Digital Signal Processing 22 (2012) 786–794
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Digital Signal Processing www.elsevier.com/locate/dsp
Time domain signal enhancement based on an optimized singular vector denoising algorithm Hamid Hassanpour a,∗ , Amin Zehtabian b , S.J. Sadati c a b c
School of Information Technology and Computer Engineering, Shahrood University of Technology, Shahrood, Iran Faculty of Electrical and Computer Engineering, Tarbiat Modares University, Tehran, Iran Department of Computer and Electrical Engineering, Noshirvani University of Technology, Babol, Iran
a r t i c l e
i n f o
Article history: Available online 4 April 2012 Keywords: Time series Noise reduction Singular values Singular vectors Savitzky–Golay filter Genetic algorithm
a b s t r a c t This paper presents a new time domain noise reduction approach based on Singular Value Decomposition (SVD) technique. In the proposed approach, the noisy signal is initially represented in a Hankel Matrix. Then SVD is applied on the Hankel Matrix to divide the data into signal subspace and noise subspace. Since singular vectors are the span bases of the matrix, reducing the effect of noise from the singular vectors and using them in reproducing the matrix leads to considerable enhancement of information embedded in the matrix. The noise-reduced singular vectors from the signal subspace are utilized to reconstruct the data matrix. This matrix is finally used to obtain the time-series signal. The results of applying the proposed method to different synthetic noisy signals indicate a better efficiency in noise reduction compared to the other time series methods. © 2012 Elsevier Inc. All rights reserved.
1. Introduction Noise is an undesired phenomenon that often appears in signals and systems. The interference of noise may cause some troubles in system analysis. As a result, signal enhancement and noise reduction have wide applications in signal processing and they are often employed as a pre-processing stage in various applications [1–4]. There are several methods for noise reduction that can be categorized into time, frequency and time–frequency domains. Some of the existing methods reduce the noise by considering prior assumptions [5–7]. In other words, these methods are suitable only for specific applications and conditions. For example, when using a typical Low-Pass Filter (LPF), it is assumed that the noise is placed at the high frequency regions of the noisy signal. It means that the frequency bands of noise and clean signal must be distinct. This assumption may not be acceptable in various conditions and may restrict its applications. In addition, the low-pass filters may cause shifting the signal in time domain and changing the shape of the signal slightly. This problem is illustrated in Fig. 1, where a Butterworth LPF filter is applied on a noisy signal. Although the noise was removed, the shape of the enhanced signal is not as same as the original noise-free signal. The Wiener filter is another existing method for reducing the effect of noise that is widely used by researchers and is utilized in technical applications. This filter is usually able to reduce the noise
*
Corresponding author. E-mail addresses:
[email protected] (H. Hassanpour),
[email protected] (A. Zehtabian). 1051-2004/$ – see front matter © 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.dsp.2012.03.009
of the signal. However, the amount of noise reduction is often accompanied by signal degradation. In other words, Wiener filter can be used to reduce the noise where the SNR is high enough (usually higher than 4). When SNR for a signal is low, using Wiener
Fig. 1. The effect of using a typical LPF for reducing the noise of a signal: top solid line represents the clean signal, dotted line shows the Butterworth low-pass filtered signal, and bottom solid line indicates the noisy signal with SNR = 2.
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filter may not be a suitable solution and may just transform the noise from one form to another [8]. This is a discouraging factor in choosing Wiener filter for noise reduction. We have recently developed a Time–Frequency (T–F) based approach for reducing the noise from a signal [9]. This approach is based on applying the SVD operation to the matrix associated with the time–frequency representation of the signal. Indeed, this technique separates noise subspace and signal subspace using singular values of the data matrix as criteria for subspace division. Although the prominency and the noticeable performance of the T–F based technique have been clearly proved yet, but there are still two problems with them which make them not recommended in some specific noise reduction applications. The main deficiency is the computational complexity of the approach. Indeed, high computational time is required for representing signal in the time– frequency domain. In addition, some of the time–frequency distributions, such as B-distribution [10], cannot be synthesized to the time series. Recently, time domain based approaches for noise reduction have received a considerable attention among researchers [11,12]. These techniques often construct a time data matrix of the noisy signal. In this paper, we utilize the Hankel matrix for constructing the data matrix and then this data matrix is divided into signal subspace and noise subspace using the SVD-based approach introduced in [13]. In this article the Savitzky–Golay low-pass filter is utilized to reduce noise from the singular vectors. The noisereduced singular vectors with the singular values of the signal subspace are used to reconstruct the matrix. Subsequently, this noisereduced matrix is used to extract the time series, representing the noise-attenuated signal. We show that the proposed method has a significantly better performance in reducing noise from a noisy signal, in comparison with other existing time-domain based approaches. 2. Noise reduction In this paper, we suppose that the clean signal has been corrupted by an additive white Gaussian noise:
Xn = X s + W n
(1)
where X n , X s and W n respectively denote the noisy signal, clean signal and additive white Gaussian noise. For X n (i ), i = 1, 2, . . . , n representing the noisy signal, the Hankel matrix is constructed as follows:
⎡ X (1) n ⎢ Xn (2) H =⎢ ⎣ .. . Xn ( L )
X n (2) X n (3)
.. .
⎤ ... Xn ( K ) . . . X n ( K + 1) ⎥ ⎥ .. ⎦ .
X n ( L + 1) . . .
(2)
Xn ( N )
(3)
where H n , H s and H wn are respectively the Hankel representations of the noisy signal, original clean signal and the additive white Gaussian noise. Generally, the singular value decomposition of a P × Q real matrix H is of the form:
H = UΣV T
Fig. 2. Normalized singular values of the Hankel matrix constructed from a given noisy signal.
S 0 Σ= 0 0
(5)
The diagonal elements of matrix S are sorted in a decreasing order. Therefore, S = diag(σ1 , σ2 , . . .) with components such that σ1 σ2 · · · > 0. For further information, see Refs. [14,15]. 2.1. Noise subspace subtraction To enhance the information embedded in the Hankel matrix, we propose dividing the data matrix into signal subspace and noise subspace using the singular value decomposition. Since singular vectors are the span bases of the matrix, reducing the effect of noise from the singular vectors and using them in reproducing the matrix, leads to further enhancement of information embedded in the matrix. Mathematically, the subspace separation of the noisy signal can be expressed as below:
T
Hn = U Σ V = ( U s
Un )
ΣS
0
0
Σn
VTS VTn
(6)
then
According to the assumption of additive noise, we can also write the following equation for the Hankel matrices:
H n = H s + H wn
787
(4)
where U P ×r and V Q ×r are orthogonal matrices and their columns are called the left and right singular vectors respectively. The matrix Σ is a r × r diagonal matrix of singular values and consequently can be expressed as:
Hn = Us ΣS VTS + Un Σn VTn
(7)
where ΣS and Σn respectively represent the singular values which belong to the clean signal subspace and noise subspace. From Eqs. (3) and (7), we can deduce that
Hs = Us ΣS VTS
(8)
and
Hwn = Un Σn VTn
(9)
As can be inferred from above equations, we must determine a threshold point in the matrix Σ where lower singular values from that point can be categorized as the singular values of the noise subspace and hence should be set to zero. To determine this point, we plot the singular values of Σ matrix for a given noisy signal with respect to their indexes (see Fig. 2). A break point can be seen clearly in Fig. 2, where slope of the curve changes drastically. Thus this threshold point can be determined by calculating derivation of the curve in each point
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Fig. 3. The effect of applying Savitzky–Golay filter to the singular vectors of a given noisy signal. From top to bottom: clean signal, noisy signal, the result of using SVD and filtering the noisy singular values per se, the result of filtering the singular vectors as well as singular values.
and finding the place where the maximum change between a two consequent derivation values is occurred. Since the noise subspace is mainly related to those singular values that are lower than this threshold point, we suggest setting these singular values to zero for space division [16]. To clarify the proposed method, we briefly describe the concept of matrix rank. The rank of a matrix can be directly determined by the number of nonzero singular values from its SVD. For example, when the clean data matrix is symmetric (Toeplitz) or per symmetric (Hankel) and the data are sinusoidal or complex exponential, with no additive noise, then the rank is equal to twice the number of real sinusoid or the number of complex exponentials presented in data [17]. But a noisy signal has much more nonzero singular values which belong to the noise subspace. These redundant singular values must be set to zero for noise reduction.
Fig. 4. The effect of using different values as Savitzky–Golay window size. From top to bottom: clean signal, noisy signal and the enhanced signals after applying n w = 5, n w = 15, n w = 35 and n w = 45, as the window sizes of the Savitzky–Golay filter.
2.2. Enhancing the singular vectors Since by merely filtering the singular values, some noisy data will still be available in the signal subspace, we propose filtering the singular vectors for further noise reduction. In this study, singular vectors are treated as time-series. To reduce the effect of noise from them we use the Savitzky–Golay smoothing filter [18]. In this approach a polynomial of degree d is fitted to n w consecutive data points from the time series, where n w is the frame or window size. Filtered singular vectors can be obtained as follows:
Uie = F Ui ,
i = 1, . . . , P
(10)
Vie = F Vi ,
i = 1, . . . , Q
(11)
where F(.) is the Savitzky–Golay smoothing filter. Fig. 3, illustrates the effects of applying the Savitzky–Golay filter to the noisy singular vectors of an arbitrary linear FM signal. In this experiment we have plotted the noisy signal, the enhanced signal after eliminating the lower singular values and the result of filtering the singular vectors using the Savitzky–Golay filter. The differences be-
Fig. 5. The effect of using different values as Savitzky–Golay polynomial degree, when the window size is set to 15. From top to bottom: clean signal, noisy signal and the enhanced signals using d = 5, d = 4, d = 3 and d = 2, as the degrees of the Savitzky–Golay filter.
tween the two signals can clearly demonstrate the importance of filtering the singular vectors. The performance of our proposed noise reduction approach depends on some parameters including the number of rows in the data Hankel matrix, the degree and the frame size of the Savitzky– Golay filter. As illustrated in Figs. 4 and 5, choosing various frame
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Fig. 6. Comparing the performance of the three different noise reduction techniques on multi-component signals with SNR = 5 dB (a) and SNR = 2 dB (b). In each sub-plot, from top to bottom: clean signal, noisy signal, output of LPF, Wiener filter, and the proposed approach.
sizes and polynomial degrees for Savitzky–Golay filter, leads to achieving different results. As mentioned before, there are three parameters affecting the performance of our proposed method. They are the number of rows l, in the data matrix, the degree d and window size n w of Savitzky–Golay filter. To set these parameters properly, we define a cost function and use the genetic algorithm to minimize the cost. Once a filter is applied to a signal, the level of sudden changes in successive samples is reduced. On the other hand, the enhanced signal should still be similar to the noisy signal after filtering. Hence, the following cost function is defined by the authors and offered for optimally tuning the parameters of the filter:
xe (k) − xn (k) J (l, d, n w ) = (1 − α ) k
xe (k + 1) − xe (k) +α
(12)
k
where xn , xe , and k represent the noisy signal, enhanced signal, and their sample number respectively. In this equation, α is a factor determining smoothness of the enhanced signal and must be between 0 and 1. In this paper it is empirically set to 0.3. At the right side of the above equation, the first term indicates the distance between the enhanced signal and the noisy signal; and the second term indicates the smoothness of the enhanced signal. We minimize this cost function using genetic algorithm. The genetic algorithm is an iterative algorithm which randomly chooses some values from the search space in each repetition. In this algorithm, the enhanced signal is initially computed using samples of the three parameters. Then the cost function in (12) is computed. In each repetition, those parameters which minimize the cost function will be stored as the optimum parameters. After several iterations, the final optimal parameters will be achieved. For more detailed information refer to reference [19].
The enhanced data matrix is then obtained using:
He = Ue Σ s VTe
(13)
where Ue and Ve are the noise-reduced versions of left and right singular vectors and Σ s is the enhanced singular values matrix after filtering the redundant singular values. The matrix He is supposed to be the best approximation of H s from Eq. (3). Finally, the enhanced signal Xe is extracted as follows:
Xe = He (1, 1) . . . He (1, Q), He (2, Q) . . . He (P, Q)
(14)
3. Performance evaluation To show the performance of the proposed approach, several experiments have been carried out on multi-component periodic signals as well as linear FM (LFM) signals corrupted by additive white Gaussian noise. The results are described in the following. 3.1. Experiment 1: Multi-component periodic signal Let
X (t ) = 0.39 sin sin(2π f t ) + 0.75 cos cos 2π (7 f )t
+ 0.93 sin sin 2π (2 f )t + 0.69 cos cos 2π (4 f )t + η(t ) (15)
represents a multi-component signal containing noise (η(t )). In this experiment, we use this signal with f = 23 Hz, and the sampling frequency f s is set to 2.5 kHz. Performance of the proposed method is compared with that of Butterworth LPF and Wiener filter. We set the Butterworth filter with pass frequency, stop frequency, pass-band ripple, stop-band ripple, respectively equal to 7f,
790
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Fig. 7. Considering PSD of the signals on 100 realizations to comparing the performance of the three different noise reduction techniques on multi-component signals with SNR = 5 dB (a) and SNR = 2 dB (b).
7f + 10, 2 dB, and 8 dB, where f = 23 Hz. The optimal number of iterations for the Wiener filter applied in these experiments is determined experimentally and equals 50. Indeed this parameter has been varied in an extensive range to obtain the best results for Wiener filtering. The signal in Eq. (15) was filtered using the three different approaches and the time domain representations of the clean, noisy and enhanced signals are shown in Fig. 6 for 5 dB and 2 dB signal to noise ratio (SNR). As the figure shows, the low-pass filter can reduce the noise, but with the cost of shifting and slightly changing the shape of the signal. This deformation is proportional to the filter window length. Although there are no such deficiencies in using the Wiener filter, the noise attenuation level is less than the proposed method. In addition, Wiener filter is not able to reduce the noise in a signal with a low SNR. To further evaluate the performance of the three different denoising approaches, it is desired to compare power spectrum density (PSD) and the phase response of the clean and enhanced multi-component signals. Hence, we have computed the PSDs of these signals using the Welch type estimation methods on 100 realizations and plotted the phase response of each signal. The results are shown in Figs. 7 and 8 respectively. As the power spectral densities show, the proposed method is able to retrieve the frequency information of the signal better than both of Wiener and LPF methods, especially where the SNR is low. On the other hand, respect to the phase plots of the signals, the proposed method can also retain the phase characteristics of the original clean signal.
with the pass frequency, stop frequency, pass-band ripple, stopband ripple, respectively equal to 150 Hz, 200 Hz, 2 dB, and 8 dB. The optimal number of iterations for the Wiener filter is experimentally set to 22. The time domain representations of the clean, noisy and enhanced signals are shown in Fig. 9. Similar to the previous experiment, the LPF shifts and deforms the signal. But deformation amount is more considerable compared to the previous experiment. This event seems reasonable, because the noise and the clean signal have more overlap in frequency regions for a LFM signal. Hence by filtering the high frequency bands of noisy signal, some noises may still be available in the filtered signal. On the other hand, Wiener filter has a good performance where SNR is not less than about 4 dB. Indeed, where the energy of noise is significant and SNR is low, the performance of Wiener filter drops drastically. As can be seen in the figure, the proposed method is able to reduce the noise from the noisy nonstationary signal, even where the SNR is low and the frequency range of the signal is wide. The PSD and phase plot of the linear FM signals on 100 realizations have been plotted in Figs. 10 and 11. It can be noticed that in the case of LFM signals, the LPF filter may converge to the clean signal’s PSD curve at very low power/frequency magnitudes (in dB) as good as the proposed method. But at higher power/frequency magnitudes, the proposed method clearly demonstrates its prominence in retrieving the frequency components of the clean signal, in comparison with other approaches.
3.2. Experiment 2: LFM signals
3.3. Experiment 3: Applying the Savitzky–Golay filter directly to the time domain noisy signal
The three approaches in the previous experiment are applied on linear FM signals considered as nonstationary signals. In this experiment, the LFM signal’s frequency begins from 1 Hz and terminates at 150 Hz. The sampling frequency is f s = 4 kHz and the number of samples is N = 600. The Butterworth filter is set
Now we directly use the Savitzky–Golay smoothing filter for reducing the noise of the multi-component and LFM signals in time domain. In this technique we obtain the optimum parameters of the smoothing filter by the genetic algorithm. In the Savitzky– Golay smoothing approach, each value of the series is replaced
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Fig. 8. Phase plot of the signals on 100 realizations to comparing the performance of the three different noise reduction techniques on multi-component signals with SNR = 5 dB (a) and SNR = 2 dB (b).
Fig. 9. Comparing the performance of the three different noise reduction techniques on LFM signals with SNR = 5 dB (a) and SNR = 2 dB (b). In each sub-plot, from top to bottom: clean signal, noisy signal, output of LPF, Wiener filter, and the proposed approach.
with a new value which is obtained from a polynomial fit to 2k + 1 neighboring points. The parameter k is equal to, or greater than the order of the polynomial. Since this approach does not utilize the SVD operator, consequently there is no subspace division. Figs. 12 and 13 respectively illustrate the results of applying the Savitzky–Golay filter, directly to the noisy multi-component and LFM signals. As can be inferred from these figures, the proposed
technique is more prominent in noise reduction compared to the use of Savitzky–Golay filter without SVD. 3.4. Experiment 4: Monte Carlo simulation In this section, the performance of the pre-mentioned approaches is compared using the two most well known criteria, SNR
792
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Fig. 10. Considering PSD of the signals on 100 realizations to comparing the performance of the three different noise reduction techniques on LFM signals with SNR = 5 dB (a) and SNR = 2 dB (b).
Fig. 11. Phase plot of the signals on 100 realizations to comparing the performance of the three different noise reduction techniques on LFM signals with SNR = 5 dB (a) and SNR = 2 dB (b).
and Euclidean distance. In the first criterion, the SNR of the enhanced signal is computed and then compared with the one in the noisy signal. SNR is a term for the ratio between the power of signal and the power of background noise:
SNR =
P Signal P Noise
(16)
The signal-to-noise ratio also can be expressed in the logarithmic decibel scale as below:
SNR(dB) = 10 log10
P Signal P Noise
(17)
As the second criterion, the Euclidean distance between the clean and the noisy signal is computed and then compared with the distance between the clean and the enhanced signals. The results of the Monte Carlo simulation on 100 realizations of different SNR values for the multi-component and the LFM signals are shown in Table 1. In this table, the proposed approach is compared with four other noise reduction techniques including
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Fig. 12. The effect of applying the Savitzky–Golay smoothing filter directly to a given multi-component noisy signal. From top to bottom: clean signal, noisy signal, the result of using the Savitzky–Golay smoothing per se, the result of applying the proposed method.
the low-pass filtering, Wiener filtering, use of Savitzky–Golay filter for directly filtering the time domain noisy signal and use of SVD without filtering the singular vectors. The two latter techniques are respectively abbreviated to SG filtering and pure SVD in the table. Indeed, the characteristics of the Savitzky–Golay filter used in the SG filtering method are optimized by the genetic algorithm. The threshold point in the pure SVD method is also determined by the technique which was introduced in the Noise Subspace Subtraction section. The gained results attest that the proposed approach which reduces the noise of the singular values as well as the singular vectors has a better performance for this noise condition compared to the other noise reduction approaches. The results in this table indicate that while the SNR of a noisy
793
Fig. 13. The effect of applying the Savitzky–Golay smoothing filter directly to a given LFM noisy signal. From top to bottom: clean signal, noisy signal, the result of using the Savitzky–Golay smoothing per se, the result of applying the proposed method.
signal highly affects the performance of the Wiener filter, the proposed method can enhance the signal even at the presence of very energetic noise (SNR < 2 dB). In addition, since the white noise has a wide frequency activity; LPFs cannot effectively reduce the noise at their pass-band frequency area. 4. Conclusions The technique proposed in this paper is a new approach for enhancing the noisy signals in time domain. In this paper, using the SVD-based technique the singular values of the noise subspace are initially eliminated from the singular values matrix. Then the singular vectors are filtered utilizing the Savitzky–Golay smoothing filter. The optimal number of Hankel matrix’s rows, the polynomial degree and window size of the Savitzky–Golay filter are
Table 1 The Monte Carlo simulation on 100 realizations of different SNR values for the multi-component and LFM signals. Initial SNR (dB)
Method
5
Wiener LPF SG filtering Pure SVD Proposed approach
2
Final SNR (dB) Multi-component
Initial Euclidean distance
Final Euclidean distance
LFM
Multi-component
LFM
12.44 7.92 9.25 16.29 18.02
11.21 6.14 8.12 11.46 14.33
228 228 228 228 228
206 206 206 206 206
Multi-component 122 209 165 78 56
LFM 107 174 236 109 88
Wiener LPF SG filtering Pure SVD Proposed approach
0.55 4.57 6.12 10.66 12.10
0.24 4.20 5.05 8.10 11.89
316 316 316 316 316
293 293 293 293 293
569 265 189 157 125
506 227 215 163 111
1
Wiener LPF SG filtering Pure SVD Proposed approach
−3.11 2.90 3.08 7.90 10.01
−6.09 2.96 3.01 5.17 8.15
351 351 351 351 351
325 325 325 325 325
904 305 302 168 150
857 273 307 201 156
0
Wiener LPF SG filtering Pure SVD Proposed approach
−23.50 1.47 2.55 7.06 8.86
−8.48 2.14 2.46 4.23 6.82
470 470 470 470 470
459 459 459 459 459
4660 338 314 190 164
1088 302 326 221 161
794
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determined using the genetic algorithm. Results in this paper indicate the considerable advantages of the proposed approach over the existing approaches for noise reduction in time domain. Further works on this approach will include the development of the proposed technique at the presence of colored noise. References [1] J. Btocker, U. Parlitz, M. Ogorzalek, Nonlinear noise reduction, Proc. IEEE 90 (2002) 898–918. [2] K. Hermus, P. Wambacq, Assessment of signal subspace based speech enhancement for noise robust speech recognition, in: IEEE International Conference on Acoustics, Speech, and Signal Processing, 2004, pp. 945–948. [3] S. Winder, Analog and Digital Filter Design, NEWNESS Pub. Co., 2002. [4] B.T. Lilly, K.K. Paliwal, Robust speech recognition using singular value decomposition based speech enhancement, in: IEEE TENCON – Speech and Image Technologies for Computing and Telecommunications, 1997, pp. 257–260. [5] H. Hanssanpour, M. Mesbah, B. Boashash, Time–frequency feature extraction of newborn EEG seizure using SVD-based techniques, EURASIP J. Appl. Signal Process. 16 (2004) 2544–2554. [6] C.H. You, S.N. Koh, S. Rahardja, Subspace speech enhancement for audible noise reduction, in: IEEE International Conference on Acoustic, Speech and Signal Processing, 2005, pp. 145–148. [7] S.V. Vaseghi, Advanced Digital Signal Processing and Noise Reduction, third edition, John Wiley & Sons Ltd., 2006. [8] J. Chen, J. Benesty, Y. Huang, S. Doclo, New insights into the noise reduction Wiener filter, IEEE Trans. Audio, Speech Language Process. 14 (2006) 1218– 1234. [9] H. Hassanpour, A time–frequency approach for noise reduction, Digit. Signal Process. 18 (2008) 728–738. [10] B. Barkat, B. Boashash, A high-resolution quadratic time–frequency distribution for multicomponent signals analysis, IEEE Trans. Signal Process. 49 (10) (2001) 2232–2239. [11] H. Hassanpour, S.J. Sadati, A. Zehtabian, An SVD-based approach for signal enhancement in time domain, in: IEEE International Workshop on Signal Processing and Its Applications, 2008. [12] K. Shin, J.K. Hammond, P.R. White, Iterative SVD method for noise reduction of low-dimensional chaotic time series, Mech. Syst. Signal Process. 13 (1) (1999) 115–124, Article No. mssp.1998.0184. [13] H. Hassanpour, Improved SVD-based technique for enhancing time–frequency representation of signals, in: The IEEE International Symposium on Circuits and Systems (ISCAS), New Orleans, USA, 2007, pp. 1819–1822. [14] L.L. Scharf, The SVD and reduced rank signal processing, Signal Process. 25 (1991) 113–133. [15] Y. Hua, W.Q. Liu, Generalized Karhunen–Loève transform, IEEE Signal Process. Lett. 5 (1998) 141–143.
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Hamid Hassanpour received the B.S. degree in Computer Engineering from Iran University of Science and Technology, Tehran, Iran, in 1994, the M.S. degree in Computer Engineering from Amirkabir University of Technology, Tehran, Iran, in 1997, and the Ph.D. degree in signal processing from Queensland University of Technology, Brisbane, Australia. He is currently an Associate Professor at Department of Computer Engineering, Shahrood University of Technology, Iran. Amin Zehtabian was born in Mazandaran, Iran, in 1984 and received his B.Sc. and M.Sc. degrees in Electronics Engineering respectively in 2006 and 2009 at University of Mazandaran, Iran. He won the Biomedical Ph.D. Scholarships in France and Italy, respectively in 2010 and 2011 and he is currently a Ph.D. student in Communications Engineering at University of Tarbiat Modares, Iran. Amin also serves as one of the owners as well as Deputy-in-Chief at Teta Electronics Company in Iran. Jalil Sadati was born in Behshahr, Iran, on November 13, 1977. He received the B.Sc. degree in Control Engineering in 2002 from K.N. Toosi University of Technology, Tehran, Iran and M.Sc. in Control Engineering at Ferdowsi University of Mashhad in 2005. Currently he is a Ph.D. student in Babol University of Technology, Babol, Iran. His research interests are Fractional Order Control, Nonlinear Control, and Artificial Intelligence.
