Modificar Estructura Comb

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Modified Comb Nonrecursive Structure

Gerardo Molina Salgado, and Gordana Jovanovic Dolecek

Department of Electronics

Institute INAOE

Puebla, [email protected], [email protected]

AbstractThis paper introduces the modified comb

nonrecursive structure for the decimation factor which can be

represented as a power of two and three. The structure has an

improved alias rejection characteristic, decreased passband

droop and a low power consumption. The implementation

results are also included.

I. INTRODUCTION

Oversampled Sigma-Delta (SD) converters have becomewidely used as a valid alternative to conventional A/D(Analog/Digital) converters [1]. In contrast to conventional

A/D converters, an oversampling SD-A/D converter samplesthe analog input signal with a frequency much larger than theNyquist frequency. The SD-A/D converter consists of asigma-delta modulator and a decimation stage whichdecreases the oversampled frequency of the modulator to theNyquist frequency. The key part of the decimation stage is adecimation filter which is responsible for the aliasing rejectionintroduced in the process of the decreasing of sampling rate.CIC (Cascaded-Integrator-Comb) filter [2] is widely used asthe decimation filter due to its simplicity; it requires nomultiplication or coefficient storage but rather onlyadditions/subtractions.

The CIC integrator section works at the higher input datarate thereby resulting in higher chip area and higher powerdissipation for this section. In order to resolve this problem thenon-recursive structure can be used [3],

[ ]KMK

zzzM

zH)1(21 ...1

1)( ++++

= . (1)

The most popular nonrecursive structure uses the decimation

factor which can be represented as a power of two,M=2n1,

Kn

i

i

zzH

+=

=

1

0

22

1

)1(2

1)( . (2)

The number of additions per output sample, (APOS) is

)12(2)2...22( 1111

2 =+++= nnn

KKAPOS . (3)

However, like CIC structure, the nonrecursive structure has a

poor magnitude characteristic. Recently, the magnituderesponse improvements of nonrecursive structures, where thedecimation factor is a power of 2, have been reported [4], [5].A non-recursive comb structure where a decimation factor is

a power of three,M= 23n , is introduced in [6].

Kn

i

iK

i

i

n

in

zzzzH

=

=

++=

=

1

0

32313

03

22

2 3

1

3

1)( (4)

The number of APOS is:

)13(3)3...33(2 222 13 =+++= nnn KKAPOS . (5)

In this paper we consider the nonrecursive structure wherethe decimation factor is power of two and three.

2132 nnM = . (6)

Combining (2) and (4) we have

)()()()()( 2332 zHzHzHzHzHcomb == . (7)

The goal is to keep the low power consumption and toimprove the magnitude characteristic in the first folding bandof the filter (7).

The rest of the paper is organized in the following way.Section II describes the nonrecursive structure. Theimprovement of the alias rejection is discussed in Section III.The choice of the design parameter is discussed in Section IV.Section V presents the passband compensation. Finally, theimplementation is presented in Section VI.

II. NONRECURSIVE STRUCTURE

The section H2(z) has n1 stages of the filters (1+z-1)

followed by the decimation factor of two. Similarly, thesection H

3

(z) has n2

stages of the filters (1+z-1+z-2) followedby the decimation factor of 3. The nonrecursive structure inwhich the sectionH2(z) is coming before the section H3(z) isdenoted as Structure 1. Similarly, in the Structure 2 the sectionH3(z) comes first. The numbers of APOS for the Structure 1

This work is supported by CONACYT.

978-1-4673-2527-1/12/$31.00 2012 IEEE 1128


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and 2 are denoted as APOS2,3 and APOS3,2, respectively.TakingK=1 we have:

)13(33)12(2 2213,2 +=nnn

APOS . (8)

)12(22)12(3 1122,3 +=nnn

APOS . (9)

The corresponding values are shown in Fig.1 for n1=1, 2, 3and n2=1, 2, 3.

Figure 1. Numbers of APOS for Structures 1 and 2.

Note that the number of APOS for the Structure 1 is generallyless than that of the Structure 2 for the same values of n1andn2:

2,33,2 APOSAPOS . (10)

Therefore in this work we will consider the Structure 1 whichis presented in Fig. 2 with more details.

Figure 2. The nonrecursive structure with less number of APOS.

III. IMPROVEMENT OF FIRST FOLDING BAND

ATTENUATION

The first zero of the comb filter is at the frequency 2/M.Consider thecomb filterforM=3. It has the first zero at2/3.The corresponding expanded filter

N

NN

comb

z

zzH

=

1

1

3

1)(

3

(11)

has the first zero at 2/3N. Therefore the comb filter and the

expanded filter (11) will have the first zero in the same point

if the following condition is satisfied:

N=M/3. (12)

Observing the structure from Fig.2 and the condition (12) wecan easily conclude that the additional zero in the first folding

band is obtained by cascading K2 additional filters in the last

stage. The number of the APOS in the proposed structure is

23,2 6KAPOSAPOSp += . (13)

The system function and the magnitude response of the

proposed filter are given in the following:

21

3/1 1

1

3

1

1

11)(

K

M

MK

M

pz

z

z

z

MzH

=

(14)

21

)6/sin(

)2/sin(

3

1

)2/sin(

)2/sin(1)(

KK

jp

M

MM

MeH

= . (15)

The proposed structure is shown in Fig.3, where K1=K. Next

issue is the choice of the parameterK2.

Figure 3. The proposed structure.

IV. CHOICE OF THE PARAMETERK2

The worst case aliasing is at the frequency [7]

RMMA

=

2, (16)

where R is the decimation factor which follows the combdecimation stage. We will consider hereR=2.

Placing (16) into (15) and taking sinfor small value of ,we arrive at:

21

)4/sin(

)4/3sin(

3

1

4/3

)4/3sin(1)(

KKj

pM

eH A

. (17)

From here we easily find the increase of the worst case

attenuation (WCA) for each added cascaded filter at the laststage:

dBA 5424.9)4/sin(3

)4/3sin(log20 =

=

. (18)

Denoting the desired WCA in dB asAdes, we have:

5424.92

combdes AAK , (19)

where is the ceiling function and Acomb is the comb

attenuation.However, in order to avoid the decrease of the attenuations

in the sub sequential folding bands, the following conditionhas to be satisfied,

K2


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Example 1: Consider the comb filter with M=36 and K=4which has the WCA of -41.8181dB. However, if we need at

least the attenuation of -55dB, from (19) we find:

23824.15424.9

8181.41552

==

=K

The corresponding magnitude response along with that of the

comb filter is shown in Fig. 4. The zoom in the first folding

band is also shown.

Figure 4. Magnitude responses in Example 1.

V. PASSBAND COMPENSATION

The proposed structure has a high passband droop. In thissection we consider the passband compensation. To this endwe modify the compensator from [8],

,)]1([)( 2 cKMMc zAzSzH

++= (21)

whereS=-2-4, andA=-(24+2), (22)

andKcis the number of the cascaded compensators.

Using the multirate identity the filter (21) can be moved to a

low rate which is M times less than the high input rate. Thecorresponding magnitude response is given as,

cKjc MASeH )cos2()(

+= . (23)

The choice ofKcdoes not depend onMbut only on theK1and K2. The values for Kc, obtained by the MATLABsimulations, are given as.

>++

=++=

31

3

2121

2121

KKforKK

KKforKKKc . (24)

However, the introduced compensator (Kc=1) decreases the

WCA by, (see (16) and (23))

( ) 0231.1log20)(log20 === ASKeHA cj

ccA dB. (25)

From (18) and (25) it follows that for each added filters H3(z)

at last stage and for each added compensator at the low rate

the improvement of the worst case attenuation is:

Acomp=A-Ac=8.5193dB. (26)

Example 2: Consider the proposed filter with M=36, K1=4

andK2=2 from Example 1. From (24) we have Kc=5 and the

decreased worst case attenuation is of 5.1153dB. The

increased number of APOS is 3 5=15. The magnituderesponses of the uncompensated and the compensated filters

are shown in Fig.5.a. along with the zoom at the first folding

band. The passband zoom is shown in Fig.5.b.

a. Overall magnitude responses and the first folding band zoom.

b. Passband zoom.

Figure 5. Illustration of Example 2.

VI. IMPLEMENTATION

In this section we present the implementation of the filterspresented in Example 1 and 2. We have described the filters inVHDL (Very High Speed Integrated Circuit HardwareDescription Language) at the Register Transfer Level.

The VHDL models of the filters, including the frequencydivider, have been synthesized into CMOS standard cells of0.18um technology. The obtained transistor level models offilters, without parasitic effects, were used in Synopsys Hspicein order to simulate power consumption with a power supplyof 1.8V. The input for the filters was an ideal first order onebit sigma delta modulator, in which the input is a sine wave of

78.25Khz and the modulated output has a frequency of10MHz. The obtained layouts were used to measure the usedarea of each filter.

A. Without Compensator

In the comb filter the first two stages are implemented bycascadingK1=4 basic filters shown in Fig. 6 (a) while the lasttwo stages are implemented by cascading K1=4 basic filtersshow in Fig. 6 (b). As a difference with the comb filter in the

0 0.2 0.4 0.6 0.8 1

-200

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

/

MagnituderesponsesindB

Comb, M=36, K=4

Proposed: M=36, K1=4, K2=2

0.045 0.05 0.055 0.06 0.065-100

-80

-60

-40

0 0.2 0.4 0.6 0.8 1-200

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

/

Magnituderesponsesin

dB

Proposed

Proposed compensated

0.0450. 050.0550.060.065-100

-80

-60

-40

Proposed

Proposed compensated

0 0.002 0.004 0.006 0.008 0.01 0.012-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

/

MagnituderesponsesindB

noncompensated

compensated, Kc=5

1130


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proposed structure last stage has an additional cascade ofK2=2basic filters shown in Fig. 6 (b). The word length increase incomb filter is ofK1bits in the first two stages and 2K1bits inthe last two stages, but in proposed filter the last stage has anincrease of 2(K1+K2) bits.

Table 1 presents a summary of power consumption andused area of the implemented comb and proposed filterwithout the compensator, we can see that the increase inK2=2

basic filters in the last stage practically does not increase thepower consumption of proposed filter compared with that ofthe comb filter. The increase is only about 13.8%, sincedynamic power is proportional to the sampling rate [9].Additionally, proposed filter has an increase in used area ofabout 31.6% due to the extra K2=2 basic filters added in thelast stage.

Figure 6. Basic filters to implement (a) the first two stages and (b) lasttwo stages.

TABLE I. SUMMARY OF AREA AND POWER CONSUMPTION OF COMBAND PROPOSED FILTER WITHOUT COMPENSATOR.

FilterWCA

(dB)

Total

Power

(W)

Extra

Power

(%)

Total

Area

(m2)

Extra

Area

(%)

CombM=36

K=4 -41.8 333 0 194,402 0

ProposedM=36

K1=4,K2=2-60.8 379 13.8 255,850 31.6

B. With Compensator

The proposed compensation filter is implemented bycascading five stages shown in fig. 7, where the coefficientsare normalized with 2

-4, in order to use left hard-wire shifts in

the sum of products.

Table II presents a summary of power consumption andused area of proposed filter with and without the compensator.

The compensated filter has a relative increase in powerconsumption of only 15.5 % compared with theuncompensated filter, due to the fact that the compensationfilter is working at the low sampling rate. Additionally, thecompensated filter has a relative increase in used area about109% compared with the uncompensated one, since each stageof the compensation filter has a word length increase of Kc=5bits.

Figure 7. Stage to implement the proposed compensation filter.

TABLE II. SUMMARY OF AREA AND POWER CONSUMPTION OFPROPOSED FILTER WITH AND WITHOUT THE COMPENSATOR.

Proposed FilterWCA

(dB)

Total

Power

(W)

Extra

Power

(%)

Total

Area

(m2)

Extra

Area

(%)

Uncompensated -60.9 379 0 255,850 0

Compensated

Kc=5-55.8 438 15.5 536,085 109

VII. CONCLUSION

We presented here the new nonrecursive comb basedstructure which exhibits the low power characteristic and animproved alias rejection. We considered here the case where

the decimation factor is a power of two and three. Theimplementation results confirm that the proposed structure hasa low power consumption. However, the proposed structurehas an increased passband droop. This problem is solved byadding a simple cascaded compensator after the lastdecimation stage. The increased number of APOS is of 3Kc,where Kc is the number of the cascaded compensators.However, the compensators introduce the decrease of theworst case attenuation of about 1 db for each cascadedcompensator. As a result, the number of the cascadedcompensators has to be less than 6.

REFERENCES

[1] [Pervez M., Henrik V. and Jan V., An Overview of Sigma-DeltaConverters: How a 1-bit ADC achieves more than 16-bit resolution,

IEEE Signal Processing Magazine, Volume 13, Issue 1, September1996, pages 61-84.

[2] E. Hogenauer, An Economical Class of Digital Filters for Decimationand Interpolacion, IEEE Transactions Acoustic, Speech and Signal

Processing, Vol.ASSP-29, (Apr.1981), pp.155-162.

[3] Y. Gao, L. Gia, and H. Tenhunen, A Comparison Design of CombDecimators for Sigma-Delta Analog-to-digital Converters, Analog

Integrated Circuits and Signal Processing, 1999, Vol. 22, pp. 5160.

[4] G. Molina Salgado, G. Jovanovic-Dolecek, Non-recursive comb-decimation filter with an improved alias Rejection, 3rd IEEE Latin

America Symposium on Circuits and Systems, February 2012.(In press).

[5] G. Jovanovic-Dolecek, G. Molina Salgado, Low Power Non-Recursive Comb-Based Decimation Filter Design, Accepted forISCCSP, Roma Italy.

[6] C. Zhang, E. Ofner, Low Power Non-Recursive Decimation Filter.

Electronics circuits and system, ICECS07, 2007, pp. 804-807.[7] A. Kwentus, A. Willson,Application of Filter Sharpening to Cascaded

Integrator-Comb Decimation Filters, IEEE Transactions on SignalProcessing, Freb. 1997, vol. 45, No. 2, pp. 457-467.

[8] G. Jovanovic Dolecek, Simple Wideband CIC Compensator,Electronics Letters, Vol.45, No.24, November 2009, pp.1270-1272.

[9] N. Weste, D. Harris, and A. Banerjee, CMOS VLSI design, 3rdedition,Pearson International Edition, 2005.

(a) (b)

BIN +2

BIN BIN

BIN +1

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