## 4 Signal Evaluation Approach under the Local Despreading Waveform Mismatch  
### Formulation and Statistical Properties of Correlator Output  
To evaluate the code tracking accuracy, the mathematical formulation at every module in the diagram of Figure 1 should be obtained, particularly the discriminator output. In the process, correlator output is an intermediate result, which can be used in the evaluation of acquisition and demodulation performance. Different from the existing literature, the evaluation methods provided in this section support unmatched receiving. Initially, the correlation of the local despreading signal \(\hat{s}(t)\) and received signal \(r(t)\) after integrate and dump (I & D) is provided as:  
\[\delta_{i}=\frac{1}{T}\int_{(k-1)T}^{kT}r(t)\,\hat{s}^{\mathrm{*}}(t-\tau_{0}+ \tau_{i})dt \tag{7}\]  
which is derived from the \(k\)-th integration cycle with the duration \(T\), where the subscript \(i\in\{E,P,L\}\) stands for the branches of early, prompt and late, with the code phase delays of \(\tau_{\mathrm{E}}=\Delta/2\), \(\tau_{\mathrm{P}}=0\) and \(\tau_{\mathrm{L}}=-\Delta/2\), respectively, where \(\Delta\) stands for early-late correlator spacing. \(\hat{s}(t-\tau_{0}+\tau_{i})\) stands for the local signal replica, where \(\tau_{0}\) is the estimated signal delay. Then the error of TOA estimation can be expressed as \(\varepsilon=\tau_{0}-\tau_{0}\). It should be noted that in traditional matched receiving, the local despreading replica is selected as the conjugate form of the received spreading signal. In this model, however, the local despreading signal \(\hat{s}(t)\) and the received signal \(s(t)\) are allowed to be different in the spreading/despreading code chip waveform, i.e., \(\varphi(t)\) and \(\hat{\varphi}(t)\) are allowed to be different, thereby unmatched receiving is supported. Denote \(w(t)\) as Gaussian noise and interference:  
\[w(t)=n(t)+\iota(t) \tag{8}\]  
Define \(s(t)\) as the effective part of signal, then Equation (1) can be rewritten as:  
\[r(t)=s(t-\tau_{0})+w(t) \tag{9}\]  
Substituting this into Equation (7), the correlator output becomes:  
\[\delta_{i}\doteq c_{\mathrm{s}\hat{s}}(\tau_{i})+c_{\mathrm{s}\hat{\mathrm{ *}}}(\tau_{i}) \tag{10}\]  
where:  
\[c_{\mathrm{s}\hat{\mathrm{*}}}(\tau_{i})=\frac{1}{T}\int_{(k-1)T}^{kT}e^{ \mathrm{i}\hat{\mathrm{*}}}s(t-\tau_{0})\hat{s}^{\mathrm{*}}(t-\tau_{0}-\tau_{i })dt \tag{11}\]\[c_{ws}(\tau_{i})=\frac{1}{T}\!\int_{(k-1)T}^{kT}w(t)\hat{s}^{*}(t-\hat{\tau}_{0}- \tau_{i})dt \tag{12}\]  
The two terms are studied individually for their distinct characteristics. Equation (11) can be reformulated as follows, where inverse Fourier transform is applied:  
\[ \begin{array}{ll}c_{ss}(\tau_{i})&=\frac{1}{T}\!\int_{(k-1)T}^{kT}e^{i\theta} s(t-\tau_{0})\hat{s}^{*}(t-\hat{\tau}_{0}-\tau_{i})dt\\ &=\frac{e^{i\theta}}{T}\!\int_{(k-1)T}^{kT}s(t-\tau_{0})\hat{s}^{\rho_{s}/2}_{ -\beta_{r}/2}\,\hat{\[PERSON]}^{*}(f)\,e^{-j2\pi f(t-\tau_{0}-\tau_{i})}dfdt\\ &=\frac{e^{i\theta}}{T}\!\int_{-\beta_{r}/2}^{\beta_{r}/2}\hat{\[PERSON]}^{*}(f) \int_{(k-1)T}^{kT}s(t-\tau_{0})e^{-j2\pi f(t-\tau_{0})}dt\,e^{-j2\pi f(\epsilon -\tau_{i})}df\\ &=\frac{e^{i\theta}}{T}\!\int_{-\beta_{r}/2}^{\beta_{r}/2}\hat{\[PERSON]}^{*}(f) \Phi(f)\,e^{-j2\pi f(\epsilon-\tau_{i})}df\\ &=e^{i\theta}\!\int_{-\beta_{r}/2}^{\beta_{r}/2}\mathcal{X}_{s,\hat{s}}(f)\,e ^{-j2\pi f(\epsilon-\tau_{i})}df\end{array}  \tag{13}\]  
where \(\mathcal{X}_{s,\hat{s}}(f)\) is the cross-spectrum defined in Equation (5), which is of the essence to support unmatched receiving. The spectrum-based representation facilitates the evaluation under different front-end bandwidth, which is commonly conducted in the recent researches. Similarly, by utilizing the properties of the zero-mean Gaussian process of noise and interference \(w(t)\), the variance of Equation (11) can be obtained:  
\[\text{Var}(c_{ws})=\frac{C_{s}}{T}\!\int_{-\beta_{r}/2}^{\beta_{r}/2}G_{w}(f)G _{\delta}(f)df \tag{14}\]  
where \(G_{w}(f)\) is the power spectral density of the noise and interference \(w(t)\).