 # Design of Compensators for Comb Decimation Filters

Gordana Jovanovic Dolecek (Institute INAOE Puebla, Mexico)
DOI: 10.4018/978-1-5225-7598-6.ch060

## Abstract

This chapter 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.

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)

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