Structures for Interpolation, Decimation, and Nonuniform Sampling Based on Newton s Interpolation Formula

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1 Structures for Interpolation, Deciation, and Nonunifor Sapling Based on Newton s Interpolation Forula Vesa Lehtinen, arkku Renfors To cite this version: Vesa Lehtinen, arkku Renfors. Structures for Interpolation, Deciation, and Nonunifor Sapling Based on Newton s Interpolation Forula. Laurent Fesquet and Bruno Torrésani. SAPTA 09, ay 2009, arseille, France. Special session on efficient design and ipleentation of sapling rate conversion, resapling and s, <hal > HAL Id: hal Subitted on 30 Jan 200 HAL is a ulti-disciplinary open access archive for the deposit and disseination of scientific research docuents, whether they are published or not. The docuents ay coe fro teaching and research institutions in France or abroad, or fro public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de docuents scientifiques de niveau recherche, publiés ou non, éanant des établisseents d enseigneent et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Structures for Interpolation, Deciation, and Nonunifor Sapling Based on Newton s Interpolation Forula Vesa Lehtinen and arkku Renfors Departent of Counications Engineering, Tapere University of Technology P.O.Box 553, FI-330 Tapere, Finland {vesa.lehtinen,arkku.renfors}@tut.fi Abstract: The variable fractional-delay (FD) filter structure by Tassart and Depalle perfors Lagrange interpolation in an efficient way. We point out that this structure directly corresponds to Newton s interpolation (backward difference) forula, hence we prefer to refer to it as the Newton FD filter. This structure does not function correctly when the fractional delay is ade tie-variant, e.g., in saple rate conversion. We present a siple odification that enables tie-variant usage such as fractional saple rate conversion and nonunifor resapling. We refer to the new structure as the Newton (interpolator) structure. Alost all advantages of the Newton FD structure are preserved. Furtherore, we suggest that by transposing the Newton interpolator we obtain the transposed Newton structure which can be used in deciation as well as reconstruction of nonuniforly sapled signals, analogously to the transposed Farrow structure. The presented structures are a copetitive alternative for the Farrow structure faily when low coplexity and flexibility are required.. Introduction In [][2][3], Tassart and Depalle as well as Candan derive an efficient ipleentation structure for FD filters, depicted in Fig., fro Lagrange s interpolation forula. It turns out that the obtained filter structure directly corresponds to Newton s (backward difference) interpolation forula [4] (with soe subexpression sharing) which indeed is equivalent with Lagrange interpolation [5]. Newton s backward difference forula is τ f ( t + τ) ( ) f () t ----, () = 0! where τ ( ) = ( τ + k) k = 0 is the rising factorial, and is the backward difference operator such that f () t = f () t f ( t ) and 0 f () t = f () t, resulting in f () t = ( ) k f ( t k). (3) k = 0 k Newton s backward difference forula provides an efficient eans to realise piecewise-polynoial interpolation for DSP. Its coplexity is only O() (where is the interpolator order) cf. equivalent Lagrange ipleentations based on the Farrow structure [6] having O( 2 ) coplexity [3]. The subfilters are ultiplier-free and extreely siple. The structure is odular, as highlighted by the grey shading in Fig., and the interpolator order can be changed in real tie [3]. Unfortunately, the structure presented in Fig. does not function correctly in saple rate conversion (SRC). Because the ultiplications are perfored between the subfilters, aking the tie-variant will result in incorrect output. This is because each output saple should only depend on the current value of the delay paraeter D; in Fig., past values of D contribute to the output through the delayed paths through the subfilters. Therefore, the structure in Fig. is only useful in single-rate, tie-invariant or slowly-varying fractional-delay filtering. We propose a slightly odified structure that allows arbitrary resapling, including increasing the saple rate by arbitrary, also fractional, factors (fractional interpolation). We also point out that the structure can be transposed to obtain a deciator structure that possesses all the advantages of the Newton interpolation structure. (2) This work was supported by the Graduate School in Electronics, Telecounications and Autoation (GETA).

3 z D D D D+ Figure. The fractional-delay filter structure proposed in [][3], based on Newton s interpolation forula. 2 3 Dt () Dt () Dt () 2 Dt () + Figure 2. The Newton interpolator structure suitable for saple rate conversion. The hold & saple () blocks perfor the sapling at the output saple instants. 2. The Newton structure for interpolation In order to allow fractional SRC and arbitrary resapling, the Newton structure ust work correctly with a tie-variant fractional delay. This is achieved through two siple steps: (i) We invert the suation order at the output part of the structure fro that presented in [][3] (this was already done in [2]). (ii) The tie-varying ultiplications can now be ipleented in the high-rate part between the adders. The iproved structure is shown in Fig. 2. We refer to it as the Newton interpolator structure or the Newton structure for short. Also the iproved structure is odular, peritting changing the interpolator order in real tie. In single-rate FD filtering, the iproved structure is equivalent to [][2][3]. In Fig. 2, the blocks stand for hold & saple, i.e., each output saple obtains the value of the previously arrived input saple. In fractional interpolation, i.e., increasing the saple rate by a fractional factor, we use the coon notation illustrated in Fig. 3. The tie interval between the previous input saple and the next output saple to be generated is expressed using the fractional interval variable µ which is noralised with respect to the input saple interval so that µ [0, ). Interpolation of uniforly spaced input saples can be odelled as convolution [5], leading to the generic odel depicted in Fig. 4 [7]. The continuous-tie (CT) linear tie-invariant (LTI) odel filter is piecewise polynoial, with + pieces, each with duration equal to the input saple interval T in. Hence the ipulse response length is ( + )T in. Input saples µ l T in ( k )T in ( l ) µ l+ T in Output saples kt in ( k +)T in l ( l+) Figure 3. Definition of the fractional interval µ for interpolation. xn [ ] DT x CT () in Figure 4. factors. CT H CT ( f ) The coposite transfer function of cascaded subfilters is T in y CT () t n x CT () t = xn [ ]δ t F in n F in yn [ ] = y CT ( n out The generic odel for SRC by arbitrary ( z ) = ( ) n z n n = 0 n cf. (3). The output of the interpolator is, (4)

4 Dt () Dt () Dt () 2 Dt () z z Figure 5. The transposed Newton structure for deciation and reconstruction of signals fro nonuniforly spaced saples. y( ( k + µ )T in ) h( ( n + µ )T in )xk [ n] n = 0 xk [ n] ( ) n ( ) ( D 0 µ ) n = 0 = n n! (2.) Input saples µ l Output saples µ l+ where for 0, and = 0, n < 0 n> n ( x) = ( x k) (6) k = 0 is the falling factorial. The delay of the interpolator is D 0 T in. The paraeter D 0 can be chosen quite freely, but the best aplitude response and linear phase response are obtained with D 0 = ( + ) 2 []. The continuous-tie odel ipulse response of the interpolator is then (cf. the expression of the filter input in Fig. 4) h( ( n + µ )T in ) ( ) n+ ( D 0 µ ). = n n! T in The reversed suation order in the high-rate part coes with a price: the structure is ore costly to pipeline than those in [][3] because the signal paths cannot share pipeline registers. 3. The transposed Newton structure There exists a duality between deciation and interpolation that allows transforing a deciator into an interpolator and vice versa through network transposition [7]. By transposing the Newton interpolator, we obtain the structure depicted in Fig. 5. We refer to this as the transposed Newton structure. The transpose is obtained by inverting the flow direction of all signals and replacing each block with its dual. For instance, the block is replaced with the accuulate & dup. There exist a nuber of definitions for duality, including the adjoint. Here we use the generalised duality/transpose as defined in [7]. (5) (7) ( k ) k ( l )T in lt in ( l+)t in ( k +) Figure 6. Definition of the fractional interval µ for the transposed structure (dual of interpolation). () block, which sus up all its input saples since the previous output saple. This is also the ost straightforward way to obtain the transposed Farrow structure fro the Farrow structure 2 [9]. The output saples of the transposed Newton structure are uniforly spaced, but the input saples ay arrive at arbitrary tie instants. The generic SRC odel (Fig. 4) is valid also for the transposed Newton structure. The odel ipulse response is again piecewise-polynoial, now with the piece duration equal to the output saple interval. The odel ipulse response is obtained by replacing T in with in (7) and redefining µ according to Fig. 6 (reflecting the duality between deciation and interpolation). For an input saple arriving at tie instant t, the fractional interval is µ () t = t t [0, ). For fractional deciation, the fractional interval for the l th input saple is lt µ in l = lt in The ipulse response in the generic odel is now h( ( n + µ ) ) ( ) n + ( D 0 µ ) = n n! 2. The structure in [8] (transposed structure I in [9]) is not the true transpose of the Farrow structure even though the duality of responses holds. (8) (9) (0)

5 with integer n. Again, D 0 = ( + ) 2 for the best response. In the frequency response, the odel filter has + zeros at each (nonzero) integer ultiple of the output saple rate, hence realising antialiasing regardless of the deciation factor. The transposed Newton structure is able to receive input saples at arbitrary tie instants, which akes it a potential building block for reconstruction of signals fro nonuniforly spaced saples (e.g., in algoriths like [0][]), as earlier suggested for the transposed Farrow structure in [2]. The transposed Newton structure shares the advantages and disadvantages of the Newton interpolator, such as odularity, O ( ) coplexity and the inefficient zero locations. 4. Coputational coplexity In interpolation by factor R, the Newton structure will perfor ( + R) additions and ( + R) ultiplications per input saple on average. In deciation by R, the transposed Newton structure will perfor ( R ) ( + ) + 2 additions and ( + R) ultiplications per output saple. The first ter in the addition count coes fro the block. ultiplication by a constant inverse of a sall integer requires only few additions/subtractions. Unabiguous coplexity coparison between the proposed structures and alternatives, ainly the Farrow faily, would require specifying the ipleentation technology and the SRC factor. However, the following points can be ade: (i) The basis ultipliers are ore coplex in the Newton structures (integer part present in the tie-variant coefficients) than in Farrow structures (no integer part). Hence, large SRC factors are unfavourable to the Newton faily. (ii) If the Lagrange response suffices, the ultiate siplicity of the subfilters akes the Newton faily superior to the Farrow structure when the SRC factor is sall. (iii) The response of the Newton structures can be iproved only by increasing the order (i.e., nuber of stages). In designs with a low oversapling factor and/or strict perforance requireents, this ay lead to a very high filter order. In such cases, an optiised Farrow design with a non-lagrange response will have a lower coplexity and saller delay. 5. Conclusions The proposed structures allow efficient piecewise Newton interpolation for SRC and arbitrary resapling as well as its dual for deciation and reconstruction of nonuniforly sapled signals. The advantages of the proposed structures include low, O() coplexity (high orders are feasible at the cost of a long delay), very siple subfilters and run-tie adjustability of the filter order. As a drawback, the basis ultipliers running at the high-rate end of the filter have longer wordlengths than in the Farrow counterparts. Due to their siplicity, the Newton structures ay be useful as building blocks of ore coplicated algoriths for interpolation, deciation, and reconstruction of nonuniforly sapled signals. References: [] S. Tassart and Ph. Depalle, Fractional delays using Lagrange interpolators, in Proc. Nordic Acoustic eeting, Helsinki, Finland, 2 4 June, 996. [2] S. Tassart, Ph. Depalle, Analytical approxiations of fractional delays: Lagrange interpolators and allpass filters, in Proc. IEEE Int. Conf. Acoust., Speech, and Signal Proc. (ICASSP 97), 2 24 Apr 997, pp [3] Ç. Candan, An efficient filtering structure for Lagrange interpolation, IEEE Signal Processing Letters, Vol. 4, No., Jan 2007, pp [4] E.W. Weisstein, "Newton s Backward Difference Forula." Available: NewtonsBackwardDifferenceForula.htl. Visited: 22 Jan [5] E. eijering, A chronology of interpolation: Fro ancient astronoy to odern signal and iage processing, in Proc. of the IEEE, Vol. 90, No. 3, ar 2002, pp [6] C.W. Farrow, A continuously variable digital delay eleent, in Proc. IEEE Int. Syp. Circ. Syst. (ISCAS 88), Espoo, Finland, June 988, pp [7] R.E. Crochiere, L.R. Rabiner, ultirate Digital Signal Processing, Prentice-Hall, 983. [8] T. Hentschel, G. Fettweis, Continuous-tie digital filters for saple-rate conversion in reconfigurable radio terinals, in Proc. European Wireless, Dresden, Gerany, Sep 2000, pp [9] D. Babic, J. Vesa, T. Saraäki,. Renfors, Ipleentation of the transposed Farrow structure, in Proc. IEEE Int. Syp. Circ. Syst., ay 2002, pp. IV-5 IV-8. [0] F. arvasti,. Analoui,. Gashadzahi, Recovery of signals fro nonunifor saples using iterative ethods, IEEE Trans. Signal Proc., Vol. 39, No. 4, Apr 99, pp [] F.A. arvasti, P.. Clarkson,.V. Dokic, U. Goenchanart, C. Liu, Reconstruction of speech signals with lost saples, IEEE Trans. Signal Proc., Vol. 40, No. 2, Dec 992, pp [2] D. Babic and. Renfors, Reconstruction of non-uniforly sapled signal using transposed Farrow structure, in Proc. Int. Syp. Circ. Syst. (ISCAS), Vancouver, Canada, ay 2004, Vol. III, pp