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Third IEEE Signal Processing Workshop on Signal Processing Adavances in Wireless Comm unications,Taoyuan, Taiwan, March 20-23. 001An Efficient Decimation Sinc-Filter Design for Software RadioApplications

M. Laddomada, L. Lo PrestiDep. Elettronica,Politecnico d i Torino,C.so Duca degli Abruzzi 24,10129 Torino, ITALY,

laddomada@lpolito.it, oprestiQpolito.it

Abs tmct - During these Last years, the software radio hasemerged as th e only way to design a multimode, m ultistan-dard transceivers that support, everywhere, any communi-cation standard. Software radio, however, requires not onlypowerful transceiver architectur es, bu t also efflcient digitalsignal processing algorithms.In this paper a method to design and implement a veryefficient decimation fllter for a Sigma-Delta converter is pro-posed.The main idea of this article is to free the positions ofthe zeros of the classical decimation Sinc-Alter so to attaina high selective frequency behaviour. Th e filtering stage isrealized by a classical sinc-fllter cascaded with two othe r fll-ters obtained from the flrst one by rotat ing its zeros by twoopposite angles. The whole cascade yields to a simple re-cursive structure with a frequency behaviour more selectivethan a Sinc-Alter ho u nd the noise fold frequency bands.Although the design method may be extended t o a multi-stage implementation, th e fllter design proposed in this ar-ticle deals with th e flrst flltering stage. Since the designedfllter eliminates the most part of he quantization noise, thefollowing filtering stag es may be designed with relaxed spec-iflcations adopting classical fllter design methods.

I. INTRODUCTIONMany papers in the literature have discussed the problem

of finding an efficient way to realize a decimation processthat follows a EA A/D converter. Some books deal withthis argument in an exhaustive way [l], 2].

In the practical applications, a common decimation filterwith a simple realization is the Sinc-filter [3),141. Althoughthis filter has a very simple structure, i t does not guaranteea satisfactory frequency response. For this reason, otherfiltering stages with a more selective frequency behaviourmust be used after a Sinc-filter [5].

The st op ba nd attenuation achieved by many Sinc-basedfilter design methods presented in the literature is con-strained by the position of the zeros of the transfer func-tion. Consequently, the only way to increase the stop-bandattenuation is to use higher order filters. This approach,however, increases the filter complexity and, at the sametime, the computational load. Based on these considera-tions, the concept proposed in this article (see [SI) onsistsin realizing the decimation filter by freeing the positions ofzeros of the transfer function. In this manner it is possi-ble to optimize the positions of the zeros so to obtain thedesired stop-band attenuation.The paper is organized as follows. Section I1 introducesthe decimation concept and outlines the notation appliedthroughout the paper. In section I11 the proposed filter

0-7803-6720-0/01/$1 O.OOQ2001 JEEE 337

M. Mondin, C. RicchiutoDep. Elettronica,Politecnico di Torino,C.so Duca degli A'bruzzi 24,10129 Torino, ITALY,

mondinOpolito.it,ricchiutoOcclix6.polito.it

Fig. 1. Structure of a EA converter.

Fig. 2. Double stage filter architecture.

design method is discussed. Section IV shows some resultsobtained by t he proposed modified Sinc-filter design andcompares them with the basic Sinc-filter. Finally, sectionV draws the conclusions.

11. GENERALONSIDERATIONS ABOUT SIGMA-DELTACONVERTERSA EA converter is made by two main stages (Fig. 1).The

first stage contains a modulator, that is an oversamplerand a quantizer, which converts the input signal z( t ) na digitized output stream. This task is accomplished byperforming an oversampling a t fs >> f ~ ,here f~ = 2B,is the Nyquist ra te and B, is the input signal bandwidth.The oversampling ratio is defined as p = f i =&. heoutput of the modulator may be considerexas a discrete-time signal y,, representing the roughly quantized versionof the sequence of samples z(n/fs). The first oversamplingstage exchanges resolution with speed. The second stage,the decimator shown in Fig. 1, causes the discrete signalyn to achieve a higher resolution and a rate less than fa.

Since the decimation function is a sampling process inthe discrete-time domain, it requires, as a first step, an anti-aliasing low-pass filter with bandwidth W, = 2 $.

A decimation filter for a EA converter must be gener-ally designed taking into account very severe specifications,due to the high level of the quantization noise, especiallyat high frequencies. For this purpose, in the real applica-tions the architecture shown in Fig. 2is implemented. Thetwo filters HD ( z )and HK ( z ) are both lowpass, but with

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different characteristics from the point of view of both spec-ifications and implementation. Both filters are designed sothat their frequency characteristics are assigned in the fre-quency range 0 < w < 112 and the oversampling ratio isp = D K .

The Sinc-filter, which is a classical solution for the firstst+ge decimation filter, has the following transfer functionand frequency response:

where D is the decimation ratio. From this transfer func-tion is possible to observe that the attenuation around anyzero is not sufficient to gua rantee a good anti-noise filter-ing. This situation verifies also using a cascade of J Sinc-filters because, in this manner, J zeros are put in the sameposition, one over the othe rs.

111. MODIFIED INC-FILTERThe design of th e proposed filter is based on t he following

considerations. If we apply a clockwise rotation by an anglea to any zeroi = l , - . . , D 1 (3 ),- ejg-i1 -

of Eq.(2), we obtain the following transfer function

(4 )In the same manner, by applyingzeros of Eq.(2), we obtain the following transfer function

opposite rotation to the

( 5 )These two filters have complex coefficients but their cas-caded yields to a filter H T ( z )with real coefficients since the

zeros of H,(z ) are complex conjugate pairs. The cascadeof the two filters has t he following transfer function

1 1 - cos(aD)z-D + z-2DH,(r) = H,(Z)Hd(Z)-2 1 - ~ 0 ~ a z - l2 -2 (6)It is straightforward to verify that this filter has null fre-quency points at

a a- f -D 2n (7)where i is an integer number whose maximum value is re-lated to the decimation factor D.

At this point, if a is chosen in such a way as to putzeros in the null intervals, this filter, cascaded to a classicalSinc-filter, can be used to greatly increase the stopbandattenuation in the null intervals. A convenient choice couldbe a = 2 rW z ,or , more in general,

a = qznw, (8)

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Fig. 3 . Modified Sinc architecture; c = 2cosa and 6 = 1+ 2 c o s ( D a ) .where q is of the order of 1 . If q < 1, the zeros are putinside the frequency band where the most significant partof the input spectrum is contained. If q > 1 more relaxedspecifications can be assigned in th e second stage. In fa,&,in thi s case some denoisingat the free intervals is performedat the first stage.

At this point the first stage decimation filter can be writ-ten as

where a = 1 + 2 cosa and b = 1 + 2 cos(Da).equation isThe frequency response corresponding to the previous

An important feature of this filter is that its phase islinear.

Several recursive structures can be found, depending onthe way of writing the transfer function. In Fig. 3a conjig-uration with two multipliers, one at high ra te and the ofherone at low rat e, is shown. In this figure c = 2cosa. As, itis evident, this structure exhibits a very low complexity,and it is very flexible, as the null frequency points can beeasily located in any point of the null intervals, by properly ,choosing the value of a.

IV . RESULTSIn this section we show some results obtained by deci-mating an oversampled da ta stre am outputs from the sigma

delta modulator with the modified Sinc-filter. The imple-mentation with a classical Sinc-filter is compared with amodified Sinc-filter.

The magnitude of the frequency response of a modifiedSinc-filter is shown in Fig. 4.

Fig. 5 shows the attenuation around the first zero o f aclassical Sinc-filter and a modified Sinc-filter for differentvalues of a . The dotted curve shows the magnitude of aclassical Sinc-filter, while the continuous curves refer tothree different modified Sinc-filters which have the param-eter a displayed. This figure shows tha t the attenuationband of a modified Sinc-filter around the zero increaseswhen a increases.

Fig. 6hows the signal to noise ratio for a modified Sinc-filter as a function of the rotation angle a.

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-=:L140Dglt.1 F w0 0 5 0 1 015 0 2 0 2 5 0 3 035 0 4 Q45 O S

Fig. 4. Magnit ude response of a Modified Sinc-filter. The Decima-tion factor is equal to 16 and the rotation angle a s equal to0.03.

Fig. 6. Signal to Noise ratio versus angle a achieved by a modifiedSinc with a decimat ion factor= 16.

Fig. 5 . Magnitude response for a classical Sinc-filter and a ModifiedSinc-filter aro und the first zero, for various values of th e rotationangle a.

v. CONCLUSIONIn this paper, an efficient modified Sinc-filter design has

been shown. Due to its high filtering effect into the so-called null intervals, the proposed method yields to a highnoise rejection. This characteristic is very important since,after the decimation, only the noise in the null intervalsis folded inside the signal bandwidth. Consequently, thisfilter has the effect to reject the main amount of noise thatwould aliases into the useful base-band signal. For thisreason, the second filter stage may be designed with relaxedspecifications.

REFERENCES[l ] J. C. Candy, G. C. Temes Oversampling Methods for A/D andD / A Conversion in Oversampling Delta-Sigma Data Convert-

ers edited by J. C. Candy and G. C. Temes, IEEE Press 1992S. R. Norsworthy, R. Schreier, G. C. Temes Delta-Sigma Con-verters: Theory, Design and Simulation, IEEE Press, NewYork,1997S. Chu, C. Sidney Burrus , Multirate Filter Designs Using Comb[2]

[3]

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Filters, IEEE %ns. Circuits and Sps., vol. CAS-31, pp 913-924, November 1984J . C. Candy, B. A. Wooley, 0.J. Benjam in, A Voiceband Codecwith Digital Filtering, IEEE 5% Commun., vol. COM-29,pp. 815-830, June 1981H. B.Hogenauer, A Canonical class of digital filters for deci-mation and interpolation IEEE f i n s . Acoust. Speech, SignalProcessing, vol. ASSP-29, pp. 155-162, Apr. 1981L. Lo Prest i Eficie nt Modified-Sinc Filt ers for Sigma-DeltaA/D Converters, Submitted to IEEE Transaction on Circuitsand systems I1