Analysis and Processing of One-Dimensional Signals Using Wavelet

Abstract

In recent years, wavelet analysis has become an effective and important computational tool in signal processing and image processing applications. Wavelet analysis is known for its successful approach to solving the problem of signal analysis in both the time domain and frequency domain. The analysis of the nonstationary signal generated by physical phenomena has posed a great challenge for various conversion techniques. Transformation techniques such as Fourier transform (FT) and short Fourier transform (STFT) fail to analyze nonstationary signals. But wavelet transform (WT) techniques may be able to efficiently analyze both stable and unstable signals. WT is able to analyze one-dimensional signals, such as audio signals and two-dimensional signals such as images. In this chapter, we discuss wavelet transduction techniques and their applications in detail and focus on the analysis and processing of the wave-encoded laser signal as one-dimensional electrical signals and its use in alarm systems. In the second stage, we filter the speech signal and determine the fundamental frequencies using wavelet transformation.

1. Introduction

This chapter introduces the study and realization of a laser barrier alarm system, after the laser is obtained by an electronic device, the wireless control system is connected to the control room to announce the application in real-time, the laser is used in many application fields, from industry to medicine, it uses an alarm system to detect and deter intruders. Basic security includes protecting the perimeter of a military base or a safe distance in unsafe places or near a government place. The first stage secures surrounding access points such as doors and windows; the second stage consists of internal detection with motion detectors that monitor movements, there are several types of products on the market, and the system you buy can be wired or wireless. Wired systems use cables to connect each device to the central control panel. A wireless system that runs on batteries and transmits its signals at a radio frequency, and there are no cables. In this chapter, we rely on the realization of a coded laser barrier that is sent between two units, processing the signal, and comparing the agreed conditions, and in order to be high accuracy, we suggest that we use wavelet transduction to process the received signal and know the frequencies that achieve the activation of the alarm.

2. Warning and protection systems

Alarm and protection systems have been developed in many fields, in many areas such as the military field, where lasers have been used for detection. Any attempt to break through the wall of the military barracks. In addition to the technologies in this field, we are working on the use of coded lasers, which means that we send pulses that are very limited in frequency and periodicity, as well as in the number of pulses during a pre-agreed period of time [1].

Laser pulses can be obtained through an electronic circuit with analog processing, and to eliminate any noise in the receiving circuit, we filter the signal using wavelet transformation, thanks to which we get high accuracy and an effective system that works in real-time [2].

In addition to activating the alarm, this system can also work to send information via radio waves to the control room so that the leadership can make decisions at the same time [2, 3].

2.1 Photovoltaic barriers

Photovoltaic barriers are optical or electronic systems consisting of a sensor (receiver) and a light source (emitter). The light source can be an ordinary lamp, an infrared emitter (e.g., a pulse), LEDs, or a laser emitter (Figure 1) [2].

Figure 1.

Photovoltaic barriers.

2.2 Laser barrier application

Single barriers consist of a separate interacting transmitter and receiver. Reflex barriers and detectors combine sensor and light source in a single box. In reflex barriers, the emitted light beam is returned by a reflector (prism, reflective sheet) to the receiver [4]. Light barriers register an interruption in the light beam and convert the information. If an object passes through the beam of an optoelectronic barrier, the sensor generates a predefined electrical output signal. It triggers an alarm. Detectors send a very fine infrared beam and react to the reflection of light from an object. The maximum detection distance depends largely on the reflectance rate, shape, color, and surface quality of the material [5, 6, 7].

3. System structure

The corresponding figure shows the stages of transmitting and processing information that determines with high accuracy all the electronic circuits on which this project depends, as it consists of a laser transmitter encoded between two transmitting and receiving units, the processing stage, and the activation of the alarm with the radio wave communication system (Figure 2).

Figure 2.

Configuration of the whole system.

3.1 Laser transmitter circuit

The corresponding circuit shows the electronic card responsible for producing the laser pulses, with the possibility of changing the frequency and the periodic ratio [8] (Figure 3).

Figure 3.

Laser transmission circuit.

Circuit diagram simulated in Crocodile Technology 607 (Figure 4).

Figure 4.

Simulation results of the laser transmitter circuit: (a) square electric signal, (b) pulse signal, and (c) rectangular signal.

3.2 Basic Astable 555 oscillator circuit

The 555 IC can be used to create a free-running as table oscillator to continuously produce square wave pulses (Figure 5).

Figure 5.

555 oscillator.

The previous electronic circuit generates square signals or pulses and this is related to the values of the resistors and capacitors and depends on connecting the second electrode with the sixth and separated between them by a special resistance for discharging the capacitor through the seventh electrode.

In the initial conditions, the tension between the poles of the capacitor is equal to zero, and when the generator is connected, the capacitor starts charging in an exponential equation until it reaches the value 2/3 Vcc. Then, the outlet voltage ceases. In these conditions, the capacitor starts discharging until it reaches 1/3 Vcc, and this is repeated. The process is repeated several times and this is called an unstable oscillator.

$$T=t_1+t_2$$E1

$$t_1=0.69\left(R_1+R_2\right)C$$E2

$$t_2=0.69.R_2.C$$E3

The output frequency of oscillations can be found by inverting the equation above for the total cycle time giving a final equation for the output frequency of an Astable 555 Oscillator as:

$$F=\frac{1}{T}$$E4

Applied results of the transmission circuit obtained using an oscilloscope (Figure 6).

Figure 6.

The real results of the laser transmitter circuit—square electric signal.

The output signal can be controlled by connecting a direct polarizing diode between the sixth and seventh poles so that we can determine the charging constant and the discharging constant, which in turn controls the type of output signal (Figure 7).

Figure 7.

Transmitter circuit results—pulsed signal.

3.3 Laser receiver circuit

One of the advantages of the practical amplifier called IC LM 741 is the comparison between the inverting and non-inverting input signals so that the output voltage is symmetrical and varies according to the comparison process, and in this case, it works in the nonlinear characteristic.

The practical amplifier may be used in the process of amplifying weak signals if it operates in the linear characteristic so that the relationship between the input signal and the output signal is fixed (Figure 8).

Figure 8.

Laser receiving circuit using comparator LM 741.

After receiving and processing the laser beam, we get the following signal (Figure 9).

Figure 9.

Laser beam reception signal.

The signal obtained is either square, rectangular or pulsed, depending on the transmission signal, and accordingly, the average value can be calculated as follows:

$$\overline{E}=\frac{1}{T}\underset{0}{\overset{T}{\int }}E\left(t\right).\mathit{dt}$$E5

$$\overline{E}=\frac{1}{T}\underset{0}{\overset{t1}{\int }}E\left(t\right).\mathit{dt}+0$$E6

$$\overline{E}=\frac{1}{T}E.t_1$$E7

Knowing that the mean value is proportional to the duration t₁ then

$$\overline{E}=\eta .E$$E8

$\eta$, the cyclic report.

The following circuit is to compare the average value of the main signal and the reference voltage, to increase the accuracy by adding another practical amplifier and some electronic components (Figure 10).

$$\overline{E}=2.5V,V1=2.3V,V2=2.8V$$

Figure 10.

Comparison circuit with two amplifiers LM 741.

So $V_1\prec \overline{E}\prec V_2$ for this condition the alarm system is in the off state because no laser beam cut between the two cards (transmission and reception).

3.4 Processing circuit

The main processing circuit consists of the following electronic components (Figure 11).

Figure 11.

Electronic circuit of the project.

The corresponding figure represents the printed circuit of the project using Express PCB software and electronic components CMS (Figure 12).

Figure 12.

The mother card for the project.

The following figure shows the real picture of the project with the processing circuit and data transmission using radio waves (Figure 13).

Figure 13.

The final electronic card for the project.

3.5 Experimental results

The figures above show the transmit and receive signal with duty cycles of approximately 25 and 75%, respectively (Figures 14 and 15).

Figure 14.

Experimental results for a pulsed signal.

Figure 15.

Experimental results of a rectangular signal.

Emission signal

Reception signal +/− 12

4. Analysis of the reception signal by wavelet transformation

In recent years, wavelet transform (WT) has been relied upon as a radical alternative to signal processing, especially after the discovery of the problematic Fourier theory, which focuses on the frequency domain, and given that most physical quantities are unstable here. It can be said that we need to develop methods of processing and analysis, it can be said that in the seventies of the last century the wavelet theory was used in several areas, such as noise removal, image improvement, classification of audio signals, etc. [9]. Chu and Kim applied the Morlet wavelet transform to analyze the effect of noise.

The wavelet theory depends on important properties in processing so that the value of a window of the mother wave is proportional to the frequency of the signal to be processed. There are two types of wavelet transports: continuous (CWT) and discrete (DWT) existing transport processes. Both transformations are continuous in time (analog), and with their help, analog signals can be represented [10].

4.1 General theory of CWT

In this work, we only touched upon some of the basic equations, definitions, and concepts of the wavelet transform, and a more rigorous mathematical treatment of this topic can be found in [10, 11]. The time-continuous wavelet transform of f(t) is defined as:

$${\mathit{CWT}}_{\psi }f\left(a,b\right)=\frac{1}{\sqrt{a}}\underset{-\infty }{\overset{+\infty }{\int }}f\left(t\right).\stackrel{\ast }{\psi }\left(\frac{t-b}{a}\right).\mathit{dt}$$E9

where:

• ${\mathit{CWT}}_{\psi }f\left(a,b\right)$ is the wavelet coefficient of the function f(t)

• Ψ(t) is the analyzing wavelet

• a (> 0) is the scale parameter

• b is the position parameter

4.2 Temporal and spectral resolutions in the CWT

The signal processing in the time and frequency domain is very important so that we can know the different frequency components, where we express the time resolution in the time domain σt, and the spectral resolution in the frequency domain σw from CWT as:

$${\sigma }_t^2\left(\gamma \right)=\underset{-\infty }{\overset{+\infty }{\int }}{\left(t-\mathit{u\gamma }\right)}^2{\left|{\phi }_{\gamma }\left(t\right)\right|}^2\mathit{dt}$$E10

Wavelets perform and deliver a high-accuracy scale-based analysis of specific data [12]. They then find a wide range of applications and uses for these waves including signal processing, mathematics, and numerical analysis, and for their good use in signal and image processing, they are a radical alternative to the fast Fourier transform where the DWT provides a time-frequency representation when a non-stationary tool is needed for processing and analysis, DWT can be used. The study showed that discrete wave delivery has a high performance in processing speech signals so far [9, 13].

A computer-assisted experiment (CAM) is not fundamentally different from a laboratory experiment as it was traditionally performed using different measuring instruments and laboratory equipment, but the computer integration in the processing of the laser pulse receiving signal brings several advantages. The data acquisition process can be automated, and the measurement results can be saved and processed easily and in a very short time by various software tools. In addition, the presentation of the results in graphic form is greatly simplified and their scientific analysis, which facilitates the analysis and use of the wavelet transform on the obtained signal.

4.3 Acquisition of reception signal with CoolEdit

CoolEdit program is used to record electrical signals that are proportional to the physical quantities to be processed and will be recorded in a one-dimensional matrix at a sampling frequency of 64 kHz using a 16 transducer called the sampling period.

$$T_e=\frac{1}{F_e}=\frac{1}{64\times {10}^3}=0.015\times {10}^{-3}s$$E11

The following design represents how to record the electrical signal from the receiving circuit using computer-assisted experiments (Figure 16).

Figure 16.

Diagram of the proposed method.

The signal obtained is considered to be a non-stationary signal and it is also made up of several signals which cannot be recorded using the oscilloscope because it is technically unable to track and oscillate instantaneous signals of very high frequency. Response time, although it is very infinite, also take a slow-motion until it stabilizes, and this is called differential vibration.

The molar figure shows the difference between the signal recorded by the cathode oscilloscope and the one recorded by a computer-assisted experiment (Figures 17 and 18).

Figure 17.

The results of the transmitter circuit by the oscilloscope.

Figure 18.

Real results of the transmitter circuit by CoolEdit.

In the normal case, the signals resulting from the receiving circuit can be drawn using the cathode oscilloscope, but it does not give us rapid changes that can only be detected by using alternative devices. Computer-supported experiments were selected and then we process the data, which is represented in one-dimensional arrays using the MATLAB program. For these calculations we used the MATLAB cwt function.

After processing the stored matrix, we get the following figure, after applying the wavelet transform with the selection of the Haar wavelet algorithm to analyze the signal into several levels, where we can know the basic frequencies of the received laser signal (Figure 19).

Figure 19.

Simulation results of wavelet transformation on the receiving signal at an increased frequency.

The following figure shows how to deconstruct the signal obtained in the receiving circuit, showing the frequency resulting from the differential vibration, which we would not have obtained without applying the wavelet theory (Figure 20).

Figure 20.

The Haar wavelet transform on the received signal.

5. Denoising and enhancement speech signal using wavelet

After the development of the field of information processing and the discovery of wavelet analysis, it became common to process signals and unstable physical quantities, such as speech analysis, and sound signature discovery by recognizing the basic frequencies of letters. Wavelets proved successful in processing such signals, which is an alternative to all methods. It was there before where it processes data in real-time using time-wave resolution, waveforms rely on the Henning window [7]. Recognition performance depends on frequency domain coverage. The goal of good speech recognition is to increase the bandwidth of the wavelength without significantly affecting the time accuracy. This can be done by collecting the white noise of the wave, which is difficult to detect and remove by traditional methods.

5.1 Speech enhancement methods

There are many methods available to improve speech, reduce noise, and the quality of audio signals, and each algorithm has a principle that it depends on in the processing methodology, and this depends on the goals we want to reach. In this chapter, we propose the corresponding layout, which enables us to filter the audio signals and identify the frequencies that make up them, as shown in Figures 2 and 21.

Figure 21.

Speech enhancement method.

5.2 Detection of singularity of the impulse noise signal in CWT

For noise evaluation, the oscillation of the acoustic signals is regarded as a considerable important metric. The CWT is often applied to detect the singularities of a transient signal (Figures 22 and 23; Table 1).

Figure 22.

Vowel A time domain.

Figure 23.

The vowel A with the spectrum and the CWT.

Frequency Hz Vowel	Frequency F1	Frequency F2	Frequency F3
A	625	1491	2356
E	387	1985	2875
I	246	2018	3196
U	312	750	2079
O	313	756	2271

Table 1.

Shows the various frequencies for the vowel “A, E, I, U, O.”

In the general case, we can rely on three main frequencies to define letters and words, after removing noise and applying wavelet transform. Practically, the third frequency can be neglected because it may be close in several letters, and we are satisfied with only the first and second frequencies, especially if the noise is removed at an acceptable rate.

The figure opposite shows the letter E after noise cancelation (Figure 24).

Figure 24.

The vowel E with the spectrum and the CWT.

5.3 Enhancement vowel “A,” by the wavelets

The following figure shows the resulting multi-resolution vowel “A,” by the wavelets (Figure 25).

Figure 25.

Vowel “A,” levels of decomposition by the wavelets: (A) original speech signal, (a1) appro 1, (d1) level 1, (a2) appro 2, and (d2) level 2. $\mathrm{A}=\text{a1}+\text{d1}+\text{a2}+\text{d2}$

6. Conclusion

In this work, we relied on processing the data sent between a transmitter and receiver, which were coded laser pulses that control a sophisticated warning system that can be used in a military field. In order for this system to be effective, we decided to analyze the received signal by converting waves that have proven successful in several fields. After deconstructing the received signal using wavelet transformation, it can be said that the pulses are not as perfect as we expected, because in fact they consist of several signals with different frequencies, and this leads to the lack of dependence on this signal in controlling any system, especially if it requires high accuracy or efficiency in Performance such as controlling the speed of the DC motor or transmitting digital information by wireless communications, especially if it is a high frequency.

The use of wavelet transform is not limited to electrical signal processing but can be applied to improve sound quality, and this method depends on the signal threshold that each waveform parameter of the signal is compared to a certain threshold. Using wavelet transform to remove noise from a signal may require identifying which components contain the noise, and then reconstructing the signal without those components after eliminating them.

In contrast to the STFT transformation which has constant accuracy at all times and at all frequencies, we can say that WT has good temporal accuracy and low-frequency accuracy at high frequencies, and good frequency accuracy and low temporal accuracy at low frequencies.

Wavelet decomposition is very similar to the Gabor transform: the speech signal is written as a superposition of the displacement and expanding waves. The main frequencies can be recognized in a short time and with great accuracy especially in word recognition programs if we get rid of the noise in the audio signal and this is what we discussed in this section where the wavelet transform can be used to reduce noise.

After applying the proposed algorithm, noise from speech signals was successfully reduced by using wavelet transform. This method gives us a hands-on approach on how to filter sound and recognize different frequencies.