# A New Method for Least-Squares and Minimax Group-Delay Error Design of Allpass Variable Fractional-Delay Digital Filters

- Cheng-Han Chan
^{1}, - Soo-Chang Pei
^{2}and - Jong-Jy Shyu
^{3}Email author

**2010**:976913

https://doi.org/10.1155/2010/976913

© Cheng-Han Chan et al. 2010

**Received: **28 February 2010

**Accepted: **22 December 2010

**Published: **4 January 2011

## Abstract

A double-loop iterative method is proposed to design allpass variable fractional-delay (VFD) digital filters basing on the minimization of root-mean-squared group-delay error. In the inner loop, an iterative quadratic optimization is proposed to replace the original nonlinear optimization for the minimization of root-mean-squared group-delay error, while an iterative weighting-updated technique is applied in the outer loop to further reduce the maximum group-delay error. Several examples will be presented to demonstrate the effectiveness and good convergence of the proposed method.

## Keywords

## 1. Introduction

For the past decade, the design of variable fractional-delay (VFD) digital filters became an important topic in digital signal processing due to their wide applications in signal processing and communication systems such as comb filter design, sample rate conversion, tunable modulator and acoustic system [1–5]. Since Farrow proposed an effective structure for implementing variable digital filter [6], several works concerning VFD filter design have been presented, including an excellent tutorial paper by Laakso, and so forth [7], FIR-based design [8–11], IIR-based design [12, 13] and allpass-based design [14–24] with their respective feature.

In this paper, the design of allpass VFD digital filters is investigated on the possible minimization of root-mean-squared group-delay error. Among the existing literature in which allpass structure is applied, most applications concern the minimization of phase-oriented error, and only [23] focuses on the minimization of root-mean-squared group-delay error by converting a nonlinear optimization problem to a linear least-squares (LS) optimization problem.

In this paper, an alternative method will be presented with comparable performance. Likely, the direct approximation of group-delay response is a highly nonlinear problem, so an iterative quadratic optimization will be proposed to overcome it in this paper. Then a weighting-updated technique [11, 25] is proposed to further reduce the maximum group-delay error of the designed system, which constitutes the outer loop of the overall process while the iteration stated above makes up the inner loop.

It is also pointed out in [26] that if the allpass filter design has a phase approximating error less than at it must be stable. In this paper, although there is no theoretical proof, it can be found that the designed allpass VFD filter is usually stable when mean delay of the desired response is equal to the order of the designed allpass filter and the range of adjustable parameter is properly assigned.

This paper is organized as follows. In Section 2, the review of conventional weighted least-squares (WLS) design (as Deng's method [21]) basing on the minimization of phase-oriented error and frequency-response-oriented error is given, and it will be shown that both will lead to the same solution. The formal formulation for LS group-delay error design of allpass VFD filters will be presented in Section 3, in which an iterative method is proposed to replace the original nonlinear optimization of group-delay-oriented error. Then in Section 4, a weighting-updated technique is proposed to further reduce the maximum group-delay error, and design examples will be given to demonstrate the effectiveness and good convergence of the proposed double-loop iterative method. Also, an example with a different range of the adjustable variable is given to show the significant effect on overall performance, which has also been revealed in [14, 24]. Finally, the conclusions are given in Section 5.

## 2. Review of Deng's Method of Allpass VFD Digital Filters

which is used to approximate (2) as much as possible over the region .

### 2.1. Phase-Oriented Approximation

### 2.2. Frequency-Response-Oriented Approximation

Hence, both phase- and frequency-response-oriented approximations will lead to the same solution.

### 2.3. WLS Solution of the Design Problem

## 3. LS Group-Delay Error Design of Allpass VFD Digital Filters

When is small enough, for example, smaller than , where is a preassigned very small positive constant, the iterative process can stop. In this paper, is used. As to the initial coefficient vector , we can adopt the solution in (18) by setting . The details of iterative procedures will be described in the next section.

respectively. To compute (29), the frequency and the variable are uniformly sampled at step sizes and , respectively.

Example 1.

Comparison of evaluated errors in (29).

Method | Design time (seconds) | ||||
---|---|---|---|---|---|

0.242 | 0.03145 | 0.001205 | 0.0001788 | 0.38 | |

Lee, Caccetta, and Rehbock's method [23], LS design, | 0.0992 | 0.005276 | 0.002199 | 0.0000718 | 3.19 |

0.1474 | 0.004137 | 0.002312 | 0.0000707 | 28.36 | |

0.04464 | 0.001927 | 0.000724 | 0.0000543 | 28.13 | |

Lee, Caccetta and Rehbock's method [23], WLS design, | 0.155 | 0.002836 | 0.00307 | 0.0000838 | 58.63 |

0.1964 | 0.002966 | 0.003235 | 0.0000834 | 148.76 | |

0.0664 | 0.001189 | 0.001141 | 0.0000365 | 196.56 |

## 4. Minimax Group-Delay Error Design of Allpass VFD Digital Filters

In this section, a weighting-updated technique is proposed to minimize the maximum group-delay error of an allpass VFD filter obtained in Section 3, which constitutes the outer loop of the overall process while the iteration in Section 3 makes up the inner loop. The overall iterative process is described in detail below.

Step 1.

Given , , , and , set , and find the initial coefficient vector by (18).

Step 2.

Set the inner iterative counter .

Step 3.

Increase the inner iterative counter by 1, and calculate , , , and .

Step 4.

Find the coefficient vector by (27).

Step 5.

If the condition is satisfied, go to the next step; otherwise go to Step 3.

Step 6.

occurs for the first outer iteration only. Find the absolute error ripples of , and denote the th ripple with ripple interval by , , where is the number of ripples in . Then search the maximum value and the minimum value of , .

Step 7.

where is a preassigned very small positive constant. If the condition is satisfied, stop the process; otherwise go to the next step.

Step 8.

Step 9.

Calculate
**,**
in (17) and replace
by
. Then go to Step 2.

Example 2.

Following Example 1, the allpass VFD filter is continuously designed with minimax group-delay error. If is used, the design took thirteen outer iterations and the respective inner iterations are three and two in the first and second outer iterations, and one in the others. Figure 3(d) presents the final group-delay errors, and the errors computed by (29) are also listed in Table 1. To illustrate the stability of the designed filter, the maximum pole radius is shown in Figure 3(e), which shows that the designed filter is stable since the poles are all inside the unit circle for .

Example 3.

Filter coefficients for the proposed LS design in Example 3.

| |||||
---|---|---|---|---|---|

| 1 | 2 | 3 | 4 | 5 |

1 | −0.995911478379215 | 0.003037237182070 | 0.000674977600074 | 0.002203931411874 | −0.001547094931521 |

2 | 0.491860988660958 | 0.489906840770088 | −0.004126440722118 | −0.004968604465653 | −0.000451455145871 |

3 | −0.321238701261896 | −0.480959682281854 | −0.155527315538002 | 0.009336078160601 | 0.002875779605052 |

4 | 0.234086131719820 | 0.429265271544100 | 0.228966726293102 | 0.025840785057596 | −0.005934120518170 |

5 | −0.180442437563948 | −0.376964115092541 | −0.258474768641364 | −0.058885621354534 | 0.001432724569099 |

6 | 0.143669792117880 | 0.329957770218435 | 0.265616427088872 | 0.083096234713679 | 0.005227747340939 |

7 | −0.116657162172812 | −0.288508807455575 | −0.260564239814727 | −0.098781179155452 | −0.011492189916140 |

8 | 0.095861039125088 | 0.251939854553747 | 0.248549917352264 | 0.107475439484688 | 0.016441996382781 |

9 | −0.079319256800620 | −0.219538064590932 | −0.232515100469996 | −0.110719224055686 | −0.019869040384123 |

10 | 0.065857103009291 | 0.190720077751894 | 0.214249655962206 | 0.109824100478193 | 0.021858526125015 |

11 | −0.054726435065962 | −0.165033191450544 | −0.194914864048044 | −0.105865858472555 | −0.022611233785938 |

12 | 0.045425714251763 | 0.142128041436134 | 0.175304729171998 | 0.099719296033186 | 0.022361719532284 |

13 | −0.037603022298150 | −0.121728204225754 | −0.155979945905486 | −0.092096431614038 | −0.021344479562458 |

14 | 0.031001228444685 | 0.103608784007633 | 0.137345198030118 | 0.083574882334005 | 0.019774852983403 |

15 | −0.025425012312619 | −0.087578010616568 | −0.119690472293712 | −0.074621308011672 | −0.017844113508204 |

16 | 0.020720915083929 | 0.073467158589459 | 0.103220353275063 | 0.065607022593667 | 0.015713946630399 |

17 | −0.016764317587889 | −0.061120833690438 | −0.088069015325893 | −0.056822176308913 | −0.013518649356740 |

18 | 0.013451438220377 | 0.050393631010479 | 0.074315411373739 | 0.048485405849704 | 0.011363357432515 |

19 | −0.010693672377723 | −0.041145191376987 | −0.061990029855892 | −0.040754079903763 | −0.009328571849153 |

20 | 0.008414281412575 | 0.033240476434287 | 0.051085766424762 | 0.033731771771784 | 0.007469846563140 |

21 | −0.006545751741101 | −0.026547275109895 | −0.041561893474693 | −0.027476917314175 | −0.005823266602704 |

22 | 0.005028471552094 | 0.020938283213411 | 0.033353508964552 | 0.022009510808459 | 0.004405809839568 |

23 | −0.003809384590832 | −0.016289718620921 | −0.026374265528517 | −0.017318832133697 | −0.003221101051906 |

24 | 0.002841516201697 | 0.012484128343807 | 0.020525052306191 | 0.013369584615560 | 0.002259936792143 |

25 | −0.002083164017217 | −0.009409316711960 | −0.015695742339824 | −0.010108579318228 | −0.001505805273695 |

26 | 0.001497755497971 | 0.006961232471637 | 0.011773041636065 | 0.007470070929259 | 0.000935170532813 |

27 | −0.001053228383405 | −0.005042751228794 | −0.008641133167121 | −0.005381083473594 | −0.000522366521324 |

28 | 0.000721982433284 | 0.003566279107085 | 0.006188479311859 | 0.003765621798479 | 0.000239360156968 |

29 | −0.000480297868477 | −0.002452163386710 | −0.004307224739879 | −0.002548399099865 | −0.000059778977985 |

30 | 0.000308285023293 | 0.001630854379561 | 0.002898818696283 | 0.001657749575799 | −0.000041970396936 |

31 | −0.000189289826257 | −0.001040887457275 | −0.001872166300857 | −0.001027714173915 | 0.000087596983292 |

32 | 0.000109818709357 | 0.000630593764138 | 0.001148101727283 | 0.000599713352933 | −0.000096409825026 |

33 | −0.000058932273691 | −0.000355717936032 | −0.000656467669958 | −0.000323254739076 | 0.000082969883834 |

34 | 0.000028159894437 | 0.000180788324236 | 0.000339826063819 | 0.000157105923634 | −0.000058660087578 |

35 | −0.000010912672360 | −0.000076734250069 | −0.000151603214037 | −0.000073350022415 | 0.000026749510599 |

Filter coefficients for the proposed minimax design in Example 3.

| |||||
---|---|---|---|---|---|

| 1 | 2 | 3 | 4 | 5 |

1 | −0.995993596236449 | 0.002951361938129 | 0.000056522430938 | 0.003227364563976 | −0.002459241277563 |

2 | 0.492019060719535 | 0.490165494401252 | −0.002681420024658 | −0.006345896805414 | −0.000444700949259 |

3 | −0.321471225422751 | −0.481418973949900 | −0.157782729637916 | 0.010673985821541 | 0.003558502797986 |

4 | 0.234388948779710 | 0.429936634960839 | 0.232014003186862 | 0.024803935644968 | −0.007082100578430 |

5 | −0.180809553898889 | −0.377848777076039 | −0.262282154813875 | −0.058330113715328 | 0.002858896290434 |

6 | 0.144093799282764 | 0.331049886098540 | 0.270138023365234 | 0.083140840476517 | 0.003671971963968 |

7 | −0.117129820320514 | −0.289799197810240 | −0.265745946995025 | −0.099497954673552 | −0.009920842069164 |

8 | 0.096372854062878 | 0.253409909828576 | 0.254315329105907 | 0.108894154915106 | 0.014947469673871 |

9 | −0.079860248869537 | −0.221163954074996 | −0.238774548835977 | −0.112836471202399 | −0.018523661245126 |

10 | 0.066417235371802 | 0.192475127776987 | 0.220905131511245 | 0.112605063283127 | 0.020710959749977 |

11 | −0.055295700680418 | −0.166886921064048 | −0.201858125775743 | −0.109247481340844 | −0.021690925658959 |

12 | 0.045994562958711 | 0.144048641408887 | 0.182424468016734 | 0.103618709513285 | 0.021681689765273 |

13 | −0.038162626126081 | −0.123683810542076 | −0.163165397984585 | −0.096416215678346 | −0.020903696245568 |

14 | 0.031543613480551 | 0.105568113889007 | 0.144488637747372 | 0.088206732231356 | 0.019558989262805 |

15 | −0.025943254665904 | −0.089511445430816 | −0.126690976298732 | −0.079454047897343 | −0.017831187066804 |

16 | 0.021209234821121 | 0.075347332574665 | 0.109985232980534 | 0.070527891068195 | 0.015873812332757 |

17 | −0.017218205011616 | −0.062923918561836 | −0.094518367293347 | −0.061725532030323 | −0.013818324456679 |

18 | 0.013867583703075 | 0.052099069080072 | 0.080381481983181 | 0.053273169499334 | 0.011767529430927 |

19 | −0.011069999914064 | −0.042736331042533 | −0.067618950223505 | −0.045340098661654 | −0.009802870961721 |

20 | 0.008749913929641 | 0.034705233910870 | 0.056239697296178 | 0.038045151627557 | 0.007982131561675 |

21 | −0.006840717660722 | −0.027876032814163 | −0.046213102316074 | −0.031459203715713 | −0.006345583922439 |

22 | 0.005284001340328 | 0.022127902740351 | 0.037494541869108 | 0.025622074058447 | 0.004913569954169 |

23 | −0.004027273226697 | −0.017338667620351 | −0.030005448553794 | −0.020536165254725 | −0.003696194040458 |

24 | 0.003024238308363 | 0.013394444480610 | 0.023658566786054 | 0.016180658703661 | 0.002688753419761 |

25 | −0.002233814412328 | −0.010187262429483 | −0.018357061617550 | −0.012518726197641 | −0.001880234103510 |

26 | 0.001619612427739 | 0.007614467619845 | 0.013993527079620 | 0.009494927822061 | 0.001251724331496 |

27 | −0.001149754797462 | −0.005581013645482 | −0.010458788127664 | −0.007046111678278 | −0.000781330341624 |

28 | 0.000796663861173 | 0.004000709213585 | 0.007645263913791 | 0.005101208642714 | 0.000441439263909 |

29 | −0.000536633718021 | −0.002795599745994 | −0.005449025857033 | −0.003589487278191 | −0.000206255609100 |

30 | 0.000349545529838 | 0.001895830247705 | 0.003770427414620 | 0.002441776327828 | 0.000053417213621 |

31 | −0.000218498674617 | −0.001239482351210 | −0.002517028536055 | −0.001596601179870 | 0.000032333826084 |

32 | 0.000129607420879 | 0.000773720821702 | 0.001606542109353 | 0.000996204580929 | −0.000067169661349 |

33 | −0.000071635910779 | −0.000454306657023 | −0.000967504900344 | −0.000587397249461 | 0.000068485823988 |

34 | 0.000035791801597 | 0.000245674851842 | 0.000539971198304 | 0.000321069026938 | −0.000055829905847 |

35 | −0.000015815837778 | −0.000125519073336 | −0.000300465948474 | −0.000202997633437 | 0.000015539277740 |

## 5. Conclusions

In this paper, a double-loop iterative method has been proposed to minimize the root-mean-squared group-delay error in LS and minimax senses for the design of allpass VFD digital filters. For the LS design, an iterative quadratic optimization is used in the inner loop, while a weighting-updated technique is further applied to minimize the maximum group-delay error in the outer loop. From the presented experiments, it has been shown that the performance in group delay and phase for the proposed systems can be improved drastically by appropriately specifying the range of fractional delay. For the computational complexity, although the design time of the proposed method is much more than the existing methods, an alternative method has been revealed in this paper for further research in the future.

## Authors’ Affiliations

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