Gaussian filter:
1. continuous-time impulse response of the Gaussian filter:
2. the frequency respone can be shown as,
cause
then
get
the 3dB bandwidth should equal to the bandwith of singal to satisfy the requirement of filter design, so,
then
GMSK
the GMSK process:
the sequence of M-ary data symbols is shaped with Gaussian filter, the symbol do convolution with h(t), shown as,
the CDF function is shown as,
so the phase response can be present as,
GMSK is kind of Continuous-phase modulation(CPM), to make the phase continuity, the q(t) should be normalized according to below.
In CPM, the baseband representation of the modulated signal is
h is modulation index, for the MSK the h = 0.5 that make the minum frequency offset to keep the carrier orthogonal.