# Frequency-Domain Characterization of Finite Sample Linear Systems with Uniform Window Inputs
## 4 General Case for L and N  
**Lemma 2**. : _Given integers \(L\), \(N\geq 2\), \(i_{L}\), and \(\Delta_{i}=N_{\infty L}+i_{L}L\),_  
\[P_{2}[k]=Z[k](1\ -\ W^{-\Delta_{i}k})(1\ -\ W^{Lk}) \tag{22}\]  
_and_  
\[p_{2}[n]=\delta[n,N]-\delta[n-1,N]-\delta[n+\Delta_{i},N]+\delta[n+\Delta_{i}- 1,N] \tag{23}\]  
Figure 3: Example of \(h[n]\), \(h_{1}[n],h_{2}[n]\), and \(h_{3}[n]\) for a complex output with \(L=10\), \(M=4\), and \(N=23\). The upper and lower plots are for the real and imaginary parts, respectively. are Fourier transform pairs with \(k,n\in\mathbb{N}\)._  
Proof. : Let \(G_{LN}\) be the greatest common factor of \(L\) and \(N\), \(k_{1}=\frac{N}{G_{LN}}\), \(\mathbf{k_{0}}\)= {\(k_{1}\),... (\(G_{LN}-1\))\(k_{1}\)}. The IDFT of \(P_{2}[k]\) is  
\[\mathcal{F}^{-1}[P_{2}[k]]=\frac{1}{N}\sum_{k=0,\
eq\mathbf{k_{0}}}^{N-1}\frac {1-W^{k}}{1-W^{lk}}W^{-nk}\Big{(}1-W^{-\Delta_{i}k}\Big{)}\Big{(}1-W^{Lk}\Big{)}\]  
\[=\frac{1}{N}\sum_{k=0}^{N-1}\Big{(}1-W^{k}\Big{)}W^{-nk}\Big{(}1-W^{-\Delta_{i }k}\Big{)}-\frac{1}{N}\sum_{k\in\mathbf{k_{0}}}\Big{(}1-W^{k}\Big{)}W^{-nk} \Big{(}1-W^{-\Delta_{i}k}\Big{)}. \tag{24}\]  
For any integer m,  
\[\sum_{k\in\mathbf  
_(ii) For \(0\leq n\leq L_{\beta}-1\) and \(0\leq m\leq M-1\), we have_  
\[\delta\big{[}n-m+\Delta_{\beta},\ N\big{]}=\delta\big{[}n-m+\Delta_{\beta}-1,\ N \big{]}=0; \tag{27}\]  
\[\delta\big{[}n-m-L_{\beta},\ N\big{]}=\delta\big{[}n-m-L_{\beta}-1,\ N\big{]}=0. \tag{28}\]  
Proof. : To prove the first property, we rewrite \(\Delta_{\beta}\) as  
\[\Delta_{\beta}=\left\{ \begin{array}{cc}N_{\forall L}+(M_{\#L}+1)L,&M_{ \forall L}\geq N_{\forall L}\\ N_{\forall L}+M_{\#L}L,&M_{\forall L}<N_{\forall L}\end{array} \right.=\left\{ \begin{array}{cc}M+L+N_{\forall L}-M_{\forall L},&M_{\forall L}\geq N_{ \forall L}\\ M+N_{\forall L}-M_{\forall L},&M_{\forall L}<N_{\forall L}\end{array} \right.\]  
From the first and second expressions in the above, we see that \(\Delta_{\beta}>M\) and \(\Delta_{\beta}<N\), respectively. \(L_{\beta}=\left\{ \begin{array}{cc}(N_{\#L}-M_{\#L})L,&M_{\forall L}\geq N_ {\forall L}\\ (N_{\#L}-M_{\#L})L,&M_{\forall L}<N_{\forall L}\end{array} \right.\) is a multiple of L. Since \(0<\Delta_{\beta}<N\), we have \(L\leq L_{\beta}\leq N-L\). The ranges of \(m\) and \(n\) in the second property lead to \(\min(n_{m}-1)=-M\) and \(\max\,(n_{m})=L_{\beta}-1\), where \(n_{m}=n-m\). The property \(\Delta_{\beta}>M\) in the above leads to  
\[\min\big{(}n_{m}+\Delta_{\beta},n_{m}+\Delta_{\beta}-1\big{)}=\min\big{(}n_{m} +\Delta_{\beta}-1\big{)}=\Delta_{\beta}-M>0 \tag{29}\]  
Further,  
\[\max\big{(}n_{m}+\Delta_{\beta},n_{m}+\Delta_{\beta+1}-1\big{)}=\max\big{(}n_{ m}+\Delta_{\beta}\big{)}=L_{\beta}-1+\Delta_{\beta}=\ N-1<N. \tag{30}\]  
Thus,  
\[\delta\big{[}n_{m}+\Delta_{\beta},\ N\big{]}=\delta\big{[}n_{m}+\Delta_{\beta }-1,\ N\big{]}=0. \tag{31}\]  
Other expressions of this property include  
\[\delta\big{[}n_{m}-L_{\beta},\ N\big{]}=\delta\big{[}n_{m}-L_{\beta}-1,\ N \big{]}=0,\]  
\[mod\big{(}n_{m}+\Delta_{\beta},\ N\big{)}\
eq 0,\ mod\big{(}n_{m}+\Delta_{\beta}-1,\ N \big{)}\
eq 0.\]  
If \(i\leq n\leq i+N-\Delta_{\beta}-1\),  
\[\delta\big{[}n_{m}-i+\Delta_{\beta},\ N\big{]}=\delta\big{[}n_{m}-i+\Delta_{ \beta}-1,\ N\big{]}=0.\]  
Theorem 2.: _Given integer \(L\geq 1\), \(M\geq 1\), and \(N\geq L+M+1\), let_  
\[g_{1}[n]\equiv h_{1}[n]-h_{1c}\big{[}n+\Delta_{\beta}\big{]}, \tag{32}\]  
\[g_{2}[n]\equiv g_{1}[n]-g_{1c}[n-L], \tag{33}\]  
\[g_{3}[n]\equiv g_{2}[n]+g_{3}[n-L], \tag{34}\]  
\[g_{4}[n]\equiv g_{3}[n]+g_{4}\big{[}n-L_{\beta}\big{]}, \tag{35}\]  
_where \(n\in\mathbb{N}\), \(h_{1c}\) and \(g_{1c}\) are circular expansions of \(h_{1}\) and \(g_{1}\), respectively, and all other functions are zero at negative indices.