# A Low Sampling Rate Receiver Design for Multi-Antenna Multi-User OFDM Systems
###### Abstract
[26] discussed a time interleaved (TI) architecture with slower sub-ADCs in parallel, but the mismatch among the sub-ADCs could give rise to error floors in the receiver performance. The authors also proposed an online iterative approach for the joint estimation of mismatch and channel parameters, which leveraged training sequences that were provided for the channel estimation. Ref. [27] presented a DFT/IDFT-free receiving scheme for spread-OFDM signals, which allowed each user to receive the signal without DFT and IDFT using sub-Nyquist sampling and a proper sampling delay. Ref. [28] investigated fronthaul optical links using sub-Nyquist sampling rate ADCs for beyond fifth generation (B5G) and 6G sub-THz massive multiple-input multiple-output (Ma-MIMO) beamforming. Researchers also investigated other possible low sampling rate schemes in different scenarios, such as under-sampling restoration digital predistortion (USR-DPD) [29], frequency comb OFDM radar systems [7] and sub-Nyquist sampling for bandwidth- and hardware-efficient mobile fronthaul [30]. In an OFDM system with \(N\) subcarriers spaced by \(1/T\), the dimension of the signal space of one OFDM signal is \(N\) (or \(2N\) in real dimension) and thus, the Nyquist sampling rate is \(N/T\). When \(N\) subcarriers are uniformly allocated among \(K\) users, then the space is divided into \(K\) subspaces, each with the dimension of \(V=N/K\). In other words, in multi-user OFDM systems, the information intended for each user is roughly constrained within a much smaller subspace. Theoretically, such sparsity implies that fewer measurements (samples) would be needed to retrieve the information, at least for a high SNR regime. Based on this principle, this paper proposes a low sampling rate receiver for multi-antenna multi-user OFDM systems. With the aid of zero-forcing precoding, the sampling rate of each receiver can be reduced to \(f_{\rm s}=f_{0}/K\). The simulation results show that the performance loss is insignificant and acceptable in terms of bit error rate, mutual information and peak-to-average power ratio. The rest of this paper is organized as follows. Section 2 introduces the conventional multi-antenna multi-user OFDM system. Based on this system, we propose a low sampling rate receiver design in Section 3. Section 4 presents the simulation results and comparisons of bit error rate, mutual information and peak-to-average power ratio. Section 5 concludes the paper. _Notation_: Throughout this paper, we use bold lower case letters to denote vectors, such as \(\mathbf{a}\), and bold upper case letters to denote matrices, such as \(\mathbf{A}\). \(\mathbf{A}^{\rm t}\) is the conjugate transpose of matrix \(\mathbf{A}\). \(\mathbf{A}^{*}\) is the element wise conjugate. \(\|\mathbf{a}\|\) is the Euclidean norm. \(\operatorname{vect}(\mathbf{a}_{1},\mathbf{a}_{2},\cdots,\mathbf{a}_{n})\) is the concatenation of \(n\) column vectors. \(\operatorname{tr}(\mathbf{A})\) is the trace of \(\mathbf{A}\). \(\operatorname{diag}(\mathbf{a})\) is a diagonal matrix constructed from \(\mathbf{a}\). \(\mathbb{E}[\cdot]\) represents mathematical expectation.