# Power-Efficient Random Access Design for Machine Type Communication
## 3 LTE-nMTC Random Access Design
### Physical Layer Mapping and Timing Estimation  
As noted before, the use of a preamble is mainly for the purpose of indicating the presence of a device and providing a means for estimate the round-trip propagation delay. The former is achieved by decoding the random access message with the help of CRC check. Then the random access message  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Coverage Class \(\alpha\)** & **4 (140 dB)** & **3 (145 dB)** & **2 (150 dB)** & **1 (155 dB)** \\ \hline
**TDM channels \(L\)** & 1 & 36 & 1 & 12 & 1 & 3 & 1 \\
**Optimal bandwidth \(w^{\text{tt}}\) (kHz)** & 12 & 90 & 9 & 30 & 6 & 9 & 3 \\
**Channel bandwidth \(w\) (kHz)** & 12 & 90 & 9 & 30 & 6 & 9 & 3 \\
**Number of subcarriers per channel \(s\)** & 4 & 30 & 3 & 10 & 2 & 3 & 1 \\
**Total channels** & 15 & 72 & 20 & 72 & 30 & 60 & 60 \\ \(N_{\text{RACH}}=LW/w=LS/s\) & 130 & 9 & 60 & 9 & 27 & 11 & 10 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Random access channel configuration for the total bandwidth of 180 kHz according (17).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Coverage class \(\alpha\)** & **4 (140 dB)** & **3 (145 dB)** & **2 (150 dB)** & **1 (155 dB)** \\ \hline
**TDM channels \(L\)** & 1 & 36 & 1 & 12 & 1 & 3 & 1 \\
**Optimal bandwidth \(w^{\text{tt}}\) (kHz)** & 12 & 90 & 9 & 30 & 6 & 9 & 3 \\
**Channel bandwidth \(w\) (kHz)** & 12 & 90 & 9 & 30 & 6 & 9 & 3 \\
**Number of subcarriers per channel \(s\)** & 4 & 30 & 3 & 10 & 2 & 3 & 1 \\
**Channel bandwidth \(w\) (kHz)** & 12 & 90 & 9 & 30 & 6 & 9 & 3 \\
**Number of subcarriers per channel \(s\)** & 4 & 30 & 3 & 10 & 2 & 3 & 1 \\
**Total channels** & 90 & 432 & 120 & 432 & 180 & 360 & 360 \\ \(N_{\text{RACH}}=LW/w=LS/s\) & 125 & 9 & 59 & 8 & 26 & 10 & 10 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Random access channel configuration for the total bandwidth of 1.08 MHz according (17).  
Figure 7: Illustrative diagram of the resource configurations for random access channels (RACHs).  
can be treat as a known signal after successful decoding. So we can use it as the \"preamble\" to estimate the uplink timing.  
According to the analysis in above, we know that the random access signal contains \(s\) subcarriers and the optimal \(s\) for different coverage classes are different. As we mention before, the timing accuracy is related to the signal bandwidth. We therefore use a hopping pattern to increase the timing accuracy. For the random access signals which contain more than one subcarrier, we can treat them as the combination of several standalone subcarriers. Therefore, we just need to define the subcarrier level hopping pattern.  
Figure 8 shows an example of the resource element mapping of our proposed random access signal with one subcarrier (e.g., CC1). In our design, every five OFDM symbols is defined as a symbol group, the middle resource element is the pilot for demodulation and decoding, the rest are used to convey the random access message \(M\). The hopping is done in adjacent symbol groups with the hopping step \(\Delta q\) subcarriers. In order to take full advantage of resources, we define a bi-direction hopping pattern. That is, for the two subcarriers whose gap is \(\Delta q\), their hopping direction is opposite as shown in Figure 8 (the tones with the same color is corresponding to a logic subcarrier of random access signal). Therefore, for the random access signal contains several subcarriers, we can map these subcarriers to several standalone subcarriers in physical resource to further increase the virtual signal bandwidth for timing estimation.  
According to our previous work [21], the hopping step \(\Delta q\) should satisfy  
\[2\pi\Delta qw_{0}\Delta t<2\pi, \tag{20}\]  
where \(w_{0}\) is the subcarrier spacing, \(\Delta t\) is the maximum round-trip propagation delay. For example, the maximum value of \(\Delta q\) is 9 for 35 us propagation delay with the subcarrier spacing 3.75 kHz.