Position, Navigation, and Timing Technologies in the 21st Century. Группа авторов
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38.5.3.1 Discriminator Statistics
In order to study the discriminator statistics, the received signal noise statistics must first be determined. In what follows, the received signal noise is characterized for an additive white Gaussian noise channel.
Received Signal Noise Statistics: To make the analysis tractable, the continuous‐time received signal and correlation are considered. The transmitted signal is assumed to propagate in an additive white Gaussian noise channel with a power spectral density
and the continuous‐time matched‐filtered baseband signal x(t) is given by
The resulting early and late correlations in the DLL are given by
where
where
Coherent Discriminator Statistics: The coherent baseband discriminator function is defined as
The normalized signal component of the discriminator function
It can be seen from Figure 38.18 that for small values of
where α is the slope of the discriminator function at Δtk = 0 [57], which is obtained by
Since Rc(τ) is symmetric,
and the linearized discriminator output becomes
It is worth noting that Rc(τ) and Rc ′ (τ) are obtained by numerically computing the autocorrelation function of the pulse‐shaped short code. Since the FIR of the pulse shaping filter h[k] is defined over only 48 values of k, the autocorrelation function Rc(τ) will be defined over 95 values of τ. However, interpolation may be used to evaluate Rc(τ) and Rc ′ (τ) at any τ. The mean and variance of Dk can be obtained from Eq. (38.12), and are given by
(38.14)
Now that the discriminator statistics are known, the closed‐loop pseudorange error is characterized next.
38.5.3.2 Closed‐Loop Analysis
In order to achieve the desired loop noise‐equivalent bandwidth, K in Eq. (38.11) must be normalized according to
Figure 38.18 Output of the coherent baseband discriminator function for the CDMA short code with different correlator spacings (Khalife et al. [12]).
Source: Reproduced with permission of IEEE.
In cellular CDMA systems, for a teml of 1.2, the loop filter gain becomes K ≈ 4Bn, DLL; hence, the choice of K in Section 38.5.2.3. Assuming a zero‐mean tracking error, that is,
At steady state, var{Δtk + 1} becomes