1 Introduction

Speech is the earliest method of communication between persons, since it carries imperative information about the identity, gender, emotional state, language more than messages or words. The main objective of implementing artificial intelligence in the speech processing field is to create a system that is capable of identifying the person from his voice (Abimbola 2007). A speaker recognition (SR) system comprises two step: feature extraction and classification. Feature extraction is the procedure of finding out the essential characteristics of the speaker from his speech signals, while rejecting redundancy. It abbreviates the speech signal into a small feature vector that is sufficient for good speaker distinction (Reynolds 2002). There are various feature extraction methods that have been used such as linear prediction coefficients (LPCs), cepstral linear prediction coefficients (CLPCs), and mel frequency cepstral coefficients (MFCCs). The MFCCs method is the most prevalent feature extraction method. It is based on low-level features (Das 2014).

The classification stage includes two sub-classes: training and testing. In the training, after extracting features from the speech signals, a model for each speaker is made and kept in the database. In the testing phase, when an unknown speaker is enrolled to the system, the corresponding model is matched to those stored in the database and a decision is made. An Artificial Neural Network (ANN) can be used for classification (Hossain et al. 2007).

Extraction of the features in the existence of noise is the basic problem in the SR systems, and it is still a challenging task. The circumstantial noise and channel mis-representations are the main causes of distortion in speech signals. A prior step in SR is introduced to lessen distortion to achieve better performance of the system. There are many speech enhancement methods that have been used. Spectral subtraction is a well-known method that was used by Kaladharan in 2014 for speech denoising (Kaladharan 2014). In 2015, El-Moneim et al. 2015 used a hybrid method based on EMD and spectral subtraction with some transform domains and achieved the best accuracy with the DWT.

In the framework presented in this paper, two proposed filtering methods: Adaptive Noise Canceller (ANC) and SG filter are used in a pre-processing step in the SR systems with features extracted from transform domains. The ANC (Shanmugam et al. 2013) is a filter that requests two inputs: a primary input containing the distorted version of the signal and a reference input containing noise correlated with the primary signal. The reference signal is adaptively filtered and subtracted from the primary signal to get the enhanced version of the signal. The obtained results show good performance of the SR system with the ANC, although it has some limitations such as the requirement of prior knowledge about the noise and the channel. The SG filter is another filter that is used for smoothing prior to the identification process. A comparison is presented to show how the system is affected by the different speech enhancement methods.

The paper is structured as follows. In Sect. 2, a summary of the SR system and its main steps is given. The proposed SR system with enhancement methods is presented in Sect. 3. The used pre-processing methods in the SR system are investigated in Sect. 4. The performance of the suggested methods with the SR system is substantially verified by experimental results and compared with those of the spectral subtraction and the hybrid methods with the SR system in Sect. 5. Finally, in Sect. 6, the conclusion is presented.

2 Speaker recognition system

As mentioned above, the SR system aims at recognizing the speaker based on discriminative attributes extracted from the speech signals (Campbell 1997). The SR has two stage: feature extraction stage and classification stage as in Fig. 1.

Fig. 1
figure 1

Speaker recognition system

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2.1 Feature extraction

Feature extraction is the main phase of the SR system. It retains the most important information, while discarding redundancy. The MFCCs are the features used in this paper with polynomial coefficients.

2.1.1 Mel frequency cepstral coefficients (MFCCs)

The main procedure of estimating the MFCCs is shown in Fig. 2. The MFCCs are extracted with short-term analysis of speech (Muda et al. 2010; Abd El-Samie 2011). The speech signal is sub-divided into short segments called frames. At first, the speech is pre-emphasized to promote high-frequency components. Framing and windowing is then performed. After that, discrete fourier transform (DFT) is taken to get the speech spectrum. The speech signal spectrum is warped on Mel scale to approximate the human hearing perception for speech, which is linear below 1 kHz and logarithmic above 1 kHz.

Fig. 2
figure 2

MFCCs extraction procedure

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$$ Mel\left( f \right) = 2595\log_{10} \left( {1 + \frac{{f_{linear} }}{700}} \right) $$
(1)

Finally, the DCT is performed on the log of the warped spectrum to get the MFCCs:

$$ c_{g} = \sqrt {\frac{2}{{N_{f} }}} \mathop \sum \limits_{m = 1}^{{N_{f} }} \log \left( {\hat{S}\left( m \right)} \right)\cos \left[ {\frac{g\pi }{{N_{f} }}\left( {m - 0.5} \right)} \right] $$
(2)

where cg is the MFCC, \( {\hat{S}\left( m \right)} \) is the mth Mel filter output, g = 0,1,2,…,G−1, and G is the number of MFCCs. It is chosen between 12 and 20. The Nf is the number of Mel filters. The first few coefficients are taken as they give the most specific information about the speaker.

2.1.2 Polynomial coefficients

The MFCCs are not sufficiently robust in noisy environments, and hence polynomial coefficients are added to reduce sensitivity to mismatch amongst training and testing data. Polynomial coefficients are calculated to represent the slope and curvature of the time waveform for each cepstral coefficient. Polynomial coefficients are computed by expanding the time waveform by orthogonal polynomials. Two orthogonal polynomials are used (Furui 1981):

$$ p_{1} \left( i \right) = i - 5 $$
(3)
$$ p_{2} \left( i \right) = i^{2} - 10i + 55/3. $$
(4)

The polynomial coefficients can be computed as;

$$ a_{g} \left( t \right) = \frac{{\mathop \sum \nolimits_{i = 1}^{9} p_{1} \left( i \right)c_{g} \left( {t + i + 1} \right)}}{{\mathop \sum \nolimits_{i = 1}^{9} p_{i}^{2} \left( i \right)}} $$
(5)
$$ b_{g} \left( t \right) = \frac{{\mathop \sum \nolimits_{i = 1}^{9} p_{2} \left( i \right)c_{g} \left( {t + i + 1} \right)}}{{\mathop \sum \nolimits_{i = 1}^{9} p_{2}^{2} \left( i \right)}} $$
(6)

where ag(t) and bg(t) are the slope and curvature of the gth cepstral coefficient in tth frame, respectively. The vectors containing cg, ag and bg are concatenated together to shape one feature vector for each frame.

2.2 Classification

Classification has two stages: training and testing. In the training, a model for each registered speaker is made using the MFCCs, and then it is stored in the database. In testing, when a new speaker enters the system, similar features are extracted and associated with those already existing in database, and a decision is then taken. The used classifier here is an ANN as in Fig. 3. The ANN is an intelligent mathematical algorithm that entails three main parts: input layer, middle or hidden layer(s), and output layer. Firstly, the input layer receives the system input. The hidden layer acts as the innermost layer of the ANN. It consists of many units called neurons. Inside the neurons, the main mathematical calculations are performed to process the inputs and submit the appropriate outputs (Dreyfus 2005; Sharma et al. 2005).

Fig. 3
figure 3

ANN structure

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A neuron in the hidden layer receives and sends a group of values from the previous layer to the subsequent layers. The ANN entails a huge number of simple processing units called neurons that influence each other via connections called weights. Every layer consists of cells, and these cells are linked by the weights for the information to flow from the input layer to the output layer through the hidden layer. The ANN is trained by adjusting the weights between neurons via the learning rules (Dreyfus 2005; Sharma et al. 2005; Galushkin 2007; Love et al. 2004).

In Fig. 3, if x is an m-element vector containing x1, x2, x3, …., xn as the neuron inputs, and the weights are w1,1, w1,2, w1,3, …..,w1,n, the input of the net v will be:

$$ v_{j} = \mathop \sum \limits_{i = 1}^{m} \mathbf{x}_{i} \mathbf{w}_{ji} + b $$
(7)

where b is the bias, and wji is the weight from the unit i to unit j. The input is then the argument of the activation function. Once the net input is computed, the output activation is computed as a function of vj.

$$ y_{j} = f\left( {v_{j} } \right) $$
(8)

where f is the activation function, y is the output of the neuron, b is the bias that performs affine transformation of the output. The training algorithm sets the weights by minimizing the sum of the squared error between the desired output dn and the real output yn of the output neurons specified by:

$$ E = \frac{1}{2}\mathop \sum \limits_{n = 1}^{N} \left( {d_{n} - y_{n} } \right)^{2} $$
(9)

where N is the number of output neurons. The error back-propagation algorithm is used for that purpose until the error is diminished via an iterative method.

3 Proposed SR system with pre-processing

Speech enhancement is a preliminary step in the SR system, since the speech signal is sometimes contaminated with noise. The enhancement process aims at reducing or removing this unpleasant noise. Different filters are used such as ANC and SG filters. Their performance is compared with the well-known spectral subtraction method and also with the hybrid method. For robust extraction of features, discrete transforms are used in the SR system. The DWT, DCT, and DST are investigated for this purpose. The DWT decomposes the speech into approximation and details. The effect of noise is less on the approximation. On the other hand, the DCT and the DST have a brilliant energy compaction property (Nasr et al. 2018). The block diagram of the proposed SR system with pre-processing is illustrated in Fig. 4.

Fig. 4
figure 4

Proposed SR system with pre-processing

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4 Enhancement methods

4.1 Spectral subtraction method

Spectral subtraction is a common enhancement method. Its general idea is the estimation of noise during the non-activity of speech and the subtraction of this estimated noise from the noisy speech to get an estimate of the original speech. If s(k) is the original speech signal, y(k) is the degraded speech signal and v(k) is the noise, then (Karam 2014; Pawar 2013);

$$ y\left( k \right) = s\left( k \right) + v\left( k \right) $$
(10)

Taking the fast Fourier transform (FFT), we get

$$ Y\left( \omega \right) = S\left( \omega \right) + V\left( \omega \right) $$
(11)

where \( {{Y(}}\omega ) { = }\left| {{{Y}}(\omega )} \right|\mathop e\nolimits^{{j\theta_{y} }} \) and \( {{V(}}\omega ) { = }\left| {{{V(}}\omega )} \right|\mathop e\nolimits^{{j\theta_{y} }} \). Hence,

$$ \hat{S}\left( \omega \right) = \left\{ {\left| {Y\left( \omega \right)} \right| - \left| {V\left( \omega \right)} \right|} \right\}e^{{j\theta_{y} \left( \omega \right)}} $$
(12)

where \( \hat{S}(\omega) \) is the expected spectrum of the clean speech signal, and V(ω) is estimated by averaging of spectra during non-active periods of speech. Taking the inverse FFT, an estimate of the original signal is obtained. This method is efficient, but it introduces musical noise (Evans 2006).

4.2 Hybrid method with EMD and spectral subtraction

The EMD is a signal decomposition method that decomposes the signal to a number of components, called intrinsic mode functions, which must satisfy the criteria of a symmetric envelope with zero mean (Rilling et al. 2003). The EMD is performed in time domain. The IMFs are deduced from the signal itself. The method of deriving the IMFs is called sifting, and its steps are as follows (Kim and Oh 2009; Alotaiby et al. 2014):

  1. 1.

    For a signal s(t), all local extrema (minima, maxima) are specified.

  2. 2.

    Interpolation is performed between minima to form the lower envelope l(t), and between maxima to form the upper envelope u(t).

  3. 3.

    The local mean m(t) of the envelopes, m(t) = \( \frac{u\left( t \right) + l\left( t \right)}{2} \), is estimated.

  4. 4.

    The mean m(t) is subtracted from the original signal s(t) to get the primary IMF,

    $$ h\left( t \right) = s\left( t \right) - m\left( t \right) $$
  5. 5.

    The primary IMF h(t) must satisfy the condition that the numbers of extrema and zero crossings are the same or vary by one at most and the zero mean condition. If this is not achieved, steps 1 to 4 are repeated on h(t) as the basis input till the resultant IMF satisfies the conditions.

  6. 6.

    The residual is estimated between the IMF component ci(t) and the signal as \( r\left( t \right) = s\left( t \right) - c_{i} \left( t \right) \).

  7. 7.

    Steps 1 to 6 are repeated on the residue r(t).

  8. 8.

    Sifting process stops, when the residual has only one extremum.

After obtaining a number of IMFs, spectral subtraction is performed on each individual component, and then the original or enhanced speech signal is reconstructed by the superposition of the enhanced IMFs and the residual as shown in Fig. 5. This method decreases the musical noise owing to the averaging effect (El-Moneim et al. 2015).

Fig. 5
figure 5

Flowchart of the hybrid method

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4.3 Proposed adaptive noise canceller (ANC)

Noise is any objectionable distortion in the speech signal that causes reduction in the signal strength, and difficulty in hearing and reduces intelligibility of the speech. Consequently the performance of the SR system will be reduced. Removal of the noise corrupting the speech signal is the greatest shared problem in speech processing techniques. The ANC is a noise reduction filter whose parameters are changed to meet the required performance. The used criterion is the minimization of the MSE (Shanmugam et al. 2013) by using a normalized least mean square (NLMS) algorithm. Two input signals are applied to the ANC simultaneously: the contaminated signal as the primary input and the noise as the reference signal. The reference signal is filtered to get an approximation of noise corrupting the speech signal (Jafari and Chambers 2003). This estimated noise is subtracted from the corrupted speech signal to obtain an error signal, which is fed back to the adaptive filter to adjust the filter coefficients through the NLMS algorithm to minimize total MSE (Dhubkarya 2012).

If s is the original speech signal and no is the noise corrupting the speech, then the degraded speech will be,

$$ s_{n} = s + n_{o} $$
(13)

The error signal will be,

$$ e = s_{n} - y $$
(14)

where y is the filter output

The MSE will be,

$$ E\left[ {e^{2} } \right] = E\left[ {\left( {s_{n} - y} \right)^{2} } \right] $$
(15)
$$ E\left[ {e^{2} } \right] = E\left[ {s^{2} } \right] + E\left[ {\left( {n_{o} - y} \right)^{2} } \right] + 2E\left[ {s\left( {n_{o} - y} \right)} \right] $$
(16)

no is uncorrelated with s,

$$ E_{{min}} \left[ {e^{2} } \right] = E\left[ {s^{2} } \right] + E_{{{\text{min}}}} \left[ {\left( {n_{o} - y} \right)^{2} } \right] $$
(17)

The term (no−y) is minimized to lower the MSE (Anand 2013). The error is the improved version of the speech signal, which is used as the input to SR system as shown in Fig. 6.

Fig. 6
figure 6

Adaptive noise canceller

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Despite the ability of the ANC to remove noise from a noisy speech signal, it has some limitations such as the requirement of prior information about the signal and the noise characteristics.

4.4 Proposed Savitzky Golay (SG) filtering approach

The SG filter is a smoothing polynomial digital filter (Schafer 2011). For the signal samples, approximated coefficients are found during fitting with the help of the neighboring samples of the windowed signal. The SG filter is thought to be a generalization of the moving average filter that better preserves the features of the signal such as the height and width of peaks (Awal et al. 2011). The data samples are windowed with equal numbers of samples to the right and the left of the central sample. This central sample is replaced by the approximated coefficient, and then the window is lifted by one sample to the right to fit the next central sample. This process is repeated for the whole data samples (Guiñón et al. 2007; Shajeesh et al. 2012).

The output signal of the SG filter is calculated by:

$$ g_{m} = \mathop \sum \limits_{{k = -n_{L} }}^{{n_{R} }} c_{{k + n_{L} }} .s_{m + k} $$
(18)

where gm is the output waveform, \( c_{{k + n_{L} }} \) is the SG filter coefficients, nL is the number of samples to the left of the data sample m, and nR is number of the samples to the right of data sample m,

$$ {\mathbf{c}}_{{M + n_{L} }} = \left[ {\left( {(\mathbf{A})^{T} .{\mathbf{A}}} \right)^{ - 1} .\left( {(\mathbf{A})^{T} } \right)} \right]_{{0,M + n_{L} }} $$
(19)

where

$$ M = - \,n_{L}, \cdots, n_{R} $$
(20)

The SG filtering is simple with high computational speed, but some of the first and last data samples cannot be smoothed.

5 Simulation results

A database of 8 speakers has been created. Each speaker repeated a certain sentence 10 times to obtain 80 samples of speech. Feature vectors consisting of 13 MFCCs and 26 polynomial coefficients have been created from these samples. 39 coefficients (13 MFCCs +26 polynomial coefficients) for each frame are fed to train the ANN. In the testing phase, each speaker is asked to say the same sentence, and additive white Gaussian (AWGN) is added to the speech signal with different signal-to-noise ratios (SNRs) from 0 to 30 dB. Features are extracted from the noisy speech signals and fed to the ANN for matching. Spectral subtraction is applied to reduce the noise and features are extracted from different transforms (DST, DCT, DWT) of the resultant enhanced speech signals. Moreover, the speech signals are also pre-processed with the hybrid method. Comparisons are presented between different scenarios: feature extraction from noisy speech, feature extraction after spectral subtraction, feature extraction after the hybrid method, feature extraction after the ANC and feature extraction after the SG filter. The recognition rate defined as the ratio between the number of success identifications and the total number of identification trials is estimated. The results are given in Figs. 7, 8, 9, 10 and 11. It is clear that the proposed methods achieve good results, and the DCT is a proper candidate with filtering methods in the cases of AWGN, while the DWT is the optimum transform for feature extraction, when the hybrid method and the spectral subtraction method are used.

Fig. 7
figure 7

Recognition rate vs. SNR from different transforms of the noisy speech

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Fig. 8
figure 8

Recognition rate vs. SNR from different transforms after spectral subtraction method

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Fig. 9
figure 9

Recognition rate vs. SNR after the hybrid method

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Fig. 10
figure 10

Recognition rate vs. SNR from different transforms after ANC method

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Fig. 11
figure 11

Recognition rate vs. SNR from different transforms after SG filter

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6 Conclusion

This paper presented some filtering methods for noisy speech to be used for robust SR systems. The suggested filters are the ANC and the SG. The MFCCs as features and the ANN as a classifier have been considered in the paper. Simulation results have shown a great success of the proposed filters compared to the spectral subtraction and the hybrid methods. In addition, the suggested methods, when applied prior to SR, lead to a noticeable improvement of performance. For the spectral subtraction method and the hybrid method, the DWT is the optimum transform for feature extraction, while the DCT is the optimum transform in the cases of ANC and SG filtering.