# Two-Step Multiuser Equalization for Hybrid mmWave Massive MIMO GFDM Systems
## 3 Transmitter Design
### Transmitter Model  
The block diagram of the transmitter side is based on the low complexity modulation scheme GFDM and has two main parts, digital and analog, as depicted in Figure 1. In the digital part, first the input bit stream \(\mathbf{d}_{u}=\left[d_{u,0^{\prime}},d_{u,1},\ldots,d_{u,KM-1}\right]^{T} \in\mathbb{C}^{KM}\) is modulated by a _M-ary_ QAM (M-QAM) constellation and then divided into \(K\) vectors of complex valued data symbols \(\mathbf{d}_{u,k}=\left[d_{u,t}\right]_{j=KM}^{(k+1)M}\in\mathbb{C}^{M}\). Each vector is spread on the \(K\) subcarriers and \(M\) time slots. The converting process from time to frequency domain can be applied by \(\mathbf{Z}_{M}\in\mathbb{C}^{M\times M}\) to each \(\mathbf{d}_{u,k}\), where the result \(\mathbf{Z}_{M}\mathbf{d}_{u,k}\) represents the data at subcarrier \(k\) in the frequency domain [34]. The transformed vector \(\mathbf{Z}_{M}\mathbf{d}_{u,k}\) is passed through the second frequency domain processing block that contains three stages. These stages in Figure 1 correspond to the three blocks (up-sampling, filtering, and up-conversion operations). Each operation may be represented by a matrix, thus the signal at the output of each block is equal to the signal at the input multiplied by the corresponding matrix. Therefore, the repetition matrix \(\mathbf{R}^{(L)}\) that has the identity matrices \(\mathbf{I}_{M}\) is applied to the obtained frequency samples,  
\[\mathbf{R}^{(L)}=\underbrace{\left[\mathbf{I}_{M},\ldots,\mathbf{I}_{M} \right]^{T}}_{L}\in\mathbb{C}^{LM\times M}, \tag{4}\]  
in order to duplicate the samples \(L\) times. Then, to each subcarrier, the diagonal of the pulse shaping filter matrix \(\mathbf{\Gamma}_{tx}=\text{diag}(\mathbf{Z}_{LM}\mathbf{g})\in\mathbb{C}^{LM \times LM}\) that contains the frequency samples of the filter \(\mathbf{g}\) will be applied. Herein, the functionality of the permutation matrix \(\mathbf{P}^{(k)}\in\mathbb{C}^{NM\times LM}\) on the \(k\)th subcarrier signal is used to circular up-convert the signal from base-band to band-pass, where \(\mathbf{P}^{(1)}=\left[\mathbf{I}_{LM},0_{LM},0_{LM},\ldots\right]^{T}\), \(\mathbf{P}^{(2)}=\left[0_{LM},\mathbf{I}_{LM},0_{LM},\ldots\right]^{T}\),... with \(0_{LM}\), a square matrix that contains zero elements [34]. At last, in the second domain conversion block, all subcarriers signal are superimposed and transformed to the time domain \(\mathbf{c}_{u}=\left[c_{u,0},\ldots,c_{u,NM-1}\right]^{T}\in\mathbb{C}^{NM}\) by applying inverse discrete Fourier transform (IDFT) matrix \(\mathbf{Z}_{NM}^{H}\), where  
\[\mathbf{c}_{u}=\mathbf{Z}_{NM}^{H}\underset{k=0}{\overset{K-1}{\sum}}\left( \mathbf{P}^{(k)}\mathbf{\Gamma}_{tx}\mathbf{R}^{(L)}\mathbf{Z}_{M}\mathbf{d}_ {u,k}\right) \tag{5}\]  
Figure 1: Block diagram of the user terminal (UT). FFT (fast Fourier transform), IFFT (inverse fast Fourier transform), DAC (digital to analog converter), RF (radio frequency). It is of paramount importance to mention that the transmitter model of GFDM can be represented by a single matrix in order to facilitate the application of standard receiver methods such as matched filter (MF), ZF, and minimum mean square error (MMSE). In this case, the matrix representation Equation (5) can be rewritten as  
\[\mathbf{c}_{u}=\mathbf{Z}_{NM}^{H}\mathbf{A}\mathbf{d}_{u}, \tag{6}\]  
where \(\mathbf{A}=[\mathbf{A}_{0},\mathbf{A}_{1},\dots,\mathbf{A}_{K-1}]\in\mathbb{ C}^{NM\times KM}\) is the complex-valued modulation matrix that contains all transmitted signals as \(\mathbf{A}_{k}=\mathbf{P}^{(k)}\mathbf{\Gamma}\mathbf{R}^{(L)}\mathbf{Z}_{M} \in\mathbb{C}^{NM\times M}\). Finally, transmitted signal \(\mathbf{x}_{u,l}\in\mathbb{C}^{N_{tx}}\) is obtained after applying the analog precoder \(\mathbf{f}_{a,u}\in\mathbb{C}^{N_{tx}}\), as  
\[\mathbf{x}_{u,l}=\mathbf{f}_{a,u}e_{u,l}\ \,\ l=0,\dots NM-1 \tag{7}\]  
The analog precoder elements are phase shifters, that is, \(\left|\mathbf{f}_{a,u}(i)\right|_{2}=N_{tx}^{-1}\), and because of the hardware constraints, its entries have the same amplitude.