# A Low-Complexity Pilot-Based Frequency-Domain Channel Estimation for ICI Mitigation in OFDM Systems
## 2 System Model  
### OFDM Modulation and Sub-Carrier Coupling Matrix  
We consider an OFDM system with perfect time and frequency synchronization. We also assume that the power amplifiers are ideal, i.e., they always work in the linear region, and thus, there is no distortion nor out-of-band emission due to the high PAPR exhibited by OFDM [26]. We have considered \(N\) sub-carriers with \(G\) samples reserved for the CP; hence, the \(m\)-th time domain OFDM symbol can be expressed as follows:  
\[x_{n}^{(m)}=\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}X_{k}^{(m)}\mathbf{e}^{j\frac{ 2\pi}{N}kn},\quad-G\leq n\leq N \tag{1}\]  
where \(X_{k}^{(m)}\) represents the transmitted complex symbol on sub-carrier \(k\) and \(j=\sqrt{-1}\). It is assumed that the CP is longer than the maximum delay spread of the channel, \(L\), in order to avoid ISI. Hence, the samples of the received OFDM symbol are  
\[y_{n}^{(m)}=\sum_{l=0}^{L-1}h_{n,l}^{(m)}x_{n-l}^{(m)}+w_{n}^{(m)},\quad-G\leq n \leq N \tag{2}\]  
being \(h_{n,l}^{(m)}\) the time-varying CIR sample at the time instant \(n\) of the \(m\)-th OFDM symbol in the sample lag \(l\), whereas \(w_{n}^{(m)}\) is a sample of additive white Gaussian noise. We can write the above output/input relation in matrix form. Let us define the vector of transmit and receive time-domain samples as \(\mathbf{x}^{(m)}=[x_{0}^{(m)},..,x_{N-1}^{(m)}]^{T}\) and \(\mathbf{y}^{(m)}=[y_{0}^{(m)},..,y_{N-1}^{(m)}]^{T}\). Then, the received vector can written as  
\[\mathbf{y}^{(m)}=\mathfrak{h}^{(m)}\cdot\mathbf{x}^{(m)}+\mathbf{w}^{(m)}= \mathfrak{h}^{(m)}\mathbf{F}^{H}\mathbf{X}^{(m)}+\mathbf{F}^{H}\mathbf{W}^{(m)} \tag{3}\]  
where \(\mathfrak{h}^{(m)}\) denotes the (time-variant) circular convolution matrix whose \((n,l)\)-th element is expressed as  
\[\mathfrak{h}^{(m)}(n,l)=h^{(m)}_{n,(n-l)_{N}} \tag{4}\]  
where \(\langle\bullet\rangle_{N}\) stands for the modulo operation. The matrix \(\mathbf{F}\in\mathbb{C}^{N\times N}\) stands for the discrete Fourier transform (DFT) matrix, whose \((k,n)\)-th element can be expressed as \(\mathbf{F}(k,n)=\frac{1}{\sqrt{N}}\mathrm{e}^{j\frac{2\pi}{N}kn}\). Hence, the OFDM received symbol in the frequency domain is expressed as  
\[\mathbf{Y}^{(m)}=\mathbf{F}\mathbf{y}^{(m)}=\mathbf{F}\mathfrak{h}^{(m)} \mathbf{F}^{H}\mathbf{X}^{(m)}+\mathbf{W}^{(m)}=\mathbf{H}^{(m)}_{\mathrm{df} }(d,k)\mathbf{X}^{(m)}+\mathbf{W}^{(m)} \tag{5}\]  
where \(\mathbf{H}^{(m)}_{\mathrm{df}}(d,k)=\mathbf{F}\mathfrak{h}^{(m)}\mathbf{F}^{H}\) stands for the sub-carrier coupling matrix, which models the output/input relation between transmit and receive symbols in the frequency domain. The \((d,k)\)-th element of the coupling matrix can be computed as  
\[\mathbf{H}^{(m)}_{\mathrm{df}}(d,k)=\frac{1}{N}\sum_{n=0}^{N-1}\sum_{l=0}^{N- 1}\mathfrak{h}^{(m)}(n,l)\mathrm{e}^{-j\frac{2\pi}{N}(dn-lk)} \tag{6}\]  
It can be proven that this element can be also expressed as [16]\(\mathbf{H}^{(m)}_{\mathrm{df}}(d,k)=h^{(m)}_{\mathrm{df}}(d-k,k)\) being \(h^{(m)}_{\mathrm{df}}(d,k)=\frac{1}{N}\sum_{n=0}^{N-1}\sum_{l=0}^{N-1}h^{(m)} _{n,l}\mathrm{e}^{-j\frac{2\pi}{N}(dn+lk)}\). The proof follows from the fact that \(h^{(m)}_{n,l}=0\) for \(l\geq L\) and the relation between time-varying CIR and circular convolution matrix given by (4). For low mobility conditions, the channel can be considered as time-invariant in a given OFDM symbol, i.e., \(h^{(m)}_{n,l}=h^{(m)}_{l}\). Therefore, the coupling matrix is a diagonal matrix whose elements are expressed as  
\[H^{(m)}_{k}=\frac{1}{\sqrt{N}}\sum_{l=0}^{N-1}h^{(m)}_{l}e^{-j\frac{2\pi}{N} lk}, \tag{7}\]  
and hence, there is no ICI. In this latter case, the output/input relationship of the channel at a given sub-carrier \(k\) and symbol \(m\) can be expressed as:  
\[\gamma^{(m)}_{k}=H^{(m)}_{k}\cdot X^{(m)}_{k}+W^{(m)}_{k}. \tag{8}\]  
As a consequence, channel estimation based on pilots can be done in the frequency domain with a small computational complexity.