Abstract. 1 Introduction. 2 Continuous-Time Digital Filters

Transcription

1 Continuous-Time Digital Filters for Sample-Rate Conversion in Reconfigurable Radio Terminals Tim Hentschel Gerhard Fettweis, Dresden University of Technology, Mannesmann Mobilfunk Chair for Mobile Communications Systems, D-0062 Dresden, Germany Abstract Reconfigurable radio terminals must cope with a multitude of master clock rates of diverse mobile communications air interfaces. Digital sample rate conversion (SRC) is an elegant way to enable the processing of signals with sample rates incommensurate to the clock rate of a nonsynchronized analog-to-digital converter. SRC is a process of resampling thus, requires anti-aliasing filtering. The well-known Farrow-structure provides a means of implementing digital SRC on a parameterizable hardware platform enabling the adaptation to different rate change factors. Still, the Farrow-structure can only implement filters with poor anti-aliasing characteristics. The transposed Farrow-structure introduced in this article overcomes these problems thus, represents a perfect means for SRC in reconfigurable radio terminals. A companion article comprehensively covers applications. Introduction Reconfigurable radio is supposed to be a solution for both equipment manufacturers mobile network operators, to keep pace with the fast evolution of mobile communications stards. The ultimate goal of this idea is the software radio which is completely independent from the air-interface. Different air-interfaces are based on different master clock rates which a reconfigurable radio needs to cope with. A solution to this is to exchange the master crystal when reconfiguring the radio. If this is unacceptable one could fill up the empty space of a terminal with a selection of available crystals just to be on the safe side always to have an appropriate crystal available. Both ideas are neither elegant nor cost-efficient. A way-out from this dilemma is the implementation of one fixed master crystal which mainly controls the analog-to-digital the digital-to-analog converters, to calculate the data at a stard-specific clock rate by means of digital samplerate conversion (SRC). It is not within the scope of this article to go into detail about SRC. However, it should be stressed that SRC is not interpolation. SRC is a process of resampling thus, causes spectral repetition known as imaging. If there are no restrictions with respect to the signal (e.g., regarding oversampling) also aliasing must be expected from SRC. Since aliasing destroys signal contents it can be concluded that anti-aliasing is the most important property to be obeyed in any SRC system [6, 7]. Based on this spectral interpretation of SRC it can be concluded that interpolation is an appropriate method for SRC only if the signals are considerably oversampled i.e., if the interpolation filter attenuates all possible aliasing-components by attenuating the (relatively narrow) image components. An equation describing the filtering resampling of SRC is (see also e.g., [4]) y(m )= k= x(k ) h(m k ) () It describes a convolution-like operation between the samples of the input signal x(k ) samples of the continuous-time impulse response h(t) of the required filter, delivering the signal y(m ) at a new sample rate. Ideally, h(t) is an ideal low-pass reconstruction filter []. Each calculation of an output sample requires another set of samples of h(t). Hence, Eq. () describes a time-varying system. Since can be arbitrary, it is necessary to know the continuous-time course of the filter s impulse response h(t), rather than some dedicated samples. Therefore, the filter can be regarded as a continuous-time filter within a time-varying discrete-time system. Restricting leads to rational factor SRC, if = L M, with L,M N + (2) integer factor SRC, if either L =, or M =, the well-known discrete-time convolution (i.e., no SRC) for L = M =. We shall not apply these restrictions in this article. Still, solutions for rational or integer factor SRC can be obtained by substituting Eq. (2) to the results of this article. 2 Continuous-Time Digital Filters Eq. () suggests that SRC requires a continuous-time digital filter. The filter is digital since certain (quantized) samples of the input signal the impulse response are involved in the computation of one output sample. Still, since

2 order of the polynomials x(k ) y(m ) c 0 (0) c 0 () c 0 (2) c 0 (N ) x 0 (m ) c (0) c () c (2) c (N ) x (m ) c n (0) c n () c n (2) c n (N ) x n (m ) number of polynomial pieces µ m Figure : Farrow-Structure ( = ) the samples of the impulse response of the respective filter depend on the actual values of,,m, there is no way of selecting storing a certain set of samples of h(t). Generally, the complete continuous-time impulse response h(t) must be known. For rational integer factor SRC the system described by Eq. () becomes periodically time-varying. Hence, only a certain set of samples of h(t) is involved in the computations. This set of coefficients can be stored employed in the typical systems for rational factor SRC e.g., polyphase filters []. If L M get large, the necessary memory size might be infeasible. Thus, also for rational factor SRC it is very attractive to calculate the necessary samples of h(t) on dem as it is required for arbitrary factor SRC. In order to keep the effort low for calculating the samples of h(t), simple functions describing continuous-time impulse responses are sought for. Polynomials are such simple functions. Therefore, polynomial filters are promising cidates to be employed in systems for SRC. We shall limit the class of polynomial filters to piecewise polynomial impulse responses composed from pieces of equal length. Given polynomial pieces of degree n length { n h j (t)= c i( j) ( ) t i 0 t < (3) 0 else a piecewise impulse response composed from N polynomials h j (t) is N h(t)= j=0 h j (t j ) (4) An equivalent description is h(t)=h t ( t ) t, 0 t < N (5) where denotes the floor-operation i.e., the largest integer smaller or equal than ( ). Eq. (5) might seem to be a somewhat odd description. Still, it enables to give up the usual limitations on t for the polynomial pieces in Eq. (3) by shifting them to the description of h(t) itself (Eq. (5)). Thus, it becomes possible to directly substitute Eq. (3) to Eq. (5). n ( t ) ( t t ) i h(t)= c i, 0 t < N (6) An open question is the choice of. Therearetwo choices for which Eq. (6) can be simplified considerably a hardware structure can be derived, namely = =. The next two sections are dedicated to these simplifications. 3 The Farrow-Structure Substituting Eq. (6) to Eq. () setting = yields n ( ) mt2 k y(m )= x(k ) c i k= ( m T ) 2 mt2 kt i (7) k, 0 m k < N

3 ˆx 0 (k ) ˆx (k ) ˆx n (k ) µ k x(k ) c 0 (0) c (0) c n (0) y(m ) number of polynomials c 0 () c () c n () c 0 (N ) c (N ) c n (N ) order of the polynomials Figure 2: Transposed Farrow-Structure for = ( sts for Integrate--Dump) which can be simplified to where x i (m )= n y(m )= k= µ m = m x i (m ) (µ m ) i (8) ( x(k ) c i m T ) 2 k mt2 (9) [0,) (0) µ m is the so-called intersample position, indicating the distance between the previous input sample the current output sample (Figure 3). Eqs. (8)-(0) describe an implementation of SRC with polynomial filters which is commonly known as the Farrow-structure [2, 3]. It is sketched in Figure. It can also be interpreted as a polyphase interpolator with an infinite number of polyphase branches that are implemented by just one reference polyphase branch, a polynomial description of how to calculate the remaining ones. The reference polyphase branch can be obtained by setting µ m = 0(i.e.,t = k for k = 0,,...,N in Eq. (6)) thus yielding c 0 (l). It represents the very samples of the impulse response h(t) which mark the starting points of the individual polynomial pieces. These starting points (i.e., the reference branch) the order of the connecting polynomial pieces determine the transfer characteristics of the SRC system. The higher the order of the polynomial pieces the better the impulse response h(t) can be matched to the application. If high order polynomials are infeasible, it is also possible to use shorter polynomials of lower order. In this case more reference polyphase branches (i.e., starting points of polynomial pieces) are required. This can be achieved by decreasing the length of the polynomial pieces by a factor = () leading to time-varying coefficients in the Farrowstructure. This approach can be seen as a generalization of the Farrow-structure has been introduced as the generalized Farrow-structure [8]. 4 The Transposed Farrow-Structure Having learned that with Eq. () a generalization was possible compared to =, we now start instantly with the more general case of setting =. Substituting Eq. (6) to Eq. () yields y(m )= k= n x(k ) ( m k 0 m k < N c i ( m k m k ) i, ) (2)

4 (m ) m (m + ) output samples: µ k+ µ k+2 input samples: µ m time (k ) k (k + ) (k + 2) Figure 3: Sample Time Relations ( = ) which can be simplified to n y(m )= k= ˆx i (k ) c i (m k T ) (3) with ˆx i (k )=x(k ) (µ k ) i (4) µ k = k T k [0,) (5) For = the quantity µ k indicates the distance between the current input sample the next output sample. Thus, it is equal to the intersample position µ m of Eq. (0) (Figure 3). Eqs. (3)-(5) describe a novel structure implementing SRC with piecewise polynomial filters with a piece length of =. It is shown in Figure 2 for the case =. Applying the concepts of generalized transposition of networks [] to the structure of Figure it can be verified that its generalized transpose is the structure of Figure 2. Therefore, it should be named the Transposed Farrow-Structure, orthegeneralized Transposed Farrow- Structure in case of >. While the Farrow-structure can be derived from the respective equations relatively easily, this is not so obvious for the Transposed Farrow-structure. The most important equation to underst is Eq. (3). Its second sum describes a convolution-like operation. For = it can be observed that for any fixed m certain consecutive samples ˆx i (k ) might be weighted with the same c i (l) before being summed up contributing to a new output sample. This comes from the fact that k does not necessarily change when incrementing k (see Figure 3 where two input samples arrive at (k + ) (k + 2) before a new output sample is generated at (m + ) ). The respective summation can be realized by means of an integrate-dump circuit. This principle does not change for >. In this case different (time-varying) c i (l) might be used for weighting the ˆx i (k ) before integration. 5 Comparison Two structures for SRC with piecewise polynomial filters have been presented. The fundamental difference about the performance of the two is the length of the polynomial pieces. A typical example for = are Lagrange interpolation filters. These filters have transfer zeros clustered about integer multiples of the input sample rate thus, attenuate the image components [9]. They have very poor anti-aliasing performance. By giving up the interpolation constraint several optimization procedures can be employed resulting in spreading the clustered transfer zeros thus, widening the stop bs [0]. Still, it is not possible to cluster the transfer zeros about the aliasing components i.e., about integer multiples of the output sample rate 2. This is the moment where the transposed Farrowstructure comes into play. By using the same polynomial functions as with the Farrow-structure, still, stretched (or compressed) to meet =, the transfer zeros are clustered about the aliasing components at integer multiples of thus, attenuate the aliasing components. Hence, the comprehensive literature on polynomial interpolators can be exploited to design polynomial sample-rate converters that perform anti-aliasing. 6 Conclusions On the basis of the description of polynomial impulse responses of Eq. (6) it became possible to derive the wellknown Farrow-structure moreover, the novel transposed Farrow-structure. The Farrow-structure is principally a polynomial polyphase-interpolator. With the introduction of the transposed Farrow-structure a polynomial polyphase-decimator has been presented. It can realize anti-aliasing which is necessary in most SRC applications. A typical application is SRC in reconfigurable radio terminals where the hardware platform for SRC must be parameterizable. The parameterizability of both the original the transposed Farrow-strutures is given by the fact that they implement continuous-time impulse responses. A companion paper [5] deals with the application of the transposed Farrow-structure for combined filtering for SRC synchronization.

5 7 Acknowledgment Parts of this work have been supported by the European Commission, ACTS project SORT (Software Radio Technology). Literature [] R.E.CrochiereL.R.Rabiner. Multirate Digital Signal processing. Prentice-Hall, 983. [2] L. Erup, F. M. Gardner, R. A. Harris. Interpolation in Digital Modems - Part II: Implementation Performanc. IEEE Transactions on Communications, COM-4(6): , une 993. [3] C. W. Farrow. A Continuously Variable Digital Delay Element. In Proc. IEEE International Symposium on Circuits Systems (ISCAS 88), pages , Espoo, Finl, une 988. [4] F. M. Gardner. Interpolation in Digital Modems - Part I: Fundamentals. IEEE Transactions on Communications, COM-4(3):50 507, Mar [5] M. Henker G. Fettweis. Combined filter for sample rate conversion, matched filtering, symbol synchronization in software radio terminals. In this conference. [6] T. Hentschel G. Fettweis. Sample Rate Conversion for Software Radio. IEEE Communications Magazine, pages 2 0, Aug [7] T. Hentschel, M. Henker, G. P. Fettweis. The Digital Front-End of Software Radio Terminals. IEEE Personal Communications, 6(4):40 46, Aug Biographies Tim Hentschel (hentsch@ifn.et.tu-dresden.de) received his MSc/Dipl.-Ing. degree in electrical engineering from King s College London, University of London, U.K., the Dresden University of Technology, Germany, in , respectively. From 995 to 996 he was with Philips Communications Industries, Nurnberg, Germany. Since May 996 he is with the Mannesmann Mobilfunk Chair for Mobile Communications Systems at the Dresden University of Technology, Germany, working towards his PhD. His current research interests include software radio, specifically the design investigation of digital signal processing algorithms for reconfigureable front-ends. Gerhard Fettweis (fettweis@ifn.et.tu-dresden.de) received his MSc/Dipl.-Ing. PhD. degree in electrical engineering from the Aachen University of Technology (RWTH), Germany, in , respectively. From 990 to 99 he was a Visiting Scientist at the IBM Almaden Research Center in San ose, CA, working on signal processing for disk drives. From 99 to 994 he was Scientist with TCSI, Berkeley, CA, responsible for signal processor developments for mobile phones. Since September 994 he holds the Mannesmann Mobilfunk Chair for Mobile Communications Systems at the Dresden University of Technology, Germany. He is an elected member of the SSC Society s Administrative Commitee, of IEEE ComSoc Board of governors, since , respectively. He has been accociate editor for IEEE Trans. on CAS II, now is associate editor for IEEE -SAC wireless series. [8] T. A. Ramstad. Fractional Rate Decimator Interpolator Design. In Proceedings of the IX European Signal Processing Conference (EUSIPCO 98), pages , Isl of Rhodes, Greece, Sept [9] R. W. Schafer L. R. Rabiner. A Digital Signal Processing Approach to Interpolation. Proceedings of the IEEE, 6(6): , une 973. [0]. Vesma. Optimization Applications of Polynomial-Based Interpolation Filters. PhD thesis, Tampere University of Technology, P.O.B. 527, FIN- 330 Tampere Finl, May 999. ISBN