# Improving Passband Characteristics in Chebyshev Sharpened Comb Decimation Filters
## 1 Introduction  
The Kaiser-Hamming sharpening [1] of comb decimation filters was first introduced in [2] to improve the passband and stopband characteristics of the filters. Later, various improvements to the Kaiser sharpening method were presented [3; 4; 5; 6; 7]. The Chebyshev sharpening of comb decimation filters was first introduced by [PERSON] in [8]. This filter provides a high predetermined attenuation in all folding bands. However, the passband has a high passband droop. The author also proposed a pre-sharpening version in which the passband droop is decreased but still needs to be improved. The authors in [9] proposed a novel method based on interval analysis, while in [10; 11], were proposed different Chebyshev sharpening polynomials and the optimal filters to compensate for the passband droop. The sharpening combs belong to the modified comb decimation filters. In [12], the method to design an optimal compensator for Chebyshev sharpening combs provides better compensation than the compensators in [11]. This work is based on particle swarm optimization (PSO) and sinusoidal magnitude response.  
This paper is dedicated to the design of compensators for Chebyshev sharpened comb filters and extends the previous work [12]. The next section elaborates on compensators' general importance and applications as well as their principal design methods, emphasizing their practicality and the potential real-world applications of the research. It then describes the detailed proposed design procedure, including the design steps and the corresponding MATLAB programs.  
The novelty of this work includes the following:  
* In addition to the Chebyshev comb decimators introduced in [10; 11], the Coleman popular Chebyshev sharpened comb [8] is also inclusively included in our research.
* Three optimal and multipliers compensators are designed, all with sinusoidal magnitude responses, with different complexity for the Chebyshev sharpened comb filters [8; 10; 11].
* Unlike the methods in [10; 11], normalization of the designed compensation filters is unnecessary.
* A generalized approach is provided since all designs are based on sinusoidal magnitude responses and optimizing the amplitudes of sinusoidal responses using PSO.
* The flexibility of the design is shown in choosing narrowband or wideband designs and a trade-off between the quality of compensation and the number of required adders.
* The design is simple since only one parameter in the narrowband case and two parameters in the wideband case should be optimized, unlike the approaches in the literature based on optimizing the compensator filter coefficients.  
This paper is organized as described in the continuation. The next section introduces compensators, importance, applications and the narrowband and wideband designs. Section 3 presents in detail the passband compensation design of the Chebyshev sharpened combs introduced in [8; 10; 11]. Section 4 compares the methods from the literature and presents the impact of the proposed designs on the possible practical applications. It also provides some further ideas for potential future work.