# Channel Estimation Based on IOTA Filter in OFDM/OQPSK and OFDM/OQAM Systems
## 3 OFDM/OQPSK System Model
### OFDM-IOTA System Model  
Figure 3 represents the OFDM-IOTA system model. The incoming binary bits are first QPSK modulated, then every OFDM/QPSK symbol is separated into real and imaginary parts, which is called real map. After real mapping, the OFDM symbols are symbol-phased by multiplying them with the phase coefficient \(\mathfrak{j}^{m+n}\). At the receiver, the OFDM symbols are symbol-dephased by multiplying them with the phase coefficient \(\mathfrak{j}^{-(m+n)}\). After inverse fast Fourier transform (IFFT) operation, the signals are convoluted by the polyphaser IOTA filter, which can be denoted as \(\xi_{\tau_{0}}(m)\). The transmitted OFDM/QQPSK signals in the time domain can be written as [15]:  
\[\mathfrak{s}^{i}_{m,n}=\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}\mathbf{a}^{i}_{m,n}\cdot \mathfrak{j}^{m+n}\cdot\xi_{\tau_{0}}(m-n\tau_{0})\cdot\mathrm{e}^{\frac{|2 nm}{ N}},0\leq m\leq M-1,0\leq n\leq N-1, \tag{3}\]  
where \(\mathbf{a}^{i}_{m,n}\) is the real or imaginary OQPSK symbol of the \(i\)-th signal frame on the \(m\)-th symbol of the \(n\)-th subcarrier. \(N\) is the number of subcarriers within one OFDM symbol and \(M\) is the number of transmitted OFDM/QQPSK symbols. In the OFDM-IOTA system, \(M=300\) and \(N=512\). \(\xi_{\tau_{0}}(m)\) is the transmit IOTA filter on the \(m\)-th OFDM/OQPSK symbol, and \(\tau_{0}\) is the duration of the OFDM/OQPSK symbol. Figure 3: OFDM-IOTA system model. AWGN: additive white Gaussian noise; FFT: fast Fourier transform; IFFT: inverse FFT. Figure 2: IOTA filter in linear and decibel formats: (**a**) Linear format, (**b**) Decibel format. Take the \(i\)-th signal frame for example, suppose \(\mathbf{S}_{m,n}^{i}\) to be the FFT of \(\mathbf{s}_{m,n}^{i}\), the received frequency-domain symbols after multipath channel \(\mathbf{Y}_{m,n}^{i}\) can be represented as:  
\[\mathbf{Y}_{m,n}^{i}=\mathbf{S}_{m,n}^{i}\cdot\mathbf{H}_{m,n}^{i}+\mathbf{W}_{m,n}^{i}, \tag{4}\]  
where \(\mathbf{H}_{m,n}^{i}\) and \(\mathbf{W}_{m,n}^{i}\) are the channel frequency response (CFR) and the frequency-domain noise of the \(i\)-th frame, respectively. After AWGN channel, the received time-domain symbols of the \(i\)-th frame on the \(m\)-th symbol and the \(n\)-th subcarrier can be denoted as \(\mathbf{y}_{m,n}^{i}\), which is an IFFT of \(\mathbf{Y}_{m,n}^{i}\). The OFDM signals are dephased by multiplying them with the coefficients \(\mathrm{j}^{-(m+n)}\). After real de-mapping, the received OFDM/OQPSK signals can be represented as [15]:  
\[\mathbf{\hat{Y}}_{m,n}^{i}=\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}\mathbf{y}_{m,n}^{i}\cdot \mathbf{\xi}_{\tau_{0}}(m-n\tau_{0})\cdot\mathrm{e}^{-\frac{\mathrm{j}^{-2 mm}}{N} }+\mathbf{\omega}_{m,n}^{i}, \tag{5}\]  
where \(\mathbf{\xi}_{\tau_{0}}(m)\) is the receive IOTA filter. \(\mathbf{\omega}_{m,n}^{i}\) is the time-domain AWGN of the \(i\)-th signal frame on the \(m\)-th symbol of the \(n\)-th subcarrier. As illustrated in Figure 3, the received OFDM/OQPSK symbols of the \(i\)-th frame \(\mathbf{\hat{y}}_{m,n}^{i}\) are QPSK demodulated, de-interleaved, and finally channel decoded into binary bits [16]. Suppose the FFT transform of \(\mathbf{\hat{y}}_{m,n}^{i}\) is \(\mathbf{\hat{Y}}_{m,n}^{i}\), after the polyphaser IOTA filter, the estimated CFR \(\mathbf{\hat{H}}_{m,n}^{i}\) of the \(i\)-th frame can be expressed as:  
\[\mathbf{\hat{H}}_{m,n}^{i}=1.4\cdot\mathbf{\hat{Y}}_{m,n}^{i}\cdot\mathrm{j}^{-(m+n)} /\mathbf{X}_{\mathrm{P}}^{i}+\mathbf{\hat{W}}_{m,n}^{i}, \tag{6}\]  
where \(\mathbf{X}_{\mathrm{P}}^{i}\) are the preamble symbols of the \(i\)-th OFDM signal frame. \(\mathbf{\hat{W}}_{m,n}^{i}\) is the frequency-domain AWGN in \(\mathbf{\hat{H}}_{m,n}^{i}\). Supposing \(\mathbf{\hat{h}}_{m,n}\) is the IFFT transform of \(\mathbf{\hat{H}}_{m,n}^{i}\), since the noise suppression process will not be carried out in the LS criterion, it degrades the estimation accuracy of channel impulse responses (CIRs).