1 Introduction

In the present era, speech enhancement plays a major role in the field of speech processing as it is related to the speaking as well as listening skills. In general, speech enhancement is employed with a desire of processing the noisy speech signals, thereby enhancing human perception [3, 38, 41]. Generally, the quality of the speech is related to the attributes of the speaker, like the naturalness and speaker recognizability, whereas the intelligibility of the speech is related to the meaning or information content that is hidden behind the words [45] [27]. Speech signals are utilized in many purposes and in recent times, COVID-19 [28, 36] has been detected from speech signals. Hence, it is vivid that, the ability to communicate (speak and listen) diminishes in the noisy environment.

The speech enhancement is performed with the intention of reducing the impact of the communication problem [22]. Most of the research proved that, it is a complex task to reduce the noise of the signal without distorting speech and this is the major reason behind the non-availability of an ideal enhancement systems [4, 30]. Beyond this, efforts to enhance the “higher quality and/or intelligibility of noisy speech” will definitely end up with a mass increment in the performance of the speech signal and hence it can be employed in the fields of “speech coding/compression and speech recognition, hearing aids, voice communication systems and so on” [21, 35]. Further, the goal behind each of the speech recognition might be different and they are application based, such as diminishing the listener fatigue, boosting the overall speech quality, enhancing the intelligibility and improving the performance of the voice communication device, etc. But, the major benchmark behind all the research is to diminish the noise level and to enhance the quality as well asthe intelligibility of the signal. Hence,“speech enhancement is necessary to avoid the degradation of speech quality and to overcome the limitations of human auditory systems” [2, 43].

A vast amount of automatic speech processing systems are playing a major role in human life, like the “mobile communication, speech and speaker recognition, hearing impaired and numerous other applications”. Moreover, the quality and intelligibility of speech areof utmost importance with the intention of enhancing the accuracy of information exchange [31]. Beyond this, in the controlled environment, human as well as automatic speech communications are found to be much more effective [5, 47]. The Spectral Subtraction algorithm suffers from the problem of restoration in the basis parameters of the speech like the power spectrum or the magnitude spectrum and here only the additive noise available in the signal can be removed [1]. Then, in the Sub-space analysis algorithm, there was a difficulty in enhancing the noise spectrum and updating the noise spectrum from period to period was a complex task. [46]. Thus, in order to override these entire problems, there is a necessity to have an optimal speech enhancement method.

Nowadays, literature works have come up with several techniques for speech enhancement as relates to speaking as well as listening skills. Tantibundhit et al. [44] proposed JT-FS with the desire of decomposing the speech signal into “transient” as well as “non-transient components” only the basis of the wavelet packets. Lee et al. [30] proffered P-SJL with the intention of enhancing single-channel speech. The phase-related information of the speech signal was represented using PSM which was similar to the T-F mask. Furthermore, the P-ASEalgorithm [48] was formulated on the basis of DNN. Shao and Chang [40] developed a framework of wavelet-based techniques with the intention of enhancing the performance of automatic speech recognition by eliminating the background noise. AKCF algorithm was introduced in [13]with the aim of enhancing speech. The noise as well as the speech parameters was estimated using the Estimate-Maximize (EM) method. Mohammadiha et al. [34] proposed SSD algorithms on the basis of NMF. In addition to this, the Bayesian Formulation of NMF (BNMF) was used for generating the novel speech enhancement method. Additionally, Chazan et al. [6] proposed the S-MSE algorithm in order to enhance the speech signal. Samuiet al. [39] proposed time-frequency masking in the basis of DNN with the intention of enhancing the speech signal and here the pre-training of the signal was accomplished using FRBM. Moreover, the advantages and challenges of the few works are listed in Table 1. These challenges have kept the main stand for motivating and accomplishing the new speech enhancement model.

Table 1 Features and Challenges of of the state-of-art Speech Enhancement models
Full size table

In addition, many optimization algorithms have been introduced recently [8,9,10, 25] and utilized in many fields for better outcomes [7]. In this research work, a modified version of a popular meta-heuristic algorithm is employed. The major contributions of this research are listed below:

  • In this research work, STFT-based noise estimation and NMF-based spectrum estimation are utilized for the estimation of the noise spectrum and signal spectrum of the noisy signal.

  • To minimize the error, a Wiener filter is employed and the tuning factor ηof Wiener filter is obtained for different signals.

  • Introducing a Fuzzy Wavelet Neural Network (FW-NN) for detecting the suited tuning factor η for the entire input signal in Wiener filter.

  • Proposed an Adaptive Randomized Grey Wolf Optimization (AR-GWO) for proper tuning of the tuning factorηreferred as tuned tuning factor(ηtuned). The proposed AR-GWO algorithm is an improved version of the traditional GWO algorithm.

The rest of the paper is organized as below: Section 2 portrays the proposed architecture of the speech enhancement model. Section 3 depicts the processed steps for enhanced speech enhancement. The results and discussions are exhibited in Section 4, and Section 5 concludes the paper.

2 Proposed architecture of speech enhancement model

2.1 Architectural representation

Figure 1 demonstrates the architecture of the proposed speech enhancement model in which the overall process takes place in “two major phases (i) training phase (ii) testing phase”. In the training phase, initially, the noise corrupted signal is fed as input to STFT-based noise estimation as well as NMF-based spectrum estimation, for estimating the noise spectrum and signal spectrum, respectively. The obtained spectrum (noise and signal) are given as input to the Wiener filter. These, filtered signals are subjected to EMD, from which the denoised signal can be obtained. Since, tuning factor η plays a key role in Wiener filter, it has to be determined for each signals, and is trained in FW-NN. Then, from the denoised signal the bark frequency is evaluated. The computed bark frequency is fed as input to the learning algorithm referred as FW-NNfor detecting the suited tuning factorη for the entire input signal in Weiner filter. The AR-GWO is employed for proper tuning of the tuning factorη. Moreover, in the testing phase of a signal, the training is accomplished initially, from which the tuning factorηis gathered for the corresponding input signal. Then, the properly tuned ηfrom FW-NNis fed as input to EMD via adaptive Wiener filter for decomposing the spectral signal and the output of EMD is denoised signal.

Fig. 1
figure 1

Proposed intelligence architecture for speech enhancement model

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Consider the clear signal as T(n), when the noise Wgets corrupted into it, and the signal becomes noisy signal \( \overline{T}(n) \). This noisy signal is fed as input to the STFT-based noise estimation and NMF-based spectrum estimation, from which the noise spectrum WT and signal spectrum \( {\overline{W}}^T \)are obtained. The obtained noise and signal spectrum are subjected to filtration using Wiener filtering process; at the end of filtration the filtered signal \( {\overline{T}}_u(n) \) is generated. Then, \( {\overline{T}}_u(n) \) is decomposed using EMD as a result of this, the bark frequency c′(u′) is obtained. This bark frequency is utilized to train FW-NN classifier. From the spectrum WT and \( {\overline{W}}^T \) as well as from FW-NN, ‘tuned η’ referred as ηtuned is acquired for all the inputs signals with AR-GWO. In the testing process, the tuned ηtuned is acquired for the corresponding signal with the aid of the AR-GWO; this ηopt is fed as input to the adaptive Wiener filtering process with the intention of tuning the input signal \( \overline{T}(n) \). The outcomes of the adaptive Wiener filter are the filtered signal \( \overline{\overline{T_u(n)}} \). Again, \( \overline{\overline{T_u(n)}} \) is decomposed using EMD and the result is the enhanced denoised signal \( \overline{\overline{T_o(n)}} \).

3 Processed steps for enhanced speech enhancement

3.1 STFT-based noise estimation

The noise power spectral density estimatorisbased onminimum statistics to track the minima from the noisy signal [26]. The STFT coefficient of the frame γ is depicted as T(γ, p) and their mathematical formula is exhibited in Eq. (1) [14].

$$ T\left(\gamma, p\right)=\tau \left(\gamma, p\right)T\left(\gamma -1,p\right)+\left(1-\tau \left(\gamma, p\right)\right){\left|T\Big(\gamma, p\Big)\right|}^2 $$
(1)

Here, the frequency bin is manifested as p. The frequency and time-dependent smoothing parameters is portrayed as τ(γ, p). With the intention of observing the mean power, the bias compensation factor is employed. The variance estimator of the smoothened PSD is represented as var{T(γ, p)} and the function corresponding to the length of minimum search interval is defined by the bias compensation factor Kmin. The variance estimator relating the smoothened PSD is indicated asvar{T(γ, p)} and this assist in evaluating the variance of T(γ, p) by fixing the length of the search interval in the algorithm. Eq. (2) depicts the mathematical formula for evaluating the variance estimator at the frame γ relating the frequency bin p. In Eq. (2), the mean smoothened periodograms is represented as \( \overline{T}\left(\gamma, p\right) \), and \( \overline{T^2}\left(\alpha, b\right) \) indicates the first-order recursive average of smoothened periodograms [14].

$$ \mathit{\operatorname{var}}\left\{T\left(\gamma, p\right)\right\}=\overline{T^2}\left(\gamma, p\right)-{\overline{T}}^2\left(\gamma, p\right) $$
(2)

This paper deal with STFT-based noise estimation and the graphical representation of the power spectrum corresponding to the noise estimated by FFT as well as STFT is exhibited in Fig. 2. The power spectrum varies by the magnitude of the frequency component. Moreover, in determining the phase content of the signal and varying sine wave frequency that alter over time are predicted using STFT. In general, the time signals which are larger in size are sub-divided into smaller equal size signals and to each of the segments the Fourier transform is employed. In addition, in the filtering process, STFT can also be interpreted. The estimation strategy is satisfied by two major properties viz. magnitude based shift invariance property and LT-FD properties. The noise spectrum WT is obtained as the resultant.

Fig. 2
figure 2

Noise power spectrum (a) estimated by FFT (b) estimated by STFT - minimum statistics

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3.2 NMF-based Spectrum estimation

In the time-frequency (γ, p) domain, the voicing of the noisy signal \( \overline{T}(n) \) takes place via STFT as per Eq. (3), to enhance the speech signal [42]. In Eq. (3), the STFT of the clear speech T(p, γ), the STFT of the noisy speech \( \overline{T}\left(p,\gamma \right) \) and the STFT of the noise signal W(p, γ) are used in pth frequency bin of γ frame. The mathematical formula for “noisy speech’s magnitude spectrum” approximation, which is most commonly, utilized assumption for NMF-based processing of speech and audio signal, is show in Eq. (4) [14].

$$ \overline{T}\left(p,\gamma \right)=T\left(p,\gamma \right)+W\left(p,\gamma \right) $$
(3)
$$ \mid \overline{T}\left(p,\gamma \right)\mid =\mid T\left(p,\gamma \right)\mid +\mid W\left(p,\gamma \right)\mid $$
(4)

The magnitude spectrum matrices of the varied signal are indicated as per Eq. (5) and magnitude spectral value corresponding to γ frame for the pth bin is depicted as jp, γ. The count of the frequency bins is represented as H and the time frames are indicated as I.

$$ J=\left[{j}_{p,\gamma}\right]\in {N}_{+}^{H\times I} $$
(5)

For the training data \( {J}_T\in {N}_{+}^{H\times {I}_T} \)as well as \( {J}_W\in {N}_{+}^{H\times {I}_W} \), the Eq. (5) is employed separately in the training stage and the outcome of these data is the basis matrices in terms of clear speech \( {F}_T=\left[{r}_{Hl}^T\right]\in {N}_{+}^{H\times {L}_T} \)and noise \( {F}_W=\left[{r}_{Hl}^W\right]\in {N}_{+}^{H\times {L}_W} \), respectively. The count of base vectors is indicated as L. In Eq. (6) ζ represents a H × I matrix, whose entities is equal to one and the transpose of the matrix, is represented as T′. In addition, the basis matrices are fixed in the enhancement stage as \( {F}_T=\left[{F}_T{F}_W\right]\in {N}_{+}^{H\times \left({L}_T+{L}_W\right)} \).The activation matrix \( {E}_{\hat{T}}={\left[{E}_T^{T\prime }{E}_W^{T\prime}\right]}^{T\prime}\in {N}_{+}^{\left({L}_T+{L}_W\right)\times {I}_{\hat{T}}} \) corresponding to the noisy speech is estimated from \( {J}_{\hat{T}}\in {N}_{+}^{H\times {I}_{\hat{T}}} \) by means of employing the NMF activation update. Further, with the assistance got from the Wiener Filter (WF), the clear speech spectrum is evaluated from the speech signal only after obtaining the activation matrix as per Eq. (7). The estimated PSD matrices corresponding of the clear speech is manifested as V′T = [V′T(p, γ)] and the evaluated PSD matrices corresponding to the noisy signal is represented as \( V{\prime}_W=\left[V{\prime}_W\left(p,\gamma \right)\right]\in {N}_{+}^{H\times {I}_{\hat{T}}} \) in Eq. (7). Further, as per Eqs. (8) and (9) the next solution is obtained via the temporal smoothing of the period grams. The temporal smoothing factor of speech ωT and noise ωW is shown in Eqs. (8) and (9), respectively.

$$ {\displaystyle \begin{array}{c}F\leftarrow F\otimes \frac{\left(J/F.E\right)E}{\zeta E},\\ {}\;E\leftarrow E\otimes \frac{F\left(J/ FE\right)}{F^{T^{\prime }}\zeta}\end{array}} $$
(6)
$$ Q=\frac{V{\prime}_T}{V{\prime}_T+V{\prime}_W}\otimes \hat{T} $$
(7)
$$ V{\prime}_T\left(p,\gamma \right)={\rho}_TV{\prime}_T\left(p,\gamma -1\right)+\left(1-{\rho}_T\right){\left({\left[{F}_T{E}_T\right]}_{p\gamma}\right)}^2 $$
(8)
$$ V{\prime}_W\left(p,\gamma \right)={\rho}_WV{\prime}_W\left(p,\gamma -1\right)+\left(1-{\rho}_W\right){\left({\left[{F}_W{E}_W\right]}_{p\gamma}\right)}^2 $$
(9)

The signal spectrum \( {\overline{W}}^T \)is obtained as the outcomes.

The obtained noise spectrum WT and signal spectrum \( {\overline{W}}^T \)are subjected to filtration using Wiener filtering process.

3.3 WienerFilter

In the signal enhancement technique, the Wiener filter has been employed in large scale [15]. The Wiener filter works on the principle of producing an estimate of the clean signal from the corrupted noise signal. The estimation is accomplished by minimizing MSE in between the desired signal and additive noise corrupted signal. The filter transfer functionis shown in Eq. (10) and it gives the solution to this optimization problem in the frequency domain. This equation is generated by considering the signal spectrum \( {\overline{W}}^T \)and the noise spectrum WT as uncorrelated and stationary signals. The power spectral density of \( {\overline{W}}^T \)is represented as GT(ω) and the power spectral density of WT is depicted as GW(ω). The mathematical formula for SNR is exhibited in Eq. (11) and the SNR formula can be incorporated in the filter transfer function as per Eq. (12). The estimated signal magnitude spectrum is indicated as \( {\hat{G}}_W\left(\omega \right) \).

$$ F\left(\omega \right)=\frac{G_T\left(\omega \right)}{G_T\left(\omega \right)+{G}_W\left(\omega \right)} $$
(10)
$$ SNR=\frac{G_T\left(\omega \right)}{{\hat{G}}_W\left(\omega \right)} $$
(11)
$$ F\left(\omega \right)={\left[1+\frac{1}{SNR}\right]}^{-1} $$
(12)

At the end of filtration the filtered signal \( {\overline{T}}_u(n) \) is generated.

The Wiener filteroften fails at all the frequencies due to the drawback of fixed frequency response and requirement of estimating the clean signal and noise signal’s power spectral density prior to filtering.

3.4 Empirical model decomposition

EMD [16] was introduced by Huang as an adaptive technique in which small number of orthogonal empirical modes referred as IMF were added to represent the complex data. The symmetric envelope is present in each of the mode in terms of local maximums and minimums. Thus at all locations of the envelope, mean is zero and in the underlying signal, there is no requirement of linearity or time invariance. Further, by the process of shifting, the riding waves are eliminated. The shifting process of EMD algorithm can be depicted as shown below. Two main properties are obeyed by EMD during the splitting of \( {\overline{T}}_u(n) \) into its IMF components. They are (a) In between two subsequent zero crossing, the IMF has only one extremum and (b) Mean value of IMF is zero.

The data set \( {\overline{T}}_u(n) \) is decomposed into IMFs ye(n) and residue q(n). The mathematical formula corresponding to this decomposition is described in Eq. (13).

$$ y(n)=\sum \limits_e{y}_e(n)+y(n) $$
(13)

Furthermore, the detailed steps of EMD are given below.

  • At first, initialization is processed i.e., d ≔ 1,q0(n) = y(n)

  • As per the following steps, dth IMF is extracted

Set k0(n) ≔ qd − 1(n),m ≔ 1 and local maxima and minima of whole km − 1(n) are identified. Then the envelope UBm − 1(n) for km − 1(n) defined by the maxima and LBm − 1(n) by the minima using the cubic splines interpolation.For both the envelopes belonging to km − 1(n), the mean zm − 1(n) is determined as \( {z}_{m-1}(n)=\frac{1}{2}\left({UB}_{m-1}(n)-{LB}_{m-1}(n)\right) \). This running mean is referred as low frequency local trend. Further, via the process of shifting, the evaluation of high- frequency local detail takes place.

Further, the mth component is formed as km(n) ≔ km − 1(n) − zm − 1(n). In case if km(n) is not found to be accordance with whole IMF criteria, then the process of shifting is continued by increasing mm + 1. In case, if all IMF criteria is satisfied by km(n), then set yd(n) ≔ km(n) and qd(n) ≔ qd − 1(n) − yd(n).

  • The shifting process can be stopped, if qd(n) represents a residuum and if not, then continue the shifting process by increasing d, d + 1 and again begin the process.

Further, EMD algorithm achieves the completeness of the decomposition process automatically as \( y(n)=\sum \limits_{d=1}^v{y}_d+q \) and this represents an identity. The locally orthogonal IMFs are generated by EMD algorithm and lacks to guarantee the global orthogonality, since identical frequencies might be utilized by neighboring IMFs at different time points. As a result of this, the bark frequency c′(u′) is obtained. This bark frequency is utilized to train FW-NN classifier.

3.5 Fuzzy wavelet neural network (FW-NN) classifier

Classification is the most frequently used prediction type [37]. Generally, the wavelet functions are combined with neural nets to provide better results [17,18,19]. In this work, a FW-NN model is employed and it is combination of fuzzy logic concepts and wavelet neural network. In FW-NN, each fuzzy rule corresponds to a WNN comprised of numerous wavelets with changeable translation and dilation parameters. The fuzzy rules are being the consequent part of theFW-NN architecture and it is described only by wavelet functions. The output of WNN is expressed as per Eq. (14).

$$ Y=\sum \limits_{j=1}^k{\delta}_j{\kappa}_j(X) $$
(14)

In which κj is jth layers wavelet activation function corresponding to the hidden layer. In addition, δj is the weight between the hidden (hid) and output layer.

The FWNN combines the wavelet functions and the TSK fuzzy system. A MF is shown by each of the region in the TSK fuzzy model. The FWNN has the properties of high precision and fast convergence. The FW-NN has six layers and they are discussed in the below section.

Layer 1 (input layer)

The input signal vector In = (In1, In2, …, Inn) is fed as input to the next layer and the whole FW-NN model is trained with the bark frequency c′(u′).

Layer 2 (fuzzification layer)

The fuzzy MFs are shown by each of the neuron in IF part of the rules. The MFs values are the outcomes’ from this layer. In the first layer there is l1 count of MFs and in the second layer there is l2 count of MFs. For the ith input variable, the Gaussian membership function is shown as per Eq. (15).

$$ {\displaystyle \begin{array}{c}{A}_{j_i}^i=\exp \left(-{\left(\frac{X_i-{\varpi}_{ji}}{\varsigma_{ji}}\right)}^2\right);\\ {}\kern1.32em i=1,2,.,n\kern0.24em and\kern0.24em {j}_i=1,2,\dots, {l}_i\end{array}} $$
(15)

Layer 3 Grey wolf (fuzzy rule layer)

In this layer, each neurons show fuzzy rule. The lth nodes outcome is denoted as per Eq. (2). Here, each of the input MFs based possible combinations describes a fuzzy rule.

$$ {\eta}^l=\prod \limits_{i=1}^n{A}_{j_i}^i\left({X}_i\right) $$
(16)

Layer 4 Grey wolf (normalization layer)

Normalization factor is computed for each of the neurons in this layer. The lth nodesnormalization factor is expressed as per Eq. (17).

$$ \overline{\eta^l}=\frac{\eta^l}{\sum \limits_{j=1}^m{\eta}^l} $$
(17)

Layer 5

The weighted output value is computed in this layer as per Eq. (18).

$$ {F}^l=\overline{\eta^l}{\chi}^l $$
(18)

Layer 6

The overall output is calculated in this layer by summing the previous layers outputs. This is mathematically shown in Eq. (19).

$$ Out=\sum \limits_{l=1}^m{F}^l $$
(19)

During the training phase, the MSE is selected as the performance index and this MSE minimization is being the major objective of the current research work. The mathematical formula for MSE based training is shown in Eq. (20). Here, the actual FWNN outcome is Act and the desired outcome is Pre.

$$ Er=\frac{1}{N}\sum \limits_{k=1}^N\left( Act- Pre\right) $$
(20)

4 Adaptive randomizatized grey wolf algorithm: solution encoding and objective function

4.1 Objective function and solution encoding

The major objective of the current research work is to minimize the error Er of the FW-NN. This is expressed mathematically in Eq. (21).

$$ Obj=\mathit{\operatorname{Min}}(Er) $$
(21)

The AR-GWO is employed for properly tuning the tuning factor η, which is accomplished by means of optimizing the hidden neurons (hid) of FW-NN. The solution fed as input to AR-GWO is exhibited in Fig. 3.

Fig. 3
figure 3

Solution encoding

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4.2 Standard GWO

GWO [11, 32] was introduced by Mirjalili on the basis of the natural behavior of the grey wolves and it belongs to the category of swarm intelligence algorithm. Three are four types of grey wolves and these wolves stay in groups. The highest authority among them is the α (alpha) and it has the responsibility of taking decision. The supporter of α in taking decisions is β (beta), the lowest among these wolves is ω (omega) and it has to bow other wolves. The leftovers are referred as δ (delta). The main phases of GWO are “hunting, chasing and approaching the prey, encircling the prey and attacking the prey”. The upcoming section portrays the mathematical model of GWO.

Mathematical model of GWO

  1. (i)

    Search for prey (exploitation): In the search process, the 1st, 2nd and 3rd best solutions are obtained during the search process of unique α, β and δ

  2. (ii)

    Encircling prey: The mathematical formula for prey encircling during the hunting process is represented in Eqs. (22) and (23). In Eq. (24) the current iteration and the localization of the prey is represented as x&Cg(x). The coefficient vectors are indicated as Y and D. In addition, C(x) represents the position of the grey wolf and the random values are manifested as b1 & b2. In addition Eqs. (24) and (25) are the mathematical formula for calculating the coefficient vectors Y and D, here there is a gradual decrease in the value of c from 2 to 0 over the course of iterations.

$$ A=\left|D.{C}_g(x)-C(x)\right| $$
(22)
$$ C\left(x+1\right)={C}_g(x)-Y.A $$
(23)
$$ Y=2.c{b}_1-c $$
(24)
$$ D=2.{b}_2 $$
(25)
  1. (iii)

    Hunting the prey: There lacks no information on the location of the prey in the search space. An assumption is made here that a better knowledge on the potential location of prey can be acquired from α, β and δ. This is the reason behind the storage first three results by discarding the others. The mathematical formula for hunting of prey is depicted in Eqs. (26) to (32) [32].

$$ {A}_{\alpha }=\left|D.{C}_{\alpha }-C\right| $$
(26)
$$ {C}_1={C}_{\alpha }-{Y}_1.\left({A}_{\alpha}\right) $$
(27)
$$ {A}_{\beta }=\left|D.{C}_{\beta }-C\right| $$
(28)
$$ {C}_2={C}_{\beta }-{Y}_2.\left({A}_{\beta}\right) $$
(29)
$$ {A}_{\delta }=\left|D.{C}_{\beta }-C\right| $$
(30)
$$ {C}_3={C}_{\delta }-{Y}_3.\left({A}_{\delta}\right) $$
(31)
$$ C\left(x+1\right)=\frac{C_1+{C}_2+{C}_3}{3} $$
(32)
  1. (iv)

    (iv) Attacking the prey (exploitation): This is the end process of hunting behaviour of grey wolf and this process take place, when the prey is stationary.

4.3 AR-GWO

The conventional GWO suffers from the drawbacks of “bad local searching ability, low solving precision and slow convergence”. So, the AR-GWO is formulated. In the conventional GWO, the random values b1and b2 are within the range [0, 1] and they are utilized to find the coefficient vectors Y and D in Eqs. (24) and (25). But, in the proposed model, instead of random numbers the proposed algorithm determines the random values bi1 and bi2 on the basis the fitness functions. The coefficient vectors are presented as Yi and Di are computed by utilizing Eqs. (33) and (34). Here i denote α, β and δ wolves. Further, the random values bi1 and bi2 are determined by using Eqs. (35) and (36), in which fitness of the best wolves either α, β or δ is represented as Fi,

$$ Yi=2.c{b}_{i1}-c $$
(33)
$$ Di=2.{b}_{i2} $$
(34)
$$ {b}_{i1}= Fi $$
(35)
$$ {b}_{21}=\frac{Fi}{\frac{1}{3}\sum \limits_{i=\alpha, \beta, \delta } Fi} $$
(36)

The resultant from AR-GWO is the properly tuned tuning factor ηtuned, which is fed as input to adaptive Wiener filtering.

4.4 Adaptive WienerFiltering

The role of tuning ratio ηtuned is highly substantiated. The estimated tuning ratio by the FW-NN, on the basis of the c′(u′) (bark frequency) of the NMF-based filtered EMD signal \( {\overline{T}}_o(n) \) is fine-tuned by AR-GWO. Mathematically, c′(u′) can be expressed as per Eq. (37)

$$ {c}^{\prime}\left({u}^{\prime}\right)=13\arctan \left(0.76{u}^{\prime}\right)+3.5\arctan \left[{\left(0.33u\prime \right)}^2\right] $$
(37)

The properly tuned tuning ratio ηtuned acquired from AR- GWO is fed as input to the wiener filter, instead of the constant η. The outcomes of the Adaptive Wiener filter are the filtered signal \( \overline{\overline{T_u(n)}} \). Again, \( \overline{\overline{T_u(n)}} \) is decomposed using EMD and the result is the enhanced denoised signal \( \overline{\overline{T_o(n)}} \).

In the training process, the training library is constructed by giving the known c′(u′) (bark frequency) and tuning ratio ηtuned as inputs. The testing process is said to be the online process, while the training process is an offline process. The appropriate tuning factor for diverse noises are identified in the offline process and with this, the FW-NN is trained. The actual enhancement process takes place in the online mechanism, where the tuning factor is identified with the trained network.

5 Results and discussion

5.1 Experimental setup

The proposed speech enhancement model using GWO with FW-NN was implemented in MATLAB and the resultant of each of the analysis is observed. The data set for the research work is gathered from [23]. In this database, the five noise types, namely, “airport noise, exhibition noise, restaurant noise, station noise and street noise” are added to the speech signals. The performance of the proposed model (AR-GWO) is compared with the extant modelslike GA [29], PSO [20], ABC [24], FF [12] and GWO [14] in terms of “SDR, PESQ, SNR, RMSE, Correlation, ESTOI and CSED”. Also, statistical analysis and computational time analysis are performed. Figure 4 exhibits the noisy and denoised signal for different approaches like GA, PSO, ABC, FF and GWO.

Fig. 4
figure 4

The noisy and denoised signal of various approaches (a) GA (b) ABC (c) PSO (d) FF(e) GWO and (f) AR-GWO

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5.2 Performce analysis of airport noise

The performance evaluation of the proposed model over the existing model for airport noise at varying SNR levels is shown in Table 2. whenSNR = 0 dB, the SDR of the proposed model is 2.13%, 1.04%, 0.67%, 0.56% and 2.4% superior to the extant models like GA based η tuning, ABC based η tuning, PSO based η tuning, FF based η tuning and GWO based η tuning, respectively.PESQ of the proposed model at SNR = 0 dB exhibits an improvement of 3.7%, 2.6%, 4.5%, 1.3% and 1% over the extant models like GA based η tuning, ABC based η tuning, PSO based η tuning, FF based η tuning and GWO based η tuning, respectively.In airport noise at SNR = 5 dB, RMSE of the proposed model is 3.4%, 2.7%, 1.6%, 3.44% and 0.9% superior to the traditional models like GA based η tuning, ABC based η tuning, PSO based η tuning, FF based η tuning and GWO based η tuning, respectively. At SNR = 10 dB, ESTOI of the proposed model is 1.11% better than GA based η tuning, 1.6% better than ABC based η tuning, 2% better than PSO based η tuning, 1.7% better than FF based η tuning and 0.9% better than GWO based η tuning. Further, at SNR = 10 dB, STOI of the proposed model shows an improvement of 0.9%, 1%, 1.2%, 1.1% and 0.7% better than classical model like GA based η tuning, ABC based η tuning, PSO based η tuning, FF based η tuning and GWO based η tuning, respectively.

Table 2 Performance evaluation of proposed model over existing for airport noise at varying SNR
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5.3 Performce analysis of exhibition noise

Table 3 exhibits the performance analysis of the proposed model over exiting for exhibition noise at different SNR levels. At SNR = 0 dB, the proposed model shows an improvement of 10.3%, 3.6% 10.2%, 3.1% and 1.6% over the classical models like GA based η tuning, ABC based η tuning, PSO based η tuning, FF based η tuning and GWO based η tuning, respectivelyin terms of SNR. Further, RMSE of the proposed model is 5.4%, 2.4%, 6.5%, 2% and 0.9% better than the extant models like GA based η tuning, ABC based η tuning, PSO based η tuning, FF based η tuning and GWO based η tuning, respectively at SNR = 0 dB. For the exhibition noise at SNR = 5 dB, the SDR of the proposed model is improvedover the existing model as 15.7%, 2.16%, 1.2%, 0.86%, and 6.83% by GA based η tuning, ABC based η tuning, PSO based η tuning, FF based η tuning and GWO based η tuning, respectively. Then, in terms of Correlation at SNR = 10 dB, the proposed model is found to be better than the existing approaches GA based η tuning, ABC based η tuning, PSO based η tuning, FF based η tuning and GWO based η tuning, respectively. ESTOI of the proposed model is 0.6% better than GA based η tuning, 0.05% better than ABC based η tuning, 0.06% better than PSO based η tuning, 0.3% better than FF based η tuningand 0.15% better than GWO based η tuningat SNR = 10 dB.

Table 3 Performance evaluation of proposed model over existing for exhibition noise at varying SNR
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5.4 Performce analysis of restaurant noise

Table 4 portrays the performance evaluation of the proposed model over the existing for restaurant noise at different SNR levels. From, which SDR of the signal at SNR = 0 dB is 8.5%, 5.7%, 9.7%, 5.7% and 4.3% superior to the classical models like GA based η tuning, ABC based η tuning, PSO based η tuning, FF based η tuning and GWO based η tuning. PESQ of the proposed modelis found to be 1.2%, 0.4% better than GA and ABC, 0.2%, 0.9% and0.7% better than PSO, FF and GWO, respectively at SNR = 0 dB. Then, for SNR = 5 dB, RMSE of the proposed model exhibits superiority to the traditional models like GA based η tuning, ABC based η tuning, PSO based η tuning, FF based η tuning and GWO based η tuning by 1.54%, 0.04%, 1.2%, 0.7% and 0.09%, respectively. Further, in terms of SNR, there is an improvement of 1.8%, 0.19%, 1.6%, 0.9% and 0.7% in the proposed model over the existing model like GA, ABC, PSO, FF and GWO, respectively at SNR = 5 dB. Moreover, from SNR = 10 dB, STOI of the proposed model is 0.8%, 0.5%, 0.7%, 0.6% and 0.76% better than state-of-art models like GA based η tuning, ABC based η tuning, PSO based η tuning, FF based η tuning and GWO based η tuning. CSED at SNR = 10 dB is 6.9%, 2%, 6.7%, 5.9% and 5.13% superior to the extant modelsGA based η tuning, ABC based η tuning, PSO based η tuning, FF based η tuning and GWO based η tuning, respectively.

Table 4 Performance evaluation proposed model over existing for restaurant noise at varying SNR
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5.5 Performce analysis of station noise

From 5 represents the performance analysis of the proposed model over exiting for station noise at different SNR values as (Table 5 shows the performance evaluation of the proposed model over existing model for station noise at varying SNR).. From the table, at SNR = 0 dB, the proposed model overtakes the extant modelsGA based η tuning, ABC based η tuning, PSO based η tuning, FF based η tuning and GWO based η tuning by 2.6%, 4.5%, 1.2%, 2.7% and 1%, respectively in terms of SDR. Moreover, SEQ of the proposed model at SNR = 0 dB is 3.7%, 1.14%, 2.2%, 1.7% and 1% better than the extant models like GA based η tuning, ABC based η tuning, PSO based η tuning, FF based η tuning and GWO based η tuning, respectively. Then, for SNR = 5 dB, the proposed model is better than extant models, 5.4% by GA, 2.4% by ABC, 1.2% by PSO, 3.1% by FF and 1.6% by GWO. SEI of the proposed model at SNR = 10 dB, an improvement of 0.3%, 0.2%, 0.1%, 0.5% and 0.02% over the state-of-art models. Then, atSNR = 15 dB, the proposed model is 0,6%, 0.02%, 0.9%, 0.3% and 0.4% better than extant modelsGA based η tuning, ABC based η tuning, PSO based η tuning, FF based η tuning and GWO based η tuning in terms of STOI. In terms of CSED at SNR = 15 dB, the proposed model is 9.5%, 4.5%, 10.9%, 9.8% and 7.7% better than the traditional models GA based η tuning, ABC based η tuning, PSO based η tuning, FF based η tuning and GWO based η tuning, respectively.

Table 5 Performance evaluation proposed model over existing for station noise at varying SNR
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5.6 Performce analysis of street noise

The performance evaluation of the proposed model over the existing model for the street noise is shown in Table 6. For SNR = 0db, the proposed model exhibits an improvement of 1.7%, 0.5%, 1.3%, 1.72% and 1.7% over the classical models like GA based η tuning, ABC based η tuning, PSO based η tuning, FF based η tuning and GWO based η tuning, respectively in terms of CSED. Then, for the same SNR, the STOI of the projected model is 0.4%, 0.3%, 0.9%, 0.7% and 0.3% superior to the state-of-art models like GA based η tuning, ABC based η tuning, PSO based η tuning, FF based η tuning and GWO based η tuning, respectively. For SNR = 5 dB, ESTOI of the proposed model is 1.3%, 0.9%, 1.4%, 0.5% and 0.4% superior to the conventional models GA based η tuning, ABC based η tuning, PSO based η tuning, FF based η tuning and GWO based η tuning, respectively. The correlation of the signal for the same SNR is 0.35%, 0.017%, 0.13%, 0.2% and 0.29% superior to the existing approaches GA based η tuning, ABC based η tuning, PSO based η tuning, FF based η tuning and GWO based η tuning, respectively. Then, in case of SNR = 15 dB, the PESQ of the proposed model is 1.8%, 0.27%, 0.8%, 0.9% and 0.89% better than GA based η tuning, ABC based η tuning, PSO based η tuning, FF based η tuning and GWO based η tuning, respectively. Then, for the same SNR, the proposed model is 2.2%, 0.3%, 2.7%, 1.7% and 2.12% better than the traditional models like GA based η tuning, ABC based η tuning, PSO based η tuning, FF based η tuning and GWO based η tuning, respectively in terms of SDR at 15 dB.

Table 6 Performance evaluation proposed model over existing for street noise at varying SNR
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5.7 Statistical analysis

The evaluation of statistical analysis of the adopted and existing approaches is depicted in Fig. 5. The outcomes are provided based on the error and thus the proposed model value is lower than the existing works. On considering the results, the best value of the adopted AR-GWO scheme is 9.41%, 3.51%, 8.99%, 3.84%, and 4.53% superior to the existing GA, ABC, PSO, FF, and GWO approaches. Furthermore, in mean case scenario, the suggested approach value is 5.21%, 2.57%, 3.37%, 2.81%, and 2.33% superior to the existing GA, ABC, PSO, FF, and GWO approaches. Moreover, the median value of the AE-GWO approach is 0.013641 and it is 5.22%, 2.99%, 3.77%, 3.92%, and 2.41% better than the existing GA, ABC, PSO, FF, and GWO methods. Therefore, the effectiveness of the proposed speech enhancement model is proved.

Fig. 5
figure 5

Statistical analysis of the proposed and existing approaches

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5.8 Computational time analysis

In this section, the computational time of the proposed and existing methods is evaluated and it is depicted in Fig. 6. From the graph, the computation time of the proposed AR-GWO method is 227.86 and it is 34.07%, 43.57%, 28.86%, 38.88%, and 16.03% better than the existing GA, ABC, PSO, FF, and GWO approaches respectively. Thus, the effectiveness of the adopted AR-GWO based speech enhancement method is validated.

Fig. 6
figure 6

Computational time analysis of the proposed and existing approaches

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5.9 Practical implications

The major aim of the proposed speech enhancement is to suppress the noise in a noisy speech signal and improve the quality and intelligibility of speech. The proposed speech enhancement approach utilizes in real-time applications such as speech recognition, mobile phones, VoIP, teleconferencing systems and hearing aids.

6 Conclusion

In this paper, an optimized fuzzy wavelet neural network based speech enhancement model is proposed. In the training phase, the input noise corrupted signal was initially provided as input to both STFT-based noise estimation and NMF-based spectrum estimation for estimating the noise spectrum and signal spectrum, respectively. The obtained noise spectrum and the signal spectrum are fed as input to the wiener filter and these filtered signals are subjected to EMD.Since, tuning factorη plays a key role in wiener filter, it has to be determined for each signals, and is trained in FW-NN. Then, from the denoised signal the bark frequency is evaluated. The computed bark frequency is fed as input to the learning algorithm referred as FW-NN for detecting the suited tuning factorη for the entire input signal in Weiner filter. The AR-GWO is employed for proper tuning of the tuning factor η referred as tuned tuning factor (ηtuned). In the testing phase, the training is accomplished initially and from which the tuning factor is gathered for each of the relevant input signal. Then, the properly tuned tuning factor (ηtuned) from FW-NN is fed as input to EMD via adaptive wiener filter for decomposing the spectral signal and the output of EMD is denoised enhanced speech signal.Theresultant acquired is compared over the existing models in terms of various measures. In case of street noise, at SNR = 0db, the proposed model exhibits an improvement of 1.7%, 0.5%, 1.3%, 1.72% and 1.7% over the classical models like GA based η tuning, ABC based η tuning, PSO based η tuning, FF based η tuning and GWO based η tuning, respectively in terms of CSED. Thus, the effectiveness of the work is validated via the result analysis. However, in statistical analysis, the standard deviation metric value is not better than the existing ones. Hence, in the future work, we enhanced our proposed work by utilizing the recent optimization algorithms and validate the work in real-time applications.