## 2 SC-FDE Communication System Model  
Figure 1 illustrates the typical SC-FDE transmission structure and the receiver equalization process. An SC-FDE block consists of a cyclic prefix (CP), pilot, and data. Unlike OFDM, which uses an Inverse Fast Fourier Transform (IFFT) at the transmitter, SC-FDE follows a single-carrier transmission structure and does not require an IFFT at the transmitter. However, a CP is appended to enable frequency domain equalization at the receiver. The length of the CP is \(L_{c}\), which corresponds to the last \(L_{c}\) symbols of the data. The lengths of the pilot and data are \(L_{p}\) and \(L_{d}\), respectively. The total length of the pilot and data is \(N\) (\(N=L_{p}+L_{d}\)), which is also the size of the FFT at the receiver used for equalization. At the receiver, channel impulse response (CIR) estimation is first performed for equalization. The CIR is estimated using the pilot. Let the received signal after passing through the wireless channel be denoted as \(y(n),\ n=0,\cdots,N-1\), the channel impulse response as \(h(n)\), and the transmitted pilot signal as \(s(n)\). Then, the received signal can be expressed as  
\[y(n)=\boldsymbol{s^{T}}(n)\boldsymbol{h}+z(n),\ n=-L_{c},\cdots,N-1 \tag{1}\]  
where the vector \(\boldsymbol{h}\) represents the channel impulse response \(\left(\boldsymbol{h}=\left[h(0),h(1),\cdots,h(L_{h}-1)\right]^{T}\right)\), where \(L_{h}\) represents the length of the channel impulse response. \(z(n)\) denotes Gaussian noise. The pilot starting point is defined as \(n=0\), so the time for the CP becomes negative. The vector \(\boldsymbol{s}(n)\) denotes the transmitted signal and is defined as \(\boldsymbol{s}(n)=\left[s(n),s(n-1),\cdots,s(n-L_{h}-1)\right]^{T}\). For channel equalization, the channel impulse response (CIR) estimation is first performed. The received signal \(y(n)\) for \(n=0,\cdots,L_{p}-1\) and the transmitted pilot vector \(s(n)\) for \(n=0,\cdots,L_{p}-1\) during the pilot period are used for the estimation. The CIR estimation is carried out using the least squares (LS) method as  
\[\boldsymbol{\hat{h}}=\left(\boldsymbol{S}^{H}\boldsymbol{S}\right)^{-1} \boldsymbol{S}^{H}\boldsymbol{y} \tag{2}\]  
where \(\boldsymbol{S}=\left[s(0),s(1),\cdots,s(L_{p}-1)\right]^{T}\) and \(\boldsymbol{y}=\left[y(0),y(1),\cdots,y(L_{p}-1)\right]^{T}\). Afterward, the estimated channel \(\boldsymbol{\hat{h}}\) is used to obtain the estimate of the channel frequency response \(\hat{H}(k),\ k=0,\cdots,N-1\) through an \(N\)-point FFT:  
\[\hat{H}(k)=\sum_{n=0}^{L_{h}}\hat{h}(n)\text{exp}\left(-j\frac{2\pi kn}{N} \right),\ k=0,\cdots,N-1 \tag{3}\]  
Figure 1: SC-FDE structure and equalization. To perform equalization, the FFT of the received signal with the CP removed is required. \[Y(k)=\sum_{n=0}^{N-1}y(n)\text{exp}\left(-j\frac{2\pi kn}{N}\right),\ k=0,\cdots,N-1 \tag{4}\]  
Channel equalization is performed using Minimum Mean Square Error (MMSE) equalization to improve performance in regions with a low Signal-to-Noise Ratio (SNR) [27]. \[\hat{X}(k)=\frac{\hat{H}*(k)}{\left|\hat{H}(k)\right|^{2}+\sigma_{z}^{2}}Y(k), \ k=0,\cdots,N-1 \tag{5}\]  
where \(\sigma_{z}^{2}\) represents the noise variance and * denotes complex conjugate. It is assumed that the noise variance is either known or can be estimated in advance. The equalized \(\hat{X}(k)\) is transformed back into a time-domain signal using IFFT:  
\[\hat{x}(n)=\frac{1}{N}\sum_{k=0}^{N-1}\hat{X}(k)\text{exp}\left(j\frac{2\pi kn} {N}\right),\ n=0,\cdots,N-1 \tag{6}\]  
However, if narrowband interference using a single frequency is present, the received signal in (1) can be expressed as  
\[y(n)=\mathbf{s}^{T}(n)\mathbf{h}+\text{exp}\left(j2\pi f_{l}nT+\theta\right)+z(n) \tag{7}\]  
where \(f_{l}\) represents the frequency of the interference, and \(\alpha\) represents the magnitude of the interference. The power ratio between the signal and the interference is referred to as the SIR, \(\eta\).