## 2 System Model  
Consider the uplink PHY layer data transmission shared channel of NB-IoT system with \(k=1,\ldots,K\), UE of \(u_{k}=(u_{0},u_{1},\ldots,u_{m-1}),u_{i}\in\{0,1\}\) transport blocks (TBs), which are simultaneously transmitted on their NPUSCH to the base station (BS) for network access as illustrated in Figure 2. The model employs a single-tone SC-FDMA with \(N_{k}\) contiguous subcarriers spaced with \(\Delta f_{n}=15\,\mathrm{kHz}\) and \(\Delta f_{n}=3.75\) kHz SCS, respectively. Since the transmission bandwidth is limited to \(180\,\mathrm{kHz}\), the NPUSCH frame in the time domain consists of a pair of slots each consisting of 7 OFDM symbols with one NDMRS as a pilot-aided symbol. For \(\Delta f_{n}=15\,\mathrm{kHz}\) SCS, the resource grid composed of 12 subcarriers per 0.5 ms slot duration forms one physical resource block (PRB). Similarly, for \(\Delta f_{n}=3.75\,\mathrm{kHz}\) SCS, 48 subcarriers are used per 2 ms slot duration forming one PRB.  
A \(u\) TB to be transported on NPUSCH is encoded through the LTE Turbo encoder. Recursive Systematic Convolutional (RSC) codes are primarily applied as the constituent component codes. At any time, \(k\), the input to the encoder is a bit \(u_{k}\), which is converted to the corresponding code bit \(c_{k}\) based on the generator polynomials defined by  
\[\mathrm{G}(D)=\left[1,\frac{g_{1}(D)}{g_{0}(D)}\right] \tag{1}\]  
where \(g_{1}=1+D^{2}+D^{3}\) and \(g_{0}=1+D+D^{3}\) and each is separated internally by inter-leaver \(\pi\) at \(1/3\) coding rate (\(R\)) resulting in an encoded sequence which is rate-matched and mapped onto a constellation of binary phase-shift keying (BPSK). In other words, mapping from \(\{0,1\}\) to \(\{-1,+1\}\) with the signal constellation of \(w=2x-1\) modulates signals and transmits over the wireless channel. The coded symbols generate four redundancy versions (RV), which are selected among the sequence {0,2,3,1} of the TB as code blocks [24]. Each RV is transmitted in one subframe available in each NPUSCH, which is configured into RU to schedule for transmission. The resultant code bits are mapped on \(a_{k}\) symbols. The resulting samples are transformed to time domain through the \(N_{k}\) Inverse Discrete Fourier Transform (IDFT) along the \(n\)-the subcarrier as follows:  
\[x_{k}=\frac{1}{\sqrt{N_{k}}}\sum_{n=0}^{N_{k}-1}a_{k}e^{\beta\pi n\Delta f_{n}},0 \leq n\leq N_{k}-1 \tag{2}\]  
where \(\Delta f_{n}\) is the subcarrier spacing (which can either be 15 kHz or 3.75 kHz).  
To block inter-block interference (IBI), a cyclic prefix (CP) is appended to the \(x_{n}\), then the samples are transmitted simultaneously over a wireless channel. Assuming perfect synchronization at the receiver, the CP is discarded, the N-point DFT is applied, then the discrete-time domain equivalent baseband received signals can be expressed as follows  
\[\mathbf{y}_{k}=\sum_{k=1}^{K-1}\mathbf{H}_{k}\mathbf{x}_{k}+\mathbf{z} \tag{3}\]  
where \(\mathrm{H}_{k}\) denotes the CIR matrix with \([h_{k,0},h_{k,1},\ldots,h_{k,L-1},0,\ldots,0]^{T}\) as its column entry and \(\mathbf{z}\) is the additive white Gaussian noise (AWGN) vector having a zero-mean and variance of \(\sigma_{z}^{2}\). After receiving the \(\mathbf{y}_{k}\) symbol, maximum a posterior probability (MAP) detection is performed to compute the likelihood for every demodulated symbol to deliver a soft symbol for decoding. The process is repeated for certain iterations to obtain a desired decoded output bit.