PSEUDONOISE CODE TRACKING IN DIRECT-SEQUENCE RECEIVERS
2.3 ACQUISITION (TRANSIENT) BEHAVIOR OF THE DLL
In this section, we discuss the transient response of the DLL and TDL with particular emphasis on their acquisition behavior in the presence of an ini- tial code rate offset of the incoming PN code relative to that of the clock that generates the local PN code replica at the receiver. When discussing the transient response behavior of a DLL or TDL, or for that matter any PN
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code tracking loop, there are primarily two questions of interest:
1. What is the maximum relative code rate offset (due to code Doppler, clock instabilities, etc.) between the received and locally generated PN codes so that the received signal can still be acquired? Equivalently, what is the maximum search rate (velocity) to achieve acquisition?
2. How long do the transients last, i.e., how long does it take to acquire?
The answer to the first question may be obtained (in a noise-free envi- ronment) by examination of the phase-plane trajectories. These trajectories are plots of normalized code delay error rate versus normalized code delay error twith normalized time as a parameter along the curve.To obtain these trajectories for the DLL, we begin by rewriting the system equation of noise-free operation, as given by (2.15), in the normalized form
(2.66) where
(2.67) with vntheradian natural frequencyof the loop.
For a second-order loop with linear closed loop transfer function [16]
(2.68) the product of the loop gain Gand the filter transfer function F(s) becomes (2.69) or
(2.70) where z is the loop’s damping factor. For a critically damped loop
GF01p02 12zs>p0
1>Gs>p0
GF1s2 12zs>vn
1>Gs>vn
vnGF1s2 svnGF1s2
H1s2 12zs>vn
111>G2z21s>vn2 1s>vn22 F01p02F0a p
vn
b ^ F1p2
G^ hSKM2>vn
y^ et
x^ t Tc
p0^ d
dt a 1 vn
b d
dt p vn
t^ vnt
P01yx2GF01p02Dn1x2 e
e#
t
, (2.70) simplifies to
(2.71) which, upon substitution into (2.66), yields
(2.72) the dot now denotes the derivative with respect to normalized time (t) and the prime denotes the derivative with respect to the normalized transmis- sion delay (x). The solution of the second-order partial differential equation in (2.66) for the phase-plane trajectories is facilitated by defining
, which results in
(2.73) For a constant search velocity,ÿ0 and the above equation simplifies to
(2.74) Statistical methods (e.g., Newton’s method [19]) of solving differential equations can now be applied to (2.74) to compute the trajectories in the plane for given initial conditions. One then looks for the maximum for which the phase-plane trajectories will eventually reach the point, i.e., the DLL will phase and frequency lock.
The acquisition trajectories for G10, ,N2 and N4 are shown in Figures 2.16 and 2.17, respectively [4], [14].7It is found that the maximum normalized search rate for N2 is , while for N4, the maximum normalized search rate is . Furthermore, the open loop gain has little effect on the trajectories for G10 [14].
The second question raised above can be dealt with by using the defini- tion of gas a starting point and computing the acquisition transients as a function of time and then finding the acquisition time. Figure 2.18 shows the transient response of the DLL for G100, ,N2 and N4.
The search rate is chosen to be for the first case and for the second case, respectively. The graphs show that the acquisition time (time required for the transient to subside within 兩x兩 0.1) for N 2 is shorter than that for N4. The actual acquisition time can be computed from the definition of the normalized time given in (2.67) and the relation between the single-sided loop bandwidth BLand the loop radian natural frequency
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y# 1.0 z1>12
10,y#>G2y# 1x#,x2
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g Dn1x2 312Dnœ1x21>G4x# y#>Gy$
x# .
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y#>Gy$x#>Gx$Dn1x2 12Dnœ1x2x#. GF01p02 1 12p0
1>Gp0
, 1z1>122
7Historically acquisition trajectories were first found in [1] and [3] but were later shown to be in error [4].
Figure 2.16.Acquisition trajectories for N2 (reprinted from [14]).
vn. Since from [16]
(2.75) then for critically damped loops vnBL/.5303 and
(2.76) Table 2.3 summarizes the results for several different values of loop band- width.
Finally, the actual maximum search velocity in chips/sec (which can be interpreted as the maximum allowable drift in the code) can be found from the definition of the normalized search velocity . As a function of the sin- gle-sided loop noise bandwidth BL, the sarch velocity (ys) is
(2.77) which for critically damped loops becomes
(2.78) Although all our attention has focussed on the DLL, a comparison of (2.15) and (2.42) in the absence of noise reveals that except for a factor of two in the equivalent gain G, the TDL and DLL have identical acquisition performance.
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BL chips>sec y# tACQ .5303tACQ
BL
. vn a 8z
4z21bBL
Figure 2.17. Acquisition trajectories for N4 (reprinted from [14]).