GMSK transmission and detection

Gaussian Minimum Shift Keying (GMSK) is MSK in which the NRZ data is first passed through a Gaussian low-pass pre-modulation filter before frequency modulation. The filter smooths the phase path, which suppresses side lobes and gives a very compact spectrum, while the envelope stays constant. GMSK with BT = 0.3 is the modulation used in GSM.

The Gaussian filter impulse response is:

h_G(t) = (√π/α) exp(−π² t² / α²),  α = 0.5887 / B

where B is the 3 dB bandwidth of the filter. The product BT (B × bit period) controls the trade-off: smaller BT gives a narrower spectrum but more ISI. BT = ∞ is plain MSK.

GMSK transmitter

Method 1 – direct FM:

NRZ data -> [Gaussian LPF] -> [FM / VCO     ] -> GMSK
            (BT = 0.3)        [h = 0.5      ]

The filtered data drives a voltage-controlled oscillator with modulation index h = 0.5. Simple, but the VCO must hold the exact index.

Method 2 – quadrature (I/Q) modulator:

data->[Gaussian LPF]->[Integrator]-> φ(t)
                                     |
              +----------------------+
              v                      v
           [cos φ]->(x) cos ωct   [sin φ]->(x) sin ωct
                     |                      |
                     +--------(+)-----------+
                               |
                             GMSK

The phase φ(t) is generated digitally; cos φ and sin φ modulate the I and Q carriers. This keeps the index exact and is used in practice.

GMSK receiver (coherent)

       +->(x)->[LPF]->[Sample]->I -+
       |  cos ωct                  |
GMSK->-+                       [Decision / ]-> data
       |  sin ωct              [ Viterbi   ]
       +->(x)->[LPF]->[Sample]->Q -+

The signal is multiplied by cos and sin of the recovered carrier, low-pass filtered and sampled; I and Q are sampled at alternate bit instants (like MSK/OQPSK detection). Because of controlled ISI, GSM uses a Viterbi (MLSE) equalizer. GMSK can also be detected non-coherently with a frequency discriminator.

Constellation / phase diagram

              Q
              • (π/2)
              |
   (π) •------+------• (0)   I
              |
              • (3π/2)
 Constant envelope: points on a circle;
 phase moves ±π/2 per bit along smooth arcs.

Features

• Constant envelope, so efficient non-linear class-C amplifiers can be used (good battery life).
• Very narrow spectrum, low adjacent channel interference (GSM: 270.833 kbps in 200 kHz).
• BER is slightly worse than MSK due to ISI from the filter.