# Two-Step Multiuser Equalization for Hybrid mmWave Massive MIMO GFDM Systems
## 4 Receiver Design
### Receiver Model  
The receiver model of the proposed hybrid analog-digital receiver is described in this section. First, let us consider the unequalized received signal, given by  
\[\mathbf{y}_{l}=\sum_{u=1}^{U}\mathbf{H}_{u,l}\mathbf{x}_{u,l}+\mathbf{n}_{l}\,= \mathbf{H}_{l}\mathbf{c}_{l}+\mathbf{n}_{l}, \tag{11}\]  
where \(\mathbf{x}_{u,l}\in\mathbb{C}^{N_{rx}}\) is the transmitted signal of the \(u\)th user, \(\mathbf{n}_{l}\in\mathbb{C}^{N_{rx}}\) is a vector of Gaussian noise (AWGN) that has zero mean, and variance \(\sigma_{n}^{2},\mathbf{H}_{l}\) is the equivalent channel defined as  
\[\mathbf{H}_{l}=[\mathbf{H}_{1,l}\mathbf{f}_{a,u},\ldots,\mathbf{H}_{U,l} \mathbf{f}_{a,u}]\in\mathbb{C}^{N_{rx}\times U}, \tag{12}\]  
and \(\mathbf{c}_{l}=\left[c_{1,l},\ldots,c_{U,l}\right]^{T}\in\mathbb{C}^{U}\). From Equation (6) we can obtain the time domain transmitted signal, given by  
\[\mathbf{C}=\mathbf{Z}_{NM}^{H}\mathbf{A}\mathbf{D}\,, \tag{13}\]  
where \(\mathbf{C}=[\mathbf{c}_{1},\ldots,\mathbf{c}_{U,l}]\in\mathbb{C}^{N\!M\! \times\!U}\) and \(\mathbf{D}=[\mathbf{d}_{1},\ldots,\mathbf{d}_{U,l}]\in\mathbb{C}^{N\!M\! \times\!U}\). We can prove that the matrix of the unequalized received signal \(\mathbf{Y}=[\mathbf{y}_{0},\ldots\mathbf{y}_{NM-1}]\in\mathbb{C}^{N_{rx}\times NM}\) in the frequency domain is  
\[\mathbf{Y}=\sum_{l=0}^{NM-1}\left(\mathbf{H}_{l}\mathbf{C}^{T}\mathbf{Z}_{NM}^ {T}\mathbf{E}_{l+1}+\mathbf{N}\mathbf{E}_{l+1}\right), \tag{14}\]  
where \(\mathbf{N}=[\mathbf{n}_{0},\ldots\mathbf{n}_{NM-1}]\in\mathbb{C}^{N_{rx} \times NM}\) and \(\mathbf{E}_{l}=\text{diag}([0,0,\ldots,1,..,0,0])\in\mathbb{C}^{NM\!\times\! NM}\) is a matrix of zeros, with \(1\) in the \(l\)th position of diagonal. Then, from Equations (11), (13), and (14), we have  
\[\mathbf{Y}=\sum_{l=0}^{NM-1}\left(\mathbf{H}_{l}\mathbf{D}^{T}\mathbf{A}^{T} \mathbf{E}_{l+1}+\mathbf{N}\mathbf{E}_{l+1}\right), \tag{15}\]  
where the received signal in the time domain is \(\mathbf{Y}\!\left(\mathbf{Z}_{NM}^{H}\right)^{T}\). The block diagram of the receiver is shown in Figure 2; it has two parts, as in the transmitter, an analog and digital part. First, the analog equalizer \(\mathbf{W}_{a}\in\mathbb{C}^{N_{rx}\times N_{rx}^{RF}}\) is applied on the received signal, as  
\[\mathbf{Y}_{a}=(\mathbf{W}_{a})^{H}\mathbf{Y}\!\left(\mathbf{Z}_{NM}^{H} \right)^{T}=\sum_{l=0}^{NM-1}\left((\mathbf{W}_{a})^{H}\mathbf{H}_{l}\mathbf{ D}^{T}\mathbf{A}^{T}\mathbf{E}_{l+1}+(\mathbf{W}_{a})^{H}\mathbf{N} \mathbf{E}_{l+1}\right)\!\left(\mathbf{Z}_{NM}^{H}\right)^{T}\in\mathbb{C}^{N _{rx}^{RF}\times NM}, \tag{16}\]  
where \(\mathbf{W}_{a}\) is just composed by phase shifters and its matrix entries must have the same amplitude, that is, \(\left|\mathbf{W}_{a}(i,j)\right|_{2}=N_{rx}^{-1}\). After that, the resulted signal passes through the RF chains in order to be processed in the digital domain. This signal will be converted to the frequency domain by DFT down conversion block \(\mathbf{Z}_{NM}\), as \(\mathbf{Y}_{z}=\mathbf{Z}_{NM}\mathbf{Y}_{a}^{T}\in\mathbb{C}^{NM\!\times\!N_ {rx}^{RF}}\), to get  
\[\mathbf{Y}_{z}^{T}=\sum_{l=0}^{NM-1}\left((\mathbf{W}_{a})^{H}\mathbf{H}_{l} \mathbf{D}^{T}\mathbf{A}^{T}\mathbf{E}_{l+1}+(\mathbf{W}_{a})^{H}\mathbf{N} \mathbf{E}_{l+1}\right). \tag{17}\]  
Then, the digital equalizer \(\mathbf{W}_{d,l}\in\mathbb{C}^{U\!\times\!N_{rx}^{RF}}\) is performed, and thus the equalized receive signal can be given by  
\[\mathbf{Y}_{d}=\sum_{l=0}^{NM-1}\left(\mathbf{W}_{d,l}(\mathbf{W}_{a})^{H} \mathbf{H}_{l}\mathbf{D}^{T}\mathbf{A}^{T}\mathbf{E}_{l+1}+\mathbf{W}_{d,l}( \mathbf{W}_{a})^{H}\mathbf{N}\mathbf{E}_{l+1}\right)\in\mathbb{C}^{U\!\times \!NM}. \tag{18}\]After that, the operations of the GFDM are the same as those performed at the transmitter (Figure 1) but in reverse order and transposed conjugate. Therefore, the transpose of the permutation matrix \(\mathbf{P}^{(k)}\) is applied on the \(k\)th subcarrier to make circular down-conversion to zero frequency.