# On the Impulse Response of Singular Discrete LTI Systems and Three Fourier Transform Pairs
## 3 Three Fourier Transform Pairs and Their Properties  
### Inverse Fourier Transform of \(Z_{0}[k]\) and Properties of \(h_{1}[n]\)  
Defining \(Z_{0}[k]\equiv Z[k](1-\delta[k])\), we can rewrite \(h_{1}[n]\), defined by (7), as  
\[h_{1}[n]=\frac{1}{NL}{\sum_{m=0}^{L_{y}-1}y[m]}+\frac{1}{N}{\sum_{m=0}^{L_{y}- 1}y[m]}y_{0}[n-m],\] (7a) where \[q_{0}[m]\equiv{\sum_{k=0}^{N-1}W^{-mk}Z_{0}[k]}={\sum_{k=1}^{N-1}W^{-mk}Z[k]}, \tag{11}\]i.e., \(q_{0}[m]\) and \(Z_{0}[k]=Z[k](1-\delta[k])\) form a DFT pair. In (7a), the index of \(q_{0}\) can be negative. Noting that \(W^{-mN}=1\) and \(Z[k]=Z^{*}[N-k]\), where * is for complex conjugate, \(q_{0}[m]\) can be alternatively expressed as follows:  
\[ \begin{array}{rl}q_{0}[m]&=\frac{1}{2}\sum_{k=1}^{N-1}\,(W^{-mk}Z[k]+W^{-m(N- k)}Z[N-k])\\ &=\sum_{k=1}^{N-1,\ \
eq k_{0}}\cos{(\pi_{N}(2m+L-1)k)}\frac{\sin{(\pi_{N}k)} }{\sin{(\pi_{N}L)}}.\end{array} \] (11a) As \(q_{0}[m]\) and \(y[n]\) are real, \(h_{1}[n]\) is thus real as well. Since \(q_{0}[m]\) is an IDFT, all the DFT/IDFT properties apply, including the periodicity property, \(q_{0}[m]=q_{0}[N+m]\). **Lemma 1**. : _If \(N\) is divisible by \(L\),_  
\[q_{0}[m]=(N_{h-}-m_{\%N})\delta[m,\ L]-(N_{h-}-m_{\%N}+1)\delta[m-1,L], \tag{11b}\]  
_where \(N_{h-}=\frac{N-L}{2}\), \(m_{\%N}=mod(m,\ N)\) is \(m\) modulo \(N\), \(\delta[m,L]\equiv\left\{ \begin{array}{ll}1,\ if\ f\ \frac{m}{L}\in\mathbb{I}\\ 0,\ otherwise\end{array} \right.\) is the comb function with a period of \(L\), and \(\mathbb{I}\) is the set of all integers._  
Proof. : Equation (11) can be written as  
\[ \begin{array}{rl}q_{0}[m]=\sum_{i=1:N/L}^{N-1-1}\ W^{-mk}\frac{1-W^{i}}{1-W^ {IL}}+\sum_{i=2:N/L}^{N-1}\ W^{-mk}\frac{1-W^{i}}{1-W^{IL}}+\ldots+\\ \sum_{i=N/L-1:N/L}^{N-1}\ W^{-mk}\frac{1-W^{i}}{1-W^{IL}},\end{array}  \tag{11c}\]  
where \(\sum_{i=1:N/L}^{N-1}\) indicates that the summation starts at \(i\) with an incremental of \(N/L\) and ends at a value up to but not larger than \(N-1\). Since \(W^{IL}=W^{(N/L+i)L}\) for \(i\in\sphericalangle\), the denominator in each summation does not vary with the summation index. The above equation can be simplified to  
\[q_{0}[m]=L\sum_{k=1}^{N/L-1}\frac{W^{-mk}\delta[m,\ L]-W^{-(m-1)k}\delta[m-1, L]}{1-W^{L}}. \tag{11d}\]  
If \(m\) is divisible by \(L\) and for \(m\in\mathbb{N}\), we have  
\[q_{0}[(i+1)L]-q_{0}[iL]=-L,\ \ \frac{N-1}{L}-1\ \geq\ i\geq\ 0. \tag{12}\]  
If \(m-1\) is divisible by \(L\),  
\[q_{0}[iL+L+1]-q_{0}[iL+1]=L,\ \ \frac{N-2}{L}-1\ \geq\ i\geq\ 0. \tag{13}\]  
Using the fact that \(W^{N-L}=W^{-L}\), \(q_{0}[0]\) through \(q_{0}[L-1]\) are  
\[q_{0}[0]=\frac{L}{2}\sum_{k=1}^{N-1}\,(\frac{1}{1-W^{L}}+\frac{1}{1-W^{L(\frac {N}{L}-k)}})=N_{h-}, \tag{14}\]  
\[q_{0}[1]=-q_{0}[0]=-N_{h-}, \tag{15}\]  
\[q_{0}[l]=0,\ \ \ 1<l\leq L-1, \tag{16}\]  
As \(q_{0}[m]=q_{0}[N+m]\), the recursive Equations (12) and (13) lead to  
\[q_{0}[m]=\left\{ \begin{array}{ll}N_{h-}-m_{\%N},&m/L\in\mathbb{I}\\ -N_{h-}+m_{\%N}-1,&(m-1)/L\in\mathbb{I}\\ 0,&otherwise\end{array} \right. \tag{11e}\]  
The expression can be alternatively written as Equation (11b). \(\square\)  
Expression (11b) consists of two linear functions in one period for \(m=[0,\ N-1]\). Two example plots of \(q_{0}[m]\) expressed as the sum of (11) and the closed form of (11a) are shown in Figure 3 for \(N=20\), \(L=2\) and \(N=4\)\(5\), \(L=5\). Lemma (1) shows that the sum expressed by Equation (11a), which happens to be the Fourier transform of \(Z_{0}[k]\), has a simple expression. For instance, letting \(m=0\), we have \(\sum_{k=1}^{N-1}\
icefrac{{\
eq k_{o}}}{{\sin(\pi_{N}k)}}=\frac{N-L}{2}\). Similarly, letting \(m=1\) leads to \(\sum_{k=1}^{N-1,\ \
eq k_{o}}\cos{(\pi_{N}(L+1)k)}\frac{\sin(\pi_{N}k)}{\sin(\pi_{N}k)}=- \frac{N-L}{2}\). Other than being real and periodic, for \(i,k,m\in\mathbb{I}\), \(q_{0}\) has the following properties:  
1. \(\sum_{m=0}^{N-1}q_{0}[m]=\)\(\sum_{m=i:L}^{i+(N-1)}q_{0}[m]=0\);
2. \(q_{0}[iL]+q_{0}[N-L-iL]=0\);
3. \(q_{0}[iL]=-q_{0}[iL+1]=N_{h-}-(iL)m_{\%\text{N}}\);
4. \(q_{0}[iL+1]+q_{0}[N-L-iL+1]=0\);
5.