An Improved Rational Approximation of Bark Scale using Low Complexity and Low Delay Filter Banks

This paper proposes an algorithm to obtain the sampling factors to model any frequency partitioning so that it is realized using low complexity rational decimated non-uniform ﬁlter banks (RDNUFBs). The proposed al-gorithm is employed to approximate the Bark scale to ﬁnd a rational frequency partitioning that can be realized using RDNUFBs with less approximation error. The proposed Bark frequency partitioning is found to reduce the average deviation in center frequency and bandwidth by 65.04% and 48.50% respectively, and root mean square bandwidth deviation by 54.46% when compared to those of the existing perceptual wavelet approximation of Bark scale. In the ﬁrst ﬁlter bank design approach discussed in the paper, a near perfect reconstruction partially cosine modulated ﬁlter bank is employed to obtain the improved Bark frequency partitioning. In the second design approach, the hardware complexity of the PCM based RDNUFB is reduced by deriving the channels with diﬀerent sampling factors from the same prototype ﬁlter by the method of channel merging. There is a considerable reduction of 40.83% in the number of multipliers when merging of partially cosine modulated channels is employed. It is also found that both the proposed approaches can reduce the delay by 95.17% when compared to the existing rational tree approximation methods and hence, are suitable for real time applications.


Introduction
Multirate filter banks are widely used in many signal processing applications such as speech and audio processing systems, communications systems and biomedical systems.In most of the real time applications, delay and power of these systems become very critical.
The filter banks that model the frequency partitioning of the human ear can improve the speech intelligibility and comfort of the hearing aid [1], [2] and are used in bio-medical researches for the assessment of hearing damage [3] and the development of cochlear implants [4].These filter banks are also used in audio processing applications such as speaker recognition [5], emotion recognition [6], speech enhancement [7], [8] and coding.This work proposes a novel algorithm that approximates any non-uniform frequency partitioning to a frequency partitioning that can be realized by low complex near perfect reconstruction (NPR) filter banks.The performance of the proposed algorithm is illustrated with respect to the Bark scale.
The ability of the human ear to differentiate between frequency tones is measured using critical bands [9].In the Bark scale based frequency partitioning, the human hearing range, 20 to 20,000 Hz is divided into 24 critical bands [9], [10] as shown in Table 1.The most widely used approximation of Bark scale is the perceptual wavelet approximation realized by a 24 channel tree structured filter bank [7], [11].The frequency partitions obtained using perceptual wavelet approximation of Bark scale is shown in Table 1.Even though there are many integer decimated filter bank designs to model the Bark scale [8] - [12], since, the sampling factors are limited to integers, they cannot accurately model the Bark scale.The filter banks that realize rational sampling factors [13] are better choices to reduce the approximation error in modelling the Bark scale.
One of the popular methods to realize the rational frequency partitioning is by employing tree structure based non-uniform filter banks (NUFBs) [14].This binary tree decomposition based on wavelet filter bank is a popular method to partition the audible frequency spectrum into Bark scale.The perceptual wavelet filter bank structure that approximates the Bark scale using 24 channels is discussed in papers [7] and [11].Other approximations of the Bark scale using 21 and 17 channel wavelet filter banks are mentioned in [6].An oversampled 84-channel bank of non-uniform filters based on wavelet decomposition that models the Bark scale is discussed in [15].Since, the frequency partitioning that can be achieved by the wavelet tree decomposition is limited to the powers of two, Bark scale can not be accurately modelled using these filter banks.The deviation between the bands obtained by the filter bank and the original bands in the Bark scale is found to be high, which may affect the performance of speech processing systems.Another tree structured nonuniform filter bank, that uses two channel rational filter banks with sampling factors 5/6 and 1/6 discussed in [16], reduces the approximation error of Bark scale.However, all these tree structured filter banks suffer from high delay, and delay increases with increase in the number of decomposition levels.Hence, they can not be employed for real time speech processing applications.
Other than the tree structured filter banks, cosine modulation [17], recombination method [18], direct method [19] and partial cosine modulation [20] can also be used to realize near prefect reconstruction (NPR) filter banks with the rational frequency partitioning.In the recombination method discussed in [18], RDNUFBs can be designed by adding the analysis filter responses of a cosine modulated uniform filter bank using the synthesis filters of another filter bank.Because of this two-stage structure, here also, the system delay is found to be high.In [17], different prototype filters are cosine modulated to derive each channels of RDNUFB.However, the analysis and synthesis filters of the resultant filter bank do not possess the linear phase property in this approach.A direct structure to realize linear phase RDNUFBs, is discussed in [19], where each filter has to be separately implemented, resulting in a very high implementation complexity.In paper [20], a modified version of cosine modulation known as partial cosine modulation (PCM), is used for the design of linear phase RDNUFBs.In paper [23], the design and analysis of RDNUFBs designed using PCM and merging is discussed, where the channels with the same interpolation factors are derived from the same prototype filters and thereby reducing the hardware complexity of the RDNUFBs.
In this work, an algorithm is proposed to find out a suitable set of rational sampling factors for any frequency partitioning so that it can be realized by low complexity RDNUFBs.The proposed algorithm is used to find an improved approximation of Bark scale frequency partitioning, which is realized using low complexity RDNUFBs based on PCM and merging.
The paper is organized as follows: Section II reviews the design of RD-NUFBs using PCM and merging.Section III proposes an algorithm to find sampling factors to model any frequency partitioning to be realized using low complexity RDNUFBs.Sections IV and V respectively discuss low delay RD-NUFBs designed using PCM and merging for the rational approximation of Bark scale.Section VI discusses the adaptability of the proposed method to other frequency partitioning used in speech processing.Section VII concludes the paper.

Review of RDNUFB Design using Partial Cosine Modulation and Merging
An M -channel RDNUFB is shown in Fig. 1, where, l k and d k respectively are the interpolation and decimation factors and H k (z) and F k (z), respectively are the analysis and synthesis filters of k th channel, for k = 0, 1, 2, ...(M −1).The partial cosine (PC) modulation is an efficient technique to design RDNUFBs with linear phase.The complexity of these PCM designed RDNUFBs can be reduced by the method combining the channels.These two techniques for the RDNUFB design are reviewed below.
where, m k is the parameter used to select the locations of h k (n) and f k (n), and θ k is the modulation phase [20].Since, h 0 (n) is a low-pass filter, m k = 0 for k = 0.The filter banks designed using the PCM method use Eq. ( 1), where, θ k is selected as 0 or π and π 2 or − π 2 alternately [20].The analysis filters h k (n) are designed alternately as symmetric and anti-symmetric filters.This selection of analysis filters in the PCM based filter banks helps to cancel the aliasing between the neighbouring channels.However, the selection of θ k as 0 or π and π 2 or − π 2 alternately, results in a significant amplitude distortion in the NUFBs at ω = π and ω = 0, which is reduced by separately designing the low-pass analysis filter h 0 (n) and high-pass analysis filter h M −1 (n) [20].The remaining band-pass analysis filters h k (n), and synthesis filters f k (n) for 1 ≤ k ≤ M − 2, are designed by modulating the prototype filter p k (n) of cut-off frequency π 2d k as given in Eq. ( 1).In a signal processing system, phase distortion occurs when phase response is not linear over the frequency range of interest.The group delay is defined as the negative of the first derivative of the phase w.r.t the angular frequency ω [21] and for the filter bank in Fig. 1, it is obtained as N k l k [19].Hence, the group delay of all the channels of a filter bank should be constant for it to be free of phase distortion, i. e.
where, C is a positive integer, N k is the order of h k (n), l k is the interpolation factor of the k th channel and M is the number of channels.
The left and right cut-off frequencies of H k (z) for the RDNUFB are derived in [22] as where, e k = 0, 1, ..., l k − 1.It is shown in [22] that, the pass-band overlapping in the filter bank can be avoided, if the cut off frequencies given by Eq. ( 3), are integer multiples of π d k , for any value of e k = 0, 1, ..., l k − 1.This feasibility condition has to be satisfied for the realization of NPR RDNUFBs [22].
Here, to obtain channels with different sampling factors, different prototype filters need to be implemented.In the next method of RDNUFB design that is explained below, the complexity of the PCM designed RDNUFB is reduced by deriving the channels with different sampling factors also from the same prototype filter, reducing the number of prototype filters to be implemented.

Design of Low Complexity RDNUFB by Channel Merging
When the PCM is employed for the design of linear phase RDNUFB, different prototype filters are required for the channels with different interpolation factors, l k or decimation factors, d k .The band-pass analysis filters h k (n), corresponding to the channel with the sampling factor l k d k , 1 ≤ k ≤ M − 2, are derived from a low-pass prototype filter with cut-off frequency π 2d k and the order given by Eq. ( 2).In the RDNUFB, since different channels have different interpolation factors, it requires more prototype filters to be implemented, unlike in integer decimated NUFB (IDNUFB), where the interpolation factor is always 1. Also, from Eq. (2), it is clear that, as the interpolation factor increases, the order of the filter also increases.Hence, the reduction in the number of prototype filters helps to reduce the hardware complexity of the overall RDNUFB system.A method to reduce the number of prototype filters required to design the RDNUFB is discussed in [23], where the channels with the same interpolation factors and different decimation factors are derived from the same prototype filters.Thus, different prototype filters need to be designed only for the channels with different interpolation factors.This approach is reviewed below.
Consider an M -channel NPR RDNUFB designed using PCM with decimation and interpolation factors as shown in Fig. 1.A new RDNUFB consisting of channels that can extract another set of bandwidths can be derived from the original RDNUFB by combining the neighbouring channels with the same interpolation factors [23].Assume r adjacent channels in Fig. 1, starting from n through n + a − 1, have the same interpolation factors.Let b out of these a channels with the same interpolation factors be merged to form the n th channel in the derived RDNUFB with sampling factor ln dn given by [23], The analysis and synthesis filters of the derived channel in the new RD-NUFB are given by [23] Hn The distortion due to pass-band overlapping in the derived RDNUFB can be removed by keeping the cut-off frequencies of the analysis filters as integer multiples of π dk .Also, for the new RDNUFB to be free of phase and amplitude distortions, the interpolation factor of the derived channel should be the same as that of the original channels that are merged as given by Eq. ( 4) [23].
The first part of this work proposes an algorithm to obtain an improved frequency partitioning for Bark scale that can be realized using low complex RDNUFBs.The later section discusses the design of filter bank that realizes the frequency partitioning obtained from the proposed algorithm by PCM and channel merging.
3 Proposal-1: An Algorithm to Find a Feasible Frequency Partitioning Realizable using Low Complexity RDNUFB Modelling a non-uniform frequency decomposition requires RDNUFB with sampling factors of the k th channel as, where, BW (k) is the bandwidth of k th channel, M is the number of channels and d k and l k respectively are the decimation and interpolation factors and are relatively prime.This work proposes an algorithm to find out the sampling factors to approximate any frequency partitioning such that it can be realized using low complexity rational decimated filter banks.If the interpolation factors of the channels are high, it results in high order analysis filters as given by Eq. ( 2).It is shown in [25] that for reducing amplitude distortion, the prototype filters of the cosine modulated filter bank have to be designed in such a way that the transition width of k th prototype filter ≤ π 2d k , where d k is the decimation factor of the k th channel.When the decimation factor increases, the required transition width decreases, which also causes an increase in the order of the filters.Thus, an algorithm to find out a non-uniform frequency partitioning, which can be realized using low complexity filter banks is discussed in this section.The steps of the algorithm are explained below.
1. From the set of bandwidths to be realized, find the approximated smallest bandwidth, ASB and approximated total bandwidth according to the following constraints.
(a) Approximated bandwidth of each channel, BW appr (k), where 0 ≤ k ≤ M − 1 and approximated total bandwidth, Appr.T otal BW are integer multiples of ASB, i.e, Appr.T otal BW = P * ASB where, P and n k are integers.While fixing the approximate smallest bandwidth, ASB, the deviations of BW appr (k) and Appr.T otal BW from their actual values and the complexity of the prototype filters need to be considered.(b) The deviations from the original bandwidths have to be kept minimum.
Let n k0 be the value of n k , which closely approximates the original bandwidth of k th channel, BW (k) to the multiple of the smallest bandwidth, which is given by, 2. Generate an array of at least two integers adjacent to n k0 , for which the maximum deviation from the actual bandwidth of each channel is less than or equal to 2*ASB.Now, check for the complexity of these solutions by finding out the sampling factors for each of these solutions using Eq. ( 10) and by checking if the corresponding interpolation factor is less than a limit value.
It is assumed that when the interpolation factor obtained is less than 4, the corresponding solutions are considered as low complexity solutions.3. Starting from zeroth channel, the algorithm searches for feasible solutions in the array.The feasible solutions are the sampling factors corresponding to the bandwidths, which satisfy the condition given in Eq. ( 3).This feasibility condition is the necessary condition to be satisfied for the RDNUFB to be realizable [22].The algorithm search starts from the solution, which is the closest to the actual bandwidth.4. If realizable solutions are found, the algorithm evaluates the error parameter of the k th channel, dev k , which is defined as the number of channels from 0 through k, whose feasible solution is not the closest solution.Let the maximum limit of dev k be taken as dev max .
-If dev k is greater than dev max the algorithm re-iterates using the alternate feasible solutions.-If dev k is less than or equal to dev max the algorithm performs step 3 for the next channel.5.If realizable solutions are not found, the algorithm reiterates step 3 from the previous channel.6.If realizable solutions are found for all the M channels, the algorithm checks whether the solution selected is maximally decimated or not.7. If not maximally decimated, the algorithm reiterates step 3 for alternate solutions starting from M − 1 through 0.
The proposed algorithm is applied for the Bark frequency partitioning, so that it can be realized using low complexity rational decimated filter bank, as illustrated in the next section.
3.1 Illustration: Finding an improved approximation of Bark scale to be realized using low complex RDNUFB using the proposed algorithm For the Bark scale shown in Table 1, the total bandwidth is found to be 15500 Hz.Hence, the required sampling factors of the 24 channels to model the exact Bark scale using Eq.(6) It can be seen that, this requires high values of interpolation factors and hence, high order filters, to realize the exact Bark Scale frequency partitioning.Not all channels obey the feasibility conditions.Hence, to improve the accuracy compared to that of the wavelet scale and to reduce the complexity in modelling the exact Bark scale, the frequency partitioning of Bark scale is approximated to a modified scale using the proposed algorithm.The steps of the proposed algorithm are illustrated below w.r.t Bark scale.
1.The ASB, Appr.T otal BW are selected as 120 Hz and 15600 Hz respectively.This ASB when compared with 125 Hz in the wavelet approximation is closer to the smallest bandwidth of the actual Bark scale.Similarly, Appr.T otal BW = 15600Hz, which is a multiple of the 120 Hz and is closer to that of the Bark scale frequency partitioning than that of 16000 Hz in the wavelet approximation.Using these values of the smallest bandwidth, total bandwidth and P , Eq. ( 7), (8), and ( 9), are simplified to The approximated bandwidth corresponding to n k0 deviates from the original bandwidth only by a maximum of 60Hz.
The bandwidths corresponding to the proposed algorithm for the Bark scale are shown in Table 1.The sampling factors obtained are shown in Table 2.
It can be seen that maximum deviation of bandwidths is reduced from 500 Hz in the wavelet approximation to 260 Hz in the proposed rational approximation and thus, the proposed algorithm gives a more accurate frequency partitioning for the Bark scale than the wavelet filter bank in terms of deviations from center frequencies and bandwidths of each band.
4 Proposal-2: Design of RDNUFB using PCM for the Proposed Approximation of Bark Scale A partially cosine modulated RDNUFB is designed to get the proposed 24 rational frequency bands of the Bark scale shown in Table 1.The cut-off frequencies of various prototype filters to satisfy the feasibility conditions as explained in Section 2.1 are listed in Table 2.As mentioned in Section 2.1, the low-pass and high-pass analysis filters, H 0 (z) and H 23 (z) respectively are designed separately.The remaining band-pass analysis filters of the channels with the sampling factors l k d k are obtained by modulating low-pass filters with cut-off frequency π 2d k using Eq. ( 1).The analysis filters of channels 1 to 7 are obtained by modulating the low-pass prototype filter p 1 (n), the analysis filters of the channels 8 to 14 by modulating p 2 (n), the analysis filters of the channels 18 to 19 by modulating p 6 and the analysis filters of the channels 15, 16, 17, 20, 21 and 22 are obtained by modulating p 3 , p 4 , p 5 , p 7 , p 8 and p 9 respectively.Thus, 11 prototype filters need to be implemented to design the partially cosine modulated RDNUFB that extracts bandwidths as per the proposed approximation of Bark scale and the structure is shown in Fig. 2.
When the FIR filters are designed with the cut-off frequencies as listed in Table 2, the order and complexity of these filters are found to be very high.To alleviate this issue of hardware complex FIR filters, the prototype filters are implemented using interpolated FIR (IFIR) [26] technique in this work.
The next section illustrates the accuracy, delay and hardware efficiency of this PCM based RDNUFB in obtaining the proposed rational approximation of the Bark scale using a design example.
The design specifications of the 24-channel RDNUFB using PCM for the proposed rational approximation of Bark scale are given below.
Pass-band ripple = 0.01dB Stop-band attenuation = 60dB Transition width = 0.004π Amplitude distortion ≤ 0.05dB From the above specifications, the transition widths of the prototype filters are obtained as 0.004 l k π, where l k is the interpolation factor of k th channel.All the filters are designed using IFIR approach.The design of various filters of the RDNUFB are discussed below.
1.As discussed in Section 3, the low-pass analysis filter p 0 (n) is designed as symmetric filter with cut-off frequency of π 130 .
Figure 2: Structure of 24-channel partially cosine modulated RDNUFB to realize rational approximation of Bark Scale 2. The analysis filters of the remaining channels are band-pass filters, which are derived by modulating the corresponding low-pass prototype filters as shown in Table 2. Thus, the band-pass analysis filters of the channels with the same sampling factors can be derived from the same prototype filter.3.As mentioned in Section 2.1, the analysis filters need to be designed as symmetric and anti-symmetric alternately with the first analysis filter h 0 (n) being symmetric always to reduce the aliasing between the adjacent channels.Hence, the high-pass filter h 23 (n) in this case is an anti-symmetric filter with a cut-off frequency of 12π 13 .This high-pass filter is obtained from a symmetric even length low-pass prototype filter p 23 (n) of cut-off frequency π 13 by applying the relation h 23 (n) = (−1) n p 23 (n).The length of p 23 (n) and h 23 (n) are found to be 1854.
The magnitude responses of the prototype filters are shown in Fig. 3(a).The maximum pass-band ripple and minimum stop-band attenuation of the analysis filters are found to be 0.01566dB and 54.63dB respectively.The channel responses of the filter bank are shown in Fig. 3(b).The amplitude distortion of the RDNUFB is found to be less than 0.04dB.The Bark scale, wavelet scale, and the proposed rational scale are compared in terms of critical band rates and critical bandwidths as a function of center frequency in Fig. 4(a) and 4(b) respectively.
From these figures, it can be seen that the frequency partitioning of the proposed rational scale is much closer to the Bark scale compared to that of the wavelet scale.Fig. 4(c) illustrates the difference between the calculated values of critical band rates (z) obtained from the wavelet and the proposed approximations of Bark scale and those in the Zwicker's table [27].Using the proposed rational scale, the maximum error in critical band rates is reduced when compared to that of the wavelet tree decomposition from 1.1 to 0.82.The proposed Bark frequency partitioning is found to reduce the average deviation in center frequency and bandwidth by 65.04% and 48.50% respectively, when compared to those of the existing perceptual wavelet approximation of Bark scale.
The approximation error of any given scale can be analysed in terms of absolute value of deviations in the center frequencies and bandwidths between the original Bark scale and the approximated scale of each channel, absolute relative bandwidth deviation (RBD) of each channel and root mean square bandwidth deviation (RMSBD).RBD(k) and RM SBD are computed as [28] where, M = 24 is the number of channels, BW appr (k) and BW Bark (k) are bandwidths of the k th channel, 0 ≤ k ≤ M − 1, in the approximated scale and original Bark scale respectively.Table 3 compares the deviations in the center frequencies and bandwidths of approximated scale from the Bark scale, RBD(k) and RMSBD for various approximations of Bark Scale.The proposed Bark frequency partitioning is found to reduce the average deviation in center frequency and bandwidth by 65.04% and 48.50% respectively, and root mean square bandwidth deviation by 54.46% when compared to those of the existing perceptual wavelet approximation of Bark scale.The proposed approximation achieves 62.87% reduction in RMSBD compared to the rational tree approximation [16] of Bark Scale.
Table 4 compares the deviation in the critical band rate z of the proposed rational approximation of Bark scale with those of the existing approximations.The table also compares the hardware complexity and delay of the PCM based RDNUFB that realizes the proposed approximation with those of the following existing methods and the inferences are summarized below.
1. Comparisons with 17-channel [6], 21-channel [6], 24-channel [7] non-uniform filter banks and 24-channel PCM based IDNUFB [12] show that there are substantial reductions in the average and maximum deviations when the Bark scale is modelled using the proposed rational frequency partitioning.2. The comparisons with the 21-channel [15] and 84-channel [8] oversampled filter banks show that the PCM RDNUFB based design of the proposed frequency partitioning gives better delay along with improved average and maximum deviations in modelling the Bark scale.3. Comparisons with direct structure [19] and cosine modulated filter bank [17] are also included in Table 4.It can be seen that the design using PCM offers better delay and deviations than these techniques in approximating the Bark scale.In direct structure, all analysis/synthesis filters have to be implemented separately, which makes its hardware complexity very high.In the case of cosine modulated RDNUFB, the individual analysis or synthesis filters do not possess linear phase property, whereas in the PCM based RDNUFB, all the filters have linear phase property.4. When the filter bank to approximate the Bark scale is realized using the tree based structure with rational sampling factors (5 6, 1 6) [16], it also results in a structure with very high delay.
As mentioned earlier, only 11 separate filters need to be implemented to realize the RDNUFB for the proposed approximation.However, the low-pass filter h 0 (n) and the prototype filter p 2 (n) corresponding to the analysis filters of the channels from 8 to 14 are found to have the same cut-off frequency of π 130 and transition width of 0.004π.Thus, the analysis filters of the channels from 8 to 14 can also be derived from h 0 (n) itself, reducing the total number of filters to be implemented to 10.It is found that only 1893 multipliers are needed to implement these 10 filters.Even though, the hardware complexity of the RDNUFB for the proposed approximation is found to be higher in some of the cases in Table 4, the approximation error and delay are considerably reduced, making it suitable for high quality real-time applications.The next section proposes the method of merging the channels generated by PCM so that the overall complexity of the RDNUFB is reduced.
5 Proposal-3: Rational approximation of Bark scale using low complexity RDNUFB realized by PCM and merging When the partially cosine modulated RDNUFB is used for the proposed rational approximation of Bark scale, the analysis filters of the channels with the same sampling factors are derived from the same prototype filter.As discussed in Section 2.2, the method of merging of adjacent channels of a partially cosine modulated RDNUFB can be employed to reduce the hardware complexity.In this proposal, this low complexity RDNUFB used to obtain the proposed rational approximation of Bark scale.
In the original RDNUFB as given in Table 2, the band-pass channels 1 to 14, 16 and 20 have the same interpolation factor of 1.Hence, as discussed in Section 2.2, the analysis filters of the channels with the same interpolation factors can be merged without introducing aliasing distortion, to derive channels with new different bandwidths.Thus, the band-pass channels can be derived as follows: The analysis filters of the channels 1 to 7 are obtained directly by modulating the low-pass prototype filter p 1 according to Eq. ( 1).The analysis filters of the channels 8 to 14 are obtained by modulating p 1 according to Eq. ( 1), and then merging two adjacent modulated filters according to Eq. ( 5).The analysis filter of the channel 16 is obtained by modulating the prototype filter p 1 , and then merging five adjacent modulated filters.Similarly, the analysis filters of the channel 20 is obtained by modulating the prototype filter p 1 , and then merging 13 adjacent modulated filters.The values of position parameters m k are given in Table 5.Similarly, the band-pass channels 15, 17 and 21 have the same interpolation factor of 3 and hence, can be derived from the low-pass prototype filter p 3 .To derive the channel 15, p 3 is modulated with a position parameter m k = 94.To derive the channel 17, p 3 is modulated with position parameters m k = 10, and 11, and the resultant filters are merged together.To derive channel 21, p 3 is modulated with position parameters m k = 60, 61, 62, 63, 64, and these five modulated filters are merged together.The band-pass channels 18 and 19 have the same interpolation factor of 4 and are derived from the low-pass prototype filter p 6 .The band-pass analysis filter of channel 22 have the interpolation factor of 2 and is derived from the low-pass prototype filter p 9 .
The upper cut-off frequency of the prototype filters to be designed to derive the band-pass analysis filters of the channels with the same interpolation factor, is decided by the channels with the largest decimation factor.For example, channel 15, 17 and 21 have the same interpolation factor 3, but different decimation factors.Among these, channel 15 has the largest decimation factor of 130. Hence, the upper cut-off frequency of the prototype filter is decided by this channel and is given by π 2d k = π 260 .The same idea is used for other channels given in Table 5 also.
The low-pass and high-pass analysis filters are designed separately as in PCM based RDNUFB.The summary of the prototype filters, interpolation factors and their cut-off frequencies required for the derived RDNUFB are listed in Table 5.Thus, all band-pass analysis filters with the same interpolation factor, but same or different decimation factors can be obtained from a single prototype filter.Hence, apart from low-pass and high-pass filters, only 4 separate prototype filters are designed and implemented.The structure of this derived RDNUFB is shown in Fig. 5.

Designed directly
A design example is discussed in the following section, to illustrate the hardware efficiency and delay of this RDNUFB derived by PCM and merging in realizing the proposed rational approximation of the Bark scale.

Illustration
The design specifications of the 24-channel RDNUFB, shown in Fig. 5, to be designed using PCM and merging for the proposed rational approximation of Bark scale are given below.
Pass-band ripple = 0.01dB Stop-band attenuation = 60dB Transition width = 0.004π Amplitude distortion ≤ 0.05dB The transition widths of the prototype filters are obtained as 0.004 l k π, where l k is the interpolation factor of k th channel.The values of l k are given in Table 5.The design of various filters in the RDNUFB designed using PCM and merging are discussed below.
1.The low-pass and high-pass analysis filters are designed separately as IFIR filters as discussed in Section 4.1.
Figure 5: Structure of 24-channel RDNUFB to realize rational approximation of Bark Scale using PCM and merging.
2. The band-pass analysis filters of the channels with the same interpolation factors can be designed from the same low-pass prototype filters as discussed in Section 2.2.Since, according to Table 5, four different interpolation factors, 1, 2, 3 and 4 are required except for low-pass and high-pass analysis filters, 4 different IFIR prototype filters are designed and implemented to realize all the band-pass analysis filters.
The magnitude responses of all the prototype filters are shown in Fig. 6(a).The minimum stop-band attenuation and maximum pass-band ripple of the analysis filters are found to be 52.21dB and 0.026dB respectively.The individual channel responses of the 24-channel derived RDNUFB, which realizes the proposed approximation is shown in Fig. 6(b) The maximum amplitude distortion is found to be 0.047dB.6 compares the implementation complexities and delays of the RD-NUFBs designed in Proposal-2 and 3.It is clear from the table that the delays are the same in both the RDNUFBs discussed in this work.It requires only 6 prototype filters to be realized to design the derived RDNUFB to achieve the proposed rational approximation of Bark scale when compared to the 10 prototype filters in the PCM based RDNUFB.The number of multipliers to realize the derived RDNUFB is found to be 1120, which is much less than that of the PCM based RDNUFB.

Conclusion
This paper proposes an algorithm to find out the decimation and interpolation factors so that any frequency partitioning can be approximated using low complex RDNUFBs.The proposed algorithm is employed to obtain an improved approximation of the Bark Scale.Two techniques to realize RDNUFBs for this improved rational frequency partitioning are also discussed in this paper.In the first method, the RDNUFB is designed using PCM.The hardware complexity of this RDNUFB is reduced in the second proposal, where, the different channels are derived by merging the adjacent channels with the same interpolation factors of another low complex RDNUFB.RDNUFBs designed by both the methods offer less approximation error in modelling the Bark scale.The hardware complexity of the second RDNUFB design method is found to be less compared to that of the first method and other existing methods.Since, the delay is also less in both the methods discussed in this paper compared to the existing methods, they can be employed for real-time speech processing applications.
Author Contributions: All authors contributed to the study conception and design.Material preparation, data collection and analysis were performed by all.The first draft of the manuscript was co-written and all authors commented on previous versions of the manuscript.All authors read and approved the final manuscript

Figure 3 :
Figure 3: 24-channel RDNUFB corresponding to Bark frequency partitioning using PCM (a) Magnitude responses of prototype filters (b) Individual channel responses

Figure 4 :
Figure 4: Comparison of Perceptual wavelet approximation and proposed rational approximation of Bark scale (a) Critical band-rate, z as a function of center frequency (b) Critical bandwidth as a function of center frequency (c) Plot of deviation from Bark scale

Figure 6 :
Figure 6: 24-channel RDNUFB designed using PCM and merging for the proposed frequency partitioning (a) Magnitude responses of the prototype filters (b) Individual channel responses

Table 2 :
The design parameters of partially cosine modulated RDNUFB π 13

Table 3 :
Comparison of Bark scale, perceptual wavelet approximation of Bark scale and proposed rational scale

Table 4 :
Comparison of delay and hardware complexity of various implementation techniques of Bark frequency partitioning

Table 5 :
The design parameters to realize merged RDNUFB π 13

Table 6 :
Comparison of delay and hardware complexity of RDNUFBs realized in Proposal-2 and 3 for the proposed rational approximation of Bark scale