)

Coordinates $s_{i1} = \sqrt{E}\cos[(2i-1)\pi/4]$, $s_{i2} = -\sqrt{E}\sin[(2i-1)\pi/4]$:

Signal space diagram

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

The four points lie on a circle of radius $\sqrt{E}$. Gray coding makes adjacent points differ by one bit. The distance between adjacent points is $d = \sqrt{2E} = 2\sqrt{E_b}$ (with $E = 2E_b$), the same as BPSK, so the bit error probability is the same:

$$P_e(\text{bit}) = \frac12\,\text{erfc}\sqrt{\frac{E_b}{N_0}}$$

QPSK modulator

          +-> a1(t) -->(x)----+
binary    |   (odd bits)   ^  |
data --[serial to]         |  |
 b(t)   [parallel ]  phi1(t)  (+)--> QPSK s(t)
          |   (even bits)  |  |
          +-> a2(t) -->(x)----+
                           ^
                        phi2(t)

1. The input bit stream $b(t)$ is converted to polar NRZ form ($1 \to +\sqrt{E_b}$, $0 \to -\sqrt{E_b}$).
2. A demultiplexer (serial-to-parallel converter) splits it into odd bits $a_1(t)$ and even bits $a_2(t)$, each of duration $T = 2T_b$.
3. $a_1(t)$ multiplies the in-phase carrier $\phi_1(t)$ and $a_2(t)$ multiplies the quadrature carrier $\phi_2(t)$, giving two BPSK signals.
4. The sum of the two BPSK signals is the QPSK signal:

$$s(t) = a_1(t)\phi_1(t) + a_2(t)\phi_2(t)$$

Since each branch runs at $R_b/2$, the null-to-null bandwidth is $B = 2/T = R_b$ (BPSK needs $2R_b$).

QPSK demodulator (coherent receiver)

          +->(x)->[integrate 0..T]->[decide]->a1^-+
  x(t)    |   ^                     thresh 0      |
 --------+  phi1(t)                         [parallel]-> b^
          |                                 [to serial]
          +->(x)->[integrate 0..T]->[decide]->a2^-+
              ^                     thresh 0
            phi2(t)

The received signal is $x(t) = s_i(t) + w(t)$, with $w(t)$ white Gaussian noise of PSD $N_0/2$.

1. Two correlators multiply $x(t)$ by locally generated coherent references $\phi_1(t)$ and $\phi_2(t)$ and integrate over one symbol:

$$x_1 = \int_0^T x(t)\phi_1(t)\,dt = s_{i1} + w_1, \qquad x_2 = \int_0^T x(t)\phi_2(t)\,dt = s_{i2} + w_2$$

Because of orthogonality, the in-phase branch sees only $s_{i1}$ and the quadrature branch only $s_{i2}$; $w_1, w_2$ are independent Gaussian with variance $N_0/2$. 2. Each output is compared with a threshold of zero: $x_1 > 0 \Rightarrow$ 1, else 0 (same for $x_2$). This is the minimum-distance decision in the signal space; the decision regions are the four quadrants. 3. A multiplexer (parallel-to-serial converter) combines the two decided bit streams into the original sequence.

Error performance

Each branch is a BPSK detector with $E_b = E/2$:

$$P_e(\text{bit}) = \frac12\,\text{erfc}\sqrt{\frac{E_b}{N_0}}, \qquad P_e(\text{symbol}) \approx \text{erfc}\sqrt{\frac{E}{2N_0}}$$

So QPSK gives the error performance of BPSK with twice the bandwidth efficiency (2 bits/s/Hz in theory).