## 3 Simplified Beamforming by High Efficient CSI Estimation  
As discussed in the previous section, the 1 st eigenmode is mostly formed by the LoS component (NLoS components are suppressed). Two key challenges are reviewed here; CSI estimation in the short link budget situation and computation complexity for SVD, which is required for each subcarrier. This section presents a simplified beamforming scheme by approximately extracting CSI from the LoS component in order to alleviate the calculation cost as well as the additive noise effect. The proposed scheme is illustrated in Figure 3. First, a training signal is sent by a few antenna elements and subcarriers, e.g., 16 of the 256 elements at equal intervals and four of the 2048 subcarriers per antenna element without overlap. When total transmission power is constant, power density is concentrated on the limited number of subcarriers. Signal-to-noise power ratio (SNR) for the selected subcarriers is improved by \(10\text{log}_{10}(2048/4/16)=15.1\) dB before beamforming. In addition, reducing the number of subcarriers can suppress PAPR per antenna element, which alleviates the need for input backoff (IBO) for training signal transmission. Second, each UE antenna element receives the training signals as if they had been transmitted from a single antenna element. UE then obtains CSI for received subcarriers using known training sequence and then calculates relative CSI, which represents the relationship of the CSI between the reference antenna element and the CSI of the other ones. Next, the phase component of the relative CSI is extracted. Third, phase information for null subcarriers is interpolated by linear regression. Let \(c\), \(\lambda\), \(f_{c}\) (=\(c/\lambda\)), \(f_{s}\) and \(d_{m}\) be the light speed, wavelength, carrier frequency, subcarrier spacing, and distance between the \(m\)-th UE antenna element and the BS, respectively; relative phase component of LoS channel at the \(k\)-th subcarrier, \(\exp\{j\varphi_{m}(k)\}\), is expressed as,  
\[ \begin{array}{rcl}e^{j\varphi_{m}(k)}&=&e^{-j2\pi\frac{\xi_{c}+\xi_{c}}{c}(d_ {m}-d_{1})}\\ &=&e^{-j2\pi\frac{\xi_{c}}{c}\Delta d}\cdot e^{-j2\pi\frac{\xi_{c}}{c}\Delta d }\end{array}  \tag{16}\]  
where \(\Delta d\) = \(d_{m}-d_{1}\). Here, phase fluctuation in the frequency domain is due to the term \(\exp\{-j2\pi(k\!f_{s}/c)\Delta d\}\). In millimeter wave communication, inter-element space becomes much smaller than BS-UE distance so the relative phase information can be expressed as an array factor. Assuming  
Figure 2: Channel time variation: (**a**) channel correlation fluctuation of four eigenmodes when \(Nr\) = 16; and (**b**) channel correlation of the 1 st eigenmode with increased number of UE antenna elements, \(Nr\). a linear array with half wavelength spacing, \(\Delta d=(\lambda/2)(m-1)\mathrm{sin}\theta\) where \(\theta\) is AoA. From this we can derive:  
\[e^{i\phi_{m}(k)} = e^{-j2\pi\frac{1}{2}(m-1)\mathrm{sin}\theta}\cdot e^{-j2\pi\frac{kf_{ c}}{2}\frac{1}{2}(m-1)\mathrm{sin}\theta}\] \[= e^{-j\pi(m-1)\mathrm{sin}\theta}\cdot e^{-j\pi\frac{kf_{c}}{2} \frac{1}{2}(m-1)\mathrm{sin}\theta}\]  
where \(B\) is the channel bandwidth. It is obvious that \(-1/2\leq(kf_{c}/B)\leq 1/2\). When the system parameters are set to \(f_{c}=20\) GHz and \(B=400\) MHz, fraction bandwidth, \(B/f_{c}\), is \(0.02\). We consider that \(m=16\) and \(\theta=45^{\circ}\) at the limit, so in-band phase fluctuation is around \(10\%\) of \(360^{\circ}\). Since this condition can be satisfied, linear regression can precisely extract the LoS component using extremely reduced training subcarriers. Relationship between \(\phi_{m}(k)\) and \(k\) can be described as:  
\[\phi_{m}(k)=a_{m}+b_{m}k \tag{18}\]  
\(a_{m}\) and \(b_{m}\) can be obtained by the well-known least squares method (LSM) as follows:  
\[\left[ \begin{array}{c}a_{m}\\ b_{m}\end{array} \right]=(\mathbf{X}^{H}\mathbf{X})^{-1}\mathbf{X}^{H}\mathbf{Y} \tag{19}\]  
where  
\[\mathbf{X}=\left[ \begin{array}{cc}1&\vdots\\ 1&k^{\prime}\\ 1&\vdots\end{array}