BPSK modulation and signal space diagram

Binary Phase Shift Keying (BPSK) sends a bit by shifting the carrier phase by $180^\circ$: phase $0$ for symbol 1 and phase $\pi$ for symbol 0. Amplitude and frequency stay constant.

$$s_1(t)=\sqrt{\frac{2E_b}{T_b}}\cos 2\pi f_ct,\qquad s_2(t)=-\sqrt{\frac{2E_b}{T_b}}\cos 2\pi f_ct,\qquad 0\le t\le T_b$$

$f_c=n_c/T_b$ so each bit holds a whole number of cycles. $s_1$ and $s_2$ are antipodal signals.

Waveform (data 1 0 1)

 data:   1        0        1
       /\  /\   \  /\  /   /\  /\
      /  \/  \   \/  \/   /  \/  \
              phase flips at 1->0 and 0->1

Modulator

 binary --> [NRZ polar encoder] --> b(t) = +-1
                                     |
                        (X) <-- sqrt(2/Tb) cos 2pi fc t
                         |
                       BPSK s(t)

The polar NRZ signal (+1 for 1, -1 for 0) multiplies the carrier, so the product modulator flips the phase whenever the data sign changes.

Coherent demodulator

 x(t) -->(X)--> [integrate 0..Tb] --> x1 --> [decision]
          ^                                  x1>0 -> 1
     phi1(t) from carrier recovery           x1<0 -> 0

The correlator gives $x_1=\int_0^{T_b}x(t)\phi_1(t)dt$, which equals $\pm\sqrt{E_b}$ plus noise.

Signal space diagram

One basis function is enough ($N=1$):

$$\phi_1(t)=\sqrt{\frac{2}{T_b}}\cos 2\pi f_ct,\qquad s_1(t)=\sqrt{E_b}\,\phi_1(t),\quad s_2(t)=-\sqrt{E_b}\,\phi_1(t)$$

   region Z2        |        region Z1
      s2            |            s1
 -----*-------------+-------------*-----> phi1
   -sqrt(Eb)        0         +sqrt(Eb)
                 boundary

• Distance between points: $d=2\sqrt{E_b}$, the largest possible for a given energy.
• Decision threshold at zero.
• $P_e=\tfrac12\,\text{erfc}\sqrt{E_b/N_0}$, the best among binary schemes.
• Bandwidth $\approx 2/T_b=2R_b$ (null to null).