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 first introduced Coleman 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 structure for Chebyshev sharpening, while in [10,11], are proposed different Chebyshev sharpening polynomials and the optimal filters to compensate for the passband droop. In [12], the method to design an optimal compensator for Chebyshev sharpening combs provides better compensation than the compensators in [11]. The method is based on particle swarm optimization (PSO) and sinusoidal magnitude response.

This paper extends the previous paper [12]. Three compensators for Chebyshev sharpened comb filters from the literature are introduced to enhance passband compensation and increase the flexibility of the design. The novelty of this work includes the following:

• Besides Chebyshev comb decimators introduced in [10,11], the Coleman Chebyshev sharpened comb [8] is also included.
• Design three optimal and multipliers compensators, all with sinusoidal magnitude responses, with different complexity for the Chebyshev sharpened comb filters [8] and [10,11], providing a trade-off between the quality of compensation, the number of the required adders, and the width of the passband.
• The flexibility of the design.

The paper is organized as described in continuation. Next Section introduces three compensators of different complexity and describes the design procedure. Section 3 presents the passband compensation of the Chebyshev sharpened combs introduced in [8] and [11]. Section 4 presents the comparisons with the methods from the literature.

2. Design of Compensators

Compensators are a key area of research in the field of comb filters. They are designed to improve the passband characteristics of comb filters [13,14,15,16,17] or modified comb filters [18] at a low sample rate, i.e., after the decimation. Typically, compensators are designed as finite impulse responses (FIR) filters using various optimization methods to ensure that the cascade of the filter and the compensator provides as low as possible deviation in the passband of interest. We will adopt the denotations, as shown in the continuation.

The comb filter of order K and the decimation factor M is denoted as:

$$H\left(z,K,M\right)={\left[{\displaystyle \frac{1-z^{-M}}{M(1-z^{-1})}}\right]}^K.$$

(1)

The Chebyshev sharpened comb is denoted as G(z) and the compensator as C(z).

Since the compensator works at a low rate, using the multirate identity [19] the cascade of the Chebyshev sharpened comb and compensator is given as:

Gc(z)=G(z)C(z^M).

(2)

As the comb compensator introduced in [13,14,15,16] has sinusoidal magnitude characteristics while exhibiting high-quality compensation, we also propose to use the sinusoidal magnitude responses for the proposed compensators. We consider narrowband and wideband designs, as described in the following.

2.1. Narrowband Design

In this design, the passband frequency ω_p <1/4M. The magnitude characteristic is chosen as in [13],

$${ C}_1\left(e^{jM\omega }\right)=1+C_{11}{sin}^2\left({\displaystyle \frac{\omega M}{2}}\right),$$

(3)

where C₁₁ is the parameter of the design.

The system function of this filter is given as [13],

$$C_1\left(z\right)=2^{-2}\left[\left(-1+2z^{-1}-z^{-2}\right)C_{11}+2^2{z}^{-1}\right].$$

(4)

The compensator requires three adders and one multiplier. In a multiplierless design, the number of required adders equals:

$$N_1=3+N_{11},$$

(5)

where N₁₁ is the number of adders in the signed-power-of-two (SPT) form of C₁₁.

2.2. Wideband Design

In the wideband design the passband frequency ω_p equals 1/4M≤ ≤1/2M.

We consider two sinusoidal compensators.

2.2.1. Extension of Narrowband Design

The magnitude response is given as [14]

$$C_2\left(e^{jM\omega }\right)=\left[1+C_{21}{sin}^2\left({\displaystyle \frac{\omega M}{2}}\right)\right]\times \left[1+C_{22}{sin}^4\left({\displaystyle \frac{\omega M}{2}}\right)\right],$$

(6)

where C₂₁ and C₂₂ are the parameters of the design.

The system function of this compensator is given as:

$$\begin{gathered} C_2\left(z\right)=2^{-2}\left[\left(-1+2z^{-1}-z^{-2}\right)C_{21}+2^2{z}^{-1}\right]\times \\ \left[2^{-4}{C}_{22}\left[\left(1+z^{-4}-4(z^{-1}+z^{-3}\right)+{(2}^2+2)z^{-2}\right]+z^{-2}\right]. \end{gathered}$$

(7)

The compensator requires nine adders and two multipliers C₂₁ and C₂₂. The number of the required adders in a multiplierless design is given as

$$N_2=9+N_{21}+N_{22},$$

(8)

where N₂₁ and N₂₂ are the number of adders in SPT forms of C₂₁ and C₂₂, respectively.

To increase the flexibility of the design, we also consider the less complex compensator from [16] introduced in the next subsection.

2.2.2. Simplified Version of Compensator in (6)

The magnitude response is given as [16]:

$$C_3\left(e^{jM\omega }\right)=1+C_{31}{sin}^2\left({\displaystyle \frac{\omega M}{2}}\right)+C_{32}{sin}^4\left({\displaystyle \frac{\omega M}{2}}\right),$$

(9)

where C₃₁ and C₃₂ are parameters of the design.

The system function is given as [16],

$$C_2\left(z\right)=2^{-4}{C}_{32}\left(1-2z^{-2}\right)-\left(C_{31}+C_{32}\right)2^{-2}\left(z^{-1}-2z^{-2}+z^{-3}\right)+z^{-2}.$$

(10)

The compensator requires six adders and two multipliers C₃₁ and C₃₂. The number of the required adders in a multiplierless design is,

$$N_2=6+N_{31}+N_{32},$$

(11)

where N₃₁ and N₃₂ are the number of adders in SPT forms of C₃₁ and C₃₂, respectively.

2.3. How to Obtain the Compensator Design Parameters?

We first consider optimum design to get the parameters in compensators (3), (6), and (9) in a way that a minimum of the maximum passband deviation δ_o in the passband of interest, defined by the passband edge ω_p equals

$$20\times {log}_{10}\left|G_c\left(e^{j\omega }\right)\right|\le {\delta }_{opt}\mathrm{for} 0\le \omega \le {\omega }_p,$$

(12)

where $G_c\left(e^{j\omega }\right)$ is given in (2), and the compensators are given in (3), (6), and (9).

The optimal design is obtained using particle swarm optimization (PSO) in MATLAB [20].

3. Passband Compensation of Chebyshev Sharpened Combs from [8] and [10,11]

We present the compensation of Coleman Chebyshev comb [8] and Chebyshev combs in [10,11].

3.1. Coleman Chebyshev Sharpened Comb [8]

The fifth-order Chebyshev polynomial coefficients in [8] are [0,5,-20,16]. We decimation factor M=16, and the attenuation in the stopbands is -100 dB [8].

3.1.1. Narrowband Compensation

We consider ω_p=0.22/M and the compensator C₁ in (3). Using PSO in MATLAB, we get C₁₁o=0.9089 resulting in the optimum passband deviation of δ_o=0.0141dB. Next, C₁₁o is presented in the SPT form varying the number of adders from four to seven, as shown in Table 1. The results are shown in Table 1. It can be observed that we have approximately optimum compensation with seven adders, denoted in bold in Table 1.

Figure 1.a compares overall magnitude responses of non-compensated Coleman filter and compensated with the optimum compensator (3). The compensation slightly deteriorates the stopband, as shown in Figure 1a. Figure 1b compares passbands of the Coleman noncompensated filter, compensated with the optimum filter, and compensated with the SPT filter from Table 1with N₁=5.

3.1.2. Wideband Compensation

The passband frequency ω_p=1/2M, and M=16.

Compensator C₂

We first consider the compensator C₂ from (6). The optimal compensator coefficients are: C₂₁o=0.7904, and C₂₂o=0.8596, resulting in δo=0.0239 dB. The corresponding SPT coefficients are shown in Table 2. It can be observed that with fifteen adders, we have approximately optimum compensation.

Figure 2 contrasts the passband magnitude responses of the Coleman non-compensated, optimum compensated, and the SPT compensated with N₂=12.

Compensator C₃

The optimal compensator coefficients are C₃₁o=0.7250 and C₃₂o=1.3138, resulting in δo=0.045 dB. The corresponding SPT coefficients are shown in Table 3. With twelve and thirteen adders, we have approximately optimum compensation.

Figure 3 compares the Coleman non-compensated filter passbands, compensated with the optimum filter, and compensated with the SPT filter, N₃=11.

3.2. Chebyshev Sharpened Comb Filters [11]

We consider M=20 and Chebyshev polynomial from [11],

p(x)=1-2⁹x²+2¹⁵x⁴,

(13)

where x is a comb filter (1)

$$x=H\left(z,\mathrm{1,20}\right)$$

(14)

3.2.1. Narrowband Compensation

We chose ω_p=0.22π/M and the compensator C₁ in (3). The optimum parameter C₁₁o=0.7213, and δ_o=0.0111dB. The SPT coefficients are given in Table 4, changing N₁ from four to eight. For N₁ =8, we get approximately the optimum compensation.

3.2.2. Wideband Compensation

The passband frequency ω_p = π/2M, while the decimation factor M and the Chebyshev sharpening polynomial are the same as in Section 3.2.1. We will consider the wideband compensators C₂ and C₃.

• Compensator C₂

The optimum values of the compensator parameters obtained using the PSO method are C₂₁o=0.6337 and C₂₂o=0.6264, while the passband deviation equals δo=0.0171 dB. To get a multiplierless design the optimum parameter values are presented in the SPT form and the results are shown in Table 5. It can be observed that with N₂=15 we can get almost optimum compensation.

Figure 5 contrasts the magnitude responses in passband of the non-compensated, optimum compensated, and the SPT compensated (N₂=12) Chebyshev sharpened comb filter [11].

• Compensator C₃

The optimum parameters are obtained using PSO, resulting in C₂₁o=0.5946, C₂₂o=0.8939, and δ₂o=0.0302 dB. The SPT parameter values are given in Table 6., along with the corresponding passband deviations and the number of compensator adders.

As observed in Table 6, the approximate optimal optimization can be obtained with N₃=12.

Figure 6 compares the passband magnitude responses of the Chebyshev comb filter [11], non-compensated, compensated with the optimal C₃ filter, and compensated with the SPT C₃ filter with N₃=9.

4. Results

In this section, we present the benefits of the proposed method and compare it with the methods in the literature.

4.1. Comparisons

Only a few works propose compensators for Chebyshev sharpening combs [10,11,12]. However, we also include a comparison with one of the most recent works, in which another type of sharpening is applied to improve the passband and stopband of comb filters.

4.1.1. Comparison with Method in [10]

In Table 1 [10], the authors proposed four compensated Chebyshev sharpened combs. In continuation, we then present a comparison of all the cases.

• Chebyshev sharpening polynomial p(x)=1-2⁹x+2¹⁵x², x=H(z,1,32)

The passband edge is ω_p=0.226π/M where M =32.The design in [10] requires N=9 adders, resulting in δ_p=0.06 dB. We used the narrowband compensator C₁. The PSO design results in C_11o=0.3549 and δo=0.0043 dB. The SPT design results are shown in Table 7.

The results in Table 7 highlight the efficiency of the proposed designs (denoted in bold), which outperform those in [10], requiring fewer adders and providing better compensation.

• Chebyshev sharpening polynomial p(x)=1-2¹⁰x+2¹⁷x², x=H(z,1,32)

The passband edge is ω_p=0.164π/M where M =32.The design in [10] requires N=8 adders, resulting in δ_p=0.02 dB. In the proposed design we choose the compensator C₁. The optimum PSO design gives C_11o =0.3443 and δo=0.0012 dB. The results of the SPT design are shown in Table 8.

The proposed designs (denoted in bold) confidently present superior compensation, requiring fewer adders.

• Chebyshev sharpening polynomial p(x)=-1+27(2⁶x-2¹⁴x²+2²⁰x³), x=H(z,1,32)

The passband frequency equals ω_p=0.187π/M and M=32 resulting in N=13 adders and δ_p=0.05 dB in the design [10]. We applied the compensator C₁ and the PSO design, obtaining C_11o=0.5248 and δo=0.0034 dB. The results of the SPT design are shown in Table 9.

Once again, the proposed design shown in Table 9 reassures us with its significantly improved compensation and fewer adders.

• Chebyshev sharpening polynomial p(x)=-1+27(2⁴x-2¹⁰x²+2¹⁴x³), x=H(z,1,32)

The passband edge equals ω_p=0.354π/M and M=32.The design in [10] resulted in N=13 adders and δ_p=0.05 dB. In our design, we chose the wideband compensator C₃. The PSO design results in C_33o=and δo=dB, while the results of the SPT design are shown in Table 10.

In Table 10, presented in bold, the proposed design demonstrates better compensation and fewer adders in all cases.

4.1.2. Comparison with Method in [11]

The authors in [11] proposed four Chebyshev sharpened combs and the corresponding optimum compensator designs to improve the passband while keeping the number of adders as low as possible. We compare our designs, taking narrowband and wideband cases.

• Chebyshev sharpening polynomial p(x)=1-2⁹x²+2¹⁵x⁴, x=H(z,1,32)

The passband edge equals ω_p=0.226π/M where M=32. The design in [11] requires N=3 adders resulting in δ_p=0.02 dB. In our design, we chose the narrowband compensator C₁. The results of the PSO design are C_11o=0.3543 and δ₁o=0.0043 dB, while the results of the SPT design are shown in Table 11.

The proposed design in the first row of Table 11 exhibits better compensation at the expense of one additional adder. Unlike [11], the proposed design's flexibility offers further advantages and improved passband compensation by slightly increasing the adders.

• Chebyshev sharpening polynomial p(x)=-1+27(2⁴x-2¹⁰x²+2¹⁴x³), x=H(z,1,32)

The passband edge equals ω_p=0.483π/M and M=32.The optimum design in [11] requires N=7 adders, resulting in δ_p=0.12 dB. Since C₃ requires fewer adders than C₂, we apply the wideband compensator C₃ in our design. The results of the optimal PSO design are C_31o=0.8888, C_32o=1.8668 and δ_3o=0.0554 dB, while the SPT design results are given in Table 12.

The proposed design in the first row of Table 12 exhibits much better compensation, requiring an additional adder. Unlike [11], other results in Table 12 offer a trade-off between improving compensation and slightly increasing the number of adders.

4.1.3. Comparison with Method in [12]

We compare our design with Example 2 in [12], where the compensator for Chebyshev sharpening polynomial p(x)=-1+27(2³x²-2⁸x⁴+2¹¹x⁶), x=H(z,1,32), is designed. The design's results are given in Table II in [12]. Our design, meticulously presented in Table 13, provides a clear comparison. For N₂=11, the passband deviation equals 0.0307 dB, compared to 0.0587 dB, shown in the third row of Table II in [12]. Further increase in the number of adders, a key design parameter, results in a decrease of the passband deviation, as shown in Table 13. This relationship demonstrates the design's ability to improve performance with increased complexity.

4.1.4. Comparison with Method in [7]

In [7], a two-stage comb decimation filter with a sharpened comb in the second stage was proposed. A linear optimization technique determines the coefficients of sharpening polynomial. The complexity of the design expressed in the number of adders per output sample (APOS) equals 163, while the minimum stopband attenuation is 30 dB, and the passband deviation is 0.16 dB. This work is compared in [7] with the compensated Chebyshev sharpening comb using polynomial p(x)=1-2⁹x²+2¹⁵x⁴ [11] for ω_p=0.226π/M and M=32. This filter requires 138 APOS with a minimum stopband attenuation of 38.12 dB and a passband deviation of 0.02 dB.

The proposed compensator from the first row in Table 11 is applied to the Chebyshev sharpened comb, resulting in a passband deviation of 0.0169 dB and the number of APOS equals 139, which is a much better result than in [7].

4.2. Principal Features of the Proposed Method

The principal features of the proposed method to improve the passband of the Chebyshev sharpened comb filters from the literature are summarized as follows:

• The method includes the compensation of Coleman Chebyshev comb filters, which have a favorite characteristic of providing high and equal attenuations in all folding bands.
• The choice of narrowband and wideband compensators depends on the passband of interest. For the wideband case, there is a choice between two compensators depending on the number of adders and the compensation quality.
• All designs have the option of optimal multiplier and multiplierless designs.
• Design flexibility presented as a trade-off between the number of adders and the quality of compensations.
• The comparisons with the methods from the literature demonstrate the advantages of the proposed method.