## 4 Beamforming of the Oblique Backscatter Ionogram  
To obtain ionospheric information in a large area, the oblique backscatter sounding usually works in a frequency scanning mode. With the sounding waveform modulated by m-sequence encoded by pseudo-random two-phase code, the detection is whole coherent. Thus, at every frequency, multiple coherent accumulations are often carried out to improve the SNR. If a signal detection includes \(M\) frequency points, \(K\) coherent accumulations are carried out on each frequency, the sampling echo signal sequence can be expressed:  
\[\left[s_{i,f_{m},k}(t_{1}),s_{i,f_{m},k}(t_{2}),\cdots,s_{i,f_{m},k}(t_{N})\right] \tag{18}\]  
where \(N\) is the length of the m-sequence, which is also the total number of sounding range gates. Then \(t_{1},t_{2},\cdots,t_{N}\) represent the sampling moment and also represent the distance gate of the echo. \(i,f_{m},k\) represent the sequence number of channel, frequency and coherent accumulation respectively. After the cross-correlation with m-sequence \((t_{1},t_{2},\cdots,t_{N})\), the pulse compressed signal sequence can be obtained as Equation (19). \[\left[S_{i,f_{m},k}(t_{1}),S_{i,f_{m},k}(t_{2}),\cdots,S_{i,f_{m},k}(t_{N}) \right]=\left[s_{i,f_{m},k}(t_{1}),s_{i,f_{m},k}(t_{2}),\cdots,s_{i,f_{m},k}(t _{N})\right]\left( \begin{array}{cccc}m_{1}&m_{N}&\cdots&m_{2}\\ m_{2}&m_{1}&\cdots&m_{3}\\ \vdots&\vdots&\ddots&\vdots\\ m_{N}&m_{N-1}&\cdots&m_{1}\end{array} \right) \tag{19}\]Hence, on any range gate \(t_{n}\), the covariance matrix of the sampling sequence can be obtained after \(K\) times coherent sounding as Equation (20). \[R=\left[ \begin{array}{ccccc}S_{1,f_{m},1}(t_{n})&S_{1,f_{m},2}(t_{n})&\cdots&S_ {1,f_{m},K}(t_{n})\\ S_{2,f_{m},1}(t_{n})&S_{2,f_{m},2}(t_{n})&\cdots&S_{2,f_{m},K}(t_{n})\\ \vdots&\vdots&\ddots&\vdots\\ S_{1,f_{m},1}(t_{n})&S_{1,f_{m},2}(t_{n})&\cdots&S_{1,f_{m},K}(t_{n})\\ \end{array} \right]\left[ \begin{array}{ccccc}S_{1,f_{m},1}(t_{n})&S_{1,f_{m},2}(t_{n})&\cdots&S_{1,f_{m},K}(t_{n})\\ S_{2,f_{m},1}(t_{n})&S_{2,f_{m},2}(t_{n})&\cdots&S_{2,f_{m},K}(t_{n})\\ \vdots&\vdots&\ddots&\vdots\\ S_{1,f_{m},1}(t_{n})&S_{1,f_{m},2}(t_{n})&\cdots&S_{1,f_{m},K}(t_{n})\\ \end{array} \right]^{H} \tag{20}\]  
Based on the covariance matrix \(R\), the weight vector can be determined employing an SMI or ESB beamformer:  
\[\left[w_{1,f_{m}}(t_{n}),w_{2,f_{m}}(t_{n}),\cdots,w_{1,f_{m}}(t_{n})\right] \tag{21}\]  
Therefore, the signal after beamforming can be expressed as a form of weighted summation of the signals of the channels:  
\[S_{f_{m},k}(t_{n})=\left[w_{1,f_{m}}(t_{n}),w_{2,f_{m}}(t_{n}),\cdots,w_{1,f_{ m}}(t_{n})\right]\left[S_{1,f_{m},k}(t_{n}),S_{2,f_{m},k}(t_{n}),\cdots,S_{1,f_{ m},k}(t_{n})\right]^{T} \tag{22}\]  
After \(k\)-times coherent accumulation, the amplitude sequence of the signal for the frequency point \(f_{m}\) at \(t_{n}\) can be obtained as Equation (23). \[A_{f_{m}}(t_{n})=\left|\sum_{k=1}^{K}S_{f_{m},k}(t_{n})\right|=\left|\sum_{k= 1}^{K}\left\{\left[w_{1,f_{m}}(t_{n}),w_{2,f_{m}}(t_{n}),\cdots,w_{1,f_{m}}(t_ {n})\right]\left[S_{1,f_{m},k}(t_{n}),S_{2,f_{m},k}(t_{n}),\cdots,S_{1,f_{m},k }(t_{n})\right]^{T}\right\}\right| \tag{23}\]  
Since the weight vector is independent of the coherence number, the order of weighting and summation can be interchanged as Equation (24). In this way, the number of multiplication operations can be reduced, especially when the number of coherent accumulations is large. \[A_{f_{m}}(t_{n})=\left|\sum_{k=1}^{K}S_{f_{m},k}(t_{n})\right|=\left|\left[w_ {1,f_{m}}(t_{n}),w_{2,f_{m}}(t_{n}),\cdots,w_{1,f_{m}}(t_{n})\right]\sum_{k= 1}^{K}\left[S_{1,f_{m},k}(t_{n}),S_{2,f_{m},k}(t_{n}),\cdots,S_{1,f_{m},k}(t_ {n})\right]^{T}\right| \tag{24}\]  
So that, for the frequency point \(f_{m}\), the amplitude sequence on the oblique backscatter ionogram along the range gates can be expressed as Equation (25).