1. DIRECT DESIGN OF BANDSTOP WAVE DIGITAL LATTICE FILTERS Frequency Transformation Methods: 1- Computational complexity. 2- Impossibility to design independent two sopbands or two passbands. 3- Impossibility to design a lattice structure with degree N, N/2 = odd.

2. The Scattering Properties of Bandstop Reference Lattice Structures The Polynomial Form of S is: where:

3. The Polynomial Properties of a Stopband Reference Structure: 1- The polynomial g(ψ) is a strictly Hurwitz polynomial of even degree n. 2- The polynomial h(ψ) is an odd polynomial of degree n-1. 3- The polynomial f(ψ) is an even polynomial of degree n. 4- The three polynomials are related according losslessness:

4. The Reflection and Transmission Functions Are : The Two Allpass Functions are: Where:

5. The Resulting Transmission Function is: At Real Frequencies, the Two Allpass Functions Are: Mathematical Processing Results in :

6. Defining : The Function Δ: 1- Approximates zero within allowed error margin ε 1 in the first passband. 2- Approximates ± π/2 within allowed error margin ε 2 in the stopband. 3- Approximates zero within allowed error margin ε 3 in the 2 nd passband.

7. If a polynomial Q(ψ) is to exhibit the Δ phase function: The Approximation Procedure: 1- Translate the given amplitude specifications into corresponding phase specifications for Δ. 2- At n digital frequency points, interpolate Q(ψ).

8. 3- Apply the Remez-exchange algorithm to change the set of interpolation frequencies for optimal Q(ψ). The Synthesis Problem: 1- Factorize the resulting polynomial Q(ψ). 2- Construct the two polynomials from the left and right side roots of Q(ψ).

9. 3- Now, the two allpass functions are: 4- Factorize the two allpass functions into first and second order sections. 5- Realize each section of first order by a 2-port parallel adaptor. 6- Realize each section of 2 nd order by a two 2-port adaptors.