Here we begin with a complex Gaussian noise source that has a zero mean. It is well known that the
magnitude of such a complex signal produces a Rayleigh distribution while its phase is uniformly distributed.
In order to introduce the presence of a maximum Doppler spread, we must pass this noise
source through a low pass filter whose cutoff frequency is equal to the maximum expected Doppler
shift. Note the insertion of an LPF introduces correlation between adjacent samples. Ablock diagram
is shown in Fig. 3.57.
WIRELESS MULTIPATH CHANNEL 163
WGN
WGN
LPF
LPF
hI(t) + jhQ(t)
FIGURE 3.57 Low pass filtering approach to generate Rayleigh multipath
fading.
As shown earlier in this chapter, ideally the LPF will have a pole or asymptote at the maximum
Doppler frequency when considering the Doppler spectrum, S( f ).
The choice of the LPF is critical in order to obtain the desired autocorrelation properties discussed
above. An example of an LPF using an finite impulse response (FIR) structure is given in Fig. 3.58.
Fading LPF Frequency Response
??“180
??“160
??“140
??“120
??“100
??“80
??“60
??“40
??“20
0
??“400 ??“300 ??“200 ??“100 0 100 200 300 400
Frequency (Hz) Magnitude (dB)
FIGURE 3.58 LPF response of the WGN filters.
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