# Channel Estimation Based on IOTA Filter in OFDM/OQPSK and OFDM/OQAM Systems
## 3 OFDM/OQPSK System Model
### Design of the IOTA Filter  
The approximate time-domain IOTA function, which is denoted as \(\xi_{\tau_{0}}(t)\), can be expressed as [14]:  
\[\xi_{\tau_{0}}(t)=\frac{1}{2}\sum_{k=0}^{K-1}\left\{\overline{d}_{k,v_{0}} \left[\mathbf{h}_{\text{EGF}}(t+\frac{k}{v_{0}})+\mathbf{h}_{\text{EGF}}(t-\frac{k}{ v_{0}})\right]\right\}\cdot\sum_{l=0}^{K-1}\left[\overline{d}_{l,\tau_{0}}\cos(2\pi l \frac{t}{\tau_{0}})\right],-4\tau_{0}\leq t\leq 4\tau_{0}, \tag{1}\]  
Figure 1: Real and imaginary values of orthogonal frequency division multiplexing (OFDM)/offset quadrature phase shift keying (OQPSK) symbols. where \(t\) is the time and \(k\) represents the number of subcarriers. \(\mathbf{h}_{\text{EGF}}(t)=2^{\frac{1}{2}}\text{e}^{-\pi t^{2}}\) is the extended Gaussian filter (EGF) in the time domain. Normally, \(\tau_{0}\) and \(v_{0}\) are chosen to be \(1/\sqrt{2}\) to maintain the orthogonality characteristics between consecutive OFDM symbols in the time domain and adjacent subcarriers in the frequency domain. When \(\tau_{0}=v_{0}=1/\sqrt{2}\), \(t\in[-2\sqrt{2},2\sqrt{2}]\). The polyphase IOTA filter puts the IOTA values into the IOTA buffer with the length equals to half of the fast Fourier transform (FFT) size and the width equals to 8. The IOTA coefficients \(\overline{\mathbf{d}}_{k,v_{0}}\) can be expressed as:  
\[\overline{\mathbf{d}}_{k,v_{0}}=\sum_{q=0}^{Q-1}\mathbf{b}_{k,q}\cdot\text{e}^{-\pi(2 q+k)},0\leq k\leq K-1,0\leq q\leq Q-1, \tag{2}\]  
where \(K\) and \(Q\), which are integer numbers, are two parameters of the IOTA filter. In general, \(K=15\) and \(Q=8\) are chosen in IOTA filter design [14]. Table 1 represents a list of \(\mathbf{b}_{k,q}\) coefficients. Table 2 represents a list of important parameters in the IOTA filter, where \(k\), \(l\), and \(q\) are set to be integers. The accuracy of IOTA filter design can reach \(0.79\times 10^{-19}\)[14]. The IOTA filter which is utilized in OFDM/OQPSK and OFDM/OQAM systems increases the accuracy of channel estimation. The simulation results of linear and decibel formats of the IOTA filter are represented in Figure 2a,b. When the number of OFDM symbols is \(t=0\), the magnitude of the IOTA filter is equal to 1. The IOTA filter is orthogonal between consecutive OFDM symbols, thus it can efficiently reduce the ISI and ICI \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(\mathbf{k}\) & \(\mathbf{q}\)=0 & \(\mathbf{q}\)=1 & \(\mathbf{q}\)=2 & \(\mathbf{q}\)=3 & \(\mathbf{q}\)=4 & \(\mathbf{q}\)=5 & \(\mathbf{q}\)=6 & \(\mathbf{q}\)=7 \\ \hline
0 & 1 & 0.75 & 1.6406 & 2.6367 & 4.6529 & 6.9749 & 11.5368 & 16.8193 \\
1 & \(-1\) & \(-1.875\) & \(-3.4219\) & \(-5.9131\) & \(-9.8831\) & \(-15.7769\) & \(-24.8536\) & 0 \\
2 & 0.75 & 1.1875 & 3.0176 & 4.7681 & 9.0985 & 14.0379 & 23.4937 & 0 \\
3 & \(-0.625\) & \(-0.9609\) & \(-2.2354\) & \(-4.2567\) & \(-7.3957\) & \(-12.2028\) & 0 & 0 \\
4 & 0.5469 & 0.8320 & 1.9036 & 3.4279 & 6.6095 & 10.3934 & 0 & 0 \\
5 & \(-0.4922\) & \(-0.7451\) & \(-1.6937\) & \(-3.0167\) & \(-5.6268\) & 0 & 0 & 0 \\
6 & 0.4512 & 0.6812 & 1.5433 & 2.7365 & 5.0687 & 0 & 0 & 0 \\
7 & \(-0.4189\) & \(-0.6314\) & \(-1.4279\) & \(-2.526\) & 0 & 0 & 0 & 0 \\
8 & 0.3928 & 0.5913 & 1.3356 & 2.3594 & 0 & 0 & 0 & 0 \\
9 & \(-0.3709\) & \(-0.558\) & \(-1.2594\) & 0 & 0 & 0 & 0 & 0 \\
10 & 0.3524 & 0.5299 & 1.1951 & 0 & 0 & 0 & 0 & 0 \\
11 & \(-0.3364\) & \(-0.5056\) & 0 & 0 & 0 & 0 & 0 & 0 \\
12 & 0.3224 & 0.4843 & 0 & 0 & 0 & 0 & 0 & 0 \\
13 & \(-0.31\) & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
14 & 0.2989 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: \(\mathbf{b}_{k,q}\) coefficients of the isotropic orthogonal transfer algorithm (IOTA) filter. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Specifications** \\ \hline \(\tau_{0}\) & \(1/\sqrt{2}\) \\ \(v_{0}\) & \(1/\sqrt{2}\) \\ \(t\) & \([-2\sqrt{2},2\sqrt{2}]\) \\ \(K\) & \(15\) \\ \(Q\) & \(8\) \\ \(k\) & \([0,14]\) \\ \(l\) & \([0,14]\) \\ \(q\) & \([0,7]\) \\ \hline \hline \end{tabular}
\end{table}