Satellite Signal Acquisition, Tracking, and Data Demodulation - Understanding GPS: Principles and Applications - page 168

Table 5.2 PLL Discriminator
Discriminator
Algorithm

Output
Phase Error Characteristics

Four-quadrant arctangent.
Optimal (maximum likelihood estimator) at high and low SNR.
Slope not signal amplitude dependent.
High computational burden.
Usually table lookup implementation.

ATAN 2( Q PS , I PS ) φ

sin φ Q PS normalized by averaged prompt envelope.

Slightly outperforms four-quadrant arctangent.
Q PS approximates

Q

PS

2 2

Ave I + Q

PS PS

45º.
Normalization provides insensitivity at high and low SNR.
Also keeps slope not signal amplitude dependent.
Low computational burden.

φ to ±

Table 5.3 Common Costas Loop Discriminators
Discriminator
Algorithm

Output
Phase Error Characteristics

Classic Costas analog discriminator.
Near optimal at low SNR.
Slope proportional to signal amplitude squared A 2 .
Moderate computational burden.

Q PS × I PS sin 2 φ

Decision directed Costas.
Near optimal at high SNR.
Slope proportional to signal amplitude A .
Least computational burden.

Q PS × Sign ( I PS ) sin φ

Suboptimal but good at high and low SNR.
Slope not signal amplitude dependent.
Higher computational burden.
Divide by zero error at

Q PS / I PS tan φ

± 90º.

Two-quadrant arctangent.
Optimal (maximum likelihood estimator) at high and low SNR.
Slope not signal amplitude dependent.
Highest computational burden.
Usually table lookup implementation.

ATAN ( Q PS / I PS ) φ

amplitude (signal plus noise) following the code wipeoff and integrate and dump
process. The replica cosine function is 90º out of phase with the incoming SV carrier.
This results in a cosine

sine product at the Q output, which produces minimum Q PS
amplitude (noise only). For this reason, I PS will be near its maximum (and will flip
180º each time the data bit changes sign), and Q PS will be near its minimum (and will
also flip 180º each time the data bit changes sign).
Note that the classical complex pair carrier phase derotation scheme is not used
in this generic receiver design because the natural GPS IF signal is real. The classical
phase derotation scheme requires an I and Q input signal at IF. This, in turn,
requires the real IF signal be phase shifted 90º to produce the quadrature compo-
nent. This is a design penalty of added IF circuit complexity. If this is done on the

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