# A Low Complexity Channel Estimation and Detection for Massive MIMO Using SC-FDE
## 3 Channel Estimation  
Let us first assume the use of conventional pilots, i.e., there is no data overlapping in the training block. Conventional pilots of training sequences comprise the periodic transmission of known symbols  
Figure 2: Block diagram of m-MIMO chain with receiver and interference cancellation. utilized by the receiver to compute the channel parameters that are required for equalization purposes. In this scenario, the channel frequency response is estimated by using [11]:  
\[\widetilde{H}_{k}^{(t,r)}=\frac{Y_{k}^{(r)}X_{k}^{(t)P*}}{2\sigma_{P}^{2}}, \tag{13}\]  
where \(Y_{k}^{(r)}\) denotes the signal at the \(r\)-th received antenna \((r=1,2,\ldots,R)\), and \(X_{k}^{(t)P}\) denotes the pilot transmitted by the \(t\)-th transmit antenna \((t=1,2,\ldots,T)\). Moreover, \(\sigma_{P}^{2}\) denotes the power of the pilots, i.e., training sequences (P stands for the pilot). It is assumed that [21]  
\[\sigma_{P}^{2}=\mathrm{E}\Big{\{}\Big{|}X_{k}^{P}\Big{|}^{2}\Big{\}}/2=N \mathrm{E}\Big{(}\Big{|}x_{n}^{P}\Big{|}^{2}\Big{)}/2. \tag{14}\]  
If the pilots associated to different transmit antennas are orthogonal, then there is no interference between antennas when estimating the corresponding channels (e.g., by using disjoint sets of subcarriers for different antennas). In this case, we have  
\[X_{k}^{P(m)}X_{k}^{P(\
eq)*}=0\;m\
eq q, \tag{15}\]  
Since the channel impulse response is shorter than the cyclic prefix (which is just a fraction of the block duration), we can employ the enhanced channel estimates as [13]  
\[\widetilde{H}_{k}^{(t,r)}=\mathrm{DFT}\Big{\{}\widetilde{h}_{n}^{(t,r)}w_{n} \Big{\}}, \tag{16}\]  
where \(w_{n}=1\) if the \(n\)th time-domain sample is inside the cyclic prefix and 0 otherwise. In this case, the SNR at the channel estimates is improved by a factor \(T/T_{G}\), with \(T\) and \(T_{G}\) denoting the duration of the useful part of the block and the cyclic prefix, respectively. Let us consider now the use of superimposed pilots, i.e., \(\mathbf{X}_{k}\
eq 0\) for the subcarriers with pilots. Superimposed pilots suppress the overheads associated with channel estimation based on conventional pilot symbols, which tends to be more demanding in the case of m-MIMO, leading to a reduction in spectral efficiency. In the following, we will assume that [21]  
\[\sigma_{D}^{2}=\mathrm{E}\Big{\{}\Big{|}X_{k}\Big{|}^{2}\Big{\}}/2=N\mathrm{E }\Big{(}\Big{|}x_{n}\Big{|}^{2}\Big{)}/2, \tag{17}\]  
where \(\sigma_{D}^{2}\) stands for the power of the data. Let us assume a frame with \(N_{T}\) time-domain blocks, each with \(N\) subcarriers. If the cyclic prefix of each FFT block has \(N_{G}=NT_{G}/T\) samples, we will need \(N_{G}\) equally spaced frequency-domain pilots for the channel estimation. For pilot spacings in time and frequency \(\Delta N_{T}\) and \(\Delta N_{F}\), respectively, the total number of pilots in the frame is given by  
\[N_{P}^{Frame}=\frac{N}{\Delta N_{F}}.\frac{N_{T}}{\Delta N_{T}}, \tag{18}\]  
This means that we have a pilot multiplicity or redundancy of  
\[N_{R}=\frac{N_{P}^{Frame}}{N_{G}}=\frac{N}{N_{G}\Delta N_{F}}\cdot\frac{N_{T}} {\Delta N_{T}}, \tag{19}\]  
To avoid significant performance degradation due to channel estimation errors, the SNR associated with the channel estimation, given by \(\mathrm{SNR}_{\mathrm{est}}\approx N_{R}\sigma_{P}^{2}/\sigma_{D}^{2}\), should be much higher than the SNR for the data \(\mathrm{SNR}_{\mathrm{data}}=\sigma_{D}^{2}/\sigma_{N}^{2}\). In this scenario, assuming superimposed pilots (pilots added to data), data will represent interference to the channel estimation obtained with the pilots, and the pilots will represent interference to the channel estimation obtained with the pilots, which leads to a degradation of performance. This can be mitigated by employing pilots with relatively low power and average the pilots over a large number of blocks, so as to obtain accurate channel estimates (the window size should be such that the channel should be constant within it).