# Design of Compensators for Comb Decimation Filters

Gordana Jovanovic Dolecek (Institute INAOE Puebla, Mexico)
DOI: 10.4018/978-1-5225-2255-3.ch525

## Abstract

This article presents different methods proposed to compensate for the comb pass band droop. Two main groups of methods are elaborated: Methods that require multipliers, and multiplier less methods. The width of pass band depends on the decimation factor and the decimation of the stage which follows the comb decimation stage. In that sense the compensation can be considered as a one in the wideband, or in the narrowband. There exit methods which can be used for both: wideband and narrowband compensations (with different parameters). Usually there is a trade-off between the compensator complexity and the provided quality of compensation.
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## Background

The transfer function of comb filter is given by the following equation:

(1) where M is the decimation factor and K is the order of the filter.

The magnitude response of the filter is given as:

(2)

The comb pass band is defined by the pass band edge (Kwentus &Willson, 1997):

(3) where R is the decimation factor of the stage that follows the comb decimation stage.

For values R<4, the pass band is considered as a wideband, and in an opposite case it is a narrowband.

As an example, Figure 1 shows the wide pass band zoom (R=2), of the magnitude response of comb filter with the decimation factor M=12 and an order equal to K=3. Note that the response is not flat and has a droop, which increases with the increase of the frequency ω. The inverse comb magnitude characteristic:

(4) is also shown.

Figure 1.

Magnitude and inverse magnitude characteristics of comb, M=15, K=3, R=2

The product of the magnitude characteristics (2) and (4) results in unity:

(5)

Consequently, in order to get a flat comb magnitude characteristic it is necessary to cascade comb with a filter which has magnitude characteristic approximately equal to the inverse comb magnitude characteristic in the pass band. This filter is called a compensation filter. Denoting the magnitude characteristic of compensator as it follows:, for 0 ≤ωωp(6) where ωp is the pass band edge defined in (3).

Usually, compensation filter works at a low rate, i.e. after decimation. As a consequence, at high input rate, compensator is expanded by M.

The compensated comb is the cascade of comb and compensator. The corresponding transfer function at high input rate is:

(7)

## Key Terms in this Chapter

Multiplierless Design: The coefficients of filter are presented as powers of two which can be implemented as shifts and adders, thus avoiding multipliers.

Comb Filter: A simplest decimation filter which has all coefficients equal to unity.

Comb Compensation Filter (Compensator): The filter which has the magnitude characteristic which is an approximation of the inverse comb magnitude characteristic in the pass band. The filter is cascaded with the comb filter.

Comb Passband: The frequency band in which the decimated signal must be preserved. Ideally its characteristic is inverse of that of comb, thus resulting in approximately flat magnitude characteristic of the compensated comb in the pass band. The width of the pass band depends on the comb decimation factor, and the decimation factor of the stage which follows the comb decimation stage.

Error Function: The difference of the desired and designed magnitude responses of compensated filter.

Multipliers: Implementation of coefficients of filter transfer function which cannot be presented as a power of two.

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