Page 32 - Integrated Wireless Propagation Models

10 C h a p t e r 0 n e

to the mobile at points a and b is t:.l = d c os8 = vt:.tcos8 , where t:.t is the time required for
the mobile to travel from a to b and the incident angle 8 of the incoming wave is assumed
to be the same at points a and b since the source is assumed to be far away. The phase
change in the received signal due to the difference in path lengths is therefore
t:.< 2�/':,l 2n�t:.t
= = cos8 (1.6.3.4.1)
where A is the wavelength. The apparent change in frequency due to the motion of the
mobile, or Doppler shift, is given by f d , where

1 t:.<jl v
=
f d = 2n: -;rr X · cos8 (1.6.3.4.2)
The Doppler shift shown in Eq. (1.6.3.4.2) is a function of the mobile velocity and
the incident angle between the direction of motion of the mobile and the direction of
arrival of the wave. lt can be seen from Eq. (1.6.3.4.2) that if the mobile is moving toward
the direction of arrival of the wave, the Doppler shift is positive (i.e., the apparent
received frequency is increased), and if the mobile is moving away from the direction of
arrival of the wave, the Doppler shift is negative (i.e., the apparent received frequency
is decreased). Multipath components from a continuous-wave signal that arrive from
different directions contribute to Doppler spreading of the received signal, thus increas­
ing or decreasing the signal bandwidth.
If the baseband signal bandwidth is much greater than fd, the effects of Doppler
spread are negligible at the receiver. This is a slow-fading channel.

1.6.3.5 Coherence Time
Coherence time is the time duration over which two received signals have a strong
amplitude correlation in a Rayleigh fading environment. The coherence time and max­
imum Doppler spread are inversely proportional to each other. That is,

(1.6.3.5.1)

Coherence time Tc is used to characterize the time-varying nature of the frequency dis­
persiveness of the channel in the time domain. When the reciprocal bandwidth of the
baseband signal (e.g., symbol interval) is greater than the coherence time of the channel,
the channel will vary in time during the transmission of the signal, and distortion at the
receiver occurs. If the coherence time is defined as the time over which the amplitude
correlation coefficient is equal or greater than 0.5, then the coherence time is found as 22
9
Tc = = 0.179 / j"' (1.6.3.5.2)
16n;f
1 m
where !," = v/'A, which is the maximum Doppler shift. Equation (1.6.3.5.2) is used in
analog communication. In digital communications, the coherent time is taking a square
root of multiplication of Eqs. (1.6.3.5.1) and (1.6.3.5.2). That is,
� 9 0.423
T - 16 .f2 - - J "' f (1.6.3.5.3)
c -
TC ; "'
Comparing Eq. (1.6.3.5.3) with Eq. (1.6.3.5.2), the coherence time for digital communica­
tions is longer than that for analog communication because of the nature of code
sequences.