# 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 \(\Big{[}1-e^{j\pi k}cos(\pi_{N}Lk)\Big{]}Z[k]\)  
We define  
\[\begin{array}{c}q_{2}[m]\equiv\sum\limits_{k=0}^{N-1}W^{-mk}\Big{[}1-e^{j\pi k }\text{cos}(\pi_{N}Lk)\Big{]}Z[k]\\ =\sum_{k=1}^{N-1}W^{-mk}\Big{[}1-e^{j\pi k}\text{cos}(\pi_{N}Lk)\Big{]}Z[k], \end{array} \tag{23}\]  
i.e., \(q_{2}[m]\) and \(\Big{[}1-e^{j\pi k}cos(\pi_{N}Lk)\Big{]}Z[k]\) form a DFT pair.  
**Lemma 3**.: _If \(N/L\) is an odd integer,_  
\[\begin{array}{c}q_{2}[m]=\frac{N}{2}(\delta[m,L]-\delta[m-1,L])\delta^{ \prime}[m_{\%N}<N_{h-}]-\frac{N}{2}(\delta[m-N_{h+},L]-\\ \delta[m-N_{h+}-1,L])\delta^{\prime}[m_{\%N}\geq N_{h+}],\end{array}\] (23a) _where \(N_{h+}=\frac{N+L}{2}\) and \(N_{h-}=\frac{N-L}{2}\)._  
**Proof.** As in the proof for Lemma 2, we take the \(N\)-point DFT of \(q_{2}[m]\) as expressed by Equation (23a) for \(m\in\mathbb{N}\)  
\[Q_{2}[k]=\frac{N}{2}{\sum_{m=0L}^{\frac{N}{2}-L}}\left(W^{mk}+W^{(N_{h+}+1+m)k }-W^{(m+1)k}-W^{(N_{h+}+m)k}\right). \tag{24}\]  
The sums of the geometric sequences can be reduced to  
\[Q_{2}[k]=\begin{cases}NZ[k]\Big{(}1-e^{j\pi k}\text{cos}(\pi_{N}Lk)\Big{)},& k\
eq\mathbf{k_{o}}\\ 0,&k=\mathbf{k_{o}}\end{cases}.\] (24a) As \[Z[\mathbf{k_{o}}]=0\], the above equation can be simply written as \[Q_{2}[k]=NZ[k]\Big{(}1-e^{j\pi k}\text{cos}(\pi_{N}Lk)\Big{)}.\] (24b) The IDFT of \[Q_{2}[k]\] leads back to Equation ( 23a ) for \[m\in\mathbb{N}\], i.e., \[\sum_{k=0}^{N-1}W^{-mk}\Big{[}1-e^{j\pi k}\text{cos}(\pi_{N}Lk)\Big{]}Z[k]= \begin{cases}\frac{N}{2},&m=0:L:\frac{N-L}{2}-L;\ N_{h+}+1:L:N-L+1\\ -\frac{N}{2},&m=1:L:\frac{N-L}{2}-L+1;\ N_{h+}:L:N-L\.\\ 0,&\text{otherwise}\end{cases}\] (25) Note that \[q_{2}[N_{h-}]=q_{2}[N_{h-}+1]=0\]. Since \[q_{2}[m]=q_{2}[N+m]\], a more general expression for any \[m\] is Equation ( 23a ). Alternatively, we can express Equation ( 23a ) as \[q_{2}[m]=\frac{N}{2}[\delta(m-c^{\prime}{}_{m}-N_{h+}c^{\prime}{}_{m},L)- \delta(m+c^{\prime}{}_{m}-1-N_{h+}c^{\prime}{}_{m},L)],\] (23b) where \[c^{\prime}{}_{m}=\delta^{\prime}[m_{\%N}\geq N_{h+}]+\frac{1}{2}\delta^{\prime }[(m_{\%N}=N_{h-})\cup(m_{\%N}=N_{h-}+1)]\]. In this expression, we have assigned \[c^{\prime}{}_{m}=1/2\] at \[m_{\%N}=N_{h-}\] or \[m_{\%N}=N_{h-}+1\] to make \[q_{2}[m]\] zero at these two points. Analogous to Equation ( 20a ), Equation ( 23a ) can also be written as \[q_{2}\bigg{[}m+\frac{N+L}{2}\bigg{]}=\frac{N}{2}(-\delta[m,L]+\delta[m-1,L])b ^{\prime}_{N}[m],\] (23c) where \[b^{\prime}_{N}[m]=\delta^{\prime}[m_{\%N}\leq N_{h-}-1]-\delta^{\prime}[N-2L+ 1\geq m_{\%N}\geq N_{h-}]\]. \[\Box\] Other than the fact that \[q_{2}\] is real and periodic, additional properties include the following: (i) \[\sum_{m=0}^{N-1}q_{2}[m]=\sum_{m=iL}^{i+(N-1)}q_{2}[m]= 0,\forall i\in\mathbb{I};\] (ii) \[q_{2}[m]=q_{2}[N-L+1-m];\] (iii) \[q_{2}[iL]+q_{2}[iL+1]=0\]Note that \(q_{2}[m]=2q_{1}[m]\) for \(0\leq m\leq\ L(\min(\frac{N_{1}}{2},\frac{N_{2}-L}{2})-1)\), where \(N_{1}\) and \(N_{2}\) are used to generate \(q_{1}[m]\) and \(q_{2}[m]\), respectively, and \(min\) is the minimum operator. Like \(q_{1}[m]\), \(q_{2}[m]\) has only three values. The lengthiness in Expression (23a) is to account for the positions of the three values. The number of positive values in \(q_{1}\) per period is \(N/L\), while there are \(N/L-1\) positive values in \(q_{2}\) per period. Similar to Lemmas 1 and 2, Lemma 3 can also be used to find sums containing \(\frac{\sin(\pi k/N)}{\sin(\pi k/N)}\). Two example plots of \(q_{2}[m]\) are shown in Figure 6 for \(N=18\), \(L=2\) and \(N=25\), \(L=5\).