# Low-Computational Extended Orthogonal Matched Filter Structure for Multiuser Detection
## 2 Principle of the EOMF  
### Formulation of an EOMF  
This section describes the principle of the EOMF [18; 19]. Figure 1 shows the basic structure of an EOMF [18; 19]. In the case in which the weight vector of the array antenna when beamforming has the desired signal \(\mathbf{U}\triangleq[U_{1},\ U_{2},\ \dots,\ U_{K}]^{T}\), the signal \(\mathbf{y}_{\text{beam}}\) that enters matched filter 1 (MF\({}_{1}\)) is described by Equations (1)-(3):  
\[\mathbf{y}_{\text{beam}}=\sum_{k=1}^{K}Re\Big{\{}U_{k}^{*}\mathbf{x}_{k}\Big{\}}=\sum_{ k=1}^{K}\Biggl{(}\sum_{m=1}^{M}\sqrt{P_{m}}d_{m}\mathbf{c}_{m}Re\Big{[}e^{-j\phi_{k}( \theta_{m})}U_{k}^{*}\Big{]}+Re\Big{[}\mathbf{n}_{k}U_{k}^{*}\Big{]}\Biggr{)} \tag{1}\]  
\[\mathbf{x}_{k}=\sum_{m=1}^{M}\sqrt{P_{m}}d_{m}\mathbf{c}_{m}e^{-j\phi_{k}(\theta_{m})} +\mathbf{n}_{k} \tag{2}\]  
\[\phi_{k}(\theta_{m})=\frac{2\pi}{\lambda}R\cos\Bigl{\{}\theta_{m}-\frac{2\pi} {K}(k-1)\Bigr{\}}. \tag{3}\]  
Here, \(K\) is the number of antenna arrays, \(M\) is the total number of nodes, \(d_{m}\in\{-1,\ 1\}\) is the modulated data symbol for the binary phase shift keying (BPSK), \(\mathbf{c}_{m}=[c_{m}(1),\ c_{m}(2),\ \dots,\ c_{m}(N)]^{T}\) is the spreading sequence of the \(m\)-th node for the direct-sequence spread-spectrum (DSSS), \(P_{m}\) is the transmitted signal power of the \(m\)-th node, \(\theta_{m}\) is the angle of arrival of the signal at the \(m\)-th node, \(R\) is the radius of the circular array antenna, \(\lambda\) is the wavelength of the radio wave, \(T\) is the transposition, \(*\) is the complex conjugate, and \(\mathbf{n}_{k}\) is the noise component generated in the channel for the \(N\) chips. Similarly, in the case in which the weight vector of the array antenna when null-steering the desired signal is \(\mathbf{V}\triangleq[V_{1},\ V_{2},\ \dots,\ V_{K}]^{T}\), the signal \(\mathbf{y}_{\text{null}}\) that enters the matched filter group (MFG: MF\({}_{2},\ \dots,\ \text{MF}_{N}\)) is as follows in Equation (4):  
\[\mathbf{y}_{\text{null}}=\sum_{k=1}^{K}Re\Big{\{}V_{k}^{*}\mathbf{x}_{k}\Big{\}}=\sum_ {k=1}^{K}\Biggl{(}\sum_{m=1}^{M}\sqrt{P_{m}}d_{m}\mathbf{c}_{m}Re\Big{[}e^{-j\phi_ {k}(\theta_{m})}V_{k}^{*}\Big{]}+Re\Big{[}\mathbf{n}_{k}V_{k}^{*}\Big{]}\Biggr{)}. \tag{4}\]  
Figure 1: Basic structure of the extended orthogonal matched filter (EOMF). The adaptive array antenna is arranged in the front stage. The OMF is placed in the rear stage. Then, the output signal of the MF\({}_{1}\)\(y_{\text{MF}_{1}}\) is expressed by Equation (5):  
\[y_{\text{MF}_{1}}={\epsilon_{1}}^{T}y_{\text{beam}}=\sum_{k=1}^{K}\!\!\left(N \sqrt{P_{1}}d_{1}Re\!\left[e^{-j\phi_{\lambda}(\theta_{1})}U_{k}^{*}\right]+ \sum_{m=2}^{M}\sqrt{P_{m}}d_{m}{\epsilon_{1}}^{T}{\epsilon_{m}}Re\!\left[e^{-j \phi_{\lambda}(\theta_{m})}U_{k}^{*}\right]+{\epsilon_{1}}^{T}Re\!\left[n_{k} U_{k}^{*}\right]\right) \tag{5}\]  
In (5), the first term, the second term and the third term represent the desired signal component, the interference signal component, and the noise component, respectively. Additionally, the output signal of the MF\({}_{i}\) (\(i=2,\ \dots,\ N\)), \(y_{\text{MF}_{i}}\), is expressed as follows in Equation (6):  
\[y_{\text{MF}_{i}}={\mathbf{u}_{i}}^{T}y_{\text{null}}=\sum_{k=1}^{K}\!\!\left(\sum _{m=2}^{M}\sqrt{P_{m}}d_{m}{\mathbf{u}_{i}}^{T}{\mathbf{c}_{m}}Re\!\left[e^{-j\phi_{ \lambda}(\theta_{m})}V_{k}^{*}\right]+{\mathbf{u}_{i}}^{T}Re\!\left[{\mathbf{u}_{k}}V_{ k}^{*}\right]\right)\!. \tag{6}\]  
Here, \({\mathbf{u}_{i}}=\left[u_{i}(1),\ u_{i}(2),\ \dots,\ u_{i}(N)\right]\) is a sequence that is orthogonal to \({\mathbf{c}_{1}}\), which is generated by Gram-Schmidt orthonormalization. In fact, \(y_{\text{MF}_{1}}\) and \(y_{\text{MF}_{i}}\) must be normalized with respect to the desired signal component because the error signal to update the weight vector \({\mathbf{w}}\triangleq\left[w_{1},\ w_{2},\ \dots,\ w_{N-1}\right]^{T}\) of the linear combiner (LC) is the difference between the output of the EOMF and its hard decision signal.