PSK generation and detection; GMSK

Generation of PSK

In binary Phase Shift Keying the carrier phase is $0$ for bit 1 and $\pi$ for bit 0:

$$s(t)=\pm\sqrt{\frac{2E_b}{T_b}}\cos 2\pi f_ct,\quad 0\le t\le T_b$$

 data -->[polar NRZ encoder]--> b(t)=+-1 -->(X)--> BPSK
                                            ^
                            sqrt(2/Tb) cos 2pi fc t

The polar NRZ data multiplies the carrier in a product (balanced) modulator. When $b(t)$ changes sign, the output phase jumps by $180^\circ$.

 data:   1       0       1
 PSK : /\/\/\  \/\/\/  /\/\/\

Detection of PSK (coherent)

 x(t)-->(X)-->[integrate 0..Tb]--x-->[decision x>0 ? 1 : 0]
         ^
   carrier recovery (squaring loop / Costas loop)

1. The receiver regenerates a carrier exactly in phase with the transmitted one (e.g. square the signal, extract $2f_c$ with a PLL, divide by 2).
2. The product $x(t)\cos\omega_ct$ is integrated over the bit, giving $x=\pm\sqrt{E_b}$ + noise.
3. A zero-threshold decision device outputs 1 for $x>0$ and 0 for $x<0$.

Signal space: two antipodal points $\pm\sqrt{E_b}$ on $\phi_1$; $P_e=\tfrac12\text{erfc}\sqrt{E_b/N_0}$. Envelope detection is impossible because the envelope is constant.

GMSK

Gaussian Minimum Shift Keying is MSK in which the NRZ data is first passed through a Gaussian low-pass filter before it frequency-modulates the carrier with $h=0.5$.

 NRZ data ->[Gaussian LPF, BT]->[FM mod, h=0.5]-> GMSK

• The filter smooths the sharp transitions, so the phase changes gradually and the spectrum becomes very compact.
• The product $BT$ (filter 3-dB bandwidth × bit period) sets the trade-off; GSM uses $BT=0.3$.
• Some intersymbol interference is introduced, but it is small for $BT\ge0.3$.

Advantages: narrow main lobe and very low sidelobes (less adjacent-channel interference), constant envelope so efficient class-C power amplifiers can be used (long battery life), and simple non-coherent or coherent detection. These are why GSM uses GMSK (DECT and Bluetooth use the closely related Gaussian FSK, GFSK).