# An Uplink Channel Estimator Using a Dedicated Instruction Set for 5G Small Cells
## 3 Algorithm and Implementation of Channel Estimation
### Channel Estimation at Pilot  
In 5G communication, both the sender and the receiver agree on the known demodulation reference signal (DMRS) for both sides to measure the distortion and interference of the OFDM subspectrum to obtain the channel response values at the reference point. On this basis, the channel response values between pilots are obtained by interpolating values between two or three reference signals. Interpolated values will be used as the channel estimation for received signals. Therefore, this study used the pilot-based OFDM channel estimation method. In the channel estimation at the DMRS, the LS [19] + Wiener interpolation [20] method is suitable for small cells due to its complexity and performance trade-off, and it is widely used in OFDM channel estimation.  
Consider four users. For a certain subcarrier, let \(si\) be the data sent by the \(i\)-th user, \(h_{ij}\) is the channel from the \(i\)-th transmitting antenna to the \(j\)-th receiving antenna and \(y_{i}\) is the channel of the \(j\)-th receiving antenna. Then, the received value obtained on each antenna is shown in Equation (1) [21]. Each of the \(si^{\star}h_{ij}\) represents the reception component of the  
Figure 3: Flow chart of dedicated instruction set design.  
\(i\)-th transmitting antenna on the \(j\)-th receiving antenna after passing through its wireless channel. Each receiving antenna has four components from four different transmitting antennas. The transmission model corresponding to Equation (1) is shown in Figure 4.  
\[\left\{\begin{array}{l}s_{1}*h_{11}+s_{2}*h_{21}+s_{3}*h_{31}+s_{4}*h_{41}+n_{0}= y_{1}\\ s_{1}*h_{12}+s_{2}*h_{22}+s_{3}*h_{32}+s_{4}*h_{42}+n_{0}=y_{2}\\ s_{1}*h_{13}+s_{2}*h_{23}+s_{3}*h_{33}+s_{4}*h_{43}+n_{0}=y_{3}\\ s_{1}*h_{14}+s_{2}*h_{24}+s_{3}*h_{34}+s_{4}*h_{44}+n_{0}=y_{4}\\ \end{array}\right. \tag{1}\]  
It can be seen from the DMRS structure of Figure 2 that for each subcarrier, only two users send the DMRS, and the other two users do not send data. Taking DMRS ports 0 and 1 as an example, user 3 and user 4 do not send data that the corresponding \(s_{3}\) and \(s_{4}\) are both 0, so Equation (1) can be simplified to Equation (2).  
\[\left\{\begin{array}{l}s_{1}*h_{11}+s_{2}*h_{21}+n_{0}=y_{1}\\ s_{1}*h_{12}+s_{2}*h_{22}+n_{0}=y_{2}\\ s_{1}*h_{13}+s_{2}*h_{23}+n_{0}=y_{3}\\ s_{1}*h_{14}+s_{2}*h_{24}+n_{0}=y_{4}\\ \end{array}\right. \tag{2}\]  
As shown in Equation (2), the LS channel estimation at the pilot cannot be in the form of a simple Y/S. In order to calculate Equation (2), we can choose to divide two adjacent DMRSs on the same port into a CDM group in the time domain or frequency domain, and assume that the H value of the same CDM group is the same. Then, the two pilot symbols of the same CDM group can be combined with equations to obtain the estimation result of LS estimation with the influence of noise.  
The work of filtering the noise can be placed in the Wiener filter interpolation module between the DMRS and the MMSE channel equalization module. The method adopted in this study is to divide two adjacent DMRSs on the same port into a CDM group in the time domain. In this way, an accurate solution can be obtained in the frequency domain, and an approximate solution can be obtained in the time domain. The loss of accuracy in the time domain is negligible in low-mobility small cell scenarios. If the mobility increases, more DMRSs can be inserted in the time domain to solve the problem of decreased estimation performance due to increased mobility.  
Figure 4: Transmission model of 4 \(\times\) 4 MIMO system corresponding to Equation (1).