cascades of polynomial-based and fir filters for sampling rate conversion

CASCADES OF POLYNOMIAL-BASED AND FIR FILTERS FOR SAMPLING RATE CONVERSION BY ARBITRARY FACTORS Djordje Babic, Markku Renfors Institute of Communications Engineering, Tampere University of Technology P.O. Box 553, Tampere, FINLAND Tel: +358 3 3115 3910; fax: +358 3 31153808; E-mail: {Djordje.Babic, Markku.Renfors} @tut.fi

I INTORDUCTION The sampling rate conversion (SRC) is utilized in many DSP applications where two signals or systems having different sampling rates are to be interconnected. The SRC factor in general can be an integer, one divided by an integer, a ratio of two integers, or an irrational number. The SRC factor is determined by R = Fout Fin = Tin Tout ,

(1)

where Fin = 1/Tin (Tin) and Fout = 1/Tout (Tout) are the original input sampling rate (period) and the sampling rate (period) after the conversion, respectively. The sampling rate conversion can be divided into two general cases. For R < 1, the original sampling rate is reduced and this process is known as decimation. For R > 1, the original sampling rate is increased and this process is known as interpolation [1]. When the decimation factor 1/R or the interpolation factor R is an integer or a ratio of two relatively small prime integers, then the sampling rate conversion can be performed conveniently with the aid of fixed digital filters [1]. If these factors are irrational, then fixed digital filters cannot be directly used. Furthermore, if the factor R is ratio of two relatively large prime integers, then, in the case of the conventional polyphase implementation, the required filter orders become very large [1]. In practice this means large number of coefficients that have to be stored in coefficient memory. One way to overcome this problem is to perform calculation of the coefficients while in operation for each output sample. However, sometimes the complexity of calculation of new coefficients may exceed the complexity of filtering operation. The complexity of calculation of new coefficient can be reduced by representing the impulse response of the underlying filter in relatively simple closed mathematical form. Probably the most convenient for implementation is impulse response represented as piecewise polynomial function of low order considered here. Sampling rate conversion by non-integer factor is a typical example where it is required to determine the values between existing samples. In this case it is very convenient to use interpolation filters. Among them, polynomial-based filters This work was carried out in the project “Advanced Transceiver Architectures and Implementations for Wireless Communications” supported by the Academy of Finland. This work is also supported by Tampere Graduate School in Information Science and Engineering TISE, and NOKIA foundation.

have efficient implementation form directly in digital domain. The polynomial-based filters are efficiently implemented using the Farrow structure or its modifications [2]-[13]. The main advantage of the Farrow structure lies in the fact that it consists of fixed finite-impulse response (FIR) filters and there is only one changeable parameter being the so-called fractional interval µ. Besides this, the control of µ is easier during the operation than in the corresponding coefficient memory implementations [8], and the resolution of µ is limited only by the precision of arithmetic and not by the size of the memory. These characteristics of the Farrow structure make it a very attractive structure to be implemented using a VLSI circuit or a signal processor [8]. The non-integer SRC can be performed by using Farrow structure or its modifications directly. However, in many cases, it is more efficient to use cascaded structures consisting of modification of the Farrow structure and fixed FIR, or multistage FIR filter [5], [13]. The main advantage of using the cascaded structures instead of the direct modification of the Farrow structure lies in the fact that when jointly optimizing the two building blocks the computational complexity to generate practically the same filtering performance is drastically decreased. This has been pointed out in [5], [13] and will be seen in connection with examples of Section IV. This is due to the following facts. First, the implementation of a fixed linearphase FIR interpolator is not very costly, compared to the Farrow structure. Second, most importantly, the requirements for implementing the modification of the Farrow structure become significantly milder. This is mainly because of that the FIR filter takes care of passband and stopband shaping, where the Farrow-based structure should only take care of attenuating images of FIR filter. This paper gives an overview of several known cascaded structures for non-integer SRC. First of all, cascades of the FIR filter and direct form of the Farrow-based filter, and its dual transposed form. Further, it gives principles for some new possibilities for non-integer SRC. This paper also considers special cascaded structure for decimation that is used to decrease the overall number of operations. The presented structure can be used as the first stage in overall decimation chain. The paper is organized as follows. In Chapter I, the general principle of polynomial-based filter is described. We give also description of the several modifications of the Farrow structure. In Chapter II, the principles of conventional cascaded

0.7

m  n −1  2 t − ∑ T   n n −1   i =0 i    − 1 for ∑ Ti ≤ t < ∑ Ti f m ( n , Tn , t ) =  Tn i =0 i =0      0 otherwise. 

c 0( i ) g 0( t−iT )

(a) 0 −0.2 0.6

c 1( i ) g 1( t−iT )

(b)

0 0.17

c 2( i ) g 2( t−iT )

(c)

0 −0.14 0.1

c 3( i ) g 3( t−iT )

The cm(n)’s are the adjustable parameters being related to each other as

(d)

0

(3)

 cm (n) for m even cm ( N − 1 − n) =  − cm (n) for m odd

−0.1 1

Overall impulse response ha ( t )

(4)

(e) 0 −0.2 −6

−4

−2

0

2

4

for n = 0, 1, …, N−1. As shown in Fig. 1, the resulting ha(t) is characterized by the following properties: 1) ha(t) is nonzero for 0 ≤ t < ∑ nN=−01Tn and zero elsewhere.

6

t/T

Fig. 1. Construction of the overall impulse response ha(t) for N = 8 and M = 3. hm (t ) = ∑ nN=−01cm (n) f m (n, Tn , t )

2) In each subinterval ∑ in= 0 Ti ≤ t < ∑ in=+01Ti for n = 0, 1, …, N−1, ha(t) is a polynomial and is of degree M (or less), expressible as ha (t ) = ∑ mM= 0 cm (n) f m (n, Tn , t ) .

for (a) m = 0, (b) m = 1, (c) m = 2, (d) m = 3. (e) The resulting overall ha (t ) = ∑ mM= 0 ha (m, t ) .

3) ha(t) is symmetric around t = ∑ nN=/02 −1Tn , that is, ha( ∑ nN=−01Tn −t)

structures are explained. After that, in Chapter III, we give principle of specific power efficient cascaded structure used for sampling rate reduction. Finally, in Chapter IV, through several examples all these structures are compared in terms of number of multipliers and amount of operations. II POLYNOMIAL-BASED FILTERS This part develops the framework for the following sections by introducing the general form of the piecewise polynomial-based impulse response of the underlying continuous-time filter. It has been suggested in [3], the underlying continuous-time impulse response ha(t) is expressed in each interval of length Tn by means of a polynomial, thus it is beneficial to construct ha(t) as:

ha (t ) =

N −1 M

∑ ∑ cm (n) f m (n, Tn , t )

(2)

= ha(t) except for the time instants t = ∑ in= 0 Ti for n = 0, 1, …, N/2−1 and n = N/2+1, N/2+2, …, N. As an example, from Fig. 1, it can be seen how the impulse response ha(t) is constructed. For the piecewise polynomial impulse response in Fig. 1, the length of polynomial segments at the beginning and at the end of the impulse response (for n=1, 2, 7 and 8) is Tn=2T, thus Jn=2, while in the middle (for n=3, 4, 5, and 6) the length of polynomial segments is shorter and it is Tn=T, thus Jn=1. Based on Property 3, it is guaranteed that the resulting overall system has a linear phase that is a very attractive property in many applications. Furthermore, the generation of the above ha(t) guarantees that in the frequency domain the zero-phase frequency response, when omitting the linear-phase term, is expressible as (see [7] and [8] for details)

n=0 m=0

where N is an integer, and the polynomial basis functions fm(n, Tn, t) are given by

2µk − 1 x(k) Fin −1

−1

Z

I&D

−1

Z

Z

x(nl + N/2)

ov(l)

cM(0)

Fin

c1(0)

c0(0)

Fout v0(n,l) −1

Z

−1

cM(1)

−1

Z

c0(1)

c0(0)

c1(0)

cM(0)

−1

Z

−1

Z

Z

c0(1) cM(N − 1)

vM(n,l)

−1

Z

c1(1)

−1

Z

v1(n,l)

c1(N − 1)

v M (nl) 2µ l − 1

Fig. 2. Modified Farrow structure.

v1(nl)

c1(1)

cM(1)

v0(nl)

−1

Z

c0(N − 1)

Fout y(l)

c0(N − 1)

c1(N − 1)

Fig. 3. Transposed modified Farrow structure.

cM(N − 1)

y(l)

x(n) Fin

↑L

FIR filter z(j) HI(z) LFin

LFin

Farrow structure

y(l) Fout

N / 2 −1 M

∑ ∑ cm (n)Gm (n, Tn , f ) ,

Fin

where Gm(n, Tn, f ) is the Fourier transform of m

+ f m ( N − 1 − n, Tn , t − ∑nN=/02−1Tn ) .

(6)

The above form is a direct consequence of the symmetry properties of the cm(n)’s. Since the above approximating function is linear with respect to the unknowns, it enables one to optimize the overall filter to meet the given criteria in a manner similar to that used for synthesizing FIR filters [14]. In this contribution, we have used modified optimization routines from [8] in order to determine the polynomial coefficients cm(n). In the above, Tn, the length of polynomial segments, can be defined as Tn = JnT. Here, Jn is unity, or an integer. In general, the length of every polynomial segment may be different. The most common polynomial-based filters are those which have polynomial-segments of the same length, i.e. Tn=T for n=0,1…N−1. Depending on the selection of T, the length of polynomial segment, the resulting structures can be categorized into three main classes [5]. For the first class, T is either equal to the input or output time period, resulting in the modified [7] and transposed modified Farrow [10]-[12] structure for interpolation and decimation, respectively (see Fig. 2 and Fig. 3). For both, direct and transposed Farrow structure, there are M+1 FIR subfilters of length N, where N is number of polynomial segments and M is polynomial order. Property 3 ensures that these FIR subfilters are symmetric or antisymmetric, thus the overall number of coefficients is (N/2)(M+1). Additionally, there are also M so-called µmultipliers. For the second class, T is an integer multiple of the input or output sampling period resulting in the so-called prolonged versions [2], [5]. Finally, for the third class, T is fraction of the input or output sampling interval. The resulting structures are the cascade of the fixed linear-phase FIR interpolator and the modified Farrow structure and the cascade of the transposed modified Farrow structure and the fixed linear-phase FIR decimator [5], [13]. The cascaded forms are considered here as efficient structures for the non-integer SRC. III CASCADED STRUCTURES The cascaded direct structure, which is commonly used for interpolation, consisting of two basic building blocks is shown in Fig. 4 [5]. In the first block, the input sampling rate Fin of the input sequence x(n) is increased by an integer factor of L by using a linear-phase FIR filter transfer function of the following form: H I ( z) =

NI

∑ hI (k ) z ,

k =0

−k

↓L

y(l) Fout

where the impulse-response of HI (z) is symmetric, that is, (5)

hI ( N I − k ) = hI (k ) for k = 0, 1, K , N I .

n=0 m=0

g m (n, Tn , t ) = (− 1) f m (n, Tn , t − ∑nN=/02−1Tn )

Transposed z(j) FIR filter Farrow LFout HD(z) LFout

Fig. 5. Transposed Farrow structure in cascade with a linearphase FIR decimation filter for decimation.

Fig. 4. Linear-phase FIR interpolation filter in cascade with modified Farrow structure for interpolation.

H a ( j 2πf ) =

x(n)

(8)

It should be pointed out that in many cases, the computation complexity of the fixed linear-phase FIR interpolator can be significantly reduced by using a multistage interpolator [1], [14]. In this case, there exist several linear-phase FIR interpolator transfer functions H I(1) ( z ) , H I( 2) ( z ) , and H I( N I ) ( z ) with the first one increasing the sampling rate by an integer factor L1 followed by the second one increasing the resulting sampling rate further by an integer factor L2, etc. It is well known that this implementation is equivalent to that of Fig. 4 with L = L1⋅L2⋅LN and I

H I ( z ) = H I(1) ( z

L2 L3LLN I

) H I( 2) ( z

L3LLN I

) L H I( N I ) ( z ) .

(9)

After the fixed FIR interpolator, the sampling rate of the resulting output sequence, denoted by z(j) in Fig. 4, is equal to LFin. The second building block involves generating the desired lth output sample. This block can be implemented using: the modified Farrow structure, prolonged modified Farrow structure, modified Farrow structure of odd length, or direct polynomial-based filter having polynomial-based segments of different lengths. There exist the following two differences. First, the input sample rate is now LFin, instead of Fin. Second, the desired ha(t) is obtained from Eq. (2) by selecting T to be T = JTin/L [5]. The main advantage of using the structure of Fig. 4, instead of the direct modified Farrow structure of Fig. 2 or any modification of the direct Farrow structure, lies in the fact that when jointly optimizing the two building blocks, the computational complexity to generate practically the same filtering performance is drastically decreased [5], [13]. This is due to the following facts. First, the implementation of a fixed linear-phase FIR interpolator is not very costly. Second, the requirements for the modified Farrow structure become significantly milder. First of all, its sampling rate becomes L times higher and, after the interpolation, its relative passband region becomes L times narrower. Furthermore, this structure should only take care of attenuating narrow images that are approximately of width Fin and are centered at the frequencies f = rLFin/2 for r = 1, 2, 3... The cascaded structure presented above has two special cases for the values of given parameters. Case I corresponds to the case where the factor of prolongation J is equal to unity. This case is directly the structure presented in [13], i.e., a cascade of linear-phase FIR interpolator and modified Farrow structure of Fig. 2. The output sample is determined by using y (l ) =

(7) where

N −1 M

∑ ∑ z (nl − k + N / 2)cm (k )(2µl − 1) m

k =0 m=0

(10)

x(n) Fin

Farrow Structure

z(j) LFout

FIR filter HD(z) LFout

↓L

y(l) Fout

Decimation by Rint x(n)

FIR FILTER

Fin

Fig. 6. Modified Farrow structure in cascade with a linearphase FIR decimation filter for decimation. nl = lTout (LTin ) and µ l = lTout (LTin ) − nl .

↓Rint

xRint−1(m)

z −1 Decimation by 1+ε /Rint

(11)

Case II structure corresponds to the situation where the integer upsampling factor L is equal to unity. The presented structure in this case, consists of the cascade of FIR filter and modified Farrow structure (equally well the prolonged modified structure can be used here). This structure is always equivalent to a modified Farrow interpolator. However, it is more efficient to use the cascaded structure in practical realization. This is mainly because the implementation of a fixed linear-phase FIR filter is not very costly compared to the modified Farrow structure. In this cascade, the FIR filter takes care of passband and Farrow structure attenuates the images of the FIR filter. Thus, the used modification of the Farrow structure is very simple, the required order M and length N are small integers. A. Decimation The decimation process is dual to the interpolation. In practice it means that dual filters are used for decimation. For example, the modified Farrow structure is used mainly for interpolation, and its dual transposed modified Farrow structure is used for decimation. Therefore, the dual form of the cascaded structure from above can be used for non-integer decimation. The short explanation of the dual structure is given below as the first alternative for non-integer decimation. In this part, we also explain briefly another alternative for non-integer decimation based on the structure for interpolation. 1). Cascaded structure for downsampling between arbitrary sampling rates: First alternative This structure is motivated by the duality between the fixed digital interpolators and decimators. The cascaded transposed structure shown in Fig. 5, which is conveniently used for decimation, consists of two building blocks [2]. The first block can be any transposed modification of the Farrow structure that generates an output sequence, denoted by z(j), based on the input sequence x(n) such that the output sampling rate is an integer multiple of the desired output sampling rate Fout. In other words, the output sampling rate is LFout with L being an integer. After that, the sampling rate is further reduced by L using the fixed FIR decimator. All special cases and conclusions derived for the cascaded structure for interpolation are also valid here in dual form. 2). Cascaded structure for downsampling between arbitrary sampling rates: Second alternative It should be pointed out that the direct modification of the Farrow structure can be also usable for decimation purposes in the following manner. First, the input sampling rate is increased using the interpolation structures from above to be an integer multiple of the final desired output sampling rate, as illustrated in Fig. 6. After that, the desired sampling rate can be achieved by using a fixed linear-phase decimator reducing the sampling rate by the integer factor. The decimation structure explained

Shift by one

FIR FILTER

↓Rint

FIR FILTER

↓Rint

x1(m)

z −1 x0(m)

POLYNOMIAL y(l) -BASED Fout FILTER

µl

Fig. 7. Model of proposed decimation filter. Parallel connections of Rint FIR filters are used for integer decimation by Rint. Decimation by 1+ε /Rint is done using interpolation between the outputs of N consecutive filter branches, where N is the length of polynomial-based filter.

above is somewhat more difficult to optimize and analyze in frequency domain compared to cascaded structures from previous section. IV EFFICIENT STRUCTURE FOR DOWNSAMPLING BETWEEN ARBITRARY SAMPLING RATES The main disadvantage of the general cascaded structure described in Section III is in the fact that the polynomial-based filter is working at the high sampling rate. The polynomialbased filter performs SRC by a factor in the range (0.5, 2), thus it does not change the sampling rate drastically. However, the multipliers of the polynomial-based filter operate all the time at the high sampling rate. This fact leads to exploring the possibility of further reduction of the multiplication rate. In [15], the fractional decimator based on CIC and linear interpolator is proposed. The idea of this fractional decimator is to have linear interpolator working at the low output sampling rate, while it performs filtering if it was at the input. In this way, the multiplications of the linear interpolator are done at the low rate. In this fractional decimator structure, the CIC filter attenuates aliasing, and the linear interpolator removes images of the CIC filter. The implementation of this structure is based on realizing parallel comb stages while the integrator stages of CIC filter are shared among comb branches. In this way implementation complexity is reduced. The idea of [15] is generalized to the combination of any FIR filter and polynomial-based interpolator in [16]. The method proposed here moves the polynomial-based filter after the FIR decimator while keeping the frequency response and filtering performance of the overall structure at the same level. A. Overall structure The idea is to avoid high multiplication rate required in the cases explained above. After the modification, the polynomialbased filter does not work all the time, it is in operation according to timing determined by a control logic. The implementation of a FIR filter is not as costly as implementation of Farrow structure. The distance between

x(n) y(l)

Decimation by Rint x(n) INTEG.

COMB xRint−1(m)

↓Rint STAGES

Fin STAGES

z −1

T in

Decimation by 1+ε /Rint

0

T out = 3.3T in

2T out= 6.6Tin

3T out= 9.9Tin

4T out= 13.2Tin Shift by one

(a)

x 0(m)

COMB

↓Rint STAGES

x 2(m)

x1(m)

x1(m)

z −1 COMB

µ0

µ1

µ2

µ3

↓Rint STAGES

µ4

x0(m)

POLYNOMIAL y(l) -BASED Fout FILTER

µl 0

T out

2T out

3T out

4T out

(b)

Fig. 8. (a) The input and output samples of the proposed decimation filter for R = 3.3, using the linear interpolation filter. (b) The output samples of the two parallel FIR filter branches x0(m) and x1(m). The output samples of the overall structure (black circles) are obtained by linear interpolation between samples x0(m) and x1(m). After the output time instant 3Tout, the linear interpolation is done between the samples x1(m) and x2(m).

input samples to the polynomial-based filter is equal to the input sampling period. Therefore, the filtering is virtually performed at the high rate while the computations are done at the low rate. Using this principle, the workload can be significantly reduced at the expense of somewhat increased complexity of the overall structure. The principle explained above is based on the realization of the parallel replicas of an identical FIR decimator filter, or on the use of a non decimating FIR filter. Figure 7 illustrates the proposed structure for the decimation filter [16]. The input signal x(n) is divided into its filtered polyphase components xk(m), for k = 0, 1,· · ·, Rint −1, by using delay line and parallel identical copies of a decimating FIR filter. Therefore, the sampling rate at the output of the FIR filters is Fin /Rint. The final decimation by (1+ε/Rint) is done using polynomial-based interpolation between N signals xk(m), xk⊕1(m)… xk⊕N(m), where ⊕ denotes the modulo Rint summation, N is the length of the polynomial-based filter, and k is an index to be discussed below. Every now and then during operation, the place of interpolation block in Fig. 7 is shifted by one branch according to a certain condition. Because of the modulus Rint summation mentioned above, the next signal block for interpolation after [xRint−Ν(m), xRint−1(m)] is [xRint−Ν+1(m), x0(m)]. The fractional interval µ l is recalculated for each output sample y(l) for l = 0, 1, 2, · · ·. The shifting condition for interpolation can be determined by using the fractional interval. As can be seen from Fig. 8(b), the value of the fractional interval µ l is increased until l = 3 and the next value µ 4 is less than µ 3. This is the point where shifting is needed and, therefore, the shifting condition can be stated as follows if µ l < µ l−1 then shift sampling phase of linear interpolation by one.

(12)

Fig. 9. Model of proposed decimation filter in the case of CIC or modified comb filter. The integrator stages are shared between parallel connection of Rint comb stages. Decimation by 1+ε /Rint is done using interpolation between the outputs of N consecutive filter branches, where N is the length of polynomial-based filter.

B. Frequency response of the overall structure The overall frequency response of the decimation filter structure of Fig. 7 is the product of the frequency responses of the FIR filters of the parallel branches and polynomial-based interpolation filter. Note that the former response is periodic whereas the latter is not. The parallel FIR filters perform the polyphase decomposition of the input signal. The frequency response of this stage is simply the same as the response of one FIR filter. The polynomial interpolation is done between the samples xk(m), xk⊕1(m),…xk⊕N(m) having time interval of Tin as a mutual distance. Therefore, the polynomial interpolation is done at the higher input rate Fin. Consequently, the overall frequency response of the proposed decimation filter is given by [16]

(

)

H T e j 2πf / Fin = H (e j 2πf / Fin ) H a ( f )

(13)

Practically, Eq. (13) means that the FIR filter shapes the passband and stopband, and polynomial based filter attenuates spectral images of the FIR filter. C. Special efficient filter structures based on the presented principle The principle presented above is not generally very efficient. The advantages and efficiency depend on several factors: type and length of the FIR decimator filter, type, length, and order of polynomial-based filter, overall decimation factor, etc. There are some special filter structures whose combination in the presented way gives very efficient fractional decimator structures. This section studies the combination of CIC filters and simple polynomial-based filters. The regular structure of CIC and modified comb filter can be exploited in making the proposed fractional decimator more efficient. The integrator stages of CIC or modified comb filter can be shared between parallel comb stages, as shown in Fig. 9. Thus, it is needed only to make N+B identical parallel replicas of comb stages. After the integer decimation is done by CIC or modified comb filter, the final non-integer decimation is done by a polynomial-based filter. The implementation and control logic of this structure have been considered in details in [15].

TF PTF

0

−20

Magnitude in dB

Magnitude in dB

−20

−40

−60

−80

−100 0

FIR 1 FIR 2 Farrow Overall

0

−40 −60 −80 −100

1

2

3

4

5

6

7

8

−120 0

1

Frequency relative to Fout

3

4

5

6

7

V DESIGN EXAMPLES It is desired to convert the sampling rate between Fin = 44.1kHz and Fout = 8kHz so that the decimation factor is equal to R = 5.5125. The passband and stopband edges are located at fp = 0.4Fout and fs = 0.5Fout, respectively, whereas the minimum stopband attenuation is 60 dB and the maximum allowable magnitude deviation from unity in the passband is 0.01. For simplicity, we concentrate in the sequel on designing filters in such a manner that the passband average is scaled to be unity. When using the minimax optimization criterion proposed in [7], [8] and [5], the transposed modified Farrow structure meets the given criteria by N = 28 and M = 4. For this structure, the overall number of multipliers is (M+1)N/2+M = 74. The same requirements are met by the prolonged modified Farrow structure of Section 8 by N = 14, M = 6, and J = 2. In this case, the overall number of multipliers is (M+1)N/2+JM = 61. The magnitude responses for these two structures are shown in Fig. 10. For the structure introduced in Section III.A, the following solutions have been optimized to meet the given criteria. In the

0

FIR Farrow Overall

−20

Fig. 12. The magnitude responses for both FIR filters in the two-stage decimator, the Farrow structure, and the overall system.

first case, the sampling rate is first decreased by a non-integer factor 2.7562 by using the transposed modified Farrow structure with N = 4 and M = 3. The resulting sampling rate is two times the desired one. Therefore, the desired output sampling rate is achieved by using a fixed linear-phase FIR decimator of order KD = 52 further decreasing the sampling rate by a factor of two. The overall structure requires (M+1)N/2+M+KD/2+1 = 38 multipliers. The magnitude responses for the sub-filters as well as for the overall filter are shown in Fig. 11. In the second case, the sampling rate is first decreased by a non-integer factor 1.3781 by using the transposed modified Farrow structure with N = 4 and M = 2. The desired output sampling rate is then obtained by using a two-stage decimator with both stages decreasing the sampling by a factor two. The required orders for the first and second linear-phase FIR decimators are KD1 = 3 and KD2 = 52, respectively. 2. For this structure, the overall number of multipliers is (M+1)N/2+M+KD1/2+1+(KD2+1)/2 = 37. The magnitude responses for the sub-filters as well as for the overall filter are shown in Fig. 12. The third example corresponds to the special Case II of the cascaded structure of Section III.A. The proposed requirements TABLE I REQUIREMENTS FOR THE STRUCTURES UNDER CONSIDERATION Multiplications/s No. of multipliers

−40

TF PTF TF+FIR↓1 PTF+FIR↓1 TF+FIR↓2 PTF+ FIR↓2 TF+FIR1↓2+FIR2↓2 PTF+FIR1↓2+FIR2↓2

−60 −80 −100 −120 0

1

2

3

4

5

8

Frequency in Fout

Fig. 10. Magnitude responses for the transposed Farrow structure (TF) and the prolonged transposed Farrow structure (PTF).

Magnitude in dB

2

6

7

8

Frequency in F

out

Fig. 11. The magnitude responses for the FIR filter decimator, the Farrow structure, and the overall system.

4Fin+70Fout 12Fin+49Fout 4Fin+33Fout 8Fin+26Fout 3Fin+59Fout 6Fin+41Fout 2Fin+55Fout 4Fin+54Fout

74 61 37 34 38 39 37 38

No. of I&Ds 5 13 5 10 4 8 3 4

TF is the transposed Farrow structure, PTF is the prolonged transposed Farrow structure, TF+FIR is the transposed Farrow structure in cascade with a one-stage FIR decimator, TF+FIR1+FIR2 is the transposed Farrow structure in cascade with a two stage FIR decimator, and ↓2 is a decimation factor of FIR filter.

In this example (example is taken from [15]), the bandwidth of the input signal is fp=0.001Fin and decimation factor is R=341/34. The attenuation in aliasing bands, defined by (14), should be at least by As=80 dB, and the passband distortion is less than δp=0.01 (0.086 dB). These requirements are met by a proposed type of decimation filter having a CIC filter of order N=3 and a linear interpolation filter. Figure 14 presents the bands that cause aliasing to the desired band. As can be seen, the minimum attenuation of these bands is 84.4 dB. Because the over-sampling factor is still high after decimation, the worst case passband distortion caused by the proposed filter structure is only 0.06 dB.

FIR Farrow Overall

40

Magnitude in dB

20

0

−20

−40

−60

−80 0

0.5

1

1.5

2 2.5 3 Frequency in Fout

3.5

4

4.5

5

Fig. 13. The magnitude responses for non-decimating FIR filter, the Farrow structure, and the overall system, for Case II structure of Section VIII.

are met by using transposed modified Farrow structure with N=8 and M=4 and non-decimating fixed FIR filter of order KD= 25. For this structure, the overall number of multipliers is (M+1)N/2+M+(KD+1)/2 = 37. The relevant frequency responses are given in Fig. 13. All cascaded structure from above can be designed by using also prolonged transposed modified Farrow structure and fixed linear-phase FIR filters. The realization requirements for all above structures including also those designed using cascades with prolonged transposed modified Farrow structure are shown in Table 1. These requirements are compared in terms of the required number of multiplications per second, the required number of multipliers, and the required number I&D circuits. Structure of Section IV can be efficiently used when it is required that the fractional decimator provides enough attenuation for the frequency region which aliases to the desired baseband: Ωs =

U [Fout (r − f p ), Fout (r + f p )] . ∞

(14)

r =1

Magnitude in dB

VII REFERENCES [1] R. E. Crochiere and L. R. Rabiner, Multirate Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1983.

[2] D. Babic, T. Saramäki, M. Renfors, “Prolonged transposed Farrow structure,” in Proc of Int. Symposium on Circuits and Systems ISCAS 2003, Bangkok, Thailand, May 2003, vol. 4, pp. 317-320. [3] D. Babic, M. Renfors “Polynomial-based filters with polynomial pieces of different lengths for interpolation,” 3rd International Symposium on Image and Signal Processing and Analysis ISPA 2003, September 18-20, 2003, Rome, Italy, pp. 740-744. [4] D. Babic, A. Shahed Hagh Ghadam, M. Renfors, “Polynomial-based filters with odd number of polynomialsegments for interpolation,” to appear in IEEE Signal Processing Letters. [5] D. Babic, T. Saramäki, and M. Renfors, "Sampling rate conversion between arbitrary sampling rates using polynomial-based interpolation filter,” The Second International Workshop on Spectral Methods and Multirate Signal Processing SMMSP 2002, Toulouse, France, September 2002, pp. 57-64.

0

−50

[6] D. Babic, V. Lehtinen, M. Renfors, “Discrete-time modeling of polynomial-based interpolation filters in rational sampling rate conversion,” in Proc. of Int. Symposium on Circuits and Systems ISCAS 2003, Bangkok, Thailand, May 2003, vol. 4, pp. 321-324.

−100

−150 0

VI CONCLUSIONS We have shown here several cascaded structures for noninteger SRC. These structures are obtained by cascading polynomial-based interpolation filter with FIR or multistage FIR interpolator or decimator. The cascaded structures are effective tool for problem of non-integer conversion, where conventional fixed FIR filters cannot be directly used. We have also shown a special power efficient cascaded structure for noninteger decimation. This structure can be effectively used as the first stage in overall decimation chain.

10

20

30

40

50

60

Frequency relative to F

out

Fig. 14. Amplitude responses in aliasing bands for the overall structure in the example case.

[7] J. Vesma and T. Saramäki, “Interpolation filters with arbitrary frequency response for all-digital receivers,” in Proc. 1996 IEEE Int. Symp. Circuits and Systems, Atlanta, Georgia, May 1996, pp. 568−571. [8] J. Vesma, Optimization and Applications of Polynomial-

Based Interpolation Filters. Doctoral Thesis, Tampere University of Technology, Publications 254, 1999.

Signal Processing 2001 ICASSP 2001, Salt Lake City, USA, 2001.

[9] C. W. Farrow, “A continuously variable digital delay element,”in Proc. 1988 IEEE Int. Symp. Circuits and Systems, Espoo, Finland, June 1988, pp. 2641−2645.

[16] D. Babic, M. Renfors, “Decimation by non-integer factor in multistandard radio receiver,” submitted to EURASIP Signal Processing: An International Journal.

[10] T. Hentschel, and G. Fettweis, “Continuous-time digital filters for sample-rate conversion in reconfigurable radio terminals,” in Proc. the European Wireless 2000, Dresden, Germany, Sept. 2000, pp. 55−59.

CASCADES OF POLYNOMIAL-BASED AND FIR FILTERS FOR SAMPLING RATE CONVERSION Djordje Babic and Markku Renfors

[11] A. Gotchev, J. Vesma, T. Saramäki, and K. Egiazarian, “Multiscale image representations based on modified Bsplines,” in Proc. First International Workshop on Spectral Techniques and Logic Design for Future Digital Systems, Tampere, Finland, June 2000, pp. 431−446. [12] D. Babic, J. Vesma, T. Saramäki, M. Renfors, “Implementation of the transposed Farrow structure,” in Proc. 2002 IEEE Int. Symp. Circuits and Systems, Scotsdale, Arizona, USA, 2002, vol. 4, pp. 4−8. [13] T. Saramäki and M. Ritoniemi, “An efficient approach for conversion between arbitrary sampling frequencies,” in Proc. 1996 IEEE Int. Symp. Circuits and Systems, Atlanta, Georgia, May 1996, pp. 285−288 [14] T. Saramäki, “Finite impulse response filter design,” Chapter 4 in Handbook for Digital Signal Processing, edited by S. K. Mitra and J. F. Kaiser, John Wiley & Sons, New York, 1993. [15] D. Babic, J. Vesma, and M. Renfors, “Decimation by irrational factor using CIC filter and linear interpolation,” Proc. International Conference on Acoustics, Speech, and

Many times in DSP applications it is required to connect two systems working at two different sampling rates. The sampling rate conversion (SRC) factor can be integer, rational or even irrational factor. In the case, when the SRC factor is non-integer, it is not possible to use conventional fixed digital filters. The polynomial-based filters can be effectively used for the problem of non-integer SRC. The efficiency of the method can be increased by using various combinations of conventional fixed FIR interpolators or decimators and polynomial-based filters implemented using the modified Farrow structure or its modifications. This paper considers several different cascaded structures for non-integer SRC. First, we show generalized definition of the polynomial-based filters. After that, we explain shortly the main variants of the modified Farrow structure. Later, we introduce principle of cascaded structures for both, interpolation and decimation. We also study the special power efficient cascaded structure for non-integer decimation. The performance of the cascaded structures is illustrated here, by using several simple examples.