Impulse Response and Generating Functions of sinc N FIR Filters

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1 ICDT 20 : The Sixth International Conference on Diital Telecommunications Impulse Response and eneratin Functions of sinc FIR Filters J. Le Bihan Laboratoire RESO Ecole ationale d'inénieurs de Brest (EIB) CS BREST cedex 3 France ean.le_bihan@enib.fr Abstract Sinc finite impulse response (FIR) filters are built as a cascade of sinc filters each of lenth. The are used for diital sinal processin applications in various areas includin telecommunications. eneratin functions for the sinc FIR filters transfer functions are iven. It is shown that -Transform techniques can offer an efficient method to derive straihtforward recurrence relations for fast computation of the impulse response. oreover a simple expression valid for all the filter coefficients is also obtained. It is new and eneral compared to previousl published formula. Kewords diital filter ; impulse response ; coefficient computation ; -Transform. I. ITRODUCTIO Different structures for diital filters have been proposed for applications as decimation interpolation or noise suppression in Sima-Delta A/D conversion [ 2]. As suitable filters for hih speed operation cascadedinterator-comb (CIC) filters [] and sinc impulse finite response (FIR) filters [3 4] are amon the well nown solutions. The use of sinc filters constituted of a cascade of sinc filters each of lenth requires the calculation of all of the impulse response coefficients. So adequate recursions or expressions for rapid calculation of these coefficients have been the subect of numerous investiations as in [5-7]. Interestin computation aids have alread been published: a closed form expression for the first coefficients has been iven in [6] while a recurrence formula has been presented in [7]. It has been proved that usin -Transform techniques can be an efficient method to derive diital filter coefficients [8 ]. Here emploin these techniques new results are obtained. Simple effective recurrence relations are derived for the computation of the coefficients of sinc FIR filters. In addition a simple expression is deduced which is valid not onl for the first coefficients of the impulse response but for all of them. In addition eneratin functions for the transfer functions of sinc FIR filters are presented. In this paper the eneral form of the transfer function of the sinc FIR filters is iven in section II as well as its useful basic properties. In section III it is shown how recurrence relations for the filter coefficients can be derived usin -Transform techniques applied to the expression of the transfer function considered as a -transform. Similarl the wa to obtain an expression for the filter coefficients new and eneral compared to formula previousl published b other authors is described in section IV. Finall in section V forms of eneratin functions for the transfer functions of sinc FIR filters are expressed. II. TRASFER FUCTIO OF sinc FILTERS The transfer function H written as H or ( ) = ( ) ( ) of sinc filters can be 2 ( ) H ( ) = (2) to which corresponds the followin manitude response H ω sin( ω / 2) ( e ) = (Fi. ). sin( ω / 2) Fi.. anitude responses of H 33 (e ω ) (a) H 34 (e ω ) (b) and H 43 (e ω ) (c) versus normalied anular frequenc ω. For simplicit we will consider the scaled transfer function ( ) = H ( ) ( ) = ( ) is a polnomial with deree (-) of the form () (3) Copriht (c) IARIA 20. ISB:

2 ICDT 20 : The Sixth International Conference on Diital Telecommunications ( ) ( ) = 0 = (4) where represent the coefficients of the impulse response of the sinc filters. b h and h =. ote that = 0 ( ) 0 ( ) = ( ) are related = 0 ( < 0) = ( ) ( ) and ( ) =. In the followin we will propose simple recurrence relations useful for the computation of sinc FIR filter coefficients and especiall a new eneral expression of these coefficients. III. RECURRECE RELATIOS FOR THE FILTER COEFFICIETS In this section several relations are derived accordin to different criteria such as simplicit low order varin index. Each of these independent relations can be useful dependin on the wa chosen for computation and implemented separatel. A. Calculation of from 3 coefficients with the same values and and lower indexes - - and -(+) Differentiatin ln ( ) usin (3) and multiplin b ( ) ields ( + ) 2 d ( ) = ( ) d ( + ) d ( ) ( + ) = d + ( + ) ( ) ( ) ( ) B usin basic -Transform techniques (6) leads immediatel to ( ) [ ( ) = + + ] (7) [ ( ) ( ) ] + + ( + ) which can also be written = + ( + ) + ( ) ( ( + ) ) ( ( + ) ) ( + ) An advantae of (8) is the fact that and eep constant values. Relative drawbacs are that the number of terms is not the lowest possible and that shifts in indexes increase with and ma be lare. This recurrence relation is the same as () in [7] but here it has been straihtforwardl derived usin -Transform techniques. For example for =8 =4 and = the relation ives: = [3( 8 ) 40( )] 0 0 (5) (6) (8) = (6 ) 40(4 ) = 204 B. Calculation of from 2 coefficients with the same 84 i.e. [ ] value and indexes (- ) and ( -) Differentiatin ( ) usin (3) ields ( + ) 2 d ( ) ( ) = 2 d ( ) d ( ) ( ) = d { ( ) + ( ) ( )} B usin basic -Transform techniques it can be immediatel deduced = ( + ) + () () (0) Advantaes of () are a ver low number of terms the fact that eeps a constant value and that shifts in and indexes equal onl one and occur separatel. This new and simple relation allows a fast recursive calculation of the impulse response coefficients. Therefrom computation is quite eas even permittin to fill up the start of the table of filter coefficients simpl b hand. For example for =8 =4 and = the relation ives: = ( ) i.e. = ( ) = C. Calculation of from 3 coefficients with the same value and indexes (- ) and ( -) and (- -) Usin (3) ( ) can be lined to ( ) as follows ( ) = ( ) (2) B usin basic -Transform techniques it can be immediatel deduced from (2) = + (3) Advantaes of (3) are a low number of terms with no multiplin factors the fact that eeps a constant value and that shifts in equal onl one. Relative drawbacs are that the number of terms is not the lowest possible and that shifts in index increase with and ma be lare. This simple relation allows a fast recursive calculation of the impulse response coefficients. Computation is quite eas even permittin to fill up the start of the table of filter coefficients simpl b hand. For example for =8 =4 and = the relation ives: = i.e. = = 204 D. Calculation of from + coefficients with the same lower value - and indexes (- ) =0.. Usin (3) ( ) can be expressed as follows Copriht (c) IARIA 20. ISB:

3 ICDT 20 : The Sixth International Conference on Diital Telecommunications ( ) ( ) = + (4) ( ) = ( ) (5) = 0 B usin basic -Transform techniques (5) ields = (6) = 0 An advantae of (6) is the fact that - eeps constant in the riht side of the equation. Drawbacs are that the number of terms the number of sets of coefficients involved in the relation as well as shifts in index increase with. For example for =8 =4 and = can be easil computed usin the values of the coefficients 7 4 weihted b the binomial coefficients : = = = 0 84 i.e. 0 = = 204 = =2 =3 =4 TABLE I Values of for = 4 and = 8 (-) Copriht (c) IARIA 20. ISB:

4 ICDT 20 : The Sixth International Conference on Diital Telecommunications In this section usin -Transform techniques efficient recursive formulae for computin the impulse response coefficients of sinc filters have been derived especiall () and (3). The values of the coefficients for 8 and 4 are iven in Table I. For smmetr reasons onl the coefficients for ( ) = 0.. IT are shown. 2 Futher investiations should be undertaen in order to evaluate precisel the number of operations and the computation time when usin such or such recurrence relation or a combination of them and in a wa the more efficient strate. IV. EERAL EXPRESSIO FOR THE FILTER COEFFICIETS Let us write Y ( ) under the form ( ) = ( ) ( ) (7) with Y ( ) defined as follows Y ( ) = (8) A. Explicit expression for the coefficients of Y ( ) Differentiatin ln Y ( ) and multiplin b ( ) Y ( ) ives dy ( ) () d Basic -Transform techniques allow to obtain + = (20) ( ) = Y ( ) This relation permits eas recursive computation of from. oreover (20) leads to the followin simple explicit relation for the coefficients + = B. eneral expression for the coefficients ( ) (2) of Developin ( ) in (7) leads to the followin relation between ( ) and Y ( ) ( ) = ( ) Y ( ) (22) = 0 Then usin the -Transform translation propert the coefficients can be written [ / ] + = ( ) = 0 (23) with =0..(-) and where [/] denotes the inteer quotient of and. It is worth notin that this new expression of the coefficients applies for all values of and. This simple formula allows to calculate an coefficient numericall. It offers the obvious advantae compared to previous results to be eneral and valid for all the coefficients of the impulse response of the sinc filters. In fact in [6] (cf. ()) and in [7] (cf. (2) and (3)) explicit expressions are iven onl for the ver first coefficients of the impulse response of the sinc filters. ote that here (23) leads immediatel to these explicit expressions for =0..(- ) and for = with =0 and = respectivel. For example for =8 =4 and = ([/]=) can be easil computed usin the products of + binomial coefficients i.e multiplied b ( ) : = ( ) = 4 = i.e. = = 204 In this section a simple eneral expression (23) has been obtained for rapid computation of an of the coefficients of sinc filters. As a summar useful recurrence relations and formula (8) () (3) (6) and (23) have been derived for computin the sinc FIR filter coefficients h =. V. EERATI FUCTIOS FOR THE TRASFER FUCTIOS OF sinc FIR FILTERS Let us consider (2) ( ) = ( ) with 0 ( ) =. A. Ordinar eneratin function ultiplin ( ) b x and summin with varin from 0 to infinit leads to the followin ordinar eneratin function Γ o ( x) = ( ) x = (24) = 0 x Conversel developin Γ o ( x) into series expansion enerates the expression of the filter transfer functions ( ) as the coefficients of x ( = 0.. ). B. Exponential eneratin function ultiplin ( ) b x and summin with! varin from 0 to infinit leads to the followin exponential eneratin function x x e ( x) ( ) e = 0! Γ = = (25) Copriht (c) IARIA 20. ISB:

5 ICDT 20 : The Sixth International Conference on Diital Telecommunications Conversel developin ( x) Γ e into series expansion enerates the expression of the filter transfer functions ( ) as the coefficients of x ( = 0.. ).! VI. COCLUSIO Sinc filters constituted of a cascade of sinc filters each of lenth are useful for diital sinal processin applications in various domains includin telecommunications. In this paper -Transform has been used as an efficient tool for derivin various formulae allowin to compute the impulse response coefficients of sinc FIR filters. In particular straihtforward recursive relations have been demonstrated. Further investiations should be made to evaluate the computation time and the more efficient strate when exploitin these relations separatel or in combination. oreover a simple eneral expression new compared to previous published formula has been iven valid for all the coefficients whatever their ran and the values of and. In addition eneratin functions for the sinc FIR filters transfer functions have been iven. REFERECES [] E. B. Hoenauer An economical class of diital filters for decimation and interpolation IEEE Trans. Acoust. Speech Sinal Process. vol. ASSP-2 2 pp Apr. 8 [2] J. C. Cand A use of double interation in sima delta modulation IEEE Trans. Commun. vol. CO- 3 3 pp arch 85 [3] H. eleis and P. Le Fur "A novel architecture desin for VLSI implementation of a FIR decimation filter" Proc. ICASSP'85 pp [4] V. Friedman D.. Brinthampt D.-P. Chen and T. Deppa J. P. Edward Jr. E.. Fields and H. eleis "A bit slice architecture for sima delta analo-todiital converters" IEEE J. Select. Areas Commun. vol. 6 pp Apr. 88 [5]. Shiraishi A simultaneous coefficient calculation method for sinc FIR filters IEEE Trans. Circuits Sst. I vol pp Apr [6] S. C. Dutta Ro "Impulse response of sinc FIR filters" IEEE Trans. Circuits Sst. II (3) pp [7] E. E. Hassan "Recurrence formula for impulse response coefficients of sinc FIR filter" Electron. Letters (5) pp [8] J. Le Bihan aximall linear FIR diital differentiators Circuits Sst. Sinal Process. vol pp [] J. Le Bihan Coefficients of FIR diital differentiators and Hilbert transformers for midband frequencies IEEE Trans. Circuits Sst. II vol arch 6 pp Copriht (c) IARIA 20. ISB:

COMPARISON OF CLASSICAL CIC AND A NEW CLASS OF STOPBAND-IMPROVED CIC FILTERS FORMED BY CASCADING NON-IDENTICAL COMB SECTIONS

COMPARISON OF CLASSICAL CIC AND A NEW CLASS OF STOPBAND-IMPROVED CIC FILTERS FORMED BY CASCADING NON-IDENTICAL COMB SECTIONS FACTA UIVERSITATIS Series: Electronics and Energetics Vol. 29, o 1, March 216, pp. 61-76 DOI: 1.2298/FUEE16161M COMPARISO OF CLASSICAL CIC AD A EW CLASS OF STOPBAD-IMPROVED CIC FILTERS FORMED BY CASCADIG