## 3 Code Chip Waveform Mismatch  
Traditional despreading follows the matched receiving principle that maximizes the SNR at the correlator output [15]. This requires the local despreading signal to be exactly the same as the received signal, depicted as \(\hat{c}(t)=c(t)\). In GNSS pseudo-range measurements, besides SNR, other indicators including Gabor bandwidth, mean time to lose lock (MTLL) of tracking loop, should also be considered [14; 16; 17]. Under this background, maximizing SNR and matched receiving is no longer the only guideline to select local despreading signals. In practice, a variety of selections of despreading waveform have been applied in GNSS signal processing, for the purpose of reducing receiver complexity [3; 4; 5], multipath mitigation [2] and unambiguous tracking of BOC signals [1]. These receiving strategies, which associate with the mismatch of local despreading code chip waveform, are termed as 'unmatched receiving' hereafter. In GNSS, since the PRN spreading sequence is known on both the transmitter side and the receiver side, hereafter we consider a general case where the despreading code chip waveform \(\hat{\phi}(t)\) may be different from that of the received signal, denoted as \(\varphi(t)\). Taking the BOC\({}_{\text{sin}}\)(6,1) signal for example, the code chip waveform of the received signal is a bipolar pulse \(\varphi(t)=sign\left\{\sin(2\pi f_{sc}t)\left[u(t)-u(t-T_{c})\right]\right\}\), where \(u(t)\) is the unit step signal,\(f_{sc}\) is the subcarrier frequency. When a BPSK-like unmatched receiving algorithm is applied, the code chip waveform of local despreading signal is \(\hat{\varphi}(t)=u(t)-u(t-T_{c})\) at the central frequency of \(f_{sc}\) or \(f_{sc}\), when intermediate frequency is assumed to be zero. Accordingly, BPSK-like receiving is easy to conduct with accuracy loss. From this aspect, the local despreading signal \(\hat{s}(t)\) is:  
\[\hat{s}(t)\;=\;\hat{c}(t)=\sum_{i=-\infty}^{+\infty}a(i)\hat{\varphi}(t-iT_{c}) \tag{4}\]  
where it is reasonable to ignore the data message for its slow varying characteristics compared with the PRN code chip. To carry out the despreading of the DSSS signal, correlation is made between the received signal and the local replica. To help represent the characteristics of despreading in the spectrum domain, due to the strict limitation of band for both transmitters and receivers, a Fourier transform is conducted for the cross-correlation result. Within the observation duration (\(k\)-1)\(T_{c}<t<kT_{c}\), and for the useful signal term without noise and interference, there is:  
\[\frac{1}{T_{c}}\int_{(k-1)T_{c}}^{kT_{c}}\mathrm{E}\Big{\{}s(t)\hat{s}^{*}(t- \tau)\Big{\}}e^{-j2\pi ft}dt\;=\;f_{c}\Phi(f)\hat{\Phi}^{*}(f)\triangleq \mathcal{X}(f) \tag{5}\]  
where \(f_{c}=1/T_{c}\) is the spreading chip rate, \(\mathcal{X}(f)\) is defined as the cross-spectrum between the received signal and the local signal, \(\Phi(f)\) is the Fourier transform of the received signal \(s(t)\), and \(\hat{\Phi}(f)\)is the Fourier transform of the local signal \(\hat{s}(t)\). As a special case for the matched receiving, wherein the local code chip waveform is identical to that of the received signal, Equation (5) becomes the power spectral density \(G_{\mathrm{s}}(f)\):  
\[C_{\mathrm{s}}G_{\mathrm{s}}(f)=f_{\mathrm{c}}\Phi(f)\Phi^{\mathrm{*}}(f) \tag{6}\]  
where \(\Phi(f)\) is normalized to satisfy \(\int_{-\infty}^{+\infty}G_{\mathrm{s}}(f)=1\), by introducing the average carrier power \(C_{\mathrm{s}}\) defined on the infinite bandwidth. Likewise, the local term \(\hat{\Phi}(f)\) is commonly normalized to satisfy \(\int_{-\infty}^{+\infty}G_{\mathrm{s}}(f)=1\).