# An Unambiguous Delay-And-Multiply Acquisition Scheme for GPS L1C Signals
## 3 Results
### Noise Characteristics of the Proposed DM and Conventional Acquisition Method
#### 3.2.4 Probability of Detection with Respect to the Number of Samples  
In this section we consider the effect of the number of samples on the detection probability. It is well-known that when a longer data is used for the acquisition process, the SNR increases and the detection performance enhances. This performance improvement can be obtained through the fast sampling, which increases M in the DM method in Table 1  
The following Lemma shows that when the sampling speed increases, meaning the increase of M, the detection probability increases.  
**Lemma 7**.: _Consider the generated signal \(g(t)=g_{DM,PP}(t)\) in the DM acquisition method in Table 1. Assuming that \(c=1\) and \(\cos(2\pi f_{1}\varepsilon_{0})\approx 1\), we obtain that \(Z_{H_{0}}\ \sim N\big{(}0,\ \ M\sigma_{n}^{A}\big{)}\) and \(Z_{H_{1}}\ \sim N\big{(}\frac{3}{4}M,\ \big{(}\sigma_{n}^{A}+2\sigma_{n}^{2} \big{)}M\big{)}\), where \(M\) is the number of samples according to the appropriate sampling frequency and the coherent integration period T. Suppose that the probability of false alarm is given, then the detection probability \(P_{d}\) increases as the number of sample M increases, i.e.:_  
\[if\ M_{1}<M_{2},\ \text{then}\ P_{d}^{M_{1}}<P_{d}^{M_{2}}.\]  
Proof.: Define the probability density functions of the decision variables \(Z_{H_{0}}\) and \(Z_{H_{1}}\) as \(f_{Z_{H_{0}}}(z)\) and \(f_{Z_{H_{1}}}(z)\). When the probability of false alarm \(P_{fa}\) is given, the threshold \(\gamma\) and the probability of detection \(P_{d}\) is given as follows: \(\int_{-\infty}^{\gamma}f_{Z_{H_{0}}}(z)dz=P_{fa}\) and \(\int_{\gamma}^{\infty}f_{Z_{H_{1}}}(z)dz=P_{d}\).  
Consider two sample numbers \(M_{1}\) and \(M_{2}\) with \(M_{1}<M_{2}\), and the corresponding thresholds \(\gamma_{1}\) and \(\gamma_{2}\), and the corresponding probability of detection, as \(P_{d}^{M_{1}}\) and \(P_{d}^{M_{2}}\). We will show that \(\gamma_{1}<\gamma_{2}\) first and then \(P_{d}^{M_{1}}<P_{d}^{M_{2}}\).  
1. Suppose that the probability of false alarm \(P_{fa}\) is given. Despite of the change of variance, the normalized threshold does not change, i.e., \(\frac{\gamma_{1}}{\sqrt{M_{1}\sigma_{n}^{A}}}=\frac{\gamma_{2}}{\sqrt{M_{2} \sigma_{n}^{A}}}\), which becomes \(\frac{\gamma_{1}}{\gamma_{2}}=\sqrt{\frac{M_{1}}{M_{2}}}<1\). Hence, we have \(\gamma_{1}<\gamma_{2}\).
2. The probability of detection is given for the sample numbers \(M_{1}\) and \(M_{2}\) as: \[\int_{\gamma_{1}}^{\infty}f_{Z_{H_{1}}}  
of 5.2 kHz comes from the discussion in Section 3.1.1. For the DM approach, the generated signal \(g_{DM,PP}(t)\) in Figure 5a and \(g_{DM,PP+DD}(t)\) in Figure 6a can be precalculated and stored in the memory. Thus only one more multiplication is necessary for the DM approach to obtain \(s_{DM,\hat{\mathcal{A}}_{0}}(t)\) before correlation, compared with the conventional method in Figure 7. The correlation of the received signal \(r_{DM}(t)\) and the generated signal \(g_{DM}(t)\) is performed through the parallel code phase search acquisition using FFT [2] in this paper. However, we count it as 1 correlation for simplicity. The search time to obtain both code phase and Doppler frequency can be roughly described with the Doppler frequency resolution \(\Delta f=50\) Hz as follows:  
1. The proposed DM approach:  
2. The conventional approach:  
where K is the number of non-coherent integration.  
Thus the search time of the DM acquisition method depends on K, which depends on the noise environment.  
**Comment 1**.: _According to Lemma 7, the probability of detection can be increased as the number of sample increases, which can be obtained by sampling more fast as well as by using longer samples like non-coherent integration. So the acquisition time of the DM approach can be reduced much with higher sampling rate due to the advance of sampling technology._