Satellite Signal Acquisition, Tracking, and Data Demodulation - Understanding GPS: Principles and Applications - page 179

no loss of signal processing performance, as was the case with the TI 4100 analog
multiplexing. There is also no interchannel bias error.

5.5 Loop Filters

The objective of the loop filter is to reduce noise in order to produce an accurate
estimate of the original signal at its output. The loop filter order and noise band-
width also determine the loop filter's response to signal dynamics. As shown in the
receiver block diagrams, the loop filter's output signal is effectively subtracted from
the original signal to produce an error signal, which is fed back into the filter's input
in a closed loop process. There are many design approaches to digital filters. The
design approach described here draws on existing knowledge of analog loop filters,
then adapts these into digital implementations. Figure 5.18 shows block diagrams
of first, second, and third-order analog filters. 1 Analog integrators are represented
by 1/ s , the Laplace transform of the time domain integration function. The input
signal is multiplied by the multiplier coefficients, then processed as shown in Figure
5.18. These multiplier coefficients and the number of integrators completely deter-
mine the loop filter's characteristics. Table 5.6 summarizes these filter characteris-
tics and provides all of the information required to compute the filter coefficients
for first, second, and third-order loop filters. Only the filter order and noise band-
width must be chosen to complete the design.
Figure 5.19 depicts the block diagram representations of analog and digital
integrators. The analog integrator of Figure 5.19(a) operates with a continuous time
domain input, x ( t ), and produces an integrated version of this input as a continuous
time domain output, y ( t ). Theoretically, x ( t ) and y ( t ) have infinite numerical resolu-
tion, and the integration process is perfect. In reality, the resolution is limited by
noise, which significantly reduces the dynamic range of analog integrators. There
are also problems with drift.
The boxcar digital integrator of Figure 5.19(b) operates with a sampled time
domain input, x ( n ), which is quantized to a finite resolution and produces a discrete
integrated output, y ( n ). The time interval between each sample, T , represents a unit
delay, z -1 , in the digital integrator. The digital integrator performs discrete integra-
tion perfectly with a dynamic range limited only by the number of bits used in the
accumulator, A . This provides a dynamic range capability much greater than can be
achieved by its analog counterpart, and the digital integrator does not drift. The
boxcar integrator performs the function yn () = T xn [()] + An ( − 1 , where n is the )

discrete sampled sequence number.
Figure 5.19(c) depicts a digital integrator that linearly interpolates between
input samples and more closely approximates the ideal analog integrator. This is
called the bilinear z-transform integrator. It performs the function y ( n ) = T /2[ x ( n )] +

A ( n 1)]. The digital filters depicted in Figure 5.20 result

when the Laplace integrators of Figure 5.18 are each replaced with the digital

− 1) = 1/2[ A ( n ) + A ( n −

1. Jerry D. Holmes originally developed these analog and digital loop filter architectures and filter parameters.

They were used in the first commercial GPS receiver design, the TI 4100 NAVSTAR Navigator, Texas
Instruments, Inc., 1982.