# Pseudorange Measurement Method Based on AIS Signals
## 3 Detective Feature of Timestamp
### Amplitude Characteristics of Timestamp
#### 3.2.1 Interference of Adjacent Bits to Timestamp Amplitude Characteristics  
When the unit step response of the Gaussian filter is discussed, the bit period \(T\) is approximated to infinity. In practice, the alternately transformed sequence bits introduce amplitude distortion into the Gaussian baseband waveform. As shown in Figure 3b, the amplitude of the unit pulse signal bit hopping edge in the Gaussian baseband waveform is no longer deterministic.  
The digital message sequence can be thought of as the result of a series of shifted unit step signal alternate additions. Since the Gaussian filter is a linear time-invariant system, the output of the  
Figure 3: Relation between unit response of Gaussian Filter and parameter \(BT\). (**a**) Gaussian filter unit-step response; and (**b**) Gaussian filter unit impulse response.  
Gaussian filter is equivalent to the interleaved summation of a set of translational unit step responses of the Gaussian filter for GMSK modulation systems.  
\[O_{G}(t)=\sum\limits_{m=0}^{N}\{(-1)^{m}\times[1+erf(k(t-mT))]\} \tag{9}\]  
where \(T\) is the bit period in digital message sequence; \(m\) is the bit serial number of the bit inversion generated in the digital message sequence, and is a monotonically increasing random sequence of positive integers; \(N\) is the number of bit inversion in the finite-length digital message sequence, and its value is less than or equal to the number of the sequence.  
For the convenience of subsequent discussion without loss of generality, we defined the digital message sequence as a 0 or 1 alternating unit periodic sequence. Under such conditions, in Equation (9), \(T=1\) and \(m\in\{0,1,2,\cdots N\}\), where \(N\) is the number of bits.  
Let \(b_{j}(t)=erf[k(t-2j)]-erf[k(t-(2j+1))]\), Equation (9) can be expressed as  
\[O_{G}(t)=\left\{\begin{array}{ll}\sum\limits_{j=0}^{(N-1)}b_{j}(t)&\mbox{$N$ is odd}\\ \sum\limits_{j=0}^{(N)-1}b_{j}(t)\,+[1+erf(k(t-N))]&\mbox{$N$ is even}\end{array}\right. \tag{10}\]  
As the error function \(erf(\cdot)\) is monotonically increasing, we obtain the following conclusions: \(b_{j}(t)>0\) and \(b_{j}(t)<b_{j+1}(t)\). According to the D'Alembert discriminant method, it is known that the series \(\sum\limits_{j=0}^{\infty}b_{j}(t)\) is convergent.  
When \(t\) is a positive integer, the result \(O_{G}(n)\) is the amplitude of bit hopping edges that we want to detect timestamps in the Gaussian baseband waveform. Let \(t=0\) in Equation (10) and perform numerical analysis where the amplitude error of the first timestamp \(O_{G}(0)\) is shown in Figure 4.  
It can be seen from the results of numerical analysis that \(O_{G}(0)\) is ultimately convergent with the increase of the length of sequence \(N\). However, for different parameters \(BT\), the convergence rate and the final convergence value of \(O_{G}(0)\) are also different, where the larger the \(BT\) value, the smaller the convergence error and the faster the convergence rate.  
Assuming that sequence length \(N\) is long enough to converge the series expression, the amplitude error of each timestamp of the digital sequence in the Gaussian baseband waveform is obtained by numerical analysis. The corresponding results are shown in Figure 5.  
Figure 4: Convergence Analysis of \(O_{G}(0)\).  
From the numerical analysis, results can be drawn as follows:
* The amplitude error of the timestamp is the largest at both ends of the digital message sequence, and is the smallest in the midpoint of the sequence. The error gradually converges from the two ends to the middle.
* When \(N\) is an even number, the amplitude error of the timestamp in the digital messages sequence is in odd symmetry with respect to the midpoint of the sequence.
* When \(BT\) is smaller, the bigger the error, the slower the convergence.  
According to the above conclusions, it can be determined that the amplitude characteristic error introduced by adjacent bit disturbances in the Gaussian baseband waveform can be neglected if the appropriate timestamps are selected.