## 5 Low-Complexity Hybrid Beamforming
### Digital and Analog Beamforming Matrix Solutions
2.Initialization: estimation matrix \(\overbrace{B}\)\(\left(\cdot,\overbrace{B}\right)\), step matrix \(d^{\prime}\)\(\left(\cdot,\overbrace{B}\right)\). 3.Input matrix \(F_{opt}F_{RF}\) to generate the data sample matrix, while a row from the sample matrix is used as the data sample vector \(\overbrace{j}\). \(F_{opt}^{H}F_{RF}^{(k)}\) is obtained from Equation (78). The estimated value of column 1 of \(\overbrace{B}\)\(\left(\cdot,1\right)\) that corresponds to \(F_{opt}F_{RF}\) column 1 of the right singular matrix. ```  
**Algorithm 1**Projection Approximation Subspace Algorithm. * 4. Put the current data sample vector \(\overbrace{x}\) on the estimated value of \(\overbrace{B}\)\((,1)\). The projection on \(\overbrace{x}\) from the estimate, i.e., \(\overbrace{x}\)\(=\overbrace{x}\)\(-\overbrace{B}\)\((,i)\)\(\overbrace{y}\) and the removed projection \(\overbrace{x}\)\(\overbrace{j}\) as the new data sample vector, using Equation (78) to obtain \(F_{opt}^{H}F_{RF}^{(k)}\) the estimated value of column 2 of \(\overbrace{B}\)\((;2)\) which corresponds to \(F_{opt}F_{RF}\) column 2 of the right singular matrix. Repeat this step until the \(\overbrace{S}\) column 5. Put the current data sample vector \(\overbrace{x}\)\(\overbrace{j}\) on the estimated value of \(\overbrace{B}\)\((;1)\). The projection on \(\overbrace{x}\)\(\overbrace{j}\) from the estimate, i.e., \(\overbrace{x}\)\(\overbrace{j}\)\(=\overbrace{x}\)\(-\overbrace{B}\)\((,1)\)\(\overbrace{y}\) and the removed projection \(\overbrace{x}\)\(\overbrace{j}\) as the new data sample vector, using Equation (78) to obtain \(F_{opt}F_{RF}^{(k)}\) the estimated value of column 2 of the right singular matrix. Repeat this step until the \(\overbrace{S}\) column 6. Continue with the input data sample vector \(\overbrace{x}\)\(\overbrace{j}\)\(+1\). Combine the obtained estimates \(\overbrace{B}\)\(\overbrace{j}\)\((;\overbrace{s})\). Continue to update using Equation (78) \(F_{opt}^{H}F_{RF}^{(k)}\) column 1 of \(\overbrace{B}\)\((;1)\) which also corresponds to \(F_{opt}F_{RF}\) column 1 of the right singular matrix. * 7. Put the current data sample vector \(\overbrace{x}\)\(\overbrace{j}\)\(+1\) on the estimated value of \(\overbrace{B}\)\((;1)\). The projection on \(\overbrace{x}\)\(\overbrace{j}\)\(+1\) by removing. From \(\overbrace{x}\)\(\overbrace{j}\)\(+1\)\(\overbrace{j}\)\(+1\)\(\overbrace{B}\)\((,2)\)\(\overbrace{y}\), while using Equation (78) to continue updating \(F_{opt}^{H}F_{RF}^{(k)}\) column 2 of the column of estimated values \(\overbrace{B}\)\(\overbrace{j}\)\(+1\)\((;2)\). Repeat this step until the updated estimated value column of \(\overbrace{S}\) column of the updated estimated value column \(\overbrace{B}\)\(\overbrace{j}\)\(+1\)\((;\overbrace{s})\). * 8. Continue to enter the remaining data sample vectors and repeat the above steps until the data sample vectors are entered. * 9. **Output**\(\overbrace{B}\)\({}_{N_{T}}\), that is \(F_{opt}F_{RF}\) right singular matrix of the first \(\overbrace{S}\) column estimation result. The constraint that the columns about the digital beamformer should be orthogonal is expressed as:  
\[F_{BB,k}^{H}F_{BB,k}=\alpha F_{DD,k}^{H}\alpha F_{DD,k}=\alpha^{2}I_{kN_{s}} \tag{79}\]  
where \(F_{DD,k}\) is a matrix with the same dimension as \(F_{BB,k}\). The orthogonality constraint provides the possibility that the analog beamformer can discard the \(F_{BB,k}\) product form, which will help to greatly simplify the design of the analog beamformer. The objective function in (77) greatly simplifies the design of the analog beamformer when the AltMinmethod is applied. Since the matrix F_RF removes the product form of \(F_{BB,k}\), the \(F_{RF}\) closed-form solution is expressed as:  
\[arg(F_{RF})=arg\Big{(}F_{opt}F_{DD,k}^{H}\Big{)} \tag{80}\]  
where \(arg(F_{RF})\) generates a new matrix containing the phase of the \(F_{RF}\) matrix. The results show that the \(F_{RF}\) phase can be extracted from the equivalent beamformer \(F_{opt}F_{DD,k}^{H}\).