# Frequency-Domain Characterization of Finite Sample Linear Systems with Uniform Window Inputs
## 5 Discussion  
In the above, we have focused on marginally stable systems, where the system output is zero after \(M\) samples for a uniform testing function with a width of \(L\). When there are non-zero values beyond \(M\) samples in an IS-LTI system's output, the marginal stability of the FS-LTI system is a forced artifact. In this more general latter scenario, we can only match the first \(M\) samples of the \(h[n]\) of the FS-LTI system with that of the IS-LTI system. To be more specific, we denote the IRF of an FS-LTI(\(L,M,N\)) system as \(h[n]|_{L,M,N}\) and its output as \(y[n]|_{L,M,N}\). The IRF of the IS-LTI system is denoted as \(h_{\infty}[n]\). In the following, we show that the first \(M\) samples of \(h[n]|_{L,M,N}\) are the same as those of \(h_{\infty}[n]\). The output, \(y[n]\mid_{L,M,N}\), of an FS-LTI(\(L,\ M,\ N\)) system is the first \(M\) samples of \(h_{\infty}[n]*x_{L}[n]\). It can be written as  
\[y[n]_{L,M,N}=\left\{ \begin{array}{cc}\sum_{i=0}^{n}h_{\infty}[i],&0\leq n \leq L-1\\ \sum_{i=n-L+1}^{n}h_{\infty}[i],&L\leq\ n\leq M-1\\ 0,&M\leq n\leq N-1\end{array} \right. \tag{56}\]  
Using the more detailed notation here, applying the above \(y[n]|_{L,M,N}\) to Equation (1), we have  
\[h[n]\mid_{L,M,N}=\left\{ \begin{array}{cc}h_{\infty}[n],&0\leq n\leq M-1\\ -\sum_{i=M-L+1}^{M-1}h_{\infty}[i],&n=M\\ h[n-L]|_{L,M,N},&M+1\leq n\leq N-1\end{array} \right. \tag{57}\]  
As \(h[0:M-1]\left|{}_{L,M,N}\right.=h_{\infty}[0:M-1]\), an FS-LTI(\(L,\ M,\ N\)) system produces the first \(M\) samples of the corresponding IS-LTI authentically, irrespective of the width of the uniform testing function. Thus, although the discussion above is for marginally stable systems, they are applicable to any LTI systems up to the number of samples available. We further note that the periodicity of the FS-LTI(\(L,\ M,\ N\)) system starts at \(M-L+1\). The reason for the negative sign when \(n=M\) in Equation (57) is to satisfy the condition of \(y[M]=0\). The values in one period sum to zero, i.e., \(\sum_{i=0}^{L-1}h[n+i]\left|{}_{L,M,N}\right.=0,\ \ M-L+1<n+i<N-1\). A FIR system is a special case where the values in the oscillating part are all zeros. An essential characteristic of an LTI system is that the frequency transfer function can be used to perform the operations involving \(h_{\infty}[n]\). If the \(M\)-point DFT of \(h_{\infty}[0:M-1]\) is considered to be the FTF, the convolution sum of the output using the frequency domain operation is the result of the circular convolution in the finite-sample situation. The equivalent IRF is the periodic extension of \(h_{\infty}[0:M-1]\). This is often not desirable as the output in the first \(M\) samples is not the same as one would get in an IS-LTI system. To authentically produce the first \(M\) points using the frequency domain method, we can expand \(h_{\infty}[0,\ M-1]\) by padding it with \(M-1\) or more zeros, which is effectively \(h[0:N-1]\left|{}_{1,M,N}\right.\) with \(N\geq 2M-1\). The DFT of \(h[0:N-1]\left|{}_{1,M,2M-1}\right.\) is an appropriate FTF for an FS-LTI system constrained to \(M\) samples. With the FTF so defined, the output of an LTI system limited to \(M\) samples can be obtained from the frequency domain. Another essential characteristic of an LTI system is that we can determine the IRF using any reasonable testing function. This can be done in the time domain for an FS-LTI(\(L,M,N\)) system to obtain \(h_{\infty}[0:M-1]\) as seen in the above. In the frequency domain, however, the first \(M\) samples of the IDFT of \(H_{r}[k]=Y[k]/X[k]\), are generally not the same as \(h_{\infty}[0:M-1]\) no matter how many zeros are padded to \(x[n]\) and \(y[n]\) in obtaining \(X[k]\) and \(Y[k]\) for an IIR system. When \(x[n]\) is a uniform window function, Theorems 1 and 2 in this study and the results from [3] show that we can obtain \(h_{\infty}[0:M-1]\) through \(H_{0}[k]\) even when it has singularities, as in the cases of \(L\) and \(N\) not being coprime. Theorems 1 and 2 are also applicable to FIR systems.