# On the Impulse Response of Singular Discrete LTI Systems and Three Fourier Transform Pairs
## 1 Introduction
Despite extensive searches in the Web of Science database and the open internet, the author cannot find any references on the direct conversion from SFTF to IRF. There are possibly two reasons for this: (1) we cannot apply DFT or IDFT when there are singularities; (2) alternative ways, such as polynomial division, can be used to obtain \(h[n]\). General discussions on the zeros of the discrete-time system can be found in [5,6]. The stability of fractional oscillators is analyzed by [PERSON] et al. [7]. [PERSON] and [PERSON] [8] showed that linear dynamic controllers with integer coefficients are usually unstable. [PERSON] et al. [9] discussed uniformly marginally stable zeros of linear sampled data systems and its application to stable inversion-based control. In the following, we demonstrate that we can recover \(h[n]\) from \(H[k]\) when \(X[k]\), the DFT of the input function, is a _sinc_ function. To describe the problem more precisely, let us consider the details of \(H[k]\). To perform the discrete Fourier transform (DFT), we zero-pad \(x[n]\) and \(y[n]\) to \(N\) elements. Let \(X[k]\) and \(Y[k]\) be the DFT of the zero-padded sequence of \(x[n]\) and \(y[n]\), respectively. The \(N\)-point DFT of \(x[n]\) is defined as  
\[X[k]\equiv\sum_{n=0}^{N-1}x[n]e^{-j2\pi mk/N},\ k=0\ \ldots\ N-1. \tag{3}\]  
In this article, we restrict our input to \(x[n]=\left\{ \begin{array}{l}1,\ 0\leq n\leq L-1\\ 0,\ L\leq n\leq N-1\end{array} \right.\). For \(k\in\mathbb{N}=\{0,\ 1,\ \ldots\ N-1\}\), we have  
\[X[k]=\sum_{n=0}^{L-1}e^{-\frac{j2\pi mk}{N}}=\frac{1-W^{kL}}{1-W^{k}}=W^{(L-1) k/2}\frac{\sin(\pi_{N}Lk)}{\sin(\pi_{N}k)}. \tag{4}\]  
where \(\pi_{N}=\frac{\pi}{N}\) and \(W=e^{-j2\pi_{N}}\). \(X[k]\) has zeros at \(k=l\frac{N}{L}\), where \(l\in\{1,\ldots,L-1\}\). We denote the set of the zeros of \(X[k]\) as \(\mathbf{k_{o}}=\left\{\frac{N}{L},\ldots\frac{(L-1)N}{L}\right\}\). Denoting \(Y[k]\) as the \(N\)-point DFT of \(y[n]\), i.e., \(Y[k]=\sum_{n=0}^{L_{y}-1}y[n]W^{nk}\), \(k\in\mathbb{N}\), the frequency transfer function is thus  
\[H[k]=\frac{Y[k]}{X[k]}=Y[k]\frac{1-W^{k}}{1-W^{kL}}. \tag{5}\]The poles, which characterize the stability of a system, are at \(\mathbf{k}_{o}\), which are all on the unit cycle in the corresponding \(Z\)-transform and \(H[0]=Y[0]/L\). Such a system is considered to be marginally stable [10]. IDFT cannot be directly performed on \(H[k]\) to obtain \(h[n]\) when there are singularities. If we limit our objective to obtain \(h[n]\) from \(H[k]\), we may attempt to avoid poles in \(H[k]\). As we are concerned with discrete operations, we can sample \(H[k]\) in such a way that there are no poles. In fact, when \(N\) is not divisible by \(L\), \(H[k]\) does not necessarily have a singularity. In this first approach, the inverse transform, however, does not yield \(h[n]\), as shown in Figure 1. In this example, \(N=17,\ L=3,\ y[0:4]=0\), and \(y[5:N-1]=0\). Alternatively, when \(N\) is divisible by \(L\), we can assign a large value at \(\mathbf{k}=\mathbf{k}_{o}\) to attempt for an approximation of \(h[n]\). We show an example of this second approach in Figure 2, where \(N=16,\ L=2,\ y[0:6]=1,\ y[7:N-1]=0\), and \(H[\mathbf{k}_{o}]=10^{10}\). The approach does not work in two aspects. The \(h[n]\) computed from the IDFT method yields values proportional \(H[\mathbf{k}_{o}]\) we artificially set and it is not even proportionally correct at the beginning indices. This invalidates the notion that a large enough real value is a reasonable numerical approximation when there is a zero in the denominator. In the following, we show that when the input function \(x[n]\) is a rectangular function and the output \(y[n]\) is a finite length sequence, \(h[n]\) can indeed be recovered from \(X[k]\) and \(Y[k]\) when \(N\) is divisible by \(L\). Before we delve into the proofs, we outline our method of obtaining \(h[n]\) from \(H[k]\) in Section 2. The proof of our method is facilitated by three interesting discrete Fourier transform pairs, which are discussed in Section 3.