# Random Access for Underwater Acoustic Cellular Systems
## 3 RAP Design
### LTE-Based RAP  
The LTE RAP is constructed using a prime-length ZC sequence. The sequence is appended by a cyclic prefix (CP) to absorb both the round trip delay (RTD) and maximum delay spread (MDS) [11]. The prime length is chosen to maximize the number of RAPs with good correlation properties. However, the time-frequency specification of the LTE RAP does not match that of the underwater acoustic channel owing to a large discrepancy in the propagation speed and operational channel bandwidth. Since a straightforward application of the LTE RAP for underwater cellular systems is not possible, we attempt to modify the parameters of the LTE RAP according to the characteristics of the underwater acoustic channel. The LTE RAP occupies a channel bandwidth of 1.08 MHz with the minimum sequence duration of 800 \(\upmu\)s. The RAP slot duration (RAP format 0) including CP interval, sequence duration, and guard time interval is 1 ms for the smallest cell coverage (radius of 14 km). The lower bound of the sequence length must satisfy the following condition to facilitate unambiguous round-trip time estimation for a UE at the cell edge [11]:  
\[T_{SEQ}\geq\frac{2C_{R}}{v_{c}}+\delta_{\max} \tag{1}\]  
where \(C_{R}\), \(v_{c}\), and \(\delta_{\max}\) denote the cell radius, sound speed, and MDS, respectively. For an LTE-based RAP in an underwater channel environment, the sequence length \(T_{SEQ}\) must be greater than or equal to 6.8 s when the cell radius, sound speed, and MDS are 5 km, 1500 m/s, and 132 ms, respectively. A sequence length larger than the lower bound value in (1) must be chosen to guarantee coverage. If a sequence length of 7 s is selected for a cell radius of 5 km, the corresponding subcarrier spacing of the LTE-based RAP will be 0.1429 Hz, which is very small compared to that used in terrestrial communication systems (i.e., 1.25 kHz). However, small subcarrier spacing and long symbol duration are not favorable to underwater channels because the doubly-selective characteristic of an underwater acoustic channel destroys the orthogonality of the RAP. The RAP is also very sensitive to Doppler shift, resulting in not only inter-channel interference (ICI) caused by a fractional CFO but also the time ambiguity caused by an integer CFO. The time ambiguity results in an error in distance estimation, which is up to \(T_{SEQ}\times v_{c}\) = 7 \(\times\) 1500 = 10.5 km for a sequence duration of 7 s.  
In the LTE system, many RAPs are generated from a single ZC sequence by changing the values of cyclic shift index [11]. Therefore, cyclic shift dimensioning is important because it limits the number of RAPs for a selected root index. The lower bound of the cyclic shift, \(N_{CS}\), is given by:  
\[N_{CS}\geq\left[\left(\frac{2C_{R}}{v_{c}}+\delta_{\max}\right)\frac{N_{ZC}}{ T_{SEQ}}\right]+N_{0} \tag{2}\]  
For a sequence length, \(N_{ZC}\), of 499 and a zero-value of guard sample, \(N_{0}\), \(N_{CS}\geq\) 485. The number of possible RAPs generated by a ZC sequence with a root index is given by \(\left\lfloor N_{ZC}/N_{CS}\right\rfloor=\left\lfloor 499/485\right\rfloor=1\). Therefore, a ZC sequence can generate only one RAP in underwater acoustic channels. One of the advantages of using LTE-based RAP is the simplicity in the RAP detection process, which is composed of the following three steps: (1) the received RAP is transformed to the frequency domain via discrete Fourier transform (DFT); (2) the frequency-domain signal is multiplied by a conjugated version of the base sequence before transforming it to the time domain via inverse DFT (IDFT); (3) the power delay profile is computed and compared with a threshold value for RAPdetection and time delay estimation. An FFT/IFFT operation with a size approximately the same as the sequence length is usually used to reduce the complexity of DFT.