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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 . The continuous‐time received signal after down‐mixing and bandpass sampling is given by

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 and . Assuming the receiver is closely tracking the carrier phase [55], the early and late correlations may be approximated with

where and are zero‐mean Gaussian random variables with the following variances and covariances:

Coherent Discriminator Statistics: The coherent baseband discriminator function is defined as

The normalized signal component of the discriminator function is shown in Figure 38.18 for t_eml = {0.25, 0.5, 1, 1.5, 2}.

It can be seen from Figure 38.18 that for small values of , the discriminator function can be approximated by a linear function given by

where α is the slope of the discriminator function at Δt_k = 0 [57], which is obtained by

Since R_c(τ) is symmetric,

and the linearized discriminator output becomes

(38.12)

It is worth noting that R_c(τ) and R_c ′ (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 R_c(τ) will be defined over 95 values of τ. However, interpolation may be used to evaluate R_c(τ) and R_c ′ (τ) at any τ. The mean and variance of D_k can be obtained from Eq. (38.12), and are given by

(38.13)

(38.14)

Now that the discriminator statistics are known, the closed‐loop pseudorange error is characterized next.