## 2 Spectrum Reconstruction Model for Non-Uniform Sampling Signals
### Principle of Spectrum Reconstruction  
For an infinite-length periodic signal \(s_{c}(t)\) with a frequency band of \([-M\Delta f_{c},M\Delta f_{c}]\) and a signal period of \(T_{c}=1/\Delta f_{c}\), the value of any point \(t\) in \(s_{c}(t)\) can be accurately expressed using the following discrete inverse Fourier transform formula [21]:  
\[s_{c}(t)=\sum_{k=-M}^{M}\hat{s}_{c}(k)e^{i2\pi k\Delta f_{c}t} \tag{3}\]  
where \(\hat{s}_{c}(k)\) is the Fourier coefficient of \(s_{c}(t)\) at frequency \(f=k\Delta f_{c}\).  
However, for finite-length signals typically encountered in the real-world, they do not satisfy (3). Nevertheless, it is worth noting that (3) approximately holds for an approximate band-limited signal, with the error value depending on the spectral components outside the signal's bandwidth.  
Let \(\tilde{s}_{l}(t)\) be the periodic extension of \(s_{l}(t)\), satisfying \(\tilde{s}_{l}(t+nT_{l})=s_{l}(t)\), where \(t\in[0,T_{l})\), and \(n\) is an integer. Then, \(\tilde{s}_{l}(t)\) can be expanded using the Fourier series [22], given by  
\[\tilde{s}_{l}(t)=\sum_{n=-\infty}^{\infty}C_{n}e^{j2\pi n\frac{t}{l_{l}}} \tag{4}\]  
where \(C_{n}\) is the Fourier coefficient of \(\tilde{s}_{l}(t)\), and the expression is  
\[C_{n}=\frac{1}{T_{l}}\int_{0}^{T_{l}}\tilde{s}_{l}(t)e^{-j2\pi n\frac{t}{l_{l}} }dt=\frac{1}{T_{l}}\int_{0}^{T_{l}}s_{l}(t)e^{-j2\pi n\Delta f_{l}t}dt=\frac{1 }{T_{l}}S_{l}(n\Delta f_{l}). \tag{5}\]  
This indicates that \(C_{n}\) is equal to the value of \(S_{l}(t)\) at \(f=n\Delta f_{l}\) multiplied by a constant term. When \(t\in[0,T_{l})\), it holds that  
\[\begin{split}& s_{l}(t)=\tilde{s}_{l}(t)=\sum_{n=-\infty}^{ \infty}C_{n}e^{j2\pi n\frac{t}{l_{l}}}=\sum_{n=-\infty}^{\infty}C_{n}e^{j2\pi n \Delta f_{l}t}\\ &=\sum_{n=-M}^{M}C_{n}e^{j2\pi n\Delta f_{l}t}+\sum_{n=-\infty}^ {-M+1}C_{n}e^{j2\pi n\Delta f_{l}t}+\sum_{n=M+1}^{\infty}C_{n}e^{l2\pi n \Delta f_{l}t}\end{split} \tag{6}\]  
Let  
\[\delta_{l}^{M}(t)=\sum_{n=-\infty}^{-M+1}C_{n}e^{j2\pi n\Delta f_{l}t}+\sum_{ n=M+1}^{\infty}C_{n}e^{j2\pi n\Delta f_{l}t} \tag{7}\]  
then (6) can be rewritten as  
\[s_{l}(t)=\sum_{n=-M}^{M}C_{n}e^{j2\pi n\Delta f_{l}t}+\delta_{l}^{M}(t),\quad t \in[0,T_{l}). \tag{8}\]  
As the spectrum of \(s_{l}(t)\) is primarily concentrated within the range \([-M\Delta f_{l},M\Delta f_{l}]\), it suggests that when \(|n|>M\), the magnitude of \(C_{n}\) is close to zero. Therefore, \(\delta_{l}^{M}(t)\) is significantly smaller compared to the first term on the right side of (8). For \(N\) non-uniform sampling positions \(t_{i}\), let vector \(\delta^{M}(i)=\delta_{l}^{M}(t_{i})\), then, the norm of \(\delta^{M}\) is much smaller than the norm of \(s\), i.e., \(\|\delta^{M}\|\ll\|s\|\). We define \(\delta^{M}\) as the higher-order error term of the non-uniform sampling signal. From a time-domain perspective, \(\delta^{M}\) can be expressed as the error generated by approximating a finite-length signal using a certain order of trigonometric polynomials. As the order increases, the approximation error decreases.  
Based on (2), (5), and (8), the error term can be obtained as \(\tilde{n}=\delta^{M}\), and the expression for \(A\) is  
\[A=\frac{1}{T_{l}}\begin{bmatrix}\exp(-j2\pi M\Delta f_{l}t_{1})&\exp(-j2\pi(M -1)\Delta f_{l}t_{1})&\cdots&\exp(j2\pi M\Delta f_{l}t_{1})\\ \exp(-j2\pi M\Delta f_{l}t_{2})&\exp(-j2\pi(M-1)\Delta f_{l}t_{2})&\cdots&\exp (j2\pi M\Delta f_{l}t_{2})\\ \vdots&\vdots&\cdots&\vdots\\ \exp(-j2\pi M\Delta f_{l}t_{N})&\exp(-j2\pi(M-1)\Delta f_{l}t_{N})&\cdots&\exp (j2\pi M\Delta f_{l}t_{N})\end{bmatrix} \tag{9}\]