# A Flexible and Simple Lossless DWT Filter Bank Using a MAXFLAT FIR Half-Band Filter
## 3 Design of Lossless s-Filter Banks
### Special Design Examples of Two-Taps Symmetric Filters  
The utilization of the proposed lossless s-filter bank structure shown in Figure 2 can produce another new filter satisfying PR conditions. On a condition \(H_{U}(z)=H_{L}(-z)\) that can permit \(G_{L}(z)=H_{L}(z)\) and \(G_{U}(z)=-H_{L}(-z)\) in Figure 1, imposing Equations (2) and (4) (for given even \(m\) and \(N=2K-1\)) yields a PR condition able to characterize \(Q_{0}(z)\):  
\[Q_{0}(z)-Q_{0}(-z)=z^{-(m-2K+1)} \tag{22}\]  
This implies that \(Q_{0}(z)\) and \(Q_{0}(-z)\) are odd symmetric with respect to \(\pi/2\). Substituting Equation (8) into (22), it follows that  
\[2\sum_{n=0}^{2K-1}h_{2n}z^{-(2K-1-2n)}=z^{-(m-2K+1)} \tag{23}\]  
From Equation (23), the equivalence of both sides can be found to be established by  
\[h_{2n}\left|{}_{n=0,1,2,\ldots,\ \ 2K-1}=\left\{\begin{array}{ll} \frac{1}{2},&2n=4K-2-m\\ 0,&2n\
eq 4K-2-m\end{array}\right. \tag{24}\]  
Figure 3: The frequency responses of MAXFLAT half-band FIR filters for \(K=1,\,2,\,3,\,4,\,5\) (the order of filter is \(4K-2\)).  
Substituting Equation (24) to (8) with \(h_{2K-1}=1/2\) and using \(m=4K-2-2n\)\((n=0,\ 1,\ \ldots\ 2K-1)\) lead to a two-taps filter form  
\[Q_{0}(z)=\frac{1}{2}\Big{\{}1+z^{-(2K-1-2n)}\Big{\}} \tag{25}\]  
where \(n=0,\ 1,\ \ldots\ 2K-1\). The filter of Equation (25) also satisfies the unity gain condition of Equation (7) (i.e., \(\big{|}T_{s}(e^{i\omega})\big{|}=1,-\pi\leq\omega\leq\pi\)) for a given even \(m\), and the realization of a gain error-free s-filter bank is possible. In particular, it can be seen that Equation (25) exhibits a linear phase comb filter having \(m-2K+1\) zeros equally spaced on the z-plane's unit circle and leads to a completely symmetric comb filter of \(Q_{0}(-z)=1/2\Big{\{}1-z^{-(2K-1-2n)}\Big{\}}\) by replacing \(z\) by \(-z\). Conclusively, applying Equation (25) into the s-filter bank can be seen to produce a new two-channel orthogonal symmetric filter bank in which analysis and synthesis banks have a symmetric two-tap filter structure and each also consists of the perfect symmetric filters (of \(Q_{0}(z)\) and \(Q_{0}(-z)\)) with respect to \(\omega=\pi/2\). As an example, imposing \(n=K-1\) on Equation (25) yields a simplest linear-phase half-band lowpass filter with two taps as  
\[Q_{0}(z)=\frac{1}{2}\Big{(}1+z^{-1}\Big{)} \tag{26}\]  
Then, Equation (7) becomes \(T_{s}(z)=\frac{1}{2}z^{-3}\), and both the lower and upper bands have a half bandwidth of \(\pi/2\). Further, in the other case of \(n=K-4\) allowing \(2K-1-2n=7\), the comb filter is obtained from Equation (25) as  
\[Q_{0}(z)=\frac{1}{2}\Big{(}1+z^{-7}\Big{)} \tag{27}\]  
Figure 4 shows the frequency responses of two filters with two coefficients described in Equations (26) and (27). The analysis s-filter bank using the two-tap lowpass filter of Equation (26) leads to a symmetric subband decomposition since both filtered and subtraction-loop bands have a half bandwidth of \(\pi/2\), while using the comb filter of Equation (27) displays the frequency response that shows regularly spaced peaks giving the appearance of a comb [31]. The two-tap filter examples show that the lower and upper bands performed by the analysis s-filter bank have centrally symmetric frequency responses.