# A Low Sampling Rate Receiver Design for Multi-Antenna Multi-User OFDM Systems
## 2 OFDM Systems
### Signal Model  
The data of all users are multiplexed into an aggregate data vector \(\mathbf{d}=(d_{0},d_{1},\cdots,d_{N-1})^{\rm T}\), where \(d_{n}\in\Omega\) is the data symbol to be transmitted over the \(n\)-th subcarrier and \(\Omega\) is the signal constellation, such as QPSK, 16 QAM, etc. To transmit \(\mathbf{d}\) over \(M\) transmitting antennas, a precoder \(\mathbf{W}\) is applied that transforms \(\mathbf{d}\) into multiple vectors \(\mathbf{x}_{1},\mathbf{x}_{2},\cdots,\mathbf{x}_{M}\), where:  
\[\mathbf{x}_{m}=(x_{m,0},x_{m,1},\cdots,x_{m,N-1})^{\rm T}\in\mathbb{C}^{N}, \tag{1}\]  
Figure 1: A multi-user OFDM system. for \(m=1,2,\cdots,M\). The precoded signal is then transmitted over \(M\) transmitting antennas with \(x_{m,n}\) being transmitted over the \(n\)-th subcarrier and the \(m\)-th antenna. The time domain signal transmitted at the \(m\)-th transmitting antenna can be expressed as:  
\[s_{m}(t)=\frac{1}{\sqrt{N}}\sum_{n=0}^{N-1}x_{m,n}\mathrm{e}^{\mathrm{i}2\pi \frac{n}{T}t}, \tag{2}\]  
for \(m=1,2,\cdots,M\), \(-T_{\mathrm{cp}}\leq t\leq T\), where \(T_{\mathrm{cp}}\) is the duration of cyclic prefix (CP), \(T\) is the OFDM symbol period excluding the CP and the orthogonal subcarriers are spaced by \(1/T\). The time domain signal received at the \(k\)-th user is obtained by:  
\[y_{k}(t) =\frac{1}{\sqrt{N}}\sum_{m=1}^{M}\sum_{n=0}^{N-1}H_{k,m}^{n}x_{m,n }\mathrm{e}^{\mathrm{i}2\pi\frac{n}{T}t}+z_{k}(t) \tag{3}\] \[=\frac{1}{\sqrt{N}}\sum_{m=1}^{M}\sum_{n=0}^{N-1}\tilde{H}_{k,m}^ {n}x_{m,n}\mathrm{e}^{\mathrm{i}2\pi\frac{n}{T}t}+z_{k}(t), \tag{4}\]  
for \(k=1,2,\cdots,K\), where \(z_{k}(t)\) is the additive white Gaussian noise (AWGN), \(\tau_{k}\) represents the propagation delay and timing synchronous error and \(\tilde{H}_{k,m}^{n}=H_{k,m}^{n}\mathrm{e}^{-\mathrm{i}2\pi\frac{n}{T}t_{k}}\) is the overall channel gain. Similar to [31], we assume that the channel state information (CSI), i.e., \(\{\tilde{H}_{k,m}^{n},\forall k,n,m\}\), can be estimated by each user and that the estimates can be reported to BS reliably via the signalling channel so that all \(\tilde{H}_{k,m}^{n},\forall n,k,m\) are known to BS. We will also discuss the effect of imperfect CSI at the BS in Section 4.5. The ADC shown in Figure 1 converts the continuous waveform into a quantized time-discrete sequence. This paper focuses on the sampling rate and thus, we assume that the quantization error is negligible. Recall that the synchronous error was considered in the overall channel gain \(\tilde{H}_{k,m}^{n}\) through \(\mathrm{e}^{-\mathrm{i}2\pi\frac{n}{T}\tau_{k}}\); hence, the sampling epochs for each OFDM symbol can be unified as \(0,\Delta,2\Delta,\cdots,(N-1)\Delta\) for all users, where \(\Delta=T/N\) is the sampling interval. Then, the output samples of the ADC with a Nyquist sampling rate \(f_{0}=N/T\) can be written as:  
\[y_{k}(v)=\frac{1}{\sqrt{N}}\sum_{m=1}^{M}\sum_{n=0}^{N-1}\tilde{H}_{k,m}^{n}x _{m,n}\mathrm{e}^{\mathrm{i}2\pi\frac{n}{T}v}+z_{k}(v), \tag{5}\]  
for \(v=0,1,\cdots,N-1\), where \(z_{k}(v)\sim\mathcal{CN}(0,\sigma^{2})\). Equation (5) can be expressed in compact matrix form as:  
\[\mathbf{y}_{k}=\mathbf{F}^{\dagger}\Bigg{(}\sum_{m=1}^{M}\mathrm{diag}(\tilde{\mathbf{H} }_{k,m})\mathbf{x}_{m}\Bigg{)}+z_{k}, \tag{6}\]  
where \(\mathbf{F}\in\mathbb{C}^{N\times N}\) is the DFT transform matrix:  
\[\mathbf{F}=\frac{1}{\sqrt{N}}\left( \begin{array}{cccc}1&1&\cdots&1\\ 1&\mathrm{e}^{-\mathrm{i}\frac{2\pi}{N}}&\cdots&\mathrm{e}^{-\mathrm{i}\frac{ 2\pi(N-1)}{N}}\\ \vdots&\vdots&\ddots&\vdots\\ 1&\mathrm{e}^{-\mathrm{i}\frac{2\pi(N-1)}{N}}&\cdots&\mathrm{e}^{-\mathrm{i} \frac{2\pi(N-1)^{2}}{N}}\end{array}