## 2 System Description  
Consider a MIMO-OFDM system having \(N_{u}\) non-zero subcarriers, \(N_{R}\) receive antennas, and \(N_{T}\) transmit antennas. The frequency-domain signal is transformed by \(N\)-point inverse FFT (IFFT) to generate the time-domain signal and a guard interval (GI) with \(N_{S}\) samples is appended to the front of the OFDM symbol to combat ISI. The \(n\)-th sample transmitted from transmit antenna \(m\) during the \(l\)-th period can then be written by  
\[x_{l}^{m}(n)=\frac{1}{\sqrt{N_{T}}}\sum_{k=-N/2}^{N/2-1}X_{l}^{m}(k)e^{j2\pi kn/ N},\ n=-N_{g},-N_{g}+1,\cdots,N-1 \tag{1}\]  
where \(X_{l}^{m}(k)\) is a complex signal from transmit antenna \(m\) at subcarrier \(k\) with symbol energy \(E_{X}=|X_{l}^{m}(k)|^{2}\) and the normalization factor \(1/\sqrt{N_{T}}\) is used to keep the transmission power constant. As a consequence, one OFDM symbol of effective duration \(T_{s}=N_{s}T\) is generated, where \(T\) is the sampling time and \(N_{s}=N+N_{g}\). In the frequency domain, \(N-N_{u}\) subcarriers are zero-padded. The continuous-time baseband signal transmitted at antenna \(m\) is expressed as  
\[ \begin{split} x^{m}(t)&=\sum_{l=0}^{\infty}x_{l}^{m} (t)\\ &=\sum_{l=0}^{\infty}\sum_{k=-N/2}^{N/2-1}X_{l}^{m}(k)e^{j2\pi(k/ T_{s})(t-T_{g}-lT_{s})}s(t-lT_{s})\end{split}  \tag{2}\]  
where \(T_{g}\) is the GI duration and \(s(t)\) is the shaping pulse given as  
\[s(t)= \begin{cases}1,\ 0\leq t\leq T_{s}\\ 0,\ \text{otherwise}\end{cases} . \tag{3}\]  
Figure 1 depicts the block diagram of the MIMO-OFDM receiver including the synchronization block. Since the frequency of the local oscillator \(f_{c}^{{}^{\prime}}\) is not exactly matched to the received carrier frequency \(f_{c}\), the time-domain signal in receiver antenna \(r\) can be given by  
\[y^{\prime}(t)=e^{j2\pi(f_{c}^{{}^{\prime}}-f_{c})t}\sum_{m=1}^{N_{T}}x^{m}(t) \otimes h^{m,r}(t)+z^{r}(t),\ r=1,2,\cdots,N_{R} \tag{4}\]  
where \(f_{c}^{{}^{\prime}}-f_{c}\) means the CFO in Hz, \(h^{m,r}(t)\) is the channel impulse response (CIR) from transmit antenna \(m\) to receive antenna \(r\), and \(z^{r}(t)\) is a zero-mean additive white Gaussian noise (AWGN) process at receive antenna \(r\). In Equation (4), \(h^{m,r}(t)\) takes the form  
\[h^{m,r}(t)=\sum_{p=0}^{P-1}\alpha_{p}^{m,r}\delta(t-\tau_{p}^{m,r}) \tag{5}\]  
where \(P\) is the number of multipaths, and \(\alpha_{p}^{m,r}\) and \(\tau_{p}^{m,r}\) denote the attenuation and delay of the \(p\)-th multipath from transmit antenna \(m\) to receive antenna \(r\), respectively. Since the multiple antennas are usually co-located, only one oscillator is referenced in either the transmit side or receiver side. Moreover, the differences in Doppler shift between all transmit-receive antenna pairs are small in typical broadcasting systems. With those implications in mind, we consider scenarios where all transmit-receive antenna couples experience a single common CFO and SFO as discussed in [36; 37; 38]. The received signal \(y^{\prime}(t)\) is sampled at time instants \(t=nT^{\prime}\), which leads to a SFO \(\xi=(T^{\prime}-T)/T\). At the receiver, the sampled signal at time instants \(t=nT^{\prime}\) becomes [34]  
\[y^{\prime}(nT^{\prime})=e^{2\pi\epsilon nT^{\prime}/NT}\sum_{m=1}^{N_{T}}\sum_{ p}h_{p}^{m,r}(nT^{\prime})x^{m}(nT^{\prime}-\tau_{p}^{m,r})+z^{\prime}(nT^{ \prime}),\ r=1,2,\cdots,N_{R} \tag{6}\]  
where \(\epsilon\) is the normalized CFO by the subcarrier spacing \(1/NT\), and \(h_{p}^{m,r}(nT^{\prime})\) and \(z^{\prime}(nT^{\prime})\) are the samples of multiplicative channel and additive noise, respectively. As shown in Figure 1, the estimation of initial STO and CFO is performed during pre-FFT phase, which is usually based on the inherent redundant information in OFDM symbols. After FFT, residual CFO and SFO are estimated and tracked using either CP or SP. After the time and frequency synchronization process, the receiver performs channel estimation and MIMO detection in a sequential or iterative manner [39].