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C ONVERSION BETWEEN A RBITRARY S AMPLING R ATES : A N I MPLEMENTATION C OST T RADE - OFF S TUDY FOR THE FAMILY OF FARROW S TRUCTURES F. M. Klingler1, H. G. G¨ockler2 Robert Bosch GmbH, Hildesheim, Germany, [email protected] 2 Digital Signal Processing Group, Ruhr-Universit¨at Bochum, Germany, [email protected] 1

Abstract— We investigate and compare the computational loads of different implementations of a system for asynchronous sample rate reduction: The cascade of a preceding synchronous polyphase L-Interpolator, an asynchronous sample rate converter applying five different Farrow structures being based on Lagrange interpolation and a successive synchronous polyphase M-decimator. For a limited range of requirements, always those designs with minimum implementation cost are identified. As a result of this trade-off study, it turns out that a cascade system applying the Modified Farrow structure exploiting coefficient symmetry and with the decimation property represents the most efficient implementation. Index Terms— Asynchronous Sample Rate Conversion, Polynomial Interpolation, Farrow Structure

I. I NTRODUCTION In almost all fields of digital signal processing there is an omnipresent need to change or adapt sampling rates by an integer, rational or non-rational factor, respectively, i.e. by any arbitrary ratio. We define this unrestricted sample rate conversion factor by the ratio R=

fi fo

(1)

of the input and output sampling rates of the sample rate converter, fi and fo , respectively. Note that R may even slowly vary with time. All these requirements are, for instance, encountered in any mobile handset receiver [1–3]: The digital front end is typically operated at sample rates derived from a local oscillator, whilst the demodulation process has to be

fi

f1

L SRC u(lTi )

x(nT1 )

f2

A-SRC

fo

SRC M

y(mT2 )

w(kTo )

Fig. 1. General System for Asynchronous Sample Rate Conversion

performed at (integer multiples of) the symbol rate of the incoming data, where both clocks are relatively asynchronous, and the latter is generally time-varying due to Doppler shift. All of the aforementioned tasks of sample rate conversion (SRC) can completely be performed in the digital domain [4–6]. Obviously, the most general approach to convert the sampling frequency by a nonrational conversion factor R from one rate to another rate, where the latter is asynchronous to the former one, comprises all other cases itemized above. Hence, the investigations reported in this contribution will focus on this most general case of asynchronous SRC, subsequently denoted by A-SRC. In the digital domain, A-SRC is commonly implemented by use of the Farrow structure [7] whereof, in the meantime, a considerable variety of modifications have been proposed: The Modified [8], the Generalized [9], different Transposed [10, 11] and the Prolonged [12] Farrow structures, where their optimum applications depend on the prescribed conversion ratio R and minimum stopband attenuation A. To reduce the computational load of A-SRC, the Farrow structure is frequently combined with a preceding L-interpolator for L-fold synchronous sample rate increase (denoted by L-SRC), and/or a successive M-decimator for M-fold synchronous decima-

tion (denoted by M-SRC), respectively, as depicted in Figure 1, where L, M ∈ N. This cascade combination, subsequently denoted by LAM-SRC, has the potential of an overall system implementation with minimum expenditure by optimum allocation of subtasks to subsystems. Since no general rule for the optimum allocation is known [4], we will present an implementation cost trade-off study to approach this optimum experimentally. The state-of-the-art of the aforementioned varieties of the Farrow structure (FS), along with the underlying principle of polynomial-based interpolation [6, 13], is deployed by a concise survey in Section II. For synchronous L-SRC and M-SRC we constantly assume linear-phase FIR polyphaseinterpolators and -decimators [4, 14] where, for convenience, the overall filter length is always chosen such that all polyphase-branch filters have to perform the same number Nbr,Int (Nbr,Dec ) of multiplications. Furthermore, the memory of one polyphase-branch filter is shared with all other branch filters, calling for a single delay line of length Nbr,Int (Nbr,Dec ). As a result, coefficient symmetry cannot be exploited to reduce computation [4]. Concluding, however, it should be mentioned that we intentionally renounce the use of the generally more efficient multistage FIR polyphase approaches to synchronous SRC [4, 14], since our investigations are focused on a consistent comparison of different Farrow structures being part of an LAM-SRC cascade system. To compare the implementation effort of different LAM-SRC realizations that meet the same requirements with minimum margin, we adopt the (normalized) operation rate per output sample # "N NA M X X C= WM (m)fop (m) + WA (a)fop (a) To m=1

operation rates, as defined by (2), of various LAMSRC systems in dependence of different Farrow kernels [8–12] for a limited range of requirements and assumptions: 1) Polynomial interpolation exclusively applies symmetric Lagrange coefficients with a maximally flat linear-phase frequency response [13]. 2) Approximate equiripple design of the overall LAM-SRC system, where the linear-phase L-SRC and/or M-SRC polyphase-subsystems compensate for the decay of the magnitude response of the FS at the passband cut-off frequency. 3) We focus our investigations on the case of decimation (R > 1) of a lowpass signal of bandwidth b, where the sampling rate of the respective SRC output signal is always fixed to fo = 1/To = 3 · 2b = 6b. 4) The minimum stopband attenuation in aliasing domains is varied in a wide range, while the maximum passband deviation is kept constant. For each combination of a decimation factor R and prescribed stopband attenuation, an asynchronous SRC system is designed with the aforementioned Farrow kernels. All non-specified parameters are always determined such that the overall system performance meets the respective requirements with minimum margin. Finally, the respective designs calling for minimum implementation cost C are identified. Subsequently, in section II, a concise survey of polynomial interpolation and the Farrow structures to be compared is given. In section III, the results of our investigations are reported along with a specification of the range of requirements. Finally, the obtained results are discussed and interpreted in the concluding section IV.

a=1

(2)

as a cost function [4, 15], where NM (NA ) denotes the overall number of multiplications (additions) performed at individual clock rates fop (m) (fop (a)) and To = 1/fo is the sampling period of the output signal. Moreover, individual weighting by WM (m) (WA (a)) has the potential to consider different word lengths. Since finite arithmetic is beyond the scope of this investigation and having in mind a signal processor implementation, we assume WM (m) = WA (a) = W ∀m, a. As an additional yet secondary cost measure we consider the amount of memory for coefficients and state variables. The aim of this trade-off study is to compare the

II. P OLYNOMIAL I NTERPOLATION AND T HE FAMILY OF FARROW S TRUCTURES The A-SRC subsystem of Figure 1 can be modelled by the hybrid discrete/analogue system of Figure 2 [5,6]: The discrete-time input sequence x(nT1 ), sampled at f1 = 1/T1 , is converted into the continuoustime signal xc (t) =

∞ X

x(nT1 )δc (t − nT1 ) ,

(3)

n=−∞

where δc (t) is the continuous-time Dirac impulse. The convolution of xc (t) and the impulse response

x(nT1 )

xc (t)

yc (t)

x((n+S/2)T1)

T2

hint(t)

DAC

y(mT2 )

Fig. 2. Hybrid digital/analogue model for A-SRC

z-1

hint (t) of an analogue interpolation low pass filter yields the reconstructed signal Z ∞ yc (t) = xc (τ )hint (t − τ )dτ (4)

z-1

=

x(nT1 )hint (t − nT1 ) .

x(nT1 )hint (mT2 − nT1 )

(6)

n=−∞

is obtained, which immediately allows for an alldigital implementation of the A-SRC subsystem of Figure 2. For a general A-SRC subsystem for sample rate conversion by any arbitrary ratio R according to (1), the determination of the samples hint (mT2 −nT1 ) of the continuous impulse response must be as simple as possible, since these samples are currently changing and, hence, have to be adapted during uninterrupted system operation. This implementational requirement is best met by a polynomial definition of the impulse response hint (t). Thus, its realization only calls for the basic operations used in digital signal processing [15]. More specifically and favourably, the impulse response polynomials are defined piecewise [7] according to hint (t) =

η S−1 XX

z-1

c1 (1)

z-1

ch (S-1)

z-1

c1 (S-1) vh (nmT1)

c0 (1)

c0 (S-1) v1 (nmT1)

mm

By sampling signal (5) at f2 = 1/T2 , the discretetime A-SRC output signal y(mT2 ) =

c0 (0)

y(mT2)

Fig. 3. The (interpolating) original Farrow Structure

Hence, the original FS [7] and those four modifications thereof [8–12], to be used as an A-SRC subsystem of LAM-SRC according to Figure 1 in our trade-off study, are presented subsequently to sufficient detail. A. The (original) Farrow Structure According to the original proposal of Farrow [7], the length of the polynomial segments is set to the sampling period of the input signal (Fig. 1): τ = T1 = 1/f1 . Hence, the A-SRC output signal (6) results in y(mT2 ) =

(7)

s=0 r=0

with the basis functions r   t−sτ τ fr (t, τ, s) =  0

sτ ≤ t < (s+1)τ

(8)

else .

Here, S indicates the number of polynomial segments of identical time duration τ and η represents the order of the polynomial segments with fixed coefficients cr (s), where s ∈ {0, ..., S −1}, r ∈ {0, ..., η} and S = η+1 in the special case of Lagrange polynomials. The family of Farrow structures is the most suitable tool for an efficient implementation of (7) and (8).

η X

vr (nm T1 ) · µm r ,

(9)

r=0

where vr (nm T1 ) =

cr (s)fr (t, τ, s) ,

v0 (nmT1)

(5)

n=−∞

∞ X

z-1

ch (1)

−∞

∞ X

c1 (0)

ch (0)

S−1 X s=0


x (nm + S2 −s)T1 · cr (s) . (10)

j k Here nm = m TT21 represents the index of the input sample that directly precedes the time instant mT2 and µm = m TT21 − nm is referred to as intersample time. Obviously vr (nm T1 ) represents the output signal of the r-th of P η + 1 FIR filters with the transfer S−1 function Cr (z) = s=0 cr (s)z −s . The resulting interpolating Farrow structure (i-FS) [7] is shown in Figure 3 where, subsequently, it is taken for granted that all η + 1 FIR filters share a common delay line. If the Farrow Structure is used for decimation (d-FS), the resampling is performed prior to the multiplications and hence only the delay line is clocked at f1 , whereas all computations are executed at the reduced operation rate f2 .

The FS implements a process of resampling to f2 , where the most prominent constraint to be obeyed is anti-aliasing [16]. Hence, the desired stopband of the underlying interpolation filter is defined as [kf2 − b , kf2 + b], k ∈ Z \ {0}. However, Lagrange interpolators being based on polynomials of length τ = T1 , have zeros in the magnitude response that are periodical in f1 [13] and so these zeros do not coincide with the aliasing bands. Therefore, in combination with the preceding L-SRC, the complete frequency band outside the Nyquist interval |f | > f2 /2 is attenuated to keep this requirement: The f1 -periodical passbands of the synchronous interpolator coincide with the zeros of the Lagrange interpolator and thus the addition of both magnitude responses can result in a continuous attenuation below a predefined level. The succeeding M-decimator is used to adjust the stopband frequency f2 = Mfo and hereby to relax the requirements for this filtering and to find a system with minimum overall computational load. Two enhancements are investigated in this study, which can be used to increase the efficiency of the cascade implementation of the L-SRC and A-SRC subsystems: 1) The output sampling rate of the L-fold preinterpolator is f1 , whereas the FS processes S samples for the calculation of one y(mT2 ). Hence for f1 > Sf2 , unused samples x(nT1 ) are calculated and thus a gain in computational efficiency can be achieved if the interpolator generates only those S needed samples and omits all other computations [19]. The additional complexity for controlling the signalflow between the synchronous interpolator and the Farrow kernel however will be neglected in this study. 2) The LA-SRC subsystem has an impulse response that is equivalent to the convolution of the discrete-time impulse response of the synchronous interpolator and the impulse response hint (t) of the Lagrange interpolator. Because the discrete time impulse response of the L-interpolator is of length LNbr,Int T1 and hint (t) comprises S polynomial segments of length T1 , the combined system consists of LNbr,Int +S −1 polynomial segments of order η and length T1 . However, similar to the polyphase realization of FIR filters, only every L-th polynomial segment is needed for the interpolation of one output value

[1]. Hence the LA-SRC system can be implementedl on a FS mof order η and length LNbr,Int +S−1 . The specific set of SGF S = L polynomial segments, changes with the output time instances mT2 and hence the coefficients for this FS are time varying. Please note that for SGF S > S the number of polynomial segments for this so-called Generalized Farrow Structure (GFS) [9] is increased and that a possible symmetry of coefficients (see the MFS below) cannot be exploited. B. The Modified Farrow Structure Vesma and Saram¨aki [8, 18] introduced an alteration to the FS by using the modified basis functions r   2 t−sτ −1 sτ ≤ t < (s+1)τ τ fr (t, τ, s) =  0 else (11) instead of (8), while still keeping the length of the polynomial segments equal to the input sampling period τ = T1 . For this Modified Farrow Structure (MFS) the coefficients cr (s) can be made symmetrical/anti-symmetrical, if the number S of polynomials is even and the impulse response hint (t) is symmetrical [20]. This condition is guaranteed for Lagrange polynomials of odd orders or η = 0. The implementation is similar to that of the FS in Figure 3 if 2µm − 1 is multiplied to the output of the x((n+JS/2)T1)

c1 (0)

ch (0) z-1 z-1

z-1

ch (1)

z-1

c0 (0) z-1

c1 (1)

z-1

z-1

z-1

z-1

z-1

z-1

z-1

ch (S-1)

c0 (1)

c1 (S-1)

c0 (S-1)

z-1

z-1

2mm(0) -1

z-1 2mm(1) -1

y(mT2)

Fig. 4. The Prolonged Modified Farrow Structure for J = 2

FIR filters instead of µm . The symmetry of coefficients however, can be exploited to half the number of multiplications that are needed in the FIR filters Cr (z). As this is an apparent increase in efficiency, the MFS is be preferred to the FS and the modified basis functions (11) are used in the following.

2 mn (1) - 1

2 mn (0) - 1 x(nT1)

z-1

C. The Prolonged Modified Farrow Structure Babic [17] suggested to prolong the length of the polynomial segments by an integer number, τ = JT1 . Hence, there are J different samples x(nT1 ) within every polynomial segment, resulting in J different intersample times µm (j), j ∈ [0..J − 1]. As an example for J = 2, the Prolonged Modified Farrow Structure (PMFS) is shown in Figure 4, that can be used as kernel of a LAM-SRC system in the same way as the MFS. Due to the J-fold prolongation of the polynomial segments however, the magnitude response of this Lagrange interpolator has zeros that are periodical in f1 /J instead of f1 .

z-1

z-1

z-1 0

z-1

0

0

0

z-1

z-1

z-1

c0 (0)

c1 (0)

ch (0)

I&D 0

z-1

c0 (1)

c1 (1)

ch (1)

z-1

z-1 z-1

D. The Transposed Modified Farrow Structure The Farrow structures presented so far are based on Lagrange polynomials with length τ ∈ {T1 , JT1 } and thus, the zeros of their corresponding amplitude responses are periodic in f1 or f1 /J , respectively [13]. While choosing the length of the polynomials equal to the output sampling period τ = T2 , the zeros of the magnitude response are periodic in f2 and therefore coincide with the centre frequencies 2 mn - 1 x(nT1)

z-1

z-1 0

z-1 0

I&D 0

f1 f2

c0 (0)

c1 (0)

ch (0)

c0 (1)

c1 (1)

ch (1)

z-1

z-1

c0 (S-1)

c1 (S-1)

ch (S-1) y(mT2)

Fig. 5. The Transposed Modified Farrow Structure

c0 (S-1)

c1 (S-1)

ch (S-1) y(mT2)

Fig. 6. The Prolonged Transposed Modified Farrow Structure for J = 2

of the output signal’s aliasing spectra. Hence don’tcare-bands can be used where the attenuation can be lower. These bands get wider if f2 = Mfo is increased by the M-decimator. Still, it must be guaranteed that τ > T1 to assure that interpolation at all time instances is possible and for this, the L-fold interpolator is used to increase the sampling rate to f1 > 1/τ prior to the A-SRC. The implementing structure of (6) for τ = T2 is the transposition [10] of the original Farrow structure, where two different versions of the Transposed Modified Farrow Structures (TMFS) are known [11, 21]. It can be shown that for the structure from [11] most operations are executed on the lower output rate and therefore this version will be preferred to the one from [21]. The TMFS from [11] is depicted in Figure 5, where it should be noted that the intersample time µn = n TT21 − ⌊n TT12 ⌋ is now normalized to the output sampling period and the number of input samples, which contribute to one output sample varies because τ is not synchronous to T1 . This changing number is reflected in the so called Integrate-&-Dump-elements

TABLE I

[10], which cumulate all needed samples.

O PERATION R ATES FOR DIFFERENT SRC SUBSYSTEMS

E. The Prolonged Transposed Modified Farrow Structure Eventually the length of the polynomial segments is prolonged to τ = JT2 , J ∈ N [12]. Similar to its dual system, the PMFS, there exist J different intersample times µn (j), j ∈ [0..J − 1] and the amplitude response of the underlying Lagrange interpolator has zeros that are periodical in f2 /J. Figure 6 shows the resulting Prolonged Transposed Modified Farrow Structure (PTMFS) that can be used in a LAM-SRC system in the same way as the TMFS.

The aforementioned Farrow kernels and LAM-SRC systems are now compared with respect to their operation rate C, where Table I gives a summary of the multiplication and addition rate of all subsystems under investigation. Additionally to the assumptions already listed at the end of Section I, we set the order of the polynomial interpolator to η ∈ {0, 1, 3} and restrict the peak-to-peak deviation in the passband of the overall system to 0.5 dB. To cover a wide range of applications but yet to keep this study readily comprehensible, the conversion factor is taken out of the set R ∈ {1.01, 5.01, 10.01, 25.01}. Figure 7 shows the normalized operation rate C versus the minimum stopband attenuation A for different LAM-SRC systems with interpolating, nontransposed Farrow kernels, namely the i-MFS from (a) R = 1.01

(b) R = 5.01 4

10

10

3

3

10

10

2

2

10

10

1

10 20

1

40

60

80

10 20

(c) R = 10.01

60

80

(d) R = 25.01

4

4

10

10

3

3

10

10

2

2

10

10

1

10 20

40

1

40

60

80

10 20

40

L-Int.

(η+1) max{S, SGF S }f1 + ηf2   (η+1) S2 f1 + ηf2   (η+1) S2 f2 + ηf2   (η+1) S2 f1 + Jηf2   ηf1 + (η+1) S2 f2   Jηf1 + (η+1) S2 f2

M-Dec.

Nbr,Dec f2

i-GFS i-MFS d-MFS i-PMFS TMFS PTMFS

Nbr,Int f1

addition rate

III. C ASE S TUDY

4

multiplication rate

60

80

Fig. 7. Operation rate C versus attenuation A [dB] for LAM-SRC systems based on interpolating, non-transposed Farrow kernels ( i-MFS, ✩ i-GFS, ∇ i-PMFS with J = 2)

i-GFS

(η + 1)(max {S, SGF S } − 1)f1 +ηf2

i-MFS

(η+1)(S −1)f1 +ηf2

d-MFS

(η+1)(S −1)f2 +ηf2

i-PMFS

(η+1)(S −1)f1 +(Jη+J −1)f2

TMFS

(η+1)f1 + (ηS +S −1)f2 (Jη+1)f1 . . . +((J −1)(η+1)+ηS +S −1)f2 (Nbr,Int − 1)f1

PTMFS L-Int. M-Dec. SGF S =

l

(Nbr,Dec − 1)f2 m

LNbr,Int +S−1 L

S = η + 1 for Lagrange polynomials

Fig. 3, the i-GFS and the i-PMFS from Fig. 4. As indicated by the prefix i-, the sampling rate is always increased by the A-SRC. It can be seen that systems utilizing the i-MFS kernel are those with the lowest operation rate. The most costly implementation are those with the i-GFS, because here the number of polynomial segments is increased and no coefficient symmetry can be exploited. The comparison of LAM-Systems with transposed Farrow kernels (the TMFS from Fig. 5 and the PTMFS from Fig. 6) and the best interpolating structure (the i-MFS) is shown in Figure 8. It can be seen that for high conversion rates R, both transposed versions are much more efficient than the non-transposed structures. This is because for high oversampling wide don’t-care-bands can be used and the requirements for the M-decimator are relaxed, due to the decimating A-SRC. Similar to the comparison between the i-MFS or i-PMFS, systems comprising the TMFS kernel are more efficient than those with the prolonged version. Eventually the non-transposed Farrow kernels are

(a) R = 1.01

(b) R = 5.01

4

2 L-SRC

4

10

A-SRC

3

M-SRC

6,6

(a)

10

0 3

3

10

10

2

10

1

60

80

10 20

(c) R = 10.01

60

80

25

5

10

15

20

25

5

10

15

20

25

(c)

0

−60

10

2

0 −60

2

10

10

0 1

(d)

0 f 2 /2

3

10

(e) 5

10

15

20

25

1

40

60

80

10 20

40

60

80

Fig. 8. Operation rate C versus attenuation A [dB] for LAM-Systems based on transposed Farrow kernels (◦ TMFS, × PTMFS with J = 2 and  i-MFS for comparison) (a) R = 1.01 4

R EQUIRED M EMORY C APACITY FOR THE LAM-SRC

4

10

3

Fig. 10. LAM-SRC system with d-MFS kernel for R = 10.01 and A = 60 dB (a). Magnitude Response [dB] versus normalized Frequency f /fo for L-SRC (b), A-SRC (c), LA-SRC (d), MSRC (e)

TABLE II

(b) R = 5.01

10

S YSTEMS OF F IGURE 9 AT A FIXED ATTENUATION A = 60 D B

3

10

10

2

2

10

10

1

1

40

60

80

10 20

(c) R = 10.01

40

60

80

(d) R = 25.01

4

R = 1.01 R = 5.01 R = 10.01 R = 25.01

i-MFS 79 144 215 455

TMFS 63 39 39 61

d-MFS 67 71 89 129

4

10

10

3

3

10

10

2

2

10

10

1

10 20

20 f/f

0

10

3

10 20

15

4

10

10 20

10

−60

40

(d) R = 25.01

4

5

0

1

40

(b)

0

2

10

10 20

−60

1

40

60

80

10 20

40

60

80

Fig. 9. Operation rate C versus attenuation A [dB] for various LAM-Systems (◦ TMFS,  i-MFS, ⊳ d-MFS)

used for internal decimation. Having learned from the previous results that the prolonged and generalized versions have no advantage with respect to the minimum operation rate, only the d-MFS is compared to its interpolating counterpart i-MFS and the best transposed Farrow kernel (TMFS). It can be seen in Figure 9 that for all conversion factors, but clearest for R = 1.01, the d-MFS is the most efficient system. Apparently, sub-SRC in the LAM-system can be allocated with the most degree of freedom for the d-MFS and therefore the best partitioning can be found. As an example, the SRC sub-tasks for the case R = 10.01 and A = 60 dB are shown in more detail, where an illustration of the overall system is depicted in Figure 10 (a): First the signal is interpo-

lated to twice the input rate, i.e. 20.02 · fo by a synchronous FIR polyphase interpolator of total length L · Nbr,Int = 36 and then filtered by a Lagrange interpolator of order η = 1. The amplitude responses of the L-SRC, A-SRC and their cascade combination LA-SRC subsystem can be seen in Figure 10 (b)-(d). Apparently, the attenuation requirement A > 60 dB is fulfilled for |f | > f2 /2. The signal is resampled to f2 = 3 · fo , which corresponds to an internal decimation factor in the d-MFS of f1 /f2 ≈ 6.67. Finally the signal is 3-fold decimated by a synchronous decimator of total length M · Nbr,Int = 24 to the final output rate (Fig. 10 (e)). Besides the operation rate, the amount of memory that is needed for a SRC algorithm can be of interest for some applications as well. As an example, the sum of all coefficient memories and signal delays that are needed for the last three LAM-systems is listed in Table II for a fixed attenuation A = 60 dB. Apparently, systems using the TMFS are the ones with the lowest memory consumption. Thus, for applications where memory is a sparse good, it can be better to use the TMFS than the d-MFS at the expense of a slightly increased operation rate.

IV. C ONCLUSION Asynchronous sample rate conversion, being a fundamental task in every software radio application, is beneficially implemented by polynomial interpolation in conjunction with the Farrow Structure or one of its variations. In a cascade combination with a preceding L-fold interpolator and/or succeeding M-fold decimator, a system can be found with optimum allocation of subtasks to each stage and consequently minimum overall computational load. Our comparative investigations of different approaches to asynchronous decimation (R > 1) applying Farrow-based Lagrange interpolation have, for the adopted range of specifications, shown the following results: • The most general LAM-SRC approach to asynchronous sample rate conversion according to Figure 1 has the potential of globally highest efficiency. • This LAM-SRC represents the most efficient approach, if the A-SRC subsystem is implemented as a Modified Farrow structure with the decimation property. This result is somewhat unexpected, since the Modified Farrow structure has so far been considered to be usable for interpolation only [17, 18]. • All transposed forms of the Farrow structure require slightly more computation than the Modified Farrow structure with the decimation property. • The efficiency of both all generalized and all prolonged Farrow structures is lowest. As to future research, further studies are required to investigate the non-transposed (modified) Farrow structures with the decimation property. R EFERENCES [1] Hentschel, T.: Multirate Systems for Sample Rate Conversion in Software Radio Terminals, PhD-Thesis, Technische Universit¨at Dresden, Nov. 2000 [2] Babic D.: Techniques for Sampling Rate Conversion by Arbitrary Factors with Applications in Flexible Communications Receivers Tampere University of Technology, PhD-Thesis, 2004 [3] Jondral F.; Machauer R.; Wiesler A.: Software Radio J. Schlembach Fachverlag, 2002. – ISBN-10: 3935340176 [4] G¨ockler, H.; Groth, A.: Multiratensysteme. J. Schlembach Fachverlag, Wilburgstetten, 2004. – ISBN 3–935340–29–X [5] Evangelista, G.: Zum Entwurf digitaler Systeme zur asynchronen Abtastratenumsetzung, Ruhr-Universit¨at Bochum, Digital Signal Processing Group, PhD-Thesis, October 2001 [6] Ramstad, T.: Digital Methods for Conversion Between Arbitrary Sampling Frequencies. IEEE Transactions on Acoustics, Speech and Signal Processing ASSP-32 (1984), June, No. 3, pp. 577–591

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