De-Noising Audio Signals Using MATLAB Wavelets Toolbox

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INTRODUCTION

Based on the fact that noise and distortion are the main factors that limit the capacity of data transmission in telecommunications and that they also affect the accuracy of the results in the signal measurement systems, whereas, modeling and removing noise and distortions are at the core of theoretical and practical considerations in communications and signal processing.

Another important issue here is that, noise reduction and distortion removal are major problems in applications such as; cellular mobile communication, speech recognition, image processing, medical signal processing, radar, sonar, and  any other application where the desired signals cannot be isolated from noise and distortion.

BASIC AUDIO THEORY

Sound is the vibration of an elastic medium, whether gaseous, liquid or solid. These vibrations are a type of mechanical wave that has the capability to stimulate human ear and to create a sound sensation in the brain. In air, sound is transmitted due to pressure variations at a rate of change that is called frequency.

The difference between the extreme values of pressure represents its amplitude. Pressure variations in the range of 20 Hz to 20 kHz produce the sound which is audible to the human ear and this is more receptive when it is between 1 kHz to 4 kHz. In physical terms, the sound is a longitudinal wave that travels through the air due to vibration of the molecules. Similar to light, sound waves can be reflected, absorbed, diffracted, or refracted.

It Shows the Process of Digital Processing of Three Types of Audio Signal. Part (a) Represents a Complete Digital Audio Processing Comprising (From Left to Right) a Microphone, Amplifier, Adc, Digital Processing Material, Dac, Amplifying Section and Speaker; an Audio Recognition System in (B), and a Set of Audio Synthesis (C) .

It Shows the Process of Digital Processing of Three Types of Audio Signal. Part (a) Represents a Complete Digital Audio Processing Comprising (From Left to Right) a Microphone, Amplifier, Adc, Digital Processing Material, Dac, Amplifying Section and Speaker; an Audio Recognition System in (B), and a Set of Audio Synthesis (C) .

BASIC NOISE THEORY

Noise is defined as an unwanted signal that interferes with the communication or measurement of another signal. A noise itself is an information-bearing signal that conveys information regarding the sources of the noise and the environment in which it propagates.

Signal to noise ratio:

The signal-to-noise ratio (SNR) is commonly used to assess the effect of noise on a signal. This measurement is based on an additive noise model, where the quantized signal xq[n] is a  superposition of the unquantized, undistorted signal x[n] and the additive quantization error e[n].

White noise:

Shown in Figure 4, white noise is defined as an uncorrelated random noise process with equal power  at all frequencies. Random noise has the same power at all frequencies in the range of ∞ it would necessarily need to have infinite power, and it is therefore an only a theoretical concept.

It Shows an illustration of (a) white noise time-domain signal, (b) its autocorrelation function is a delta function, and (c) its power spectrum is a constant function of frequency .

It Shows an illustration of (a) white noise time-domain signal, (b) its autocorrelation function is a delta function, and (c) its power spectrum is a constant function of frequency .

WAVELETS THEORY

Wavelets are used in a variety of fields including physics, medicine, biology and statistics. Among  the applications in the field of physics, there is the removal of noise from signals containing  information. There are different ways to reduce noise in audio. (Johnson et al., 2007)  demonstrated  the application of the Bionic Wavelet Transform (BWT), an adaptive wavelet transform derived from a  non-linear auditory model of the cochlea, to enhance speech signal.(Bahoura & Rouat, 2006) proposed a new speech enhancement method based on time and scale adaptation of wavelet thresholds. (Ching-Ta  &  Hsiao-Chuan Wang, 2003 & 2007) proposed a method based on critical-band decomposition, which  converts a noisy signal into wavelet coefficients (WCs), and enhanced the WCs by subtracting a  threshold from noisy WCs in each subband.

It Shows the scale function (left) and wavelet function (right) of (a) Coiflet 5, (b) Daubechies 9 and (c) Daubechies 10 .

It Shows the Scale Function (left) and Wavelet Function (right) of (a) Coiflet 5, (b) Daubechies 9 and (c) Daubechies 10 .

SINE SIGNAL EXAMPLE

With MATLAB, it is possible to process noisy signals containing certain information, such as an  audio one, in order to reduce the quantity of noise contained in it. Non-periodicity characterizes an audio signal, which is composed by a large number of different frequencies signals. This feature  allows the use of conventional methods such as digital filters to eliminate noise mixed into the signal.

AUDIO SIGNAL EXAMPLE

As it has been described in section 2.1, to obtain an audio signal, we select the audio block  and we connect it to the multimedia file block. This information is saved in a user-specified file with extension . wav and then we use the program to extract this information and process it with wavelets coif5, db9 and db10.

It Shows Original Audio Signal, Audio Signal With Noise and the Recovered Signal.

It Shows Original Audio Signal, Audio Signal With Noise and the Recovered Signal.

GRAPHICAL INTERFACE USING GUIDE

For better understanding of the content of this chapter, we have developed a graphical interface, only in the case of a sine wave. We used GUIDE of MAT LAB to build this interface, you need pop-up  menu, slider, edit text, four push button and four graphics (axes), after all that is placed in your figure you need to program each item. Figure 16 shows how all of this should be seen.

CONCLUSIONS

We provide a practical approach in how to put in to practice wavelets in noisy audio data to improve clarity and signal retrieval. Since there are no books that show the code for a graphical  interface with audio processing using wavelets, this chapter presents MATLAB code to reduce the  Gaussian white noise in periodic signals (sine function) and in audio signals (composed of several  frequencies) using wavelet analysis.

We compared different wavelet families: Symlets, Daubechies and Coiflets, and we used cross-correlation to determine the best fit between an original signal and the processed one. By using Coiflet 5, Daubechies 9 and 10 we obtained the best result because they have a higher correlation at  zero.

Our signal processing technique recovers signal with a correlation higher than 99%. In analysis  for audio signal with added Gaussian white noise, while using the technique we obtained a recovered signal with a correlation of 95%. This analysis is very useful to help the reader  understand  the  know how in removing noise from a signal by using wavelets.

Therefore, when a signal shows a  periodic signal extraction from noise, it will be satisfactory. The graphical interface presented in the last section was performed while using GUIDE this one gives the readers a guideline to develop their own projects in MATLAB.

Source: Instituto Nacional de Astrofisica, Optica y Electronica
Authors: Adrian E. Villanueva- Luna | Alberto Jaramillo-Nuñez | Daniel Sanchez-Lucero |  Carlos M. Ortiz-Lima | J. Gabriel Aguilar-Soto |  Aaron Flores-Gil |  Manuel May-Alarcon

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