Automated Demodulation of Amplitude Modulated Multichannel Signals with Unknown Parameters Using 3D Spectrum Representation

In current research, new method of automated demodulation of amplitude modulated multichannel signals transmitted with unknown parameters has been proposed and tested. The method is perspective and actual because demodulation is an ill-posed problem when both the carrier and envelope signals are broadband and unknown. The innovative point of this study is the elaboration of the method that enables the evaluation of unknown parameters of the AM signal. The method is based on a new approach called the 3D spectrum representation of AM signals, where the first dimension indicates the allocation of carrier frequencies, the second indicates the band of message signal and the third is related to time. In this study, we report that the 3D representation enables to track unknown AM signal parameters, such as the carrier frequency, frequency of message signal and number of channels and apply demodulation techniques even for time-varying parameters of the signal. The results of the performed experiments indicate that the proposed method effectively performs the envelope detection and automated demodulation of multichannel AM signals.


INTRODUCTION
Amplitude Modulation (AM) is a method of transmitting signals, such as sound or digital information, in which the amplitude of the carrier wave is varied according to the message signal.AM is one of the simplest methods, in which a message signal can be modulated via a sinusoidal carrier wave, widely used in electronic communication and commercial radio broadcasting (Tokle et al., 2006).Moreover, there are two modulation schemes that can be used either with or without carrier suppression: double-sideband and Single-Sideband (SSB) (Billings, 2013).Currently, the AM multichannel signal is used in commercial radio broadcasts, where frequency ranges are allocated between radio stations.The band of the AM radio is known to be separated into three wave ranges: long range (153-279) kHz, medium (531-1,611) kHz with 9 kHz channel spacing for each range and short range (2.3-26.1)MHz; these ranges are divided into 14 broadcast bands (Forest et al., 2011).
Amplitude modulation and demodulation play an important role not only in communication technology but also in the speech recognition theory (Benzeghiba et al., 2007), electric motor faults diagnosis (Trajin et al., 2009), faults of wind turbine generators (Pons-Llinares et al., 2014) and three-phase machine fault detection (Kia et al., 2007).Thus, the development of methods for the automated detection, evaluation of unknown parameters and demodulation of AM signals is promising for further researches.
Currently, there are various approaches in demodulation techniques; coherent demodulation technique (Oppenheim and Schafer, 2009) is used in AM radio.On the other hand, an AM signal could be rectified without requiring a coherent demodulator when an envelope detector (a diode rectifier and a lowpass filter) is used.Another approach in demodulation techniques described in Benzeghiba et al. (2007), Oppenheim and Schafer (2009) is the product detector that multiplies the incoming signal by the signal of a local oscillator with the same frequency and phase as the carrier of the incoming signal.This technique will retrieve the original audio signal after filtering; moreover, it can decode both AM and SSB.If the phase could not be determined using this technique, a more complex setup would be required.
Some popular amplitude demodulation techniques include Hilbert transform (Potamianos et al., 1994) and Teager energy operator (Maragos et al., 1992;Salzenstein and Boudraa, 2009;Diop et al., 2011;Alam et al., 2013;Zeng et al., 2014).Furthermore, for the three-phase system, the Concordia transform can be employed to perform demodulation (Potamianos et al., 1994).Several demodulation techniques, such as the diode detector or product demodulator, can be used to demodulate the AM multichannel signal, but, prior to demodulation, the desired carrier frequency needs to be separated from the spectrum.A simple method to do this would be to construct a set of band-pass filters.However, constructing filters require "a priori" parameters of the AM multichannel signal, such as the number of channels, frequencies of carriers and message signal bandwidth for each channel.For unknown or time-varying parameters, constructing such filters becomes an ill-posed problem.
This study proposes an innovative technique for the measurement of unknown parameters of multichannel AM signals in real time mode using a novel transform called the 3D spectrum representation.The first dimension of this transform indicates the frequencies of carriers for all channels of the original multichannel AM signal.The second dimension carries out information about the frequencies of the message signals, whereas the third one is related to time.Thus, the proposed method enables the real-time detection and tracking of parameters of the multichannel AM signal that allows the use of traditional methods of demodulation.Note that the tracking of the carrier frequency enables the construction of the Band-pass filter and the separation of different channels of the AM signal, evaluation of the message signal bandwidth and envelope detection using the traditional diode rectification and a low-pass filter.The approached technique is noise robust because the energy of the no modulated noise has no representation in the 3D spectrum surface and is ignored during demodulation.The method is independent to the physical transmission media and can be used for both the wireless and wire communications.

This joint research was conducted in Southwest
State University that located in Kursk, Russia and in Balqa Applied University, Amman, Jordan from 12 September of 2013 until 16 March of 2015.In this study, the original continuous time amplitude modulated multichannel signal is denoted as x(t).During the sampling process, the original signal with a sample rate F s during t s seconds is transformed into discrete time signal x [n].that contains N s samples, which is given as follows: By dividing the signal x [n].into a set of no overlapped windows of size N step samples, the total number of windows N w is defined according to Eq. (2) as follows: (2) Equation (3) illustrates the segmentation process of the signal X[n] into the set of N w windows as follows: where, ":" denotes the range of indices "from: to," "d" is the index of the window, d = 0: N w .
According to Equation 4, the index d of each window is related to the current time t cur of the original signal x(t).as follows: The short-time Fourier Transform (STFT) (Paliwal and Alsteris, 2005;Al-Azzeh et al., 2015) of the signal ˲ [n] is defined according to the formula 5 as follows: where, W is the windowing function of length N segm samples.On a physical level, I Ӕ (τ, ω) is an image limited by size

X
is a 2D image in which each pixel is characterized by two dimensions: ω r is the frequency of the message signal and ω is the frequency of the carrier.
Considering the fact that ) , ( . , is a window with the number "d" of the original signal ] [n X and the index d is related to time according to Eq. ( 4), using all the ranges of d from zero to N w -1, the 2D transformation (6) can be represented as a 3D transformation (7):

d X r ω ω
describes the 3D space where the position of each point of the space is related to the frequency of the message signal, frequency of the carrier and time variable.The algorithm that implements transformation 7 is as follows: Step 1: Let x(t) be the original monochromatic AM signal in which the harmonic with the frequency ω m is modulated using the carrier with the frequency ω c .After sampling the x(t) with a sampling rate Fs, the signal was converted into a discrete form x(t). Using Eq.
(3), first, the signal N segm samples; then, each segment is multiplied by a windowing function W (Fig. 1).Assume that the original continuous time AM signal is defined as follows:

Each row of the image shown in
where, x m (t) is the message signal, f c is the frequency of the carrier.
The STFT can be expressed as the set of Fourier transforms of x i (t) signal defined as the product of x(t) and windowing functions W i as follows: The Fourier transform X i (f) for each window W i is related to the Fourier transform X m (f) of the message signal as follows: Now, consider only the positive part of X i (f) spectrum.Suppose that the message signal is represented by a single harmonic with the frequency f m , the Equation obtained is as follows: In this case, the Fourier transform of X i (f) consists of two harmonics with frequencies f c -f m (lower sideband) and f c +f m (upper sideband).Since the amplitude of the message signal varied according to Eq. ( 11), the amplitudes of the upper and lower sidebands also vary according to the same law and the STFT of x(t) signal is represented as two sinusoids.Assume that f c >>f m or the value of f m is less than the frequency resolution of the Fourier Transform.In this case, the resulting spectrogram will be represented as a single sinusoid with the frequency f m located in the position f c (Fig. 2).Step 3: Using transformation 7, the signal I " [n] transformed into a 3D representation I Ӗ (ω r , ω, 0), where ω r is the frequency of the modulating signal, ω is the frequency of the carrier and index 0 indicates the time.As shown above, the spectrogram of x(t) model contains a single harmonic with the frequency f m .Note that the result of transformation 7 for x(t) signal is represented as a single point since the energy of the signal in the position of f c is concentrated in the position f m of I Ӗ (ω r , ω, 0) space (Fig. 3).
For convenience, we have rearranged the location of the coordinate axes in the following sections of the paper according to the coordinate system of the computer images, where the vertical axis is directed from the top to bottom.
Since the parameters of the monochromatic signal x(t) are time-invariant, the position of the point is also time invariant.For the time-varying parameters x(t), the proposed 3D spectrum I Ӗ (ω r , ω, d), representation tracks the parameters of the modulating signal and carrier for each time value "d." The maximum frequencies of the carrier and message signals that can be detected by the proposed method are limited by the values ˦ and ˦ .These parameters can be evaluated using Eq. ( 12) and ( 13): The frequency resolution of the method can be expressed with reference to the properties of the DFT, where the first harmonic frequency of a signal related to its duration.The frequency resolution of the carrier is denoted as R c (Hz) and numerically described by Eq. ( 14) as follows: The frequency resolution R m (Hz) of the message signal is expressed by Eq. ( 15) as follows: When the time resolution R t (seconds) is given by 16, which is as follows: The demodulation of the signal can be performed using well-known methods of signal processing.In the current work, a model of an envelope detector is used, which is represented by a rectifier, digital Infinite Impulse Response (IIR) passband and low-pass filters that are connected according to Eq. ( 17 The impulse responses H and H can be computed using any standard procedure of the digital filter design (Libbey, 1994;Oppenheim and Schafer, 2009).
Note that signal processing according to Eq. ( 17) related to the time domain when the evaluation of AM signal parameters using the 3D spectrum representation related to the frequency domain.
The parameter of the filter design algorithm for the low-pass filter: the cutoff frequency of the low-pass filter defined according to the maximum value of the vertical position of the signal on the 3D space; for band-pass filter the cutoff frequencies can be evaluated according to Eq. ( 18): where, x is the frequency of the carrier related to the horizontal position of a point on the 3D spectral surface.
The following algorithm of the features extraction of AM multichannel signal parameters has been used in the current project: • For each window d of signal X[n], the threshold parameter tr d is defined as follows: • For each window d, define the number of channels ch max as the number of columns of the I Ӗ (ω r , ω, d) image with the brightness of pixels above tr d level.
• For each channel i, i = 1: ch max , evaluate the following features of the signal using the threshold value tr d : for the message signal parameter , , -the maximum frequency of the message signal; for the carrier: the minimum and maximum frequencies , and , .
• For each channel i, i = 1 : ch max i of the window d, provide demodulation according to Equation 17using the following parameters of the filters: for H filter-cutoff frequencies at , and , ; for H filter: -cutoff frequency , , .

RESULTS AND DISCUSSION
The ability of the suggested approach to evaluate unknown parameters of the multichannel amplitude modulated signal has been tested using the synthetic signal V(t).The signal is represented as the sum of the three components (channels) that are particularly cross linked in time: During experiments, the following parameters of the demodulation algorithm have been selected: a sample rate F s = 1 MHz, the duration of signal sampling t s -4 seconds; the size N step of the window-40000 samples and the size of the windowing function N segm -100 samples.According to Eq. ( 2), the total number of windows N w is defined as 100, the duration of the signal in each window becomes equal to 0.04 seconds, as shown in Equation 4. The selected parameters of the synthetic signal will theoretically give 10 KHz of the spectral resolution of the carrier with a percentage error of 2%, 25Hz of the message signal resolution with a percentage error of 0.5% and 0.04 seconds of the time resolution with a percentage error 0.8%.
To obtain the parameters of the multichannel AM signal, we have applied transformation 7 for various values of d from 0 to 99.For each value of d, transformation 7 returns the image with the parameters of the modulated signal.Figure 4 indicates the spectrogram of the signal 20 and the projection of the 3D spectrum into the 2D space for the values of d = 40 that related to the time moment equal to 1.6 seconds.The original size of the image shown in Fig. 4b for the selected parameters of the signal processing algorithm is defined according to the Fourier transform properties as 200×50 pixels.To display the image with a resolution of 500×500 pixels, the bilinear image scaling algorithm (Gonzales and Woods, 2008) has been applied.
According to Equation 20, the parameters of the AM signal for the mentioned time moment are as follows: the carrier frequency of the first channel x 1 (t) 340 KHz, message frequency 3KHz; carrier frequency of the second signal x 2 (t) 268 KHz, message frequency 1.92 KHz.The time-frequency domain of the signal V(t) is shown in Fig. 4a.This Figure demonstrates three components of the signal: x 1 (t) with a falling frequency, x 2 (t) with an increasing frequency and time-invariant frequency of x 3 (t) component.However, the spectrogram has no representation of the modulating frequency and visually, all three components are observed not to be modulated.Figure 4b shows the I Ӗ (ω r , ω, 40) image in which the modulated components of the signal V(t) are represented as white spots.Note that the estimated parameters of x 1 (t) and x 2 (t) components are near to the theoretical parameters.Moreover, the last unmodulated component x 3 (t) has not been presented in the image since the frequency of the message signal of this component is zero.
To study the evolution of signal parameters during time, we have applied the algorithm of the AM signal feature extraction described in the section 2. For the selected parameter d = 40, the algorithm returned the following signal parameters: the number of channels 2, parameters of the first channel:  20.The application of the algorithm for all values of d from 0 to 99 enables to illustrate the evolution of I Ӗ (ω r , ω, 0: 99) parameters in the 3D space (Fig. 5c).Furthermore, the proposed method enables analyzing the parameters In addition, note that the parameters of the carriers and message signals coincide with the parameters of the synthetic signal.
To address the of the method, the pilot study was performed to evaluate the percentage error in measuring the message frequency.During the study, the frequencies of the modulating signals were evaluated To address the accuracy of the method, the pilot study was performed to evaluate the percentage error in measuring the message frequency.During the study, the frequencies of the modulating signals were evaluated using the proposed algorithm and then comp the true values computed using the model of the signal described by Eq. ( 20).Experimentally, the error does not exceed 5% theoretical value (Table 1).
In addition, we have applied the proposed method of the 3D spectrum representation to a multichannel amplitude modulated speech sound.There is an important difference between the synthetic signal created according to Equation 20 and the real sounds, using the proposed algorithm and then compared with the true values computed using the model of the signal described by Eq. ( 20).Experimentally, the error does not exceed 5% theoretical value (Table 1).
In addition, we have applied the proposed method ion to a multichannel amplitude modulated speech sound.There is an important difference between the synthetic signal created according to Equation 20 and the real sounds,  such as speech, treated here.The synthetic sound contains a single time-scale of modulation, whereas the real sound naturally contains modulation at multiple time-scales (Turner and Sahani, 2011).
This section applies the 3D spectrum representation to the amplitude modulated signal x(t) that includes three channels x 1 (t), x 2 (t) and ε(t).The message signal x 1 (t) is a speech signal limited by a bandwidth of 5KHz that is modulated with a carrier of 400 Hz, whereas the message signal x 2 (t) is a speech signal limited by a bandwidth 3.5 KHz modulated with a time-varying carrier of x c (t) with an unknown frequency.The third channel ε(t) is a high amplitude nonmodulated noise located in the band from 290 KHz to 310 KHz.The model of the audio signal is shown as follows: Figure 6 illustrates the spectrogram of the signal x(t).
Signals x 1 (t) and x 2 (t) are English speech sounds of counting from one to ten of two different people.We have applied the following parameters of the 3D spectrum estimation: a sample rate F s = 1 MHz, duration of signal sampling t s -4 seconds; the size N step of the window: 40000 samples, size of the windowing function: N segm 100 samples.During experiments, the original message signals x 1 (t) and x 2 (t) were saved in different wave files to use it as a referenced data during the testing of the proposed method.Figure 7a shows the results of transformation 7 and the feature extraction algorithm as the I Ӗ (ω r , ω, d) image.The figure illustrates a good agreement between the measurement of the carrier frequencies using the suggested method and spectrogram 6.Furthermore, since the noise component ε(t) is unmodulated, its energy is absent in the 3D spectrum representation (Fig. 7a).To show the identity of estimating the frequency of the information signal using the proposed algorithm, we have used the spectrogram of the x 2 (t) (Fig. 7b) and projection of the I Ӗ (ω r , ω, d) image into the time-frequency domain (Fig. 7c).The analysis of Fig. 7b and 7c enables us to conclude that the measurement of the carrier frequency using the proposed method shows good agreement with the standard measurement by spectrogram.The demodulation technique was applied according to Eq. ( 17).To address the overall performance of the proposed method of the demodulation of the AM multichannel signal with unknown parameters, the Perceptual Evaluation of Audio Quality (PEAQ) algorithm (Spanias et al., 2007)  for ˲ # (t) signals.Both numerical values refer to the impairment description "Perceptible, but not annoying."Thus, the proposed method of AM signal measurements and the demodulation is an innovative approach that can be recommended in the implementation of various signal analysis algorithms and digital communication systems.

CONCLUSION
The demodulation of multichannel AM signals is a well-known problem that can be solved using numerous methods.However, the application of demodulation techniques requires the parameters such as the number of channels, frequencies of carriers and message signals for each channel of the signal.Demodulation is an illposed problem when both the carrier and envelope signals are broadband, unknown, or in the case of timevarying parameters of the signal.
This study has introduced a new perspective on demodulation, using the 3D spectrum representation approach.This perspective directly led to the development of an algorithm called the AM signal feature extraction, which enables the detection of all the required parameters of the AM signal, such as the number of channels, instantaneous frequencies of carriers and message signal and the tracking of parameters during that time.The algorithm enables real-time demodulation of all channels even for the time-varying frequency of the carrier and message signal.This plays an important role in coupled researches related to the demodulation of AM signals in the fields of data transmission, speech recognition and acoustic diagnostics of mechanism faults.Using the synthetic signal, we demonstrated that the method was robust to detect the parameters of different signals that are particularly cross linked during time.The proposed 3D spectrum representation ignores the nonmodulated components of the multichannel signal and in the case of "meeting" the frequencies of different carriers, the algorithm fails to evaluate the signal parameters only during the time when the two different channels are transmitting using equivalent carrier frequencies (Fig. 5b).In the application of the natural sounds demodulation, the method illustrates an acceptable quality of results according to the PEAQ.However, the feature extraction algorithm indicates about 5% of errors during the tracking of signal parameters, which can be explained by the fact that during pronunciation, a human speech contains pauses, which lead to the appearance of nonmodulated signal components that are ignored by the 3D spectrum approach.
The 3D spectrum representation and features extraction algorithm also lends itself naturally to changes in the assumptions about the carrier and modulation content and thus the current algorithm may form the basis of useful extensions.
τ, ω) image contains information about harmonics allocation of the signal ˲ [n] at a certain time moment τ and each row of this image indicates the evolution of harmonic ω during that time.˲ [n] is defined in the current work as the Discrete Fourier Transform (DFT) of I Ӕ (τ,:), as illustrated in Eq. (6):

Fig. 1 :
Fig. 1: Segmentation of ˲ " [n] signal into the set of rows Fig. 1 consists of N segm samples of the signal ˲ " [n] totally formed segm step N N rows.The DFT in rows provided according to Equation 5 transforms the original image into the STFT.

Fig. 2 :
Fig. 2: The spectrogram of the signal x(t) | denotes modulus operator to perform full wave rectification, ˲ [n] is the segment of the original multichannel AM signal, * is the operator of convolution and H and H are the impulse responses of the digital low-pass and band-pass filters, respectively.
the algorithm also demonstrate a valuable agreement with the theoretical values determined according to Equation

Fig. 4 :
Fig. 4: The spectrogram (a) of original signal V(t) and (b) I Ӗ (ω r , ω, 40) image signal using 3D spectrum representation; (a): Projection of I Ӗ (ω r , ω, d into time-frequency domain of message signal; (c): 3D representation of of the carriers separately from the parameters of the In addition, note that the parameters of the carriers and message signals coincide with the parameters of the d) into time frequency 3D representation of I Ӗ (ω r , ω, d)

Fig. 6 :
Fig. 6: The spectrogram of the x(t) signal Fig. 7: The 3D spectrum representation; (a) frequency extracted by proposed algorithm has been used in the current work.The PEAQ is a standardized algorithm for objectively measuring the perceived audio quality.It was originally released as International Telecommunication Union (ITU) recommendation in 1998 and last updated in 2001.Using a method of the PEAQ, we have tested a degradation of audio quality : x(t) signal; (b): spectrogram of original x 2 (t) signal; (c): Tracking the proposed algorithm The demodulation technique was applied according to Eq. (17).To address the overall performance of the proposed method of the demodulation of the AM multichannel signal with unknown parameters, the Audio Quality (PEAQ) ., 2007) has been used in the current work.The PEAQ is a standardized algorithm for objectively measuring the perceived audio quality.It was originally released as International ecommendation in 1998 and last updated in 2001.Using a method of the PEAQ, we have tested a degradation of audio quality using proposed method of demodulation.The source wave files with signals x 1 (t) and x 2 (t referenced file for the PEAQ model.The demodulated signals ˲ # (t) and ˲ $ (t) are used as the testing files.The Objective Difference Grade (ODG) is calculated by the perceptual evaluation of the audio quality algorithm.It corresponds to the subjective difference grade used in human-based audio tests.The voices of 21 volunteers (from 18 to 21 years old) are used for PEAQ evaluation.The results returned by the ODG grade show numerical values of -1.28 (standard deviation 0.048) for ˲ # (t) and -1.59 (standard deviation 0.069) Tracking the x 2 (t) signal using proposed method of demodulation.The source t) has been used as a EAQ model.The demodulated are used as the testing files.The Objective Difference Grade (ODG) is calculated by the perceptual evaluation of the audio quality algorithm.It corresponds to the subjective difference grade used in based audio tests.The voices of 21 volunteers rom 18 to 21 years old) are used for PEAQ evaluation.The results returned by the ODG grade 1.28 (standard deviation (standard deviation 0.069)

Table 1 :
Accuracy of proposed method