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Abstract—In this paper variable fractional delay filter structure which allows simple change of filter type or length has been presented. The proposed structure is a modified Farrow structure with modifications based on the extracted window method in which we obtain a single symmetric window for optimal variable delay filter design with additional gain correction dependant on required delay. In order to change a filter type only the window and coefficients of the polynomial for gain corrections computation have to be changed. The main part of the structure implementing the truncated ideal impulse response stays unchanged.
Index Terms—Digital filter, extracted window method, Farrow structure, variable fractional delay filter.
I. INTRODUCTION
RACTIONAL delay (FD) filter plays important part in many digital signal processing applications. It is necessary whenever the precise control over the signal delay, much finer than the time between two consecutive samples, is required. For example the FD filter finds its applications in sampling rate conversion [1, 2, 3], time delay estimation, symbol sampling synchronization [4, 5, 6] or modeling of musical instruments [7]. The filter impulse response in those applications usually needs to be recalculated for each output sample, therefore numerical costs of the design procedure are crucial.
Optimal filters offer the shortest impulse responses fulfilling given requirements which means lowest numerical costs of filtering. However, the design procedures require in most cases solving matrix equations [8, 9, 10] which often must be repeated several times before the optimal solution is reached [9, 10]. This makes the design procedures unacceptable for variable FD filter implementation. A common solution to this problem is the Farrow structure [6, 11, 12] which is based on a set of polynomials approximating samples of the impulse response based on the required fractional delay. In order to compute structure coefficients a set of optimal filters for different fractional delays has to be designed, and when filter length, optimality criteria or approximation band have to be
Manuscript received November 7, 2011. This work was supported by the Polish Ministry of Science and Higher Education under the research project financed from the state budget designated for science in the years 2010-2012. M. Blok is with the Department of Teleinformation Networks on the Faculty of Electronics, Telecommunications and Informatics at GdaĔsk University of Technology, (e-mail: mblok@eti.pg.gda.pl).
changed, the new coefficient set is needed.
With the proposed structure it is much simpler to change the properties of FD filter. The idea is based on extracted window method [13, 14, 15, 16, 17] which is based on the observations that optimal FD filters with different delays can be designed with single symmetric window extracted from a single optimal filter designed for an arbitrarily selected fractional delay [13, 16, 18]. The design procedure requires additional gain correction dependant on the fractional delay of the filter which can be readily computed with a low order polynomial [16]. The design procedure is fairly simple, however, it requires computation of a truncated ideal impulse response which in case of FD filter is a sampled sinc function [8]. Like the optimal FD filters, a truncated sinc FD filter can be implemented using Farrow structure, which can be incorporated into the extracted window method. It is worth noting that with such approach the Farrow structure coefficients are the same for any kind of FD filter, only the coefficients of the window and the gain correction polynomial change. Moreover, to compute those coefficients we only need to design single optimal filter for each specification needed.
II. EXTRACTED WINDOW METHOD
An FIR FD filter with impulse response h[n] and frequency response ) 2 exp( ] [ ) ( 1 0 fn j n h f H N n N =
¦
− π − = (1)approximates the ideal frequency response [8] ) 2 exp( ) ( id f j f d H = − πτ (2)
where τd is the total delay and f ∈ [−0.5, 0.5) is the normalized frequency. the frequency response (2) corresponds to the ideal infinite impulse response
) ( sinc ] [ id n n d h = −τ (3)
Because of the causality requirement, good FSD filters are characterized with nonzero integer delay D = round(τd), which for FIR filters is usually selected close to the bulk delay τN = (N–1)/2. With those two delays defined, we receive the
Versatile Structure for Variable Fractional Delay
Filter Based on Extracted Window Method
Marek Blok
F
> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 2 following formula for the total delay
ε τ
τd =D+d = N + (4)
where d ∈ [−0.5, 0.5) is the fractional delay and ε is the net delay.
As we have mentioned in the previous section, design procedures for optimal FD filters design are to complex for VFD filter implementation. However, the design process can be significantly simplified with extracted window method [14, 16]. If we take even part of the window
2 ]) 1 [ ] [ ( ] [ ext ext ref n w n w N n w = + − − , n = 0, ..., N−1 (5)
extracted from impulse response of the optimal filter designed an arbitrary fractional delay d
0 ] [ ; ] [ ] [ ] [ opt id id ext n =h n h n h n ≠ w , (6)
we can design practically optimal filters for all other delays using window method [13, 16]
] [ ] [ ) ( ] [n d wref n hid n hd =α , n = 0, ..., N−1. (7)
In comparison to standard window method we need to use additional gain correction factor which can be approximated with the following formula
¸ ¹ · ¨ © § − =
¦
− = 1 0 id ref d a( )) [ ] [ ] 2 sinc( 1 ) ( N n n h n w n f d τ α . (8)where fa is the upper frequency of the approximation band of the optimal filter. The formula (8) for real time application might be computationally too demanding, then the gain correction curve should rather be approximated with low order polynomial, with only second order polynomial required for high performance VFD filters.
The overall procedure for design and implementation of VFD filter with the use of the extracted window method is presented in Fig. 1. The most time consuming computations can be done beforehand at the design stage which consists of designing single optimal filter, extracting from it symmetric window and calculating coefficients of polynomial of the order
p approximating gain correction factor
¦
= = p k k k p d a d 0 ) ( α . (9)Next at the run time for each different fractional delay d we need to calculate truncated impulse response of the ideal filter and compute gain correction factor using polynomial coefficients obtained beforehand and use them in structure presented in Fig. 2.
III. THE PROPOSED STRUCTURE
The window method significantly reduces numerical costs of VFD filter coefficients update, though computations of truncated ideal impulse response (3) is still a problem as a sinc is a nonlinear function. The solution to this problem is the Farrow structure [6, 11, 12]. In this structure each sample of the impulse response is computed using a polynomial
¦
= = p m m m n d c n h 0 ] [ ] [ . (10)In can be shown that this leads to the following formula describing FIR FD filter
¦
¦
=− = = = p m m m N n d n y n x n h n y 0 1 0 ] [ ] [ ] [ ] [ (11) whereFig.2. VFD filter structure based on extracted window method.
Fig.1. The general diagram of the extracted window method.
Fig.2. VFD filter structure based on extracted window method. Fig.2. VFD filter structure based on extracted window method.
Fig.3. Farrow structure of order p = 2 approximating the FD filter of the length N = 6. Thick dashed box indicates structure coefficients cm[0] of the
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¦
=− = 1 0 ] [ ] [ ] [ N n m m n c n x n y . (12)Formulas (11) and (12) lead to the structure presented in Fig. 3 where each coefficients row implements separate filter with impulse response cm[n] (12) with all the filters sharing the same input buffer.
It is worth noting that as we obtain the ideal impulse response of the FD filter (3) by sampling sinc function (Fig. 4), the increase in filter length can be simply achieved by appending new samples at the beginning and the end of an impulse response of the shorter filter. In case of the Farrow structure implementing sinc function this means that we only need to design a structure for the longest filter we want to implement. With such structure in order to change the filter length we can simply switch on or off the structure segments corresponding to added or discarded samples of impulse response without need to change the coefficients set. We must remember, however, that we actually need two structures (sets of coefficients) as the polynomials approximate different sections of sinc function for even and odd filter lengths (Fig. 4).
The Farrow structure approximating the filter with truncated ideal impulse response we can be readily adapted to implement
the optimal VFD filter. We only need to replace the common input buffer from Fig. 3 with pseudo input buffer incorporating the symmetric window from Fig. 2 and multiply the structure output by the gain correction factor computed using a low order polynomial (Fig. 2).
In case of truncated sinc filter even Farrow structure order
p = 3 offers the filter performance degraded only slightly
(Fig. 6a). This is the effect of poor performance of such a filter. However, when we want to use such a filter to implement the optimal one using window method, then we must look at how well it approximates the truncated sinc (Fig. 6b) instead of how well it approximates the ideal FD filter (Fig. 6a). In Fig. 7 we can see approximation errors of the filters designed using windows extracted from optimal LS filters. For lower performance filters (N = 21, fa = 0.4) Farrow structure order p = 5 gives good results, with order p = 6 offering practically optimal performance. For the higher performance filter (N = 53, fa = 0.45) the order of Farrow structure used to approximate truncated sinc needs to be increased by one. This increases performance of sinc approximation by about 20 dB which is similar to the difference in performance between compared filters. Generally, for filters with approximation error about -100 dB
Fig.5. Magnitude of complex approximation error [dB] of FD filters with
N = 33 and d = 0.25 for different optimality criteria. Bold line – truncated
sinc, thin line – MF, dashed line – LS, dotted line – minimax.
a) b)
Fig.6. Approximation error (LS) [dB] of truncated sinc approximated with Farrow structure (a) and error between the ideal truncated sinc and the sinc approximated with Farrow structure. Farrow structure orders p = 3, 4, 5 and 6
a)
b)
Fig.4. The sinc function and impulse responses of truncated ideal FD filter with fractional delay d = −0.2. (a) N = 10, (b) N = 11. Dashed lines separate segments of sinc approximated in Farrow structure with separate polynomials.
Fig.7. Approximation error (LS) [dB] of optimal LS VFD filters versus fractional delay d. Solid line – filer length N = 21 and upper frequency of approximation band fa = 0.4; dashed line – i = 53 and fa= 0.45. Farrow
> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 4 Farrow structure order p = 7 results in practically optimal
filters.
The order required for direct implementation with Farrow structure is the same [12] as that required for the proposed structure which means that the proposed solution needs only few additional numerical operations. This small overhead in numerical costs allows for simple change of filter type and/or length which can be done by selecting a new extracted window and coefficients of polynomial approximating gain correction factor for this window, which is the advantage of the proposed structure. Fig. 5 shows examples of approximation errors of the filters obtained with different windows and the same truncated ideal impulse response. If we need to change filter length we can switching on/off branches of the Farrow structure implementing sinc which is equivalent to setting unused samples of extracted window to zero. Additionally, we can decrease numerical cost by using Farrow structure with coefficients obtained by multiplying extracted window factors by coefficients of the Farrow structure approximating truncated sinc. This would mean additional computation when changing filter type or length but at run time the only overhead would be multiplications of the filter output by gain correction factor computed from polynomial with second order polynomial suitable for most cases.
IV. CONCLUSION
The structure for VFD filter implementation proposed in this paper combines advantages of the extracted window method and the Farrow structure. Extracted window method provides simple means for designing VFD filter with symmetric window extracted from single optimal filter and additional gain correction which can be approximated by a low order polynomial. The coefficients for the optimal filter are obtained by multiplying truncated ideal impulse response (sampled sinc function) by the extracted window and the gain correction factor. This means that when we want to change the FD filter specifications, such as optimality criteria, approximation band or impulse response length, we only need to change the window and gain correction factor as the ideal impulse response is the same regardless of those parameters. The only problem is the computation of truncated impulse response and this has been solved with help of the Farrow structure.
The proposed structure can be used to implement maximally flat (MF), minimax or least squared (LS) filters [13, 16]. It is worth noting that the extracted windows we have to store in memory in order to be able to change the filter specifications are symmetric and we need to store only half of their samples.
Such versatile structure can be useful especially in multimedia application, particularly in sampling rate conversion [1, 2, 3]. Different sampling rates and quality requirements result in different specifications of VFD filter used in the processing, which with set of extracted windows prepared beforehand can be satisfied with single structure at
low numerical costs.
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