# On Channel Estimation in LTE-Based Downlink Narrowband Internet of Things Systems
## 3 NB-IoT Downlink System Model
### Analytical Downlink Signal Model  
We assume that the LTE eNB transmits a bitstream (a block of bits) \(\mathbf{b}=[b^{q}(0),b^{q}(1),\ldots,b^{q}(N_{bit}^{(q)}-1)]\), where \(N_{bit}^{(q)}\) represents the transmitted bits on the NPDSCH in one subframe for a codeword. The codeword bit is scrambled before modulation using a specified scrambling sequence for NB-IoT eNB to randomize the interference in neighboring cells and to ensure that the transmission from various cells is individual to the decoder at the NB-IoT UE receiver. Hence, we get the resulting scrambled block of bits \(\mathbf{\tilde{b}}=\left[\widetilde{b}^{q}(0),\widetilde{b}^{q}(1),\ldots, \widetilde{b}^{q}(N_{bit}^{(q)}-1)\right]\) according to [38]  
\[\mathbf{\tilde{b}}^{-(q)}(i)=(b^{(q)}(i)+c^{(q)}(i))\text{mod}2 \tag{5}\]  
where \(c^{(q)}(i)\) is the scrambling sequence defined by length-31 gold sequence and \(i=0,\ 1,\ 2,\ldots,\ N_{bit}-1\). The scrambling order will be initialized at the starting of each subframe, and the value of initialization \(c_{init}\) will be selected depending on the type of NPDSCH. If the NPDSCH contains BCCH then the scrambling sequence generator shall be initialized with  
\[\mathbf{c}_{init}=n_{RNTI}\cdot 2^{15}+(N_{ID}^{Ncell}+1)((n_{f}\text{mod}61)+1) \tag{6}\]  
Otherwise, the sequence generator shall be initialized with  
\[\mathbf{c}_{init}=n_{RNTI}\cdot 2^{14}+n_{f}\text{mod}2.2^{13}+\lfloor n_{s}/2 \rfloor\cdot 2^{9}+N_{ID}^{Ncell} \tag{7}\]  
where \(n_{RNTI}\) represents the radio network temporary identifier (RNTI), \(n_{s}\) is the initial slot of the transmitted codewords, and \(N_{ID}^{Ncell}\) represents the narrowband cell identity number which is selected from the numerical set \(\mathbf{N}_{ID}^{Ncell}\in\{0,\ 1,\ldots,503\}\). The scrambled bit sequence \(\mathbf{\tilde{b}}^{-(q)}(i)\) will be modulated with quadrature phase-shift keying (QPSK) modulation technique for improving the power efficiency at the transmitting antenna and we get the block of complex-valued modulation symbols \(\mathbf{m}=m^{(q)}(0),\ldots,m^{(q)}(M_{\text{symb}}^{(q)}-1)\), where \(M_{\text{symb}}^{(q)}\) represents the total number of modulated symbols. The modulated symbol block is further divided into \(M_{\text{symb}}/N_{RB}^{NPDSCH}\) sets, each of which corresponds to one OFDMA symbol. Here, \(N_{sc}^{NPDSCH}=M_{RB}^{NPDSCH}.M_{sc}^{RB}\) represents the total number of subcarriers for NPDSCH transmission. This symbol block will be mapped for a single antenna port to the layers \(\mathbf{x}(i)=\left[x^{(0)}(i),\ldots,x^{(v-1)}(i)\right]^{T}\), for \(i=0.1,\ldots,M_{\text{symb}}^{layer}-1\). \(v\) and \(M_{\text{symb}}^{layer}\) represent the layer and modulation symbol number, respectively. For NB-IoT single antenna transmission, the number of layers is one (i.e., \(v=1\)). Therefore, the layer mapping shall be finally defined, according to [36], by  
\[x^{(0)}(i)=m^{(0)}(i) \tag{8}\]  
where \(M_{\text{symb}}^{layer}=M_{\text{symb}}^{(0)}\). This layer mapped signal entered as input block vector to the precoder and creates the block of vectors \(\mathbf{y}(i)=\left[\ldots,y^{(p)}(i)\ldots\right]^{T}\),\(i=0,1,\ldots,M_{\text{symb}}^{ap}-1\). \(y^{(p)}(i)\) denotes the corresponding signal for antenna port \(p\). For the transmission of a single antenna port, the defined precoding is  
\[y^{(p)}(i)=x^{(0)}(i) \tag{9}\]  
where \(\mathbf{p}\in\{0,\ 4,5,7,8,11,13,107,108,109,110\}\) is the single antenna port number that will be used for NPDSCH transmission and \(\mathbf{i}=0,1,\ldots,M_{\text{symb}}^{ap}-1\), \(M_{\text{symb}}^{ap}=M_{\text{symb}}^{layer}\). The complex valued symbol block \(y^{(p)}(0),\ldots,y^{(p)}(M_{\text{symb}}^{ap}-1)\) will be mapped in the frequency-time (_k,l_) resource grid for all the antenna ports which is exploited for NPDSCH transmission. The mapping sequence will start with \(y^{(p)}(0)\) in the resource element (_k,l_) on the corresponding antenna port \(p\) in the increasing order by considering \(k\) and \(l\) consecutively.