Wavelet Based Ofdm Communication System

Published Date: 02 Nov 2017

3.0 Introduction:

In this chapter first of all we would define Wavelet, Features of Wavelets which includes Wavelet Families, different Wavelet levels and Vanishing Moments of Wavelets. Secondly, we would give an overview about Wavelet Transform, types of Wavelet Transform which including Continuous Wavelet Transform and Discrete Wavelet Transform and at last we would discuss Advantages and Disadvantages of Wavelet based OFDM Communication (WOFDM), Applications of Discrete Wavelet Transform based OFDM System(DWT-OFDM) and Future Applications of Wavelets.

3.1 What is Wavelet:

Wavelets are basically a small wave with a limited duration of time. Unlike sinusoids wavelet have a start and end point. As the wavelets have limited duration of time, so often they are irregular and non-symmetrical. Figure (3.1) shows an infinite continuous sinusoid and a finite Db-20 wavelet. As shown in the figure the sinusoid have a constant frequency overall the length of the wave but the wavelet have a pseudo frequency. A pseudo frequency is a type of frequency which varies over the length of the wavelet.[Conceptual wavelets in Digital Signal Processing by D.Lee Fugal]

3.1.1 History:

Mathematician Alfrd Haar in 1909 was the first to propose a paper regarding wavelet known as Haar Wavelet. Till 1985 the Haar wavelet was the only orthogonal wavelet the people know , and in the same year mathematician Yves Meyer constructed the second orthogonal wavelet known as Meyer Wavelet and in the year 1988 Ingrid Daubechies constructed the wavelt family known as Daubechies Wavelet.[ A Tutorial of the Wavelet Transform

Chun-Lin, Liu

February 23, 2010]

3.2 Features of Wavelet

The main features of wavelet includes a discussion on the wavelet families .Here in our project our main concern is with (haar,daubechies and symlet).Further on we would give a briefing on different wavelet levels and vanishing moments.

3.2.1 Wavelet Families:

Mainly there are three types of Wavelet Families i.e. (i)Crude Wavelets (ii)Orthogonal Wavelets (iii)Biorthogonal Wavelets

Further these wavelet families have sub families such as Crude Wavelets is divided further into five more wavelet families which are Mexican Hat, Morlet, Guassian,Meyer, Shannon, Complex frequency B-spline, Complex Morlet and Complex Guassian. The sub families of Orthogonal Wavelets are Haar, Daubechies, Symlet, Coiflets and Discrete Meyer. At last the Biorthogonal Wavelets is divided into two sub families which are Biorthogonal and Reverse Biorthogonal wavelets.

Here in our project we are using Haar, Daubechies and Symlet wavelets. So we would give an overview about Haar,Daubechies and Symlet wavelets.

Haar Wavelets:

Haar wavelet is the simplest type of wavelet. Haar wavelets do not exist in real world of digital computers but a close estimation can be obtained by up-scaling and low pass filtering of H* filter [1-1] for the production of multi-point resolution as shown in the fig 3.2.1

Haar wavelets have only one vanishing moments and the stretched versions of Haar wavelet will also have one vanishing moment. Haar wavelets are irregular, smoothless and have outright discontinuities. Haar wavelets are anti-symmetric with linear phase. Haar wavelets are used with DWT, CWT and UDWT. Haar wavelets have very good capability of dealing with edge detection and for matching binary pulses and for very short phenomenon.

Daubechies Wavelet:

Daubechies wavelet is also a family of orthogonal wavelet family.The haar wavelet is also a member of Daubechies wavelet family. A haar wavelet is also known as a Db-2.A daubechies wavelet is also smoothless and it’s regularity increases with increase in order of "n" (Db-n). The no. of Vanishing moments in Daubechies wavelet family is equal to half the no. of filter points i.e. n/2 .e.g a DB-16 wavelet have 16/2=8 vanishing moments.As compared to haar the Daubechies Wavelet donot have a linear phase. These wavlets are fast and robust .Due to it’s non-symmetric property these wavelets are used in image processing such as edge detection and airport luggage screening . Below in the figure a Db4 wavelet,Db6 and Db8 wavelet are shown.

Symlet Wavelet:

Symlet wavelets is the third wavelet family of the orthogonal wavelet family. Symlet wavelets are more symmetric as compared to Daubechies wavelets. Bieng almost symmetrical the larger symlets(Sm 16) etc have almost linear phase.Symlets wavelets have all the properties of Daubechies wavelets. They have the same no. of vanishing moments which the Daubechies wavelets have .i.e. n/2. Due to it’s cancellation capability it can be used both with DWT (Discrete wavelet transform) and CWT(Continuous wavelet transform). The orthogonality relationships of symlets are same as that of Daubechies wavelet. On the basis of symmetry, orthogonality and vanishing moments symlets wavelets are used in image processing. Symlets wavelets are also used in power load consumption signals and composite structure.C:\Users\Saleem\Desktop\SYMLET__.jpg

3.2.2 Levels of Wavelets:

Wavelet Packet Transform (WPT) and Wavelet transform(WT) is a process which is used to obtain the no. of level of the wavelets. As shown in figure 3.2.2 a Wavelet Packet is decomposed over 3 levels by passing the packet s through Low pass filters g[n] and High pass filters h[n].

C:\Users\Saleem\Desktop\Wavelets_-_WPD.png

3.2.3 Vanishing Moments of Wavelets:

The order of a wavelet transform is clearly given by the no. of vanishing moments. The Strang-Fix conditions implies that the error for an orthogonal wavelet approximation at scale a= 2-i globally decays as aL, where L is the order of the transform. This is why for a given scale the higher order wavelet transforms results in better signal approximations. Since the higher the vanishing moment is the more accurate the signal is. Vanishing moments increases the smoothness of the signal.

The basic idea is that a wavelet has p vanishing moments if and only if the wavelet scaling function can generate polynomials up to degree p−1. The "vanishing" part means that the wavelet coefficients are zero for polynomials of degree at most p−1, that is, the scaling function alone can be used to represent such functions. More vanishing moments means that the scaling function can represent more complex functions. Loosely, you can think of it as

more vanishing moments → complex functions can be represented with a sparser set of wavelet coefficients.

The "moments" part comes from the fact that this is all equivalent to saying that the first pderivatives of the Fourier transform of the wavelet filter all are zero when evaluated at 0. This is perfectly analogous to the probabilistic idea of a "moment generating function" of a random variable, which is basically the Fourier transform, and the n-th derivative evaluated at zero gives the n-th moment of the variable (i.e. the expected value, the expected value of the square, of the cube, etc.) So these Fourier transform derivative-zeros correspond to integrals back in the time/space domain that must be zero for the wavelet. In a sense, these conditions mean that the wavelet is "unbiased." It doesn't skew the function that is being transformed because the wavelet itself has no expected effect on a function until that function has a non-trivial p+1 order derivative.

3.3 Wavelet Transform:

A wavelet transform is same as the Fourier transform or much more to the windowed Fourier transform with a different merit function.The difference is that in Fourier transform the signal is decomposed into frequency domain while in wavelet transform the function is decomposed into both frequency and time domain.The wavelet transform is given by equation:

http://gwyddion.net/documentation/user-guide-en/eq-wavelet-transform-continuous.png

where the * is the complex conjugate symbol and ψ is some function. This function can be chosen arbitrarily provided that obeys certain rules.

As shown, the Wavelet transform is an infinite set of various transforms, which depends on the merit function used for its computation. Wavelet transforms give us variable size windows , so we can use short intervals(more precise time information) for higher frequencies and long time intervals for more precise low frequency information[Conceptual wavelets in Digital Signal Processing by D.Lee Fugal].

3.4 Types of Wavelet Transform:

On the basis of orthogonality wavelets are of two types. The first type is  orthogonal wavelets for discrete wavelet transform development and the second is non-orthogonal wavelets for continuous wavelet transform development. 

3.4.1 Continous Wavelet Transform:

The continuous wavelet transform was developed to overcome the resolution problem.The method of analysis of wavelet is same as it was done in Short time Fourier transform (STFT)as the signal is multiplied with a wavelet which is same as window function used in Short time Fourier Transform(STFT) and the computation of transform is separate for the different segments of the time-domain signal.However there is two main difference between Short time fourier transform(STFT) and Continous Fourier Transform(CFT).i.e.

(i). For every single spectral component the the window’s width changes as the transform is computed, which is the main characteristic of wavelet transform.

(ii) The Fourier Transform of windowed signal are not taken as a result a single peak is shown corresponding to a sinusoid, i.e. no negative frequency is computed.

[THE WAVELET TUTORIAL PART III by ROBI POLIKAR 05/11/2006 04:39 PM

http://users.rowan.edu/~polikar/WAVELETS/WTpart3.html Page 3 of 29}

The continuous wavelet transform equation is given as:

Equation 3.1

In the above equation , the transformed signal is a function of two variables, tau(á¿›) and (s) , the

translation and scale parameters, respectively. ѱpsi(t) is the transforming function, and it is called the

mother wavelet . The mother wavelet is a prototype for generating the other window functions. All the windows that are used are the compressed and shifted versions of the mother wavelet.Now let x(t) is the signal to be analyzed. After the selection of mother wavelet the continous wavelet transform is taken for all the values of s, starting with s=1 and will continue for larger values of s i.e. the the analysis of signal will start from higher frequencies and will continue towards lower frequencies. The first value of s=1 is the most compressed wavelet and with increase in value of s the wavelet will expand. The wavelet is placed at time t=0.Now the function at scale "1" is multiplied by the signal and is integrated over all time domain and the result after integration is multiplied by (1/√s).The multiplication normalize the energy such that the energy of the transformed signal remains same at every scale. The value of the transformation is the final result. It is the value which corresponds to Ί=0 and s=1 in time-scale plane. Now the wavelet which is placed at scale s=1 is shifted towards the right at location t=Ί by the Ί amount. The above equation is used to find the transform value at scale s=1 and time t=Ί in time-frequency domain. We will repeat the same process till the wavelet reaches the end of signal. Now for scale s=1 one row of points on timescale is completed. Since it is a continuous wavelet transform so both value of Ί and s is increased continuously and value of transform is computed. Further if we want to compute the value of transform by computer then both time and scale parameters are increased sufficiently by small step size which corresponds to sampling of the time-scale plane. Since this process is repeated for all value of s and each value of s fills corresponding one row of time-scale plane. At the end we will get the Continuous Wavelet Transform (CWT) when the process is completed for all the values of s.

3.4.2 Discrete Wavelet Transform:

The discrete wavelet transform provides enough information for the analysis and synthesis of original signal with a significant amount of reduction in the computation of time. Discrete Wavelet Transform(DWT) is easier to implement as compare to Continuous Wavelet Transform(CWT).

In Discrete Wavelet Transform(DWT) we pass the signal through a high pass and low pass filter for the analyzation of high and low frequencies.The signal resolution is changed by filetering option and the scale is changed by the operation of downsampling and upsampling. E.g. Downsampling by 2 means to drop every other sample of signal and downsampling by "N" reduces the no. of samples by N times in the signal.

When we upsample the signal it means that we are increasing the sampling rate of the signal by adding new samples e.g. Upsampling by 2 means adding a new sample value between every two sample of the signal and upsampling by a factor of "N" means adding "N" number of samples in the signal.

Now at the start a signal sequence is passed through a lowpass halfband digital filter which have an impulse response h[n].Filtering of a signal means convolution of signal with impulse response of filter.A discrete time convolution operation is given as:

This half band low pass filter removes all the frequencies which are above the half of highest frequency present in the signal.As the unit of frequency for the discrete time signals is radians,therefore the smapling frequency of the signal will be 2p radians and the highest frequency component that exits in the singal wil be p radians when the Nyquist’s rate is applied.

When the signal is passed through half band lowpass filter, it eliminates half of the samples accoding to Nyquist’s rule and now the highest frequency of signal will be p/2 radians instead of p radians. Now upsampling the signal by two discards every other sample and the signal will then have half the number of points. Since now the scale of signal is doubled.As, the low pass filters removes the high frequency information but doesnot changes the scale. The scale is changed only by downsampling process.And on the other hand the resolution in the signal is affected by filtering operations, so the resolution is also halved after passing through half band lowpass filter which means half of the information is lost.In short half band lowpass filter half’s the resolution but doesnot change the scale and the signal is then downsampled by two since half of the samples are redudant which doubles the scale.

The whole procedure is given by equation:

The computation of DWT(Discrete Wavelet Transform) includes the analyzation of signal at different frequency bands have different resolutions and by decompostion of signal into a detail information and coarse approximation.DWT applies two functions which are scaling functions and wavelet functions and are associated with low pass filter and high pass filters respectively.The low pass and high pass filters decomposes the signal into different frequency bands in the time domain.The signal is passed through g[n] high pass filter and then through h[n] high pass filter and after filtering since the signal have a highest frequency of p/2 instead of p due to eliminatin of half of the samples according to Nyquist’s rule, so now the signal is downsampled by 2 and every other sample is discarded.This complets the first level of decomposition and the same procedure is repeated for the level two of decompostion and so on.The equation of level one of decomposition is given as:

Here and are the outputs of high pass and lowpass filters after downsampling by 2.

This procedure is also called subband coding and can be used for further decompostion.At every level of decompostion the no. of samples is halved by the filtering and subsampling and half the frequency band spanned and hence double the frequency resolution. The following procedure is shown in the figure 3.4.2 where the x[n] is the original signal and g[n] and h[n] are the highpass and low pass filters, respectively

In Discrete Wavelet Transform(DWT) the half band low pass filter h[n] consist of the approximate information of the signal and the half band high pass filter g[n] consist of the extra details of the original signal.The length of the signal determines the no. of levels to be decomposed for example: if the length of the singal is 1024 then 10 levels of decompostion is possible .Due to successive downsampling the length of signal must be power of 2 , in order to get an efficient scheme.

The reconstruction of the original signal follows the same process as the half band filters form orthonormal bases so the above process is followed in reverse order.At every level the signals are upsampled by 2 and are then passed through g[n] high pass filter and h[n] low pass filter and then added.Since the synthesis and analysis filters are same with exception for time reversal.So the reconstruction formula is given for each layer.

If halfband filters are not ideal then perfect reconstruction is not achieved but it is impossible to get an ideal halfband filters.

3.5 Discrete Wavelet Transform Based OFDM system:

As shown in the figure 3.5, Here on the transmitter side, at first we would pass an original signal x[n] through an encoder, here the encoding of the signal is done. The encoder which we are using here is half rate convoluitonal encoder in case of convolutional coding,a Reed Solomon Encoder in case of Reed Solomon Coding and an LDPC Encoder in case of LDPC Coding. These Coding Schemes are used for the error detection and correction of the original signal.After encoding of the signal with various error detecting and correcting codes the signal is further passed through a modulator for modulation.The modulator which here we are using is a 16QAM(Quadrature Amplitude Modulator).After passing throught modulator the signal is passed through a serial to parallel converter and after that it’s inverse discrete wavelet transform is taken(IDWT) and then the signal is sent into the channel through a parallel to serial converter.The channel here is an arbiratory channel.

Now after passing through the channel the signal is received on the reciever side.Now on the reciever side the received signal containing the noise is passed through equilizer where the intersymbol interference and fading of the channel is removed and further on the (DWT) Discrete Wavelet Transform of the signal is taken and after that the signal is sent to the Demodulater through parallel to serial converter and atlast the decoding of the signal is done after which we get our desired output.

3.6 Advantages and Disadvantages of WOFDM System

Advantages:

Wavelet Based-OFDM System have the capability of representing both time and frequency information simultaneously. Flexibility with time frequency tiling which minimizes the channel disturbances at transmitter which enhance the quality of service(QOS) of wireless systems.

Wavelet Based-OFDM system gives satisfactory results for signals that are non-stationary, noisy and aperiodic.

Wavelet Based-OFDM system can reveal signal aspects which other analysis techniques cannot, which are breakdown points , discontinuities, trends etc. Wavelet Transform provides irregular wavelet shapes as analysis tool to analyze local properties of an input signal.

In Wavelet Based-OFDM system due to it’s small wavelets we can analyze particular part of data which we need to examine for error detection and correction instead of analyzing the whole signal for error detection and correction.

The wavelet scheme reduces the sensitivity of system to harmful channel effects like inter symbol interference (ISI) and Intercarrier carrier interference(ICI).

Disadvantages:

Applications of Discrete Wavelet Transform based OFDM system(DWT-OFDM) System:

Discrete Wavelet Transform have a large number of applications in engineering, computer sciences and mathematics. It’s practical applications can be found in signal processing and in data communication etc. Some of it’s applications are given below:

Audio Compression:

Human Hearing have critical bands ,fm is the masking frequency. A neighboring frequency having a magnitude lower than T(fm,f) is masked by fm and is not audible. T(fm,f) is calculated as:

M(fm) is the masking threshold which is independent of signal. Now for the compression of audio signal the frequency allocation of DWT approximates the critical bands of human ear. The frequencies (fm) having a large power are detected and the masking envelopes (Tfm,f) are computed. From here we compute the masking curve which is used for the determination of wavelet coefficients and which are removed for the compression of signal.

FingerPrint Detection:

FBI uses an algorithm for the compression of grayscale fingerprint images and the algorithm is known as a wavelet/scalar quantization(WSQ) algorithm. This algorithm is divided into three steps: a DWT, scaler quantization and Entropy Coding. A two channel linear phase filter band is used for DWT as it is symmetric and it prevents image content to shift between various subbands. The scaler quantization characteristics is used for quantization of DWT coefficients. At last the entropy coding is done using the Huffman Coding. Huffman Coding is an encoding algorithm which is used for the data compression. Now for reconstruction the entropy decoding is performed and after that quantization and finally an inverse DWT is performed.

Encoder Quantized Denoising:

DWT can filter out the noise at all frequencies. Since for the removal of noise we determine the threshold for various subbands, as the quantization error amplitude is always same so we use the same threshold for the subbands.

Real time featured Detection:

In feauture detection we distinguish signal parts with different frequency content. To distinguish between various frequencies in time we use DWT. Here we follow an on-line approach using a real time implementation of the DWT.

Repetitive Controller:

Repetitive Controllers contain a memory for the error signal and a low pass filter for stability. The DWT is used to reduce the size of the memory of Repetitive Controller. The DWT decomposes the error signal into high pass and low pass coefficients. The levels containing the low pass coefficients are used and memorized and the error signal is compressed.

Speech Recognition:

Speech processing such as speech analysis ,pitch detection and speech recognition uses DWT.

Texture Classification:

DWT is used of texture classification. For analysis and interpretation of images , the human visual system relies for a great extend on texture perception. The discrete wavelet fram analysis performs upsampling on the filters rather than downsampling of image. The edges of the image corresponds to the zero crossing of the Discrete Wavelet Transform(DWT). The zero crossings are derived from the wavelet subbands. Each level of DWT have three subbands for horizontal,vertical and diagonal directions respectively. A feature vector representing the texture is extracted from the transform. An average no. of zero crossing per pixel is determined by the no. of zero crossing of the various subbands. We assume that the texture is homogenous so now only the texture classification is required , the assumption causes no problem. For the classification of the texture we use symmetric wavelet filters which minimizes the distortion due to their linear phase.[ Wavelet Theory and Applications

A literature study

R.J.E. Merry

DCT 2005.53Prof. Dr. Ir. M. Steinbuch

Dr. Ir. M.J.G. van de Molengraft

Eindhoven University of Technology

Department of Mechanical Engineering

Control Systems Technology Group

Eindhoven, June 7, 2005]