# A Pilot-Pattern Based Algorithm for MIMO-OFDM Channel Estimation
## 2 System Modeling on MIMO-OFDM  
MIMO is available in 4G mobile communications, which is performed by equipping the transmitter and receiver with multiple antennas combined with the OFDM technology to further expand the system's capacity [13]. The combination of MIMO and OFDM forms the so-called MIMO-OFDM system which is depicted as Figure 2.  
Figure 1: General procedure of channel estimation.  
Given \(N_{T}\) denoting the number of antenna connected to transmitter, and \(N_{R}\) is the number of antennas connected to receiver. \(N\) represents the number of subcarriers, and 1/R is the ratio of pilot insertion. The impulse response in the time domain between the transmitter's \(i\)th antenna and the receiver's \(j\)th antenna is [14]:  
\[h_{i,j}(t,\tau)=\sum_{p=1}^{L}a_{p}\delta(\tau-\tau_{p}T) \tag{1}\]  
where \(L\) denotes the number of path that have maximal time delay; \(p\) represents the path number; \(a_{p}\) is the path gain for the \(p\)th path at time t; \(\tau\) is the time delay, \(\tau_{p}\) is time delay of the \(p\)th path, and \(T\) represents the signal period of the transmitted signal.  
If \(h_{i,j}[n,l]\) represents the discrete channel function in the frequency domain, then the discrete impulse response can be expressed as:  
\[H_{i,j}[n,k]=\sum_{l=1}^{L}h_{i,j}[n,l]\exp(-j2\pi\frac{kl}{N}) \tag{2}\]  
where \(l\) represents the \(l\)th path, \(n\) is the transmission moment of the OFDM symbol, \(n\) is the number of carriers that are orthogonal each other, and \(k\) represents the \(k\)th carrier.  
Suppose there are two antennas in the transmitter and receiver, respectively, i.e., \(N_{T}=N_{R}=2\). The channels undergo slow fading whose fading routine is independent of each scenario in which there are a pair of antennas that consisted of one in the transmitter and the other one in the receiver. At the time point \(n\), the data block \(\{b[n,k],k=0,1,\cdots,K-1\}\) is split into two sub-blocks denoted by \(\{t_{i}[n,k],k=0,1,\cdots,K-1,i=1,2\}\). Each sub-block data is then modulated by OFDM and is radiated by two different antennas in the transmitter. If \(K\) represents the number of sub-carriers, then the signal captured by the \(j\)th antenna in receiver will be presented as:  
\[r_{j}[n,k]=\sum_{i}H_{ij}[n,k]t_{i}[n,k]+w_{j}[n,k] \tag{3}\]  
where \(H_{ij}[n,k]\) denotes the impulse response function of the channel in the frequency domain, \(t_{i}[n,k]\) denotes the frequency-domain signal after space-time coding, \(n\) is the transmission moment of the OFDM symbol, and \(k\) is the \(k\)th carrier, as stated before. \(W_{i}[n,k]\) is the Gaussian white noises with mean values of 0 and a variance of \(\delta^{2}\).  
If each path is taken as a narrow-band broad static complex Gaussian process, then all of the paths are uncorrelated from each other. At the moment \(t\), the response function of a channel in the frequency domain can be written as the following formula in which \(\tau_{l}\) and \(\gamma_{l}\) denote the time delay and the complex amplitude of the \(i\)th path respectively:  
Figure 2: The architecture of a MIMO-OFDM system.  
\[\begin{array}{ll}H(t,f)&=\int\limits_{-\infty}^{+\infty}h(t,\tau)e^{ij2\pi f \tau}d\tau\\ &=\sum\limits_{k}\gamma_{k}(t)e^{-j2\pi f\tau_{k}}\end{array} \tag{4}\]  
The discrete form of this formula is then written as:  
\[H[n,k]=H[nT_{f},k\Delta f]=\sum\limits_{l=0}^{K_{0}-1}h[n,l]W_{K}^{kl} \tag{5}\]  
where \(W_{K}=e^{-j\frac{2\pi}{K}}\). For \(h[n,l]\), it can be rewritten as:  
\[h[n,l]=h[nT_{f},lt_{s}],l=0,1\cdots K_{0}-1\]  
where \(K_{0}\) denotes the number of non-zero sampling value of the channel's impulse response; \(t_{s}=\frac{1}{K\Delta f}\) is the sampling interval; \(\Delta f\) is the subcarrier's interval; and \(T_{f}\) is the length of the block.