A Fast Windowing-Based Technique Exploiting Spline Functions for ...

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A Fast Windowing-Based Technique Exploiting Spline Functions for Designing Modulated Filter Banks Fernando Cruz-Roldán, Senior Member, IEEE, Pilar Martín-Martín, Member, IEEE, José Sáez-Landete, Manuel Blanco-Velasco, Member, IEEE, and Tapio Saramäki, Fellow, IEEE

Abstract—A very fast technique to design prototype filters for modulated filter banks without using time-consuming multivariable optimization is introduced. In the proposed method, the prototype filter is optimized by using the windowing technique, with the novelty of exploiting spline functions in the transition band of the ideal filter, instead of using the conventional brick-wall filter. A study of the optimization techniques and three different objective functions existing in the literature has been carried out, and more suitable redefinitions of these objective functions are employed to achieve as optimized prototype filters as possible. The resulting filter banks closely satisfy the perfect reconstruction property, as is illustrated by means of examples. Index Terms—Channel bank filters, cosine-modulated filter banks (CMFBs), filter bank design, filtering theory, line-search methods, modified DFT filter banks, nearly perfect reconstruction (NPR), optimization.

I. INTRODUCTION

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ODULATED filter banks (MFBs) are used in a wide range of applications, from data compression (e.g, speech, audio, video, and biosignal coding) to data transmission [1]–[6], and their use is very attractive for the following reasons. First, high-selectivity and high-discrimination systems can be easily designed. Second, all of the analysis and synthesis subchannel filters can simultaneously generated by simply applying an appropriate modulation scheme to a single properly synthesized linear-phase finite-impulse response (FIR) prototype filter. Thirdly, in particular cases, very fast algorithms can be used to efficiently implementing the overall system [2], [7], [8]. This work focuses on describing an efficient technique for designing linear-phase FIR prototype filters for nearly perfectreconstruction (NPR) MFBs. The proposed approach is an imManuscript received June 23, 2007; revised January 31, 2008. First published May 20, 2008; current version published February 4, 2009. This work was supported in part by Comunidad Autónoma de Madrid and Universidad de Alcalá through Projects CCG06-UAH/TIC-0417 and CCG07-UAH/TIC-2034, in part under Project FIS-PI052277, and by the Spanish Ministry of Education and Science under Grant PR2007-0218. This paper was recommended by Associate Editor R. Merched. F. Cruz-Roldán, P. Martín-Martín, J. Sáez-Landete, and M. Blanco-Velasco are with the Department of Teoría de la Señal y Comunicaciones, Escuela Politécnica Superior de la Universidad de Alcalá, 28871 Alcalá de Henares (Madrid), Spain (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). T. Saramäki is with the Department Finland of Signal Processing, Tampere University of Technology, Tempere FI-33101, Finland (e-mail: [email protected]). Digital Object Identifier 10.1109/TCSI.2008.925350

proved version of that described in [9], [10], where the design of prototype filters is based on properly applying the windowing technique such that only conventional brick-wall ideal filters are used. In order to achieve better overall filter bank performances, a spline function for shaping the transition band in the ideal low-pass filter definition, as suggested in [11], has been incorporated in [12]. The use of spline functions in the filter design allows the analytical design of optimal least-squarederror FIR filters, along with a reduction of the approximation ripple. This technique eliminates the Gibbs’ phenomenon even more and allows an explicit control of the transition bandwidth. Moreover, the FIR filters are easy to calculate and to program using formulas which use the th-order transition-band spline function. In this contribution, the synthesis scheme proposed in [12] is enhanced and two efficient objective functions widely used for designing prototype filters [13], [14] are also included and modified with the purpose of even more improving the performance and the properties of the resulting NPR MFB systems. The applications of the proposed technique can be extended -channel cosine-modulated filter banks (CMFBs) to from -channel complex MFBs, which include modified discrete Fourier transform (MDFT) filter banks (FBs) [15], [16]. Fig. 1 shows the implementations of the -channel CMFB as well as those of the type-1 and type-2 MDFT FBs [15] without including the processing unit between the analysis and synthesis filter banks. Although this work focuses on the design and performance of maximally-decimated MFBs, the proposed technique can be equally well applied to the synthesis of multicarrier modulators (MCMs) based on a CMFB or a complex MFB, using the well-known mathematical equivalence for these two systems [17], [18]. The remainder of this paper is organized as follows. Section II describes how to obtain different MFBs by means of a new unified scheme of modulation. In Section III, the proposed technique for designing the prototype filter is presented. Section IV briefly reviews several efficient objective functions which are used in the suggested approach. Section V shows a description of the proposed optimization techniques and a suitable modification of the objective functions in order to increase the accuracy, speed, and reliability of the optimization process. In Section VI, several quantities are considered for evaluating the performance of the resulting MFBs when using the suggested technique. In Section VII, two examples are included for illustrating the benefits of the proposed design scheme. These exam-

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Fig. 1. Maximally-decimated MFBs under consideration. (a)

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M -channel CMFB. (b) 2M -channel type-1 MDFT FB. (c) 2M -channel type-2 MDFT FB.

ples clarify the influences of the window function, the objective function, and the order of the transition-band spline function in

the suggested filter bank design. Finally, concluding remarks are given in Section VIII.

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TABLE I REQUIRED PARAMETER VALUES IN THE PROPOSED UNIFIED MODULATION SCHEME TO ACHIEVE THE MAXIMALLY-DECIMATED MODULATED FILTER BANKS UNDER CONSIDERATION. THE COMMON DECIMATION AND INTERPOLATION FACTOR IS

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II. A UNIFIED MODULATION SCHEME TO GENERATE THE MODULATED FILTER BANKS UNDER CONSIDERATION This section shows how to generate CMFBs as well as type-1 and type-2 MDFT FBs [15] by means of applying a unified scheme of modulation to the same prototype filter. First, this scheme is described with adjustable parameters and, then, it is shown how to fix these parameters to end up with the abovementioned MFBs. A. Proposed Unified Scheme Let the form

th-order prototype filter transfer function be of the

(1) are real-valued where the impulse-response coefficients for . The proand satisfy posed unified modulation scheme is based on the following for two steps: First, the auxiliary transfer functions are defined as

is the transfer function whose corespectively, where efficient values are the complex conjugates of those of and is the coefficient whose value is the complex . In the above, and are diconjugate of that of rectly the common sampling rate conversion factor for the analysis and synthesis banks and the number of subchannels in the MFBs, respectively. The remaining parameters as well as , , , and for , in turn, should be selected such that the arrival at the desired MFB is guaranteed. These required values of the above-considered parameters are in each of the three given in Table I for the given value of MFBs under consideration. B. Modulated Filter Banks Resulting When Using the Unified Scheme Using the parameters in the second row in Table I in (2), (3a), and (3b) leads, after some manipulations, to the -channel CMFB of Fig. 1(a), where the impulse-response coefficients of [cf. the th analysis and synthesis filters for (3a) and (3b)] are given by

(4a)

(2) Then, for , the analysis and synthesis transfer functions are generated, based on the above-defined auxiliary transfer functions, as follows: for for for for

even odd

even odd (3a)

and for for for for

even odd

even odd (3b)

(4b) respectively. When comparing (4b) with the conventional definition of (see, e.g., [2]), it is seen that the additional constant is included in (4b). This is due to the following two reasons. First, for the prototype filter (and for the analysis and synthesis filters after using the above modulation scheme) resulting when applying the proposed technique to be described in Sections III–V, the maximum magnitude value in the passband is approximately equal to unity. Second, because of the energy due to the interpolation by loss of the signal by a factor of the same factor before the th synthesis filter, for compensating this loss, there is a need to include the constant of value . Similarly, using the parameters in the third and fourth rows in Table I in (2), (3a), and (3b) gives, after some manipulations, -channel type-1 MDFT FB of Fig. 1(b) and the rise to the -channel type-2 MDFT FB of Fig. 1(c), respectively, such

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that the impulse-response coefficients of the th analysis and synthesis filters [cf. (3a) and (3b)] are expressible as follows. are given For Fig. 1(b), these coefficients for by

degree of freedom for the overall synthesis compared with the . conventional brick-wall case with The frequency response of the resulting infinite-duration ideal filter is expressible as

(5a)

(9)

(5b) and for Fig. 1(c) by (6a) (6b) Again, for the same reason as in the CMFB case, the additional is included in (5b) and (6b) compared to the correconstant sponding formulae1 in [15] for the same reason as in the CMFB case. III. PROPOSED DESIGN SCHEME FOR THE PROTOTYPE FILTER

where

in the passband region and in the stopband region . In the transition , the performance of depends on the

band value2 of . After selecting the window function and the impulse-response coefficients of the ideal filter, the impulse-response coefficients of the resulting prototype filter are given by (10) as in the conventional brick-wall filter case with

.

B. Proposed Design Technique

is the th-order spline function that is used to properly shape the transition band of the ideal filter [11], thereby giving an extra

The input parameters for the optimization problem to be considered in this paper are the following: , the prototype filter order; 1) , the common sampling rate conversion factor for the 2) analysis and synthesis banks in the MFBs under consideration; recall from Section II that , the number of subchannels, is for CMFBs and for the type-1 and type-2 MDFT FBs; ; 3) the window function 4) , the integer-valued order of the transition-band spline function, as given by (8a); , the stopband edge angle of the prototype filter. 5) After knowing these input parameters, the main idea in the such proposed technique is to form an objective function that the following holds. 1) It is based on the windowing technique of the previous subsection. , the passband edge angle of the prototype filter, is the 2) only adjustable parameter. 3) The arrival at an NPR MFB is guaranteed by locating the that minimizes . value of Depending on the chosen predefined objective function to be discussed in the following section, the optimized value of can be found by using unconstrained minimization techniques. These standard techniques make the overall synthesis extremely fast compared with the case where the impulse-response coefficients of the prototype filter are directly the unknowns in the optimization. What is left is to appropriately select the objective function and the above-mentioned input data for the overall procedure in order to achieve an NPR MFB that closely approximates its PR counterpart. In the sequel, Section IV introduces three objective functions that exist in the literature and are used, after proper modifications, in this contribution, whereas Section V shows

1This contribution concentrates on the case, where the same prototype filter is used for both the analysis and synthesis filters and the overall FB delay is restricted to be equal to the order of the prototype filter.

2This contribution concentrates only on cases where is integer-valued in order to prevent  [n], as given by (8b), from taking on negative values (for more detail, see [11]).

The purpose of this section is twofold. First, it outlines the synthesis scheme to design prototype filters based on the window technique, where the spline function is utilized for properly shaping the transition band of the ideal filter. Secondly, the overall synthesis technique used in this contribution for arriving at NPR MFBs closely approximating their perfect-reconstruction (PR) counterparts is briefly described. A. FIR Filter Design Based on the Use of Transition-Band Spline Functions When generating an th-order linear-phase FIR prototype filter in the proposed technique to be described in more detail in Sections III-B, IV, and V, first, a proper window function is selected such that it is nonzero only for and satisfies for these values of the following symmetry: (7) Second, the ideal filter with the passband and stopband edge and , respectively, is generated as angles at follows: (8a) where

(8b)

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how to quickly arrive in a reliable manner, in each case of these three objective functions, at a desired NPR MFB. IV. OBJECTIVE FUNCTIONS In the existing literature, there are three main groups of techniques that have been initiated in [9], [13], [14] and have the following two attractive features that enable one to properly generate the objective function to be used in the synthesis scheme briefly discussed in Section III-B. First, as already mentioned in Section III-B, only the passband edge angle of the prototype filter is the only adjustable parameter that gives rise to NPR MFBs under consideration. Secondly, the resulting MFBs closely approximate their PR counterparts. The purpose of this section is twofold. First, these three techniques are briefly reviewed and, then, in each case, a proper objective function is generated for the above-mentioned purposes in a unified manner.

in [13], the stopband attenuation of the prototype filter should be high enough to keep aliasing between nonadjacent bands within tolerable limits. is used with the excepIn this paper, the above tion that the prototype filter is generated with the aid of the windowing technique described in Section III-A. B. Kaiser Window Approach This approach has been proposed by Lin and Vaidyanathan in [14]. The motivation4 for this approach as well as for the approach to be discussed in the following subsection is the based on the following fact. The linear-phase FIR prototype filter transfer, as given by (1), that gives rise to a -channel PR CMFB or an NPR one with no amplitude distortion as well as -channel MDFT FBs, to corresponding type-1 and type-2 is characterized by the following the property. If it is cascaded th-band with itself, then the resulting filter is a linear-phase FIR filter (see, e.g., [20]). This transfer function is given by

A. Creusere–Mitra Approach The method proposed by Creusere and Mitra [13] relies on generating the prototype filter transfer function , as given by (1), such that the corresponding frequency response satisfies (for more detail, see [13], [14] as well as references in these articles) as much as possible the following two conditions:

(13a) where

(11a)

(13b)

(11b) Meeting exactly (11a) and (11b) together with the modulation schemes3 discussed in Section II guarantees the arrival at -channel CMFBs as well as type-1 and type-2 -channel MDFT FBs without any amplitude distortion and no aliasing between adjacent and nonadjacent bands. In other words, the resulting MFBs satisfy the PR condition. The criteria of (11a) and (11b) are, however, too strict to be met by a finite-duration prototype filter. That is why Creusere and Mitra decided to use the Parks-McClellan algorithm [19] as follows: Given the prototype filter order , the stopband edge and the passband-stopband error ratio , adjust the angle passband edge angle to minimize the following objective function:

, where is the order of the prototype for filter and the impulse-response coefficients [cf. (1)] satisfy for and . th-band The impulse-response coefficients of the above for linear-phase FIR filter satisfy and5 (see, e.g., [21]) (14a) (14b) In the original Kaiser window approach, the passband edge angle of the prototype filter has been adjusted utilizing the windowing scheme of Section III-A by using the Kaiser window in the brick-wall case to minimize the following objective function: (15)

(12) where the subscript “ ” stands, for later use, for is involved in order “Creusere–Mitra Approach”. Here, , the Parks–McClellan emphasize that for each value of the algorithm optimizes the prototype filter such that on desired and weighting values for this algorithm are both unity, they are zero and . As has been pointed out whereas on

M

3It is worth emphasizing that the modulation scheme used in [13] was originally aimed at generating -band pseudo quadrature mirror filter banks, but this technique can equally well be used for the modulation schemes of this contribution.

where the subscript “ ” stands, for later use, for “Kaiser is included in the above equaWindow Approach.” Again, tion in order to emphasize its role in the optimization. Note that due to the symmetry, only the positive values of are needed in (15). In this paper, the original approach is extended to involve the use of the transition-band spline function and other windows in the overall design procedure. 4It is worth emphasizing that the motivation given here differs from that given in [14], but is very similar and applies equally well. 5In (14b), bxc stands for the integer part of x.

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C. Location of the 3-dB Cut-Off Frequency The third objective function to be considered in this paper has been originally defined in [9]. After unifying modifications, this function is expressible as dB

(16)

where the subscript “3 dB” is adopted, for later use, because this function is based on locating the 3-dB cutoff frequency of . The key idea behind using the prototype filter at this objective function is based on the frequency-domain propth-band FIR filter with the transfer erties of the linear-phase function, as given by (14a) and (14b). More details of these frequency-domain properties can be found, e.g., in [21]. As has been indicated in [10], these properties imply that if , the magnitude response of the the maximum value of prototype filter, is less than or equal to for , then the minimum and maximum and values of are in the worst case, which is very unlikely to occur in practice, and , respectively. Based on the above reasoning, it is worth pointing out the following facts. First, when designing prototype filters in practice, the stopband edge angle is in practical applications definitely less than . Second, even for , which corresponds to the 40-dB attenuation, the additional term under the square roots of the above worst-case maximum and minimum values is only , which is negligible even for a large value of . Hence, forming the objective function according to (16) is well motivated. V. UNIVARIATE UNCONSTRAINED OPTIMIZATION In order to accurately, quickly, and reliably optimize the three above-defined objective functions, as given by (12), (15), and (16), a so-called safeguarded polynomial interpolation method [22] is adopted in this contribution. The key idea for this choice is to combine a reliable method as “a golden search” with an efficient quadratic interpolation. This adopted method uses the interpolation in the regions, where the one-dimensional objective function is well-behaved, whereas it utilizes, in turn, the golden search in the regions, where this function is not so well-behaved. This compromise shows its real advantages after characterizing the properties of the objective functions and especially after their proper modifications to be described next. It has been experimentally observed that the objective function in each of the three cases is unimodal, but is not smooth in the vicinity of the minimum point of . Moreover, the function is approximately linear in this neighborhood, but when approaching the minimum point by decreasing the value of or by increasing this value for the same purpose, the resulting absolute values have different slopes. Fortunately, this problem can be easily solved by redefining the original objective functions based on the following fact. From the optimization point of view, the arrival at the same minimum point of is achievable by replacing the original objective functions, as given by (12), (15), and (16), by their squared counterparts as

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(18) (19)

dB

The use of these objective functions combined with the above-mentioned safeguarded polynomial interpolation method provides the following two attractive features. First, the problem of nondifferentiability at the minimum point is solved. Secondly, most importantly, the resulting overall algorithm is very accurate and the resulting optimization process is very quick and reliable. VI. PERFORMANCE EVALUATION Here, proper error measures are introduced for qualifying the performances of both CMFBs and MDFT FBs. When omitting the effect of the processing unit between the analysis and synthesis banks in Fig. 1, the relation between the output signal and the input signal is expressible in the -domain as (20) where

, (21)

and

and for the CMFB of Fig. 1(a), whereas and for the type-1 and type-2 -channel MDFT FB of Figs. 1(b) and (c), respectively. for , that is, In the above, (22) is the distortion transfer function determining the distortion caused by the overall system for the unaliased component , whereas the remaining transfer functions for are called the alias transfer functions and determine how well the aliased components of the input signal are attenuated. and The PR condition implies that for . Hence, the quality measurements should concentrate on both the distortion on the unaliased component . A good meaand the aliasing distortions for sure for the un-aliased distortion is the peak-to-peak amplitude distortion, given by (23) whereas a corresponding measure for the combined aliasing distortion is the peak aliasing error, defined by

(17)

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(24)

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TABLE II COMPARISON OF VARIOUS TECHNIQUES FOR DESIGNING 32-CHANNEL CMFBS WITH 511TH-ORDER SUBCHANNEL FILTERS

VII. EXAMPLE DESIGNS

In order to illustrate the benefits of the proposed design scheme, several MFBs have been designed based on prototype filters optimized by means of the three objective functions, as defined by (17)–(19), and by using various input parameters

for these functions (cf. the beginning of Section III-B). Only and there exists no intermediate the case, where processing unit between the analysis and synthesis banks, has been studied. The comparisons are summarized in Tables II and III for Example 1 and Example 2, respectively, to be considered later on. These tables include the peak-to-peak amplitude , the peak aliasing error , the minimum distortion

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TABLE III COMPARISON OF VARIOUS TECHNIQUES FOR DESIGNING 16-CHANNEL MDFT FBS WITH 128TH-ORDER SUBCHANNEL FILTERS

stopband attenuation parameter of the prototype filter , and the peak signal-to-noise ratio in the interval parameter that is defined as

(25)

Here, for is an -length input sequence of satisfying , whereas random numbers with each for is the corresponding output sequence delayed by samples that is caused by the MFB in use. parameter is a global criterion that does not Since the take into account local effects, as a local quality measure, the parameter, defined as (26) is introduced. Tables II and III also inparameter of the opticlude the minimum value mized objective functions and the convergence time of the optimization process. All the simulations were carried out on a Pentium 4 CPU that is clocked at 3 GHz and has 2 GB of an installed physical memory. The best results with respect to the above-defined numerous quality measures are indicated by the boldface letters in the tables. Example 1: Several 32-channel CMFBs with 511th-order and ) have been designed prototype filter (

with various design schemes. These design techniques involve the proposed scheme for which objective functions are given by (17) and (18) and the window functions are the Blackman and the Kaiser windows that have turned out to give the best results to the quality measures under consideration. In addition, the Kaiser window approach [14], three special designs based on the use of the brick-wall ideal filters as well as one exceptional design based on the use of the Parks-McClellan algorithm are included in Table II. As seen from Table II, the modified Creusere–Mitra objective as given by (17), with the aid of the use function of the Blackman window, along with the first-degree transition, provides the best global behaviors band spline function among the CMFB designs of this table. The effect of , the order of the spline function, on the resulting prototype filter has also been studied. When utilizing one among the three objective functions, as defined by (17)–(19), the accurate size-up of the optimum value of giving rise to the best NPR MFB, according to the quality measures of Table II, is very difficult to achieve, because this evaluation implies a thorough understanding of the relationships among itself, , , , , , and . However, and , it is easy to empirically estimate the influence given of on the design parameters through numerical simulations. and For this purpose, Fig. 2 shows, for (the given values in this example), the amplitude distortion as well as the aliasing error as functions of the interger-valued when the objective function is given by (17) and the Blackman window is in use. In Fig. 3, in turn, the influand parameters ences of the value on the are plotted. Furthermore, the resulting prototype filter responses

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Fig. 2. Variations of the amplitude distortion ( ) and aliasing error (E ) as functions of for the CMFB in the M = 32 and N = 511 case, where the objective function is given by (17) and the Blackman window is in use.

Fig. 4. Prototype filter magnitude responses jP (e )j designed using three techniques with the aid of the Blackman window in Example 1.

a performance similar to that in Example 1, that is, the filter banks resulting when using the proposed design scheme are clearly very near to the PR property due to the very low errors, in terms of the amplitude distortion and aliasing errors when compared with the PR property. Furthermore, also importantly, high reconstruction properties for the input signal, given jointly by and parameters, are achieved. means of the In addition, it is very interesting to observe from Table III that the minimum value of the optimized objective function has always been reduced with the introduction of a spline function in the transition band, which, in most cases, also corresponds to improvements in the quality parameters under consideration. VIII. CONCLUSION

Fig. 3. Influence of the value on the P SN R and M axError parameters for the CMFB in the same case as in Fig. 2.

designed using the Blackman window for and as well as that synthesized using the earlier technique described in [9], [10] are shown in Fig. 4. As seen from this figure, the integer value of has an influence on the transition band behavior of the resulting prototype filter and, thus, also on the overlap factor between the adjacent channels. According to the measurement results included in Table II for various CMFB designs, those CMFBs synthesized with the aid of proposed technique with lead to the best performances, independently of whether the objective function to be minimized is given by (17) or (18). Example 2: This example concentrates on comparing several 16-channel MDFT FBs, where all analysis and synthesis filters are obtained from the prototype filter according to (5a) and (5b), respectively. All the filters are of order 128 in order to be able to implement the entire MDFT FB by means of the fast algorithms described [16]. The comparison of proposed method with the techniques suggested in [9], [10], [13], [14] is shown in Table III. It can be observed that the proposed approach achieves

An improved windowing technique for designing prototypes for modulated analysis-synthesis multirate filter banks has been proposed. The key idea in this approach is to include a spline function to shape the transition band of the ideal filter, instead of using a conventional brick-wall filter. The design method uses an extremely fast univariate optimization technique. A study of different objective functions has been carried out and suitable re-definitions of these functions have been proposed. Simulation results have shown several advantages of this approach. First, when designing such multirate filter banks for practical applications, the main benefit of the proposed technique is a very quick design, without time-consuming multivariable optimization algorithms, even for systems with very many channels and very long prototype filters. This is very useful for multicarrier modulators implemented through the corresponding synthesis-analysis multirate filter banks or, equivalently, the corresponding so-called transmultiplexer configuration. Secondly, by properly selecting the window function and the order of the spline function for the transition band shaping of the filter, the performance of the resulting NPR multirate systems approach their PR counterparts. Another important conclusion from the reported results is the following. The lower spline order is, the lower stopband attenuation of the prototype filter, which has been measured in this , is, and simultaneously the contribution in the interval

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CRUZ-ROLDÁN et al.: FAST WINDOWING-BASED TECHNIQUE EXPLOITING SPLINE FUNCTIONS FOR DESIGNING MODULATED FILTER BANKS

wider transition band is. The above effects on the quality parameters mean that allowing a larger overlapping between adjacent channels, higher reconstruction properties are observed in the output signal of the resulting analysis-synthesis multirate filter bank, in terms of a smaller amplitude distortion and smaller aliasing errors. ACKNOWLEDGMENT The authors would like to thank the associate editor and the anonymous reviewers for their valuable comments and constructive criticisms on the manuscript which made this final contribution considerably easier for the reader to grasp. REFERENCES [1] L. Lin and B. Farhang-Boroujeny, “Cosine-modulated multitone for very-high-speed digital subscriber lines,” EURASIP J. Appl. Signal Process., vol. 2006, 2006, Article ID 19329, 16 pages. [2] P. P. Vaidyanathan, Multirate Systems and Filter Banks. Englewood Cliffs, NJ: Prentice-Hall, 1993. [3] W.-S. Lu, T. Saramäki, and R. Bregovic, “Design of practically perfect-reconstruction cosine-modulated filter banks: A second-order cone programming approach,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 51, no. 3, pp. 552–563, Mar. 2004. [4] M. B. Furtado, P. S. R. Diniz, and S. L. Netto, “Numerically efficient optimal design of cosine-modulated filter banks with peak-constrained least-squares behavior,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 52, no. 3, pp. 597–608, Mar. 2005. [5] M. B. Furtado, P. S. R. Diniz, S. L. Netto, and T. Saramäki, “On the design of high-complexity cosine-modulated transmultiplexers based on the frequency-response masking approach,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 52, no. 11, pp. 2413–2426, Nov. 2005. [6] M. Blanco-Velasco, F. Cruz-Roldán, E. Moreno-Martínez, J. I. Godino, and K. E. Barner, “Embedded filter bank-based algorithm for ECG compression,” Signal Process., vol. 88, no. 6, pp. 1402–1412, Jun. 2008. [7] F. Cruz-Roldán and M. Monteagudo, “Efficient implementation of nearly-perfect reconstruction cosine-modulated filterbanks,” IEEE Trans. Signal Process., vol. 52, no. 9, pp. 2661–2664, Sep. 2004. [8] A. Viholainen, J. Alhava, and M. Renfors, “Efficient implementation of 2xoversampled exponentially modulated filter banks,” Trans. Circuits Syst. II, Exp. Briefs, vol. 53, no. 10, pp. 1138–1142, Oct. 2006. [9] F. Cruz-Roldán, P. Amo-López, S. Maldonado, and S. S. Lawson, “An efficient and simple method for designing prototype filters for cosinemodulated pseudo-QMF banks,” IEEE Signal Process. Lett., vol. 9, no. 1, pp. 29–31, Jan. 2002. [10] P. Martín-Martín, F. Cruz-Roldán, and T. Saramäki, “A windowing approach for designing critically sampled nearly perfect-reconstruction cosine-modulated transmultiplexers and filter banks,” in Proc. Third Int. Symp. Image and Signal Process. and Anal., Rome, Italy, Sep. 2003, pp. 755–760. [11] C. S. Burrus, A. W. Soewito, and R. A. Gopinath, “Least squared error FIR filter design with transition bands,” IEEE Trans. Signal Process., vol. 40, no. 6, pp. 1327–1340, Jun. 1992. [12] F. Cruz-Roldán, P. Martín-Martín, M. Blanco-Velasco, and T. Saramäki, “A fast windowing technique for designing discrete wavelet multitone transceivers exploiting spline functions,” in Proc. 14th Europ. Signal Process. Conf., Florence, Italy, Sep. 2006, 5 pages. [13] C. D. Creusere and S. K. Mitra, “A simple method for designing highquality prototype filters for M-band pseudo-QMF banks,” IEEE Trans. Signal Process., vol. 43, no. 4, pp. 1005–1007, Apr. 1995. [14] Y.-P. Lin and P. P. Vaidyanathan, “A Kaiser window approach for the design of prototype filters of cosine modulated filter banks,” IEEE Signal Process. Lett., vol. 5, no. 6, pp. 132–134, Jun. 1998. [15] P. N. Heller, T. Karp, and T. Q. Nguyen, “A general formulation of modulated filter banks,” IEEE Trans. Signal Process., vol. 47, no. 4, pp. 986–1002, Apr. 1999. [16] T. Karp and N. J. Fliege, “Modified DFT filter banks with perfect reconstruction,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 46, no. 11, pp. 1404–1414, Nov. 1999. [17] M. Vetterli, “A theory of multirate filter banks,” IEEE Trans. Acoust., Speech, Signal Process., vol. 35, no. 3, pp. 356–372, Mar. 1987.

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[18] R. D. Koilpillai, T. Q. Nguyen, and P. P. Vaidyanathan, “Some results in the theory of crosstalk-free transmultiplexers,” IEEE Trans. Signal Process., vol. 39, no. 10, pp. 2174–2183, Oct. 1991. [19] T. W. Parks and J. H. McClellan, “A program for the design of linear phase finite impulse response digital filters,” IEEE Trans. Audio Electroacoust., vol. AU-20, no. 3, pp. 195–199, Aug. 1972. [20] T. Q. Nguyen, “Near-perfect-reconstruction pseudo-QMF filter banks,” IEEE Trans. Signal Process., vol. 42, no. 1, pp. 65–76, Jan. 1994. [21] T. Saramäki and Y. Neuvo, “A class of FIR ( th-band) Nyquist filters with zero intersymbol interference,” IEEE Trans. Circuits Syst., vol. CAS-34, no. 10, pp. 1182–1190, Oct. 1987. [22] E. K. P. Chong and S. H. Zak, An Introduction to Optimization. New York: Wiley, 2001.

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Fernando Cruz-Roldán (M’98–SM’06) was born in Baena, Spain, in 1968. He received the B.S. degree from Universidad de Alcalá (UA), in 1990, the M. S. degree from Universidad Politécnica de Madrid (UPM), in 1996, and Ph. D. degree from UA, in 2000, all in telecommunication engineering. He joined the Department of Ingeniería de Circuitos y Sistemas, UPM, in 1990, where from 1993 to 2003 he was an Assistant Professor. From 1998 to February 2003, he was a Visiting Lecturer at UA. In March 2003, he joined the Universidad de Alcalá, Spain, where he is currently an Associate Professor of signal theory and communications. His research interests include digital signal processing, filter design, and multirate systems (filter banks).

Pilar Martín-Martín (M’98) was born in Madrid, Spain, in 1973. She received the Dipl. Eng. and M.S. degrees in telecommunication engineering from the Universidad Politécnica University de Madrid, Madrid, Spain, in 1995 and 1999, respectively, and the Doctor of Technology degree (with honors) from the Universidad de Alcalá, Madrid, Spain, in 2007. She is currently working toward the Doctorl of Technology degree at Tampere University of Technology, Tampere, Finland. From 1996 to 2002, she was a docent of telecommunications in the Universidad de Alcalá, and since 2002 she has been an Assistant Professor with the same university. In 2002, she was a Visiting Researcher with the Tampere International Center for Signal Processing, and in 2004 she was Marie Curie Student Fellow, both in the Department of Signal Processing, Tampere University of Technology. Her research interests are in multirate signal processing and in multicarrier transmission.

José B. Sáez-Landete was born in Valdeganga, Spain, in 1977. He received the M.S. degree in physics from the Universidad de Zaragoza, Spain, in 2000 and the Ph.D. degree in physics from the Universidad Complutense de Madrid, Spain, in 2006. Since October 2006, he has been with the Signal Theory and Communications of the Universidad de Alcalá, Madrid, Spain, where he is an Associate Professor. His research interests include optical metrology, optimization, digital signal processing, and multirate systems.

Manuel Blanco-Velasco (M’05) was born in Saint Maur des Fossés, France, in 1967. He received the engineering degree from the Universidad de Alcalá, Madrid, Spain, in 1990, the MSc in Communications Engineering from the Universidad Politécnica de Madrid, Spain, in 1999, and the PhD degree from the Universidad de Alcalá in 2004. From 1992 to 2002, he was with the Circuits and Systems Department at the Universidad Politécnica de Madrid as Assistant Professor. In April 2002, he joined the Signal Theory and Communications Department of the Universidad de Alcalá where he is now working as Associate Professor. His main research interest is biomedical signal processing.

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Tapio Saramäki (M’98–SM’01–F’02) was born in Orivesi, Finland, in 1953. He has received the Diploma Engineer (with honors) and Doctor of Technology (with honors) degrees in electrical engineering from the Tampere University of Technology (TUT), Tampere, Finland, in 1978 and 1981, respectively. Since 1977, he has held various research and teaching positions at TUT, where he is currently a Professor of Signal Processing and a Docent of Telecommunications. He is also a co-founder and a system-level designer of VLSI Solution Oy, Tampere, Finland, specializing in efficient VLSI implementations of both analog and digital signal processing algorithms for various applications. He is also the President of Aragit Oy Ltd., Tampere, Finland, which was founded by four TUT professors and concentrates on spreading worldwide their know-how on information technology to the industry. In 1982, 1985, 1986, 1990, and 1998 Dr. Saramäki was a Visiting Research Fellow (Professor) with the University of California, Santa Barbara, in 1987 with the California Institute of Technology, Pasadena, and in 2001 with the National University of Singapore. His research interests are in digital signal processing, especially filter and filter bank design, VLSI implementations, and

communications applications, as well as approximation and optimization theories. He has written almost 300 international journal and conference articles, various international book chapters, and holds three world-wide used patents. Dr. Saramäki was a recipient of the 1987 and 2007 IEEE Circuits and Systems Society’s Guillemin–Cauer Awards as well as two other best paper awards. In 2004, he was also awarded the honorary membership (Fellow) of the A. S. Popov Society for Radio-Engineering, Electronics, and Communications (the highest membership grade i7n the society and the 80th honorary member since 1945) for “great contributions to the development of DSP theory and methods and great contributions to the consolidation of relationships between Russian and Finnish organizations”. Dr. Saramäki is also a founding member of the Median-Free Group International. He was an Associate Editor for the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-II: ANALOG AND DIGITAL SIGNAL PROCESSING from 2000 to 2001, and is currently an Associate Editor for Circuits, Systems, and Signal Processing. Dr. Saramäki has been actively taking part in many duties in the IEEE Circuits and Systems Society’s DSP Committee, namely by being a Chairman (2002–2004), a Distinguished Lecturer (2002–2003), a Track or a Co-Track Chair for many ISCAS symposiums (2003–2005). In addition, he has been one of the three chairmen of the annual workshop on Spectral Methods and Multirate Signal Processing (SMMSP), started in 2001.

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