# Power-Efficient Random Access Design for Machine Type Communication
## 3 LTE-nMTC Random Access Design
### Effective Bandwidth and Optimal Resource Configuration
Since the effective bandwidth of a low coverage class is less than that of a high coverage class, more FDM channels are available for a low coverage class for a given total bandwidth. Based on this effective bandwidth concept, the subcarrier spacing of 15 kHz for legacy LTE data channels is too large for MTC, and a finer granularity is needed. The subcarrier spacing is thus reduced by a factor of 5, i.e., \(w_{0}=3\) kHz in the new design to match the smallest effective bandwidth (CC1). This idea is consistent with the design of NB-IoT design in which 3.75 kHz subcarrier spacing is used [23]. The reduction of subcarrier spacing has the following implications: (1) Five times as many FDM channels for CC1 devices; (2) The OFDM symbol in time domain is elongated by a factor of 5, which boots OFDM symbol energy by 7 dB; and (3) The data channel CP length can also be increased by five times, i.e., 5 \(\upmu\times 5=25\)\(\upmu\)s while maintaining the same CP overhead \(\frac{25\upmu\upmu}{25\upmu+(3\upmu\mathrm{kHz})^{-1}}\approx 7\%\) as in the legacy LTE, \(\frac{5\upmu\upmu}{5\upmu\upmu+(15\upmu\mathrm{kHz})^{-1}}\approx 7\%\)[24]. Longer CP allows for not only larger multipath spread but also more tolerance to timing errors which relaxes the uplink timing requirement. This is particularly beneficial since timing estimation is less accurate under narrow band than wideband due to the reduced time resolution. However, narrower subcarrier spacing also means more vulnerability to frequency offsets due to frequency tracking errors and/or Doppler spread but it is justified by a much lower mobility (hence lower Doppler effects) in MTC applications than HTC. Note that \(\tau(w,\alpha,|M|)\) or \(\Delta\tau(w,\alpha)\) represents the transmission time for a non-contention based transmission in a collision-free environment as for the data transmissions. But the random access transmission is unscheduled or contentious. The actual transmission time is hence \(\Delta\tau(w,\alpha)\)_plus_ the re-transmissions in the case of collisions whose probability is directly related to the number of random access channels, \(N_{\mathrm{RACH}}=\frac{W}{w}\), as given in (4), and is re-written here,  
\[p_{\mathrm{c}}(w,N)=1-\left(1-\frac{1}{N_{\mathrm{RACH}}}\right)^{N-1}=1-\left( 1-\frac{w}{W}\right)^{N-1}, \tag{15}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Coverage Class \(\alpha\)** & **4 (140 dB)** & **3 (145 dB)** & **2 (150 dB)** & **1 (155 dB)** \\ \hline
**Effective bandwidth \(w^{\ddagger}(\alpha)\) (kHz)** & 94 & 30 & 9 & 3 \\ \hline
**Extra transmission time \(\Delta\tau(w^{\ddagger},\alpha)\) (\%)** & & & 10 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Effective bandwidth for different coverage classes [according to (14)]. Figure 4: The relationship between the transmission bandwidth and transmission time. where \(N\) is the number of accessing devices. Therefore, the total transmission time is also a function of the collision probability, and can be derived as follows:  
\[\tau_{\mathrm{c}}(w,\alpha,N,|M|) =\lim_{K\to\infty}\sum\limits_{i=1}^{K}\left(1-p_{\mathrm{c}}(w,N) \right)p_{\mathrm{c}}^{i-1}(w,N)(i\cdot\tau(w,\alpha,|M|)) \tag{16}\] \[=\tau(w,\alpha,|M|)\lim_{K\to\infty}(1-p_{\mathrm{c}}(w,N))\sum \limits_{i=1}^{K}i\cdot p_{\mathrm{c}}^{i-1}(w,N)\] \[=\tau(w,\alpha,|M|)\lim_{K\to\infty}\left(\frac{1-p_{\mathrm{c}}^ {K}(w,N)}{1-p_{\mathrm{c}}(w,N)}-K\cdot p_{\mathrm{c}}^{K}(w,N)\right)\] \[=\tau(w,\alpha,|M|)\lim_{K\to\infty}\frac{1-p_{\mathrm{c}}^{K}(w, N)}{1-p_{\mathrm{c}}(w,N)}\] \[=\frac{\tau(w,\alpha,|M|)}{1-p_{\mathrm{c}}(w,N)}\;.\]  
Clearly, \(\tau_{\mathrm{c}}(w,\alpha,N,|M|)>\tau(w,\alpha,|M|)\) due to the fact that \(0<p_{\mathrm{c}}(w,N)<1\) in the contention-based random access. Ultimately, we look for the \(w\) that minimizes (16). Alternatively, we search for the \(w\) that minimizes the extra time or penalty introduced by the _contention_ among \(N\) accessing devices.