The 3 dB cutoff frequency of this filter is typically normalized by the bit rate to create
an equivalent description called BTb product, where B 3 dB BW of the premodulation filter and
Tb 1/Rb is the bit time duration. We will close this section by making a few points and observations
as to how the BTb product affects the system performance.
The autocorrelation functions for both MSK and OQPSK with NRZ pulse shaping are shown in
Fig. 2.66 [26]. The figure assumes a bit time duration normalized to 1 sec (T 1 sec).
(Tb t Tb) p(t) cos a pt
2Tbb
MODULATION THEORY 89
Autocorrelation Function of MSK and OQPSK
0
0.2
0.4
0.6
0.8
1
??“2 ??“1.5 ??“1 ??“0.5 0 0.5 1 1.5 2
Time Offset (?„)
Autocorrelation Function Value
MSK
OQPSK
FIGURE 2.66 MSK and OQPSK autocorrelation functions.
The autocorrelation function for MSK is given as [30??“33]
(2.42)
The autocorrelation function for OQPSK is given as
(2.43)
The PSD is obtained by taking the Fourier transform of the above respective autocorrelation functions
(see Fig. 2.67).
The mathematical equations used to plot the above curves are given below:
(2.44)
(2.45) PSDOQPSK( f ) 2 # P # T # a sin [2pfT]
2 # p # f # Tb2
PSDMSK( f )
8 # P # T # (1 cos [4pfT])
p2 # (1 16 # T2 # f 2)2
ROQPSK(t) 1
ZtZ
2T
RMSK(t)
1p
epa1
ZtZ
2Tb # cos apZtZ
2T b sin apZtZ
2T b f
From the above equations, it is easy to see that MSK spectrum falls off at a rate of f4, while
OQPSK falls off at a rate of f 2.
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