Skip to main content

MSK as CPFSK/OQPSK; compare OQPSK, MSK, GMSK

Short answer

MSK is CPFSK with modulation index h=0.5 and tone spacing Rb/2, and is OQPSK with half-sinusoidal pulse shaping. OQPSK has a narrower main lobe but slow side-lobe roll-off; MSK has a wider main lobe (1.5Rb) with faster roll-off; GMSK has the narrowest spectrum with slightly lower power efficiency. GMSK suits GSM because of its constant envelope, narrow spectrum, and allowance for nonlinear class-C amplifiers.

How can Minimum Shift Keying (MSK) be defined as a special type of continuous phase FSK and Orthogonal Quadrature Phase Shift Keying (OQPSK)? Compare OQPSK, MSK and GMSK in terms of their bandwidth efficiency and power efficiency. Which of them are suitable for modulation in GSM and why?

MSK as continuous phase FSK

Minimum Shift Keying (MSK) is a binary continuous phase FSK (CPFSK) with modulation index h = 0.5, the minimum index for which the two tones are orthogonal over one bit (coherent detection).

s(t) = √(2E_b/T_b) cos(2π f_c t + θ(t)),
θ(t) = ± π t / (2T_b) + θ(0)
f_H − f_L = 1 / (2T_b) = R_b / 2

The phase changes linearly by exactly ±π/2 in each bit and has no jumps, so the waveform is continuous and the spectrum falls off fast.

MSK as a special OQPSK

MSK can also be written as OQPSK with half-sinusoidal pulse shaping instead of rectangular pulses:

s(t) = a_I(t) cos(πt/2T_b) cos(2π f_c t)
     + a_Q(t) sin(πt/2T_b) sin(2π f_c t)
  • a_I and a_Q are the even and odd bits, each lasting 2T_b.
  • The Q stream is offset by T_b from I (as in OQPSK).
  • The half-sine weighting makes the envelope constant and the phase continuous.

Comparison

SchemeBandwidth efficiencyPower efficiencyEnvelope
OQPSKGood: main lobe R_b, but slow side-lobe roll-offGood, BER = Q(√(2E_b/N₀))Nearly constant (no 180° jumps)
MSKWider main lobe (1.5R_b) but fast side-lobe roll-offSame BER as BPSK/QPSKConstant
GMSKBest: very narrow spectrum (BT = 0.3)Slightly lower (≈ 0.5–1 dB loss from ISI)Constant

Notes on the table: in null-to-null terms, OQPSK main lobe width is R_b (narrower than MSK's 1.5R_b), but its side lobes fall slowly (1/f²); MSK side lobes fall as 1/f⁴; GMSK shrinks side lobes further.

Suitable for GSM

GMSK is used in GSM because:

  • Its constant envelope lets mobiles use non-linear class-C amplifiers, which are power efficient and save battery.
  • Its narrow spectrum and low side lobes give low adjacent channel interference, allowing 270.833 kbps in a 200 kHz channel.
  • It can be detected coherently or non-coherently, and the small ISI is handled by GSM's Viterbi equalizer.

✓ Answer checked by Claude Opus 5.5 (high effort)

Related questions

All Wireless Communication chapter 4 questions →