# An Unambiguous Delay-And-Multiply Acquisition Scheme for GPS L1C Signals
## 2 Materials and Method
### GPS LIC DM Signal  
The received signal \(r_{n}(\mathrm{t})\) of the GPS L1C signal with an additive Gaussian noise \(n(\mathrm{t})\) can be expressed as Equation (4):  
\[r_{n}(\mathrm{t})=S_{\mathrm{L1C}}(t)\cos(2\pi(f_{IF}+f_{D})t)+n(t),\quad\text{ with}\quad n(t)\,\sim\,N(0,\sigma_{n}^{2}) \tag{4}\]  
The DM (delay-and-multiply) signal of the received signal \(r_{n}(\mathrm{t})\) is given with the delay \(\delta\) as Equation (5):  
\[\begin{array}{rcl}r_{\mathrm{DM},\delta}(\mathrm{t})&=&r_{n}(t)\cdot r_{n}( t-\delta)\\ &=&S_{\mathrm{L1C}}(t)S_{\mathrm{L1C}}(\mathrm{t}-\delta)\cos(2\pi(f_{IF}+f_{D })\mathrm{t}))\cos(2\pi(f_{IF}+f_{D})(\mathrm{t}-\delta))\\ &&+S_{\mathrm{L1C}}(t)\cos(2\pi(f_{IF}+f_{D})t)n(\mathrm{t}-\delta)\\ &&+S_{\mathrm{L1C}}(t-\delta)\cos(2\pi(f_{IF}+f_{D})(\mathrm{t}-\delta))n(t)\\ &&+n(t)n(t-\delta)\\ &&=s_{DM,\,\delta}(t)+\eta_{DM,\delta}(t)\end{array} \tag{5}\]  
For the unit of delay \(\delta\), the chip of the ranging codes \(C_{D}(\mathrm{t})\) and \(C_{P}(t)\) can be used as \(\delta=n\) chip, or the second can be used as \(\delta=n\mathrm{\,}T_{0}\) second, where \(T_{0}=\frac{1}{f_{0}}=\frac{1}{1.023\times 10^{6}}\) s.  
The first term \(s_{\mathrm{DM},\delta}(\mathrm{t})\) in Equation (5) is deterministic, and the following three terms, which are gathered in \(\eta_{DM,\delta}(t)\), are random. The first term \(s_{\mathrm{DM},\delta}(\mathrm{t})\) can be expressed as Equation (7), considering the removal of the high frequency term during the correlation process:  
\[s_{DM,\delta}(t)=r(t)r(t-\delta)\approx S_{\mathrm{L1C}}(t)S_{ \mathrm{L1C}}(t-\delta)\times\frac{1}{2}\cos(2\pi(f_{IF}+f_{D})\delta) \tag{6}\] \[= \frac{1}{2}\cos(2\pi(f_{IF}+f_{D})\delta)\cdot\frac{\epsilon}{2} \Big{[}\sqrt{3}C_{OL}(t)C_{P}(\mathrm{t})SC_{P}(t)+D(t)C_{D}(t)SC_{D}(t)\Big{]}\] \[\cdot\Big{[}\sqrt{3}C_{OL}(t-\delta)C_{P}(\mathrm{t}-\delta)SC_{ P}(t-\delta)+D(t-\delta)C_{D}(t-\delta)SC_{D}(t-\delta)\Big{]}\] \[= d\big{[}3C_{OL}(t)C_{OL}(t-\delta)C_{P}(\mathrm{t})C_{P}(t- \delta)SC_{P}(t)SC_{P}(t-\delta)\] \[+\sqrt{3}\,D(t)C_{OL}(t-\delta)C_{D}(t)C_{P}(\mathrm{t}-\delta)SC _{D}(t)SC_{P}(t-\delta)\] \[+\sqrt{3}\,C_{OL}(t)D(t-\delta)C_{P}(\mathrm{t})C_{D}(t-\delta)SC _{P}(t)\;SC_{D}(t-\delta)\] \[+D(t)D(t-\delta)C_{D}(t)C_{D}(t-\delta)SC_{D}(t)\;SC_{D}(t-\delta)\] \[\approx d\big{[}3C_{P}(\mathrm{t})C_{P}(\mathrm{t}-\delta)SC_{P}(t )SC_{P}(t-\delta)\pm\sqrt{3}\,C_{D}(t)C_{P}(\mathrm{t}-\delta)SC_{D}(t)SC_{P}( t-\delta)\] \[\pm\sqrt{3}C_{P}(\mathrm{t})C_{D}(t-\delta)SC_{P}(t)\;SC_{D}(t- \delta)+C_{D}(t)C_{D}(t-\delta)SC_{D}(t)\;SC_{D}(t-\delta)\big{]}\]where:  
\[d=\frac{c}{4}\cos(2\pi(f_{IF}+f_{D})\delta). \tag{9}\]  
We used some approximation in Equation (8) such that \(D(t)D(t-\delta)\approx\ 1,\ C_{OL}(t)C_{OL}(t-\delta)\approx 1\), \(D(t)C_{OL}(t-\delta)\approx D(t-\delta)C_{OL}(t)\approx\pm 1\), where \(\approx\) means that the term may have different values for only \(\delta\) among 10,130 chips.  
The term \(\eta_{DM,\delta}(t)\) in Equation (6) consists of three noise terms as follows:  
\[\eta_{DM,\delta}(t)=S_{\rm LIC}(t)\cos(2\pi(f_{IF}+f_{D})t)n(t-\delta)+S_{\rm LIC }(t-\delta)\cos(2\pi(f_{IF}+f_{D})({\rm t}-\delta))n(t)+n(t)n(t-\delta)\]  
where the first and second terms are zero-mean random noise, and the third one is the product of two Gaussian noises. The product of two independent Gaussian random variables \(x\) and \(y\) with zero mean and variances \(\sigma_{x}^{2}\) and \(\sigma_{y}^{2}\) has a normal product distribution, as follows:  
\[p_{xy}(u)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{e^{\frac{-u^{2 }}{2\sigma_{x}^{2}}}}{\sigma_{x}\sqrt{2\pi}}\cdot\frac{e^{\frac{-u^{2}}{2 \sigma_{y}^{2}}}}{\sigma_{y}\sqrt{2\pi}}\cdot\delta(xy-u)dxdy=\frac{K_{0}\left( \frac{|u|}{\sigma_{x}\sigma_{y}}\right)}{\pi\sigma_{x}\sigma_{y}}\]  
where, \(\delta(x)\) is the Dirac delta function, and \(K_{0}(x)\) is the modified Bessel function of the second kind and zero order, and the solution is \(K_{0}(x)=\int_{0}^{\infty}\frac{\cos(x)}{\sqrt{t^{2}+1}}dt\)[19].