Interference, Multipath, and Scintillation - Understanding GPS: Principles and Applications - page 282

and receiver is different from relative motion between satellites and receiver, causing
multipath arrivals at different Doppler shifts from the direct path. When the Doppler
differences are significant (greater than the reciprocal of the coherent integration
time in the correlator), however, they cause the received multipath signals to be
essentially uncorrelated with the direct path, and thus can often be neglected since
they do not correlate well with the reference signal used to track the direct path.
A special case of (6.43) occurs when the propagation geometry is such that the
direct path is nearly tangent to the Earth's surface (such as when the satellite is near
the horizon). Then there can be a single dominant multipath arrival that reflects
from a large object near the horizon, with excess delay orders of magnitude less than
the reciprocal of the signal bandwidth and only a small fraction of the carrier
period—often smaller than a picosecond. When the reflection coefficient is suffi-
ciently high and there are no other multipaths, then xt

~

0 1 0 . Conse-
quently, (6.43) can be approximated (when the reflection introduces a 180º rotation
of the carrier phase) as

( −− ≅ − ττ ) xt ( τ )

[ ] (

~ ~

() )

− j φ − j φ

rt ≅ α e 1 − α e xt − τ (6.44)

0 1

0 1 0

where ~

~

α 1
to be near unity, the magnitude of the quantity in square brackets is very much less
than unity. The delay of this multipath is so small that it causes negligible
pseudorange error, but by nearly canceling the direct path, it causes significant
reduction in received signal power, relative to what would be observed with
free-space propagation. This phenomenon is well known in land mobile radio [13],
and not addressed further in this section.
More general models of multipath channels [12] do not represent the fine struc-
ture as models discussed previously, but instead represent the effect of the multipath
channel—in our case, relative to the direct path, as in (6.43)—as a slowly time-vary-
ing linear system. The impulse response falls off with excess delay, and the range of
excess delays where the impulse response is essentially nonzero is called the chan-
nel's multipath spread . In turn, the multipath spread can be represented by the RMS
delay spread of the channel. This linear system has a time-varying transfer function
that describes how it passes different frequency components of the signal.
Since the transfer function at a given frequency randomly varies over time, the
correlation between transfer functions at different times and the same frequency
[12] describes the time variation of the channel. If the time variation is fast relative
to time constants in the receiver tracking loops, the multipath errors are smoothed
by the receiver processing. Otherwise, they produce a time-invariant error term. The
power spectral density resulting from the Fourier transform of this correlation is
called the Doppler power spectrum of the channel, and the range of frequencies over
which it is essentially nonzero is called the Doppler spread of the channel. The recip-
rocal of the Doppler spread is the coherence time of the channel—the time over
which the multipath structure does not change much relative to the direct path. Two
fundamental quantities introduced by this channel model—the multipath spread
and the Doppler spread—provide succinct yet useful high-level representations of
the multipath characteristics.

φπτ
1

= 2 f c is very small, so that when the reflection is strong enough for

1