Interference, Multipath, and Scintillation - Understanding GPS: Principles and Applications - page 290

BPSK-R(1), 4 MHz
BPSK-R(1), 24 MHz
BPSK-R(10), 24 MHz

15

10

5

0

− 5

− 10

− 15

0 200 400 600 800 1000

Multipath delay (ns)

Figure 6.13 Multipath ranging error envelopes showing the maximum and minimum code track-
ing error for one-path multipath with MDR -10 dB, at different multipath delays.

is the carrier
phase error introduced by the multipath. The carrier phase error is then a function of
the multipath characteristics and the delay error.
Observe

Since the carrier phase of the direct path is defined to be zero, ψ

~ ~

that when τ 1 is very small, R ( ττ −≅ ) R ( ττ −− τ ) , and

sin ~

x 0 x 0 1

~

αφ
αφ

1 1

tan ψ ≅ . When the multipath power is equal to the power in the direct

cos ~

~

1 +

1 1

path, the carrier phase error is greatest when the multipath carrier phase is 90º rela-
tive to the direct path, producing a carrier phase error of 45º. As long as the MDR is
less than or equal to unity, and the delay-locked loop maintains track on the correla-
tion function of the direct path, the magnitude of the carrier phase error is less than
or equal to 45º.
For a given MDR and excess delay, the carrier phase error varies with the
multipath phase and the error in delay estimate. The resulting minimum and maxi-
mum carrier phase errors are given respectively by

sin ~

~ ~

( )

αφ R ετ −

~

, ~

( )

1 1 x 1

ψατ = min tan

− 1

cos ~

~ ~

~

min 11

() ( )

R εα + φ R ετ −

φ π
εεατ

∈ (, 02 ]

1

~

, ~

(

∈ ∆

)

x 1 1 x 1

(6.49)

11

~ ~

( )

αφ ε sin R − τ

~

, ~

( )

x

− 1 1 1 1

ψατ = max tan

s ~

~ ~

max ~

11

() ( )

R εα + co φε R x − τ

φ π
εεατ

∈ (, 02 ]

1

~

, ~ ( )

x

∈ ∆ 1 1 1

11

ψ max ( ~ , ~ ψ min ( ~ , ~ so the resulting envelope of carrier phase error is given by ( α 11 ).

Figure 6.14 shows multipath carrier phase error envelopes for the same conditions
as

α 11 , ) )

Figure 6.13: BPSK-R(1) modulation with two different precorrelation