A low-complexity 3-level filter bank design for effective restoration of audibility in digital hearing aids

Abstract

A low-complexity method for sub-band decomposition of audio signals in digital hearing aids for audibility restoration applications is described in this paper. This 3-level filter bank is capable of generating an array of 4, 8, and 16 sub-filters from a single finite impulse response filter. The prototype low pass filter is accomplished using the Parks McClellan algorithm with a minimal number of 28 multipliers. Fractional interpolation technique is utilized to generate more number of sub-bands with narrow bandwidth from the prototype filter. This filter bank can be used for patients with any degree of hearing impairment to compensate his audiogram. The selection of filter bank is based on the rate of change of impairment recorded in the audiogram. Apart from reduced complexity, the developed filter bank has the advantage of requiring only minimal hardware, which makes the implementation of cost-effective hearing aids a reality.

Introduction

Hearing impairment is one of the most common sensory disturbances affecting humans. Estimates by World Health Organization has revealed that by 2050, one in every ten people around the globe will be suffering from disabling hearing loss [1]. Genetics, noise, diseases, drugs, ageing etc. are some of the major factors resulting in hearing impairment. There are various methods employed to overcome this defect such as, use of hearing aids and other assistive-listening devices and cochlear implants. Among these, the most effective way to compensate common hearing loss is to employ a hearing aid system which can selectively intensify the sound signals in order to suit the hearing characteristics of the patients and can also improve the speech intelligibility [2, 3]. Though hearing aids can easily compensate for majority of the hearing losses, there exists an alarming gap between the demand and supply of hearing aids. The global production of hearing aids is able to meet only less than 10% of the global need. Also, untreated hearing loss pose a greater risk of developing other diseases also, such as dementia and declining cognitive abilities as a consequence. Thus, development of efficient and affordable hearing aids with reduced complexity both in design and hardware has become a need of time.

An audiogram is used to identify the hearing capability of a patient over a varied range of frequencies by measuring the hearing threshold with respect to frequency. Since different patients have different levels of hearing impairment, the hearing aids should be tailor-made to satisfy each of their requirements. Among the various types of hearing aids available, digital hearing aid is the most popular one which provides improved performance and efficient implementation [4]. In a digital hearing aid, the filter bank—aptly called the heart of all hearing aids—should be capable of providing selective gain to each band for suiting the audiogram of the patient, thereby making the sound audible for the hearing impaired. This capability of the digital filter bank to individually adjust the gain in a particular range of frequencies as desired makes it an ideal choice in the design of a digital hearing aid [5].

Several sound wave decomposition methods have been developed to generate filter banks from the prototype filter. The ANSI S1.11 filter bank method which uses 18 numbers of 1/3 octave filters with strict specification parameters gives comparatively good matching performance, but its complexity level is very high [3]. A relaxed version of the standard ANSI filters called the Quasi-ANSI S1.11 filter bank has been developed with a decreased processing delay, but its level of complexity is also high [6]. Frequency Response Masking (FRM) is yet another technique utilized for the filter bank generation which renders sharp narrow-band digital filters from a combination of less complex sub-filters [7, 8]. The Cosine Modulation (CM) of the prototype filter is used to get an array of uniform filters and merging of adjacent bands makes it a non-uniform one [9]. A Variable Bandwidth (VBW) filter with an arbitrary sampling rate conversion technique can also be used for generating the sub-bands from the prototype filter [10]. But the sound wave decomposition structure using VBW filter requires very complex hardware for implementation [11]. Eventhough VBW filter using farrow structure has less processing delay, it also demands complex hardware [12].

Sound wave decomposition can also be achieved by the use of a Modified Discrete Fourier Transform (MDFT) method with moderate hardware complexity [5]. Filter banks designed using a combination of multirate signal processing techniques such as decimation and interpolation has moderate complexity, but the operating delay of this method exceeds the globally accepted limit in some cases [2]. This method has been modified with a two-level sound wave decomposition structure with less hardware [13]. An Interpolated Finite Impulse Response (IFIR) filter is yet another type providing a filter bank with 17 bands and moderate complexity [14]. Even though a number of techniques have been developed till date for the generation of filter banks, most of these methods are of high complexity and thereby demanding very complex hardware elements. This in turn, increases the cost of hearing aids thereby making it unaffordable for the needy. In this paper, the design of a 3-level digital filter bank for sound wave decomposition in hearing aids with very low complexity which can considerably reduce the cost of hearing aids is proposed.

The remaining part of this paper is organized as follows. Section 2 gives a description of Finite Impulse Response filters and also gives a brief introduction of Fractional Interpolated filters. The design methodology along with the structure of the proposed filter bank is illustrated in Sect. 3. The prototype filter generated, generation of 3-level filter bank using the proposed methodology and its delay analysis is given in Sect. 4. Some examples for portraying the audibility restoration capability of the proposed method and its hardware advantage is also depicted in this section along with the comparisons for matching performance and hardware complexity. Finally Sect. 5 gives the conclusion.

Finite impulse response filters

The two major classifications of digital filters are Finite Impulse Response (FIR) filters and Infinite Impulse Response (IIR) filters. Both of them have unique features. However, FIR filters are well suited for audio processing applications due to their non-recursive nature, i.e. there is no signal feedback from output to input. Due to this special feature, the impulse response of the FIR filter is limited to a particular interval. This non-recursive structure leads to a linear phase response which is essential for auditory compensation, feedback cancellation, noise reduction, and speech enhancement procedures in the hearing aid. The stability of the operation of FIR filter is higher than that of an IIR filter. Besides, FIR filters have more flexible implementation capabilities. No phase distortion is observed in FIR filters, which makes them best suited for audio processing applications [15].

The difference equation of an FIR filter of order N is given in Eq. (1). The output sequence y(n) is the convolution sum of the input sequence x(n) and the impulse response h(n). The $N+1$ coefficients of h(n) are $h(0),h(1),h(2),\dots ,h(N)$.

$$\begin{array}{c}\hfill y(n)=\sum _{k=0}^{N}h(k)x(n-k)\end{array}$$ 1

The transfer function of the above FIR filter of order N is given by

$$\begin{array}{c}\hfill H(z)=\sum _{n=0}^{N}h(n)z^{-n}\end{array}$$ 2

The coefficients of a linear phase FIR filter are always symmetric, which are given by

$$\begin{array}{c}\hfill h(n)=h(N-n)\text{for}n=0,1,2,\dots ,N\end{array}$$ 3

The direct-form realization of the Nth order FIR filter is illustrated in Fig. 1, where the input is represented by x[n] and the output is represented by y[n]. The filter coefficients are $h(0),h(1),h(2),\dots ,h(N)$ and a single sample delay element is represented as $z^{-1}$. The different multiplier outputs are summed up using an array of adders.

Fig. 1.

Direct-form structure of FIR filter

There are different methods for realizing efficient FIR filters such as window method, frequency sampling method, bilinear transformation method etc [16]. A linear phase FIR filter is obtained by the Remez exchange algorithm or Parks McClellan algorithm [17]. This method generates an optimized least order FIR filter based on the given specifications. The required specifications of an FIR filter are the passband and stopband cut-off frequencies, passband ripple, and the stopband attenuation. The generated low pass filter has an equi-ripple pass band and stop band with an optimized least order. The Parks McClellan algorithm is the solution of a minimax optimization problem which is explained as

$$\begin{array}{c}\hfill \text{minimize}{E}_{\mathit{max}}=max|E(wT)|,wT\in \mathrm{\Omega }\end{array}$$ 4

where E(wT) is a weighted error function, and $\mathrm{\Omega }$ is the union of the passband and stopband regions. The Kaiser formula based Parks–McClellan algorithm for a prototype filter of order N is given by

$$\begin{array}{c}\hfill N=\frac{-20{log}_{10}(\sqrt{{\delta }_p{\delta }_s})-13}{14.6({\omega }_s-{\omega }_p)/2\pi }\end{array}$$ 5

where ${\delta }_p$ and ${\delta }_s$ are the maximum permitted ripples in the passband and stopband respectively, and ${\omega }_p$ and ${\omega }_s$ are the passband and stopband edge frequencies. This optimized design technique requires N adders and the number of multipliers required for the realization of the FIR filter is given by [17].

$$\begin{array}{c}\hfill mult=⌈\frac{N+1}{2}⌉\end{array}$$ 6

Fractional interpolated filters

Multirate signal processing techniques are employed in digital systems for achieving a variation in the sampling rate of the system. The interpolation process will enhance and the decimation operation will diminish the sampling rate. In multirate signal processing, the interpolation and decimation operations are used to adjust the passband width of the prototype filter. The interpolation operation increases the sampling rate by an integer factor, L and the decimation operation decreases the sampling rate by an integer factor, D [18]. In the time domain, the interpolation process inserts $L-1$ number of null samples between each original sample. Similarly, the decimation process is realized by taking every Dth sample in the original sequence and discards all other samples. In the frequency domain, the interpolation and decimation processes are represented as $H(z^L)$ and $H(z^{1/D})$ respectively. Non-integer sampling rate conversions are achieved by a combination of the above interpolation and decimation operations. This fractional interpolation is described as $H(z^{L/D})$. In the fractional interpolated filter $H(z^{L/D})$, every Dth coefficients of the prototype filter are grouped while discarding the other coefficients and then inserting $(L-1)$ zeros in between the selected coefficients.

Design methodology

The prototype low pass FIR filter, H(z) is designed using the MATLAB filter design toolbox. H(z) is realized by the minimax approximation method using the Parks McClellan algorithm [17]. The coefficients of the designed prototype filter are used for further interpolation and decimation operations. The magnitude response of the prototype filter H(z) is inverted using the low pass to high pass transformation. This is achieved by inverting the sign of the alternate coefficients of the original low pass filter. The resultant complementary filter $H_c(z)$ is represented by Eq.(7), where N is the order of H(z). In hardware realization, the cost of implementation of the complementary filter can be brought down enormously by using a delay sharing technique in the transposed form representation of the FIR filter, as shown in Fig. 2. In this technique, the number of coefficient multipliers required for the realization of the prototype and the complementary filters are reduced to half of the original value [2].

$$\begin{array}{c}\hfill H_c(z)=z^{-(N-1)/2}-H(z)\end{array}$$ 7

Fig. 2.

Delay sharing method for the prototype and its complementary filter

Fractional interpolation techniques are performed on the prototype filter H(z) for generating more number of sub-filters with required bandwidth. The fractional interpolated filters are represented as $H(z^{L/D})$ with an interpolation factor, L and the decimation factor, D. The selected values of L are 4 and 8, while the value of D is 2. $H(z^{1/2})$, $H(z^{4/2})$, and $H(z^{8/2})$ are generated with fractional interpolation processes and its complementary pairs are produced using Eqs. (8–10). In the equations, the delay parameter, $\Delta$ is represented as half of the length of corresponding filter. The magnitude responses of the fractional interpolated filters and its complementary pairs are illustrated in Fig. 3a and 3b respectively.

$$\begin{array}{cc}& H_c(z^{1/2})=z^{-\Delta }-H(z^{1/2})\hfill \end{array}$$ 8

$$\begin{array}{cc}& H_c(z^{4/2})=z^{-\Delta }-H(z^{4/2})\hfill \end{array}$$ 9

$$\begin{array}{cc}& H_c(z^{8/2})=z^{-\Delta }-H(z^{8/2})\hfill \end{array}$$ 10

Fig. 3.

Magnitude responses of the fractional interpolated filters and its complementary pairs

In the proposed 3-level sound signal decomposition method for hearing aids, the incoming audio spectrum with 8 kHz bandwidth is equally divided into 4, 8, and 16 sub-bands. The bandwidth of each band in the 4-band filter bank is $\pi /4$, which is equivalent to a frequency of 2 kHz. Similarly, the bandwidth of individual bands in the 8-band and 16-band filter banks are $\pi /8$ and $\pi /16$, which corresponds to frequencies of 1 kHz and 500 Hz respectively. A suitable cascade combination of the prototype and the derived filters H(z), $H_c(z)$, $H(z^{1/2})$, $H_c(z^{1/2})$, $H(z^{4/2})$, $H_c(z^{4/2})$, $H(z^{8/2})$, and $H_c(z^{8/2})$ as shown in Fig. 3 are used to generate the 4-band, 8-band, and 16-band uniform filter banks. The transfer functions of sub-bands in the 4-band, 8-band, and 16-band filter banks are listed in Table 1. In the cascade procedure used for band selection process, the first band of the level-1 filter bank (FB) is H(z) itself with band edges $(0,\pi /4)$. In the case of level-2 filter bank, the first band is the cascade combination of H(z) and $H(z^{4/2})$ with band edges $(0,\pi /8)$. The band edges for the first band of the level-3 filter bank which uses the cascade combination of H(z), $H(z^{4/2})$ and $H(z^{8/2})$ are 0 and $\pi /16$ respectively.

Table 1.

Transfer functions of the sub-bands of 3-level filter bank

Band	Level-1 FB	Band	Level-2 FB	Band	Level-3 FB	Band edges
1	H(z)	1	$H(z^{4/2})$	1	$H(z^{8/2})$	$(0,\pi /16)$
				2	$H_c(z^{8/2})$	$(\pi /16,\pi /8)$
		2	$H_c(z^{4/2})$	3	$H_c(z^{8/2})$	$(\pi /8,3\pi /16)$
				4	$H(z^{8/2})$	$(3\pi /16,\pi /4)$
2	$H(z^{1/2})-H(z)$	3	$H_c(z^{4/2})$	5	$H(z^{8/2})$	$(\pi /4,5\pi /16)$
				6	$H_c(z^{8/2})$	$(5\pi /16,3\pi /8)$
		4	$H(z^{4/2})$	7	$H_c(z^{8/2})$	$(3\pi /8,7\pi /16)$
				8	$H(z^{8/2})$	$(7\pi /16,\pi /2)$
3	$H_c(z^{1/2})-H_c(z)$	5	$H(z^{4/2})$	9	$H(z^{8/2})$	$(\pi /2,9\pi /16)$
				10	$H_c(z^{8/2})$	$(9\pi /16,5\pi /8)$
		6	$H_c(z^{4/2})$	11	$H_c(z^{8/2})$	$(5\pi /8,11\pi /16)$
				12	$H(z^{8/2})$	$(11\pi /16,3\pi /4)$
4	$H_c(z)$	7	$H_c(z^{4/2})$	13	$H(z^{8/2})$	$(3\pi /4,13\pi /16)$
				14	$H_c(z^{8/2})$	$(13\pi /16,7\pi /8)$
		8	$H(z^{4/2})$	15	$H_c(z^{8/2})$	$(7\pi /8,15\pi /16)$
				16	$H(z^{8/2})$	$(15\pi /16,\pi )$

The structure of the proposed 3-level filter bank is illustrated in Fig. 4. The low pass output of the filter is shown by the symbol, ‘o’ and the complementary high pass output is represented by the symbol, ‘c’. The different levels of the proposed system are selected using a 3-bit control switch, namely $S_1{S}_2{S}_3$. An open switch is represented by a ‘0’ bit and closed switch is denoted by a ‘1’ bit. If the status of the control switch is made ‘100’, it will generate the 4 bands of the level-1 filter bank which is represented as $A_i$ where i denotes the band number. Similarly, a switch status ‘010’ will generate the 8 bands of the level-2 filter bank as represented by $B_i$ and when the switch assumes the status ‘001’, the 16 bands of the level-3 filter bank are generated as $C_i$.

Fig. 4.

Structure of the proposed 3-level filter bank

Results

Prototype filter

The Parks McClellan algorithm based equi-ripple low pass filter is generated using the following specifications. The sampling rate, $f_s$, is selected as 16 kHz and the cut-off frequencies of the passband, ${\omega }_p$ and stopband, ${\omega }_s$ are limited as 1.575 kHz and 2.425 kHz, respectively. The ripples in the passband, ${\delta }_p$ are limited to 0.05 dB, and the attenuation in the stopband, ${\delta }_s$ is selected as 60 dB. The transition bandwidth of the FIR filter between the passband and the stopband is set to 0.85 kHz . The order of the generated direct-form FIR filter is obtained as 55. According to Eq. (6), this linear phase prototype filter can be realized by 28 multipliers only. The prototype filter, H(z) is a low pass FIR filter with a cut-off frequency at 2 kHz and have a bandwidth of $\pi /4$. The magnitude response of the prototype low pass filter with the above parameters is shown in Fig. 5.

Fig. 5.

Magnitude response of the prototype filter, H(z)

Generation of 3-level filter bank

Uniform decompositions with 4-bands, 8-bands and 16-bands are obtained for the incoming sound signal when the proposed method is used. The multirate signal processing method is based on fractional interpolation in which the decimation and interpolation operations are employed simultaneously to get a non-integer interpolation. This gives a non-integer bandwidth reduction or expansion in the passband of the filter. The magnitude responses of the 4-band, 8-band, and 16-band filter banks are shown in Fig. 6. As given in Table 1, the 4-band filter bank is obtained from the frequency response of level-1 sub-filters. Since there are 4 bands for the entire spectrum, the bandwidth of each band in the 4-band filter bank is $\pi /4$, which is equivalent to 2 kHz in the frequency domain. The magnitude response of the level-1 filter bank is shown in Fig. 6a. Similarly, the 8-band filter bank is obtained from the cascade combination of level-1 and level-2 sub-filters. Here, each band in the level-1 filter bank is divided into two and hence the bandwidth of each band in the level-2 filter bank is $\pi /8$, which is equivalent to a frequency of 1 kHz. The magnitude response of the level-2 filter bank is shown in Fig. 6b. Furthermore, the cascade combination of level-1, level-2, and level-3 sub-filters generate the 16-band filter bank. Here, each band in the level-2 filter bank is equally divided into two and hence the bandwidth of each band in the level-3 filter bank is $\pi /16$, which is equivalent to 500Hz frequency. The magnitude response of the level-3 filter bank is shown in Fig. 6c.

Fig. 6.

Magnitude responses of sub-filters of the 3-level filter bank

Delay analysis of the 3-level filter bank

A hearing impaired listener carefully reads the lip movements of the speaker to increase the sound perception. Synchronization between the sound received in his ear and the lip movement read by him is necessary for better perception of the sound. In order to ensure proper synchronization, the processing delay of any hearing assistive device should be within the maximum limit of 20 ms. The processing delay of the fractional interpolated FIR filter is given by

$$\begin{array}{c}\hfill T={\displaystyle \frac{\mathit{NL}}{2D{f}_s}}\end{array}$$ 11

where $f_s$ is the sampling rate of the filter with order N. Besides, L and D are the interpolation and decimation factors selected for the fractional interpolation process. According to Eq. (11), the processing delays of the level-1 filter bank is 2.58 ms, level-2 filter bank is 6.02 ms, and the level-3 filter bank is 12.9 ms respectively. A detailed comparison of the design specifications of the prototype filter and the maximum processing delays of different auditory compensation methods are listed in Table 2.

Table 2.

Comparison of design specifications and delay for different methods

Filter bank design method	$f_s$ (kHz)	${\delta }_p$ (dB)	${\delta }_s$ (dB)	Maximum number of bands	Maximum processing delay (ms)
ANSI S1.11 [3]	24	1	60	18	31
Quasi-ANSI S1.11 [6]	24	1	60	18	10
Frequency response masking [7]	16	0.0001	80	8	26.6
Variable bandwidth filters [12]	16	0.05	80	10	1.1
Fractional interpolation [2]	16	0.005	50	12	21.6
2-Level fractional interpolation [13]	16	0.005	50	13	18.5
Proposed method	16	0.05	60	16	12.9

Verification of audibility restoration capability

The proposed filter banks are used to compensate the auditory defects of the hearing impaired. The hearing deficits of any patient is measured by an audiometry test conducted on single tone frequencies at 250 Hz, 500 Hz, 1 kHz, 2 kHz, 4 kHz, and 8 kHz [19]. The audiogram depicts the softest sound that can be heard by any hindered person. In the audiogram, the characteristics of the left ear is represented by the symbol ‘X’ and that of the right ear is shown by the symbol ‘O’. Selective amplification processes on the sub-filters of the filter bank results in a good hearing performance for the impaired individuals. Different persons have individual hearing thresholds at different frequencies. The thresholds up to 20 dB are considered as normal hearing while the values between 20 and 40 dB is for mild hearing loss. The hearing profiles ranging from 40 to 70 dB are the signs of moderate hearing loss and 70–90 dB indicates a severe hearing loss. The thresholds of hearing beyond 90 dB is an indicator of profound hearing loss.

The proposed sound wave decomposition filter banks are evaluated using the test audiograms which are selected from the audiometry screening process [20] and the independent hearing aid information provided by the Hearing Alliance of America, which is a public service provider [21]. The audiogram 1 in Fig. 7 is for a common ailment with mild conductive hearing loss for all frequencies in the right ear. The audiogram 2 in Fig. 7 is for age related hearing loss known as presbycusis, which affects the high frequencies badly. The audiogram 3 in Fig. 7 is for a sensorineural hearing loss, which can affect any frequency range. The hearing profiles of the left ear in Fig. 7d and 7g are considered for selective amplification procedure for the audibility restoration process using the proposed 3-level filter bank. The selective amplification procedures used in each audiogram and the corresponding matching results are also illustrated in Fig. 7. The audio spectrum is decomposed with the 4-band filter bank for correcting the mild hearing loss in all frequencies. The moderate hearing loss in high frequencies due to presbycusis is effectively compensated with a sound wave decomposition of 8 bands. The 16-band filter bank structure is essential for the matching of mild hearing loss in mid frequencies caused by sensorineural hearing loss. The filter bank with more number of bands is demanded for the compensation of audiograms with rapid variations in the hearing characteristics.

Fig. 7.

Filter banks with selective amplification procedures and matching results

A comparison of the matching performance of the proposed method with other audibility restoration methods are listed in Table 3. The matching results of audiogram 3 has been compared with the results obtained from published literature and the method proposed in this paper yielded better results. Since the audiograms 1 and 2 used in this study has not been used in the references listed in Table 3, the results obtained from the simulation of the previous methods are chosen for comparison. The results obtained have shown that the method put forth in this paper has produced better matching error in almost all cases with minimal hardware complexity.

Table 3.

Comparison of matching error in different methods

Audiogram matching method	Audiogram 1 (dB)	Audiogram 2 (dB)	Audiogram 3 (dB)
Frequency Response Masking [22]	4.23	4.16	4.82
Fractional Interpolation [2]	3.61	5.02	5.63
Interpolated FIR filters [14]	3.28	3.78	3.82
2-Level Fractional Interpolation [13]	1.49	2.54	2.44
Proposed method	2.32	2.43	2.39

Hardware advantages

As per the design specifications mentioned above, the prototype FIR filter is designed using the Parks McClellan algorithm with minimum number of multipliers. The hardware complexity of the proposed method is very meager when compared to other filter designs and filter bank generation methods in the literature. Since multiplier requires the most hardware resources, the number of multipliers used in the design is taken as the benchmark for computing the hardware complexity. This proposed linear phase FIR filter with 55 coefficients can be effectively implemented using just 28 multipliers. The fractional interpolation technique used in this proposed method curbs the increase in number of multipliers required than the original filter. Since the number of multipliers required in this proposed method is very small when compared to other similar methods, very less components and device area is required for its hardware implementation. Thus a cost-effective hearing aid of very low complexity can be achieved through this method, thereby making it affordable for needy. From Fig. 8 which shows the comparison of hardware complexities of various methods available from the literature, it can be clearly observed that the proposed method is the one with least hardware complexity.

Fig. 8.

Comparison of hardware complexity

Conclusion

A low-complexity 3-level sound wave decomposition technique for hearing aids suitable for audibility restoration applications is discussed in this paper. The sound waves are decomposed into 4, 8, and 16 bands, which can be directly used by hearing aids for auditory compensation, feedback cancellation, noise reduction, and speech enhancement applications. The linear phase FIR filter is designed using the Parks McClellan equi-ripple algorithm. The prototype low pass filter and the fractional interpolated sub-filters are implemented using only 28 multipliers, which is lowest compared to other similar techniques for sound wave decomposition. This reduces the complexity of hardware components required in the hearing aids, thereby reducing its cost significantly. Audiograms with different degrees of hearing deficiencies are also evaluated with the 3-level filter bank. The proposed filter banks are capable of compensating different types of hearing impairments such as sensorineural hearing loss, presbycusis etc with better matching errors. This proposed 3-level filter bank design of low complexity is capable of transforming the vision of affordable hearing aids into reality which can become a boon to the needy.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest in relation to the work in this article.

Ethical approval

This article does not contain any studies with human participants or animals performed by the author.