# On Channel Estimation in LTE-Based Downlink Narrowband Internet of Things Systems
## 5 Architecture of Polyphase Interpolation Filter for the Downlink NB-IoT Systems  
LMMSE channel estimation is based on an FIR filter method which contains a lower order [41]. The low order FIR filter structure remains unchanged for various channel situations and is the prime advantage of using it. However, this paper demonstrates the polyphase decomposition of the FIR filter of the conventional interpolation filter to boost the performance of the channel estimation and also reduce the complexity of the FIR filter design. Polyphase decomposition also constructs a low pass filter and minimizes the computational complexity of the system, and hence the power required will be reduced [42; 43]. In this case, the large FIR filter length is decreased to several small sections to increase the efficiency of the interpolator filter. Every smaller filter will have an extent \(Q=F/P\). Here, the FIR filter order \(F\) will be an integer multiple of polyphase \(P\). In this case, \(P\) acts as an upsampler. We know that this upsampler inserts \(P-1\) zeros in between consecutive values of the incoming signal \(\mathbf{x}(n)\)[43]. It is well explained that only \(Q\) number of values out of \(P\) values will be nonzero at a single instance. All the values are multiplied by the interpolator filter coefficients \(c(0),c(P),c(2P),\ldots,c(F-P)\). That means for \(P\) polyphase decomposition, if the FIR filter order is \(F\), the filter prototype can be illustrated according to [42] as shown below:  
\[\begin{split}\mathbf{E}(z)&=c(0)+c(1)z^{-1}+c(2)z^ {-2}+\ldots+c(P)z^{-P}+c(P+1)z^{-(P+1)}\\ &+\ldots+c(F-P+1)z^{-(F-P+1)}+c(F-1)z^{-(F-1)}\end{split} \tag{30}\]  
Figure 7: Proposed algorithm flowchart.  
where \(z^{-1}\) is the unit delay in the Z-transform notation. According to symmetry and polyphase characteristics, the LMMSE filter coefficient \(c(n)\) can be represented as \(c(F-n-1)\)[44]. Hence, polyphase decomposition in each branch \(\mathbf{E}(z)\) can be expressed as below:  
\[\mathbf{E}(z)=\left(\begin{array}{ccc}c(0)&+c(P)z^{-P}+\ldots&c(F-P)z^{(F-P)} \\ c(1)&+c(P+1)z^{-(P+1)}+\ldots&c(F-P+1)z^{(F-P+1)}\\ \vdots&\vdots&\vdots\\ c(P-1)&+c(2P+1)z^{-(2P+1)}+\ldots&c(F-1)z^{(F-1)}\end{array}\right) \tag{31}\]  
With the help of polyphase theory, every row of Equation (31) can be reduced as \(z^{-r}\mathbf{H}_{r}(z^{P})\)[42]. Here, \(\mathbf{H}_{r}\) represents the filter transfer function. Hence, Equation (31) can be represented in its compact form as given below:  
\[\mathbf{E}(z)=\mathbf{H}_{0}(z^{P})+z^{-1}\mathbf{H}_{1}(z^{P})+z^{-2}\mathbf{ H}_{2}(z^{P})+\ldots+z^{-(P-1)}\mathbf{H}_{P-1}(z^{P}) \tag{32}\]  
Figure 8 represents the proposed polyphase architecture of the interpolation filter according to the above equation. The incoming digital input stream \(\mathbf{x}(n)\) is split into P polyphase or subbands through the transfer function of each polyphase section. As a result of placing the upsampler after the filter transfer function \(\mathbf{H}_{r}\), it needs to perform \(Q\) multiplications and \(Q-1\) additions for each incoming bit sequence \(\mathbf{x}(n)\). However, in the traditional interpolation filter, the upsampler is usually placed before the transfer function, and hence it needs \(PQ\) multiplications and \(P(Q-1)\) additions to perform.