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Comb filters are a class of low-complexity filters especially useful for multistage decimation processes. However, the magnitude response of comb filters presents a droop in the passband region and low stopband attenuation, which is undesirable in many applications. In this work, it is shown that, for stringent magnitude specifications, sharpening compensated comb filters requires a lower-degree sharpening polynomial compared to sharpening comb filters without compensation, resulting in a solution with lower computational complexity. Using a simple three-addition compensator and an optimization-based derivation of sharpening polynomials, we introduce an effective low-complexity filtering scheme. Design examples are presented in order to show the performance improvement in terms of passband distortion and selectivity compared to other methods based on the traditional Kaiser-Hamming sharpening and the Chebyshev sharpening techniques recently introduced in the literature.

Efficient decimation filtering for oversampled discrete-time signals is key in the development of low-power hardware platforms for reconfigurable communication transceivers [

Comb filters are used in the first stage of the decimation chain because their system function is simple and it does not require any multiplier. However, their magnitude response exhibits a considerable passband droop in the passband

Owing to their reduced computational complexity, research on comb filters to date has been focused on (1) improving the magnitude characteristic, (2) preserving linearity of phase, and (3) having the least possible increase of computational complexity [

From the representative sample of works improving the magnitude characteristics of comb filters, we observe that the rotated-comb-based schemes [

On the other hand, the techniques relying on sharpening of comb-based filters in [

Moreover, two-stage comb-based decimation schemes have gained great popularity because the comb decimation filter in the first stage, designed in nonrecursive form, can be implemented at lower rate by polyphase decomposition, thus resulting in lower power consumption. The second-stage filtering operates at lower rate as well, but it can take advantage of CIC-like architectures for area reduction. By doing so, the overall comb-based decimation scheme achieves power and area savings. This approach has been applied to traditional comb filters [

The reasons at the very basis of this work stem from the following observations.

Sharpened compensated comb filters [

In two-stage comb-based decimation schemes, magnitude response improvements over the passband and the first folding band can be achieved by improving only the second-stage comb filter. However, in these cases, the filter in the first stage introduces a passband droop that cannot be corrected neither by resorting to traditional Kaiser-Hamming sharpening [

In the light of the previous observations, the contributions of this work are the following.

We show that, for similar magnitude characteristics, sharpened compensated comb filters guarantee lower complexity than sharpened comb filters without compensation, especially when stringent specifications must be met.

We introduce a low-complexity structure in which the simple multiplierless compensator can be embedded into the cascaded chain of comb filters working at lower rate.

We detail the optimization framework to design sharpened comb-based filters to attain given specifications on the acceptable maximum passband distortion and selectivity. The optimized sharpening coefficients are finite-precision values resulting in multiplierless structures, which are important for low-power applications. The optimization problem can be straightforwardly solved with a simple routine of the MATLAB Optimization Toolbox (available online at [

The rest of this paper is organized as follows. Section

Let

For the comb-based decimation filter, the range limits for

Let us consider as subfilter the simplest compensated comb filter, which has the following transfer function [

In this paper, we propose to use the general sharpening polynomial from (

Proposed CIC-like structure for decimation filtering with sharpened compensated comb filters.

The computational complexity of this structure measured in Additions Per Output Sample (APOS) is given by

From (

Let us discard the computational complexity introduced by the sharpening coefficients in both (

At this point, it is important to mention that we can take advantage of the frequency transformation approach [

Upon noticing that the shape of the magnitude response of comb filters changes very little with

In sharpened compensated comb filters, a lower computational complexity is obtained if

Upon comparing the sharpened comb and sharpened compensated comb filters using

Estimated degree of the sharpening polynomial to sharpen comb filters (dashed line) and compensated comb filters (solid line), with

Now, we introduce the optimization framework to obtain the discrete coefficients of

Let us consider the following notation in order to formalize the optimization problem.

The optimization problem in (

We notice in passing that a somewhat similar optimization approach was derived by Candan and made available online at [

Given the desired passband and stopband deviations

Find

Obtain

Choose the desired word-length

Create

Obtain the sharpening coefficients

Earlier in this paper we pointed out that the two-stage comb-based structure, which can be formed when the decimation factor

The transfer function of the proposed two-stage filter is

Let us identify

The following examples are discussed to show the improvement of magnitude characteristics of comb filters achieved with the proposed method in comparison to other sharpening-based schemes recently introduced in the literature.

Consider

Let us consider the following solutions.

A 1st-order comb filter (

A compensated comb filter with

Figure

Magnitude responses of the filters

When it comes to the complexities in terms of APOS, the proposed solution achieves better results too. The filter

Comparisons in terms of APOS for Examples

Method | Example |
Example |
---|---|---|

[ |
APOS > 360 | — |

[ |
APOS = 211 | APOS = 223 |

Proposed | APOS = |
APOS = 164 |

Consider a two-stage decimation filter with

Figure

Magnitude responses of the filters

This paper proposed an optimization framework to design sharpening polynomials specifically suited to comb-based decimation filtering. The goal of the optimization problem was to minimize the min-max error over the frequency bands of interest of the sharpened filter. The optimization problem can be solved straightforwardly using the MATLAB Optimization Toolbox. The sharpening coefficients are guaranteed to be integers scaled by power-of-2 terms, thus resulting in low-complexity structures. Moreover, it was shown that the use of compensated comb filters, instead of combs only as basic building blocks in the sharpened filter, results in lower complexity structures (in terms of Additions Per Output Sample) for the same magnitude characteristics. Finally, it was shown that the proposed method provides better magnitude characteristic than other sharpening-based approaches for two-stage comb-based structures since it is able to correct the passband droop introduced by the first-stage comb filter.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by CONACYT Mexico (Grant no. 179587) and by US National Science Foundation (Grant no. 0925080).

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