## 2 Preliminaries  
### Channel Estimation in UAC Systems  
In UACs, the discrete input-output relationship concerning the transmission of the signal through the channel is often converted to the following linear expression:  
\[\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{v} \tag{1}\]  
where \(\mathbf{y}\in\mathbb{C}^{M}\) denotes the received discrete signal or symbols, \(\mathbf{x}\in\mathbb{C}^{N}\) is the vector which encapsulates the unknown channel parameters to estimate, \(\mathbf{\Phi}\in\mathbb{C}^{M\times N}\) is the matrix formulated by training signal or symbols, and \(\mathbf{v}\) corresponds to a noise-related vector. All variables in (1) are presumed to be complex-valued since channel estimations is usually performed on the baseband. Specific definitions of \(\mathbf{\Phi}\) and \(\mathbf{x}\) depend on the type of the transmitted signal and the channel model adopted. For example, for CIR estimation in single-carrier transmission systems, the discrete input-output relationship of the UWA multipath channel at the \(i\)-th time instant can be expressed as:  
\[y[i]=\sum_{k=1}^{N_{\tau}}h^{*}[k]s[i-k+1]+v[i] \tag{2}\]  
where \(s[i]\) and \(y[i]\) are transmitted and received symbols, respectively, \(h[k]\) is the amplitude of the \(k\)-th path, and \(N_{\tau}\) denotes the number of delay taps or the length of the CIR vector. Let \(N=N_{\tau}\) and \(\mathbf{s}[i]=[s[i],s[i-1],\cdots,s[i-N+1]]^{T}\). The matrix form in (1) can be obtained by stacking \(M\) consecutive received symbols together, leading to  
\[\mathbf{y} \triangleq[y[i],y[i-1],\cdots,y[i-M+1]]^{T}\in\mathbb{C}^{M} \tag{3a}\] \[\mathbf{x} \triangleq[h[1],h[2],\cdots,h[N]]^{T}\in\mathbb{C}^{N}\] (3b) \[\mathbf{\Phi} \triangleq[\mathbf{s}[i],\mathbf{s}[i-1],\cdots,\mathbf{s}[i-M+1 ]]^{T}\in\mathbb{C}^{M\times N}. \tag{3c}\]  
When the assumption of time-invariant channel does not hold, the input-output relationship in (2) is changed to  
\[y[i]=\sum_{k=1}^{N_{\tau}}h^{*}[i,k]s[i-k+1]+v[i], \tag{4}\]  
where \(h[i,k]\triangleq h(i\Delta t,(k-1)\Delta\tau)\) is the sampled time-varying CIR, and \(\Delta t\) and \(\Delta\tau\) denote the sampling intervals in time and delay domains, respectively. Instead of using CIR, the time-varying channel can also be represented by its DD spread function \(u[l,k]\triangleq u[v_{l},(k-1)\Delta\tau]\)with \(v_{l}=v_{\min}+(l-1)\Delta v\) for \(l=1,\cdots,N_{v}\), and \(\Delta v\) is the sampling interval in the Doppler domain [35]. The relationship between the DD spread function and the time-varying CIR is given by the Fourier transform:  
\[h[i,k]=\sum_{l=1}^{N_{v}}u[l,k]e^{j2\pi v_{l}i\Delta t}. \tag{5}\]  
Substituting (5) into (4), we can obtain  
\[y[i]=\sum_{k=1}^{N_{v}}\sum_{l=1}^{N_{v}}u^{*}[l,k]e^{-j2\pi v_{l}i\Delta t}s[i -k+1]+v[i] \tag{6}\]  
which transforms the channel estimation problem to 2-D DD plane. Define the vectorized 2-D matrix of DD spread function \(\mathbf{u}\triangleq[u[1,1],\cdots,u[N_{v},1],\cdots,u[1,N_{\tau}],\cdots,u[N_{ v},N_{\tau}]]^{T}\) and the Fourier transform coefficient vector \(\beta[i]\triangleq\left[e^{-2\pi v_{1}i\Delta t},\cdots,e^{-2\pi v_{N_{v}}i \Delta t}\right]^{T}\). We can rewrite (6) as  
\[y[i]=(\mathbf{s}[i]\otimes\beta[i])^{T}\mathbf{u}^{*}+v[i], \tag{7}\]  
where \(\otimes\) represents Kronecker product. Similarly, the time-varying channel estimation via the DD spread function, consistent with the form in (1), can be obtained by stacking \(M\) consecutive received symbols with  
\[\mathbf{\Phi}\triangleq[(\mathbf{s}[i]\otimes\beta[i]),\cdots,(\mathbf{s}[i- M+1]\otimes\beta[i-M+1])]^{T}, \tag{8}\]  
and \(\mathbf{x}\triangleq\mathbf{u}\in\mathbb{C}^{N}\). Here, \(N=N_{\tau}\times N_{\
u}\). Channel estimation problems in multicarrier modulation schemes can also be formulated as the linear expression shown in (1). Different from the single-carrier systems, channel estimation in OFDM systems often operates in the frequency domain, turning time-domain convolutions into straightforward frequency-domain multiplications.