# Power-Efficient Random Access Design for Machine Type Communication
## 3 LTE-nMTC Random Access Design
### Effective Bandwidth and Optimal Resource Configuration  
The SNR of an OFDM resource element (RE, i.e., a subcarrier over the duration of an OFDM symbol \(T\)) of the random access channel can be represented as  
\[\rho_{\text{RE}}=\frac{pT/s}{\alpha\zeta N_{0}}=\frac{1}{s}\cdot\frac{pT}{ \alpha\zeta N_{0}} \tag{7}\]  
as a result of matched-filtering via DFT, where \(T=w_{0}^{-1}=(W/S)^{-1}\) is the length of an OFDM symbol in time domain, \(N_{0}\) is the noise density, \(\zeta\) is the noise figure, and \(\alpha\) is the coupling loss whose value depends on the MTC coverage class of the device: \(\alpha=\)155, 150, 145, and 140 dB for channel coverage  
Figure 3: Illustration of the simplified random access signaling and uplink data transmission. classes 1, 2, 3, and 4, respectively. Apparently, the total number of REs per OFDM symbol is \(s\) per channel, yielding the maximum data rate (bits/OFDM symbol) of  
\[r=s\cdot\log(1+\rho_{\text{RE}})=s\cdot\log\biggl{(}1+\frac{1}{s}\cdot\frac{pT}{ \alpha\zeta N_{0}}\biggr{)}, \tag{8}\]  
or, the maximum data rate in bits/sec of  
\[ \begin{array}{rl}r&=s\cdot\log\Bigl{(}1+\frac{1}{s}\cdot\frac{pT}{\alpha \zeta N_{0}}\Bigr{)}\biggr{/}(T+T_{\text{CP}})\\ &\approx\frac{Ws}{S}\cdot\log\biggl{(}1+\Bigl{(}\frac{Ws}{S}\Bigr{)}^{-1} \cdot\frac{p}{\alpha\zeta N_{0}}\biggr{)}\\ &=w\cdot\log\Bigl{(}1+\frac{w^{-1}p}{\alpha\zeta N_{0}}\Bigr{)}.\end{array}  \tag{9}\]  
Taking into account the imperfection of a realistic system, (9) becomes  
\[r(w,\alpha)=w\cdot\log\biggl{(}1+\frac{\beta w^{-1}p}{\alpha\zeta N_{0}} \biggr{)}, \tag{10}\]  
where \(\beta\) is a scaling factor to reflect the gap between the Shannon capacity and a realistic system. \(r(w,\alpha)\) is a monotonically increasing function of \(w\). Then the transmission time of the access request message \(M\) is given by  
\[\tau(w,\alpha,|M|)=\frac{|M|}{w\cdot\log\Bigl{(}1+\frac{w^{-1}\beta p}{\alpha \zeta N_{0}}\Bigr{)}}. \tag{11}\]  
It means the larger the \(w\) is, the larger the data rate, and the smaller the transmission time. In fact, when \(w\to\infty\),  
\[\tau(\infty,\alpha,|M|)=\lim_{w\to\infty}\tau(w,\alpha,|M|)=\frac{|M|\ln 2}{ \frac{\beta P}{\alpha\zeta N_{0}}}, \tag{12}\]  
which represents the _minimum_ transmission time for message \(M\), i.e.,  
\[\tau(w,\alpha,|M|)>\tau(\infty,\alpha,|M|),\;\forall w<\infty. \tag{13}\]  
given \(\alpha\) or the coverage class. Figure 4 plots (11) relative to \(\tau(\infty,\alpha,|M|)\), i.e.,  
\[ \begin{array}{c}\Delta\tau(w,\alpha)=\frac{\tau(w,\alpha,|M|)-\tau(\infty, \alpha,|M|)}{\tau(\infty,\alpha,|M|)}\\ =\frac{\tau(w,\alpha,|M|)}{\tau(\infty,\alpha,|M|)}-1\end{array}  \tag{14}\]  
for devices belonging to the four coverage classes, which represents the extra transmission time due to the use of a finite transmission bandwidth \(w\) (i.e., \(w<\infty\)). In Figure 4, \(p=P_{\text{UL}}=20\) dBm is the uplink transmit power, \(\zeta=5\) dB, and \(\beta=-6\) dB. From Figure 4, we can see that the transmission time is monotonically decreasing with respect to the signal bandwidth as we analyzed before. But the decreasing slope of the transmission time is not linear. There is an effective bandwidth \(w^{\dagger}\) for every coverage class such that once exceeded, increasing bandwidth has less effect on decreasing the transmission time. If we define \(w^{\dagger}\) as the effective bandwidth, at which \(\Delta\tau\bigl{(}w^{\dagger},\alpha\bigr{)}\) is equal to, e.g., 10%, the \(w^{\dagger}\) for coverage classes 4, 3, 2, and 1 are then 94, 30, 9, and 3 kHz, respectively. They are summarized in Table 1. The lower the coverage class is, the less sensitive it is to bandwidth. Clearly, devices belonging to the class whose bandwidth is less than the effective bandwidth are therefore bandwidth-limited, and those whose bandwidth is greater than the effective bandwidth are power-limited. The effective bandwidth therefore provides a good balance between the spectral efficiency (or the total number of FDM channels that a given supports) ant the transmission time.