## 4 Overall Design and System Analysis
### Theoretical Analysis
#### 4.2.2 Theoretical Basis Derivation  
To analyze feasibility of this approach. We define an overall gain function to conduction multi-objective optimization, using CRLB to construct a positioning accuracy optimization function composed of communication, timing, and some other factors. In addition to common influence factors such as positioning distance, beacon quantity, existence of communication, and timing, it can help reduce deviation to a certain degree. Given that there exists 1 positioning beacon with its location \(P=[x,y,z]^{T}\), and \(N\) positioning base stations, whose location \(r_{i}=[x_{i},y_{i},z_{i}]^{T}\), with \(i=1,2,\ldots,N\). If we know the arrival time \(t_{i}\) and \(t_{j}\) of the positioning signal \(i_{th}\) and \(j_{th}\), the arrival time difference can be calculated,  
\[\Delta_{i,j}=t_{i}-t_{j}=\frac{d_{i}-d_{j}}{v_{s}}, \tag{1}\]  
\(v_{s}\) is the propagation speed of sound in water, \(d_{i}\) and \(d_{j}\) is the distance between the beacon and the base stations, which can be represented by  
\[d_{i}=\|P-r_{i}\|=\sqrt{(x-x_{i})^{2}+(y-y_{i})^{2}+(z-z_{i})^{2}}, \tag{2}\]  
\(\|\cdot\|\) represents Euclidean distance, with more than 3 distance between base stations and beacon, the location \(P\) of the beacon can be solved by algorithms such as non-linear least squares. To analyze the influences of positioning accuracy, we define the utility function \(U(P)=Tr(CRLB(P))\), which is the mean square deviation of position estimation, to further explore how to minimize \(U(P)\). And \(CRLB(P)\) is the lower bound of Cramer-Rao, with \(Tr(\cdot)\) the trace. Assuming that the variance of measured noise is \(\sigma^{2}\), and communication rate \(R\), reliability \(Rel\), communication distance \(D\) and time accuracy \(T\) will be taken into account. 1. Communication Rate \(R\) Acoustic communication is the foundation for underwater positioning, and a higher communication rate allows more aided positioning information to be transmitted, such as the depth of the platform where the beacon is installed, speed, acceleration, attitude and some other parameters to help improve the accuracy of position estimation. Assuming that the higher the communication rate \(R\), the lower the variance of measured noise \(\sigma^{2}\) is: \[\sigma^{2}\propto\frac{1}{R}\] (3)
2. Reliability \(Rel\) Reliability will affect the packet loss rate and number of retransmissions, which will influence the effectiveness and accuracy of the measurement. Assuming that the higher the reliability, the lower the variance of measured noise \(\sigma^{2}\) is: \[\sigma^{2}\propto\frac{1}{Rel}\] (4)
3. Communication Distance \(D\)As communication distance increases, signal attenuation and noise increase, and multi-path effect is more obvious, which will affect measurement accuracy. Assuming that the measurement noise is proportional to communication distance:  
\[\sigma^{2}\propto D \tag{5}\]
4. Timing Accuracy \(T\) Timing service will directly affect the accuracy of measured time difference. And the higher the accuracy, the lower the variance of measured noise: \[\sigma^{2}\propto T\] (6)  
According to the above analysis, the variance of measured noise can be represented as:  
\[\sigma^{2}=\alpha\frac{D}{R\cdot Rel\cdot T} \tag{7}\]  
\(\alpha\) is the proportional coefficient. We utilize Fisher Information Matrix (FIM) to calculate CRLB, and elements of FIM of TDOA positioning system can be represented as:  
\[[\mathbf{I}(\mathbf{P})]_{mn}=\sum_{i<j}\frac{1}{\sigma^{2}}\bigg{(}\frac{ \partial\Delta t_{ij}}{\partial p_{m}}\bigg{)}\bigg{(}\frac{\partial\Delta t_ {ij}}{\partial p_{n}}\bigg{)} \tag{8}\]  
With \(\sigma^{2}\) substituted, we obtain  
\[[\mathbf{I}(\mathbf{P})]_{mn}=\sum_{i<j}\frac{R\cdot\mathrm{Rel}\cdot T}{ \alpha D}\bigg{(}\frac{\partial\Delta t_{ij}}{\partial p_{m}}\bigg{)}\bigg{(} \frac{\partial\Delta t_{ij}}{\partial p_{n}}\bigg{)} \tag{9}\]  
The above analysis, which is based on CRLB, provides a lower bound of the parameter estimation variance.