Introduction to Wavelet Transform Chapter 7 Instructor: Hossein Pourghassem Introduction Most of the signals in practice, are TIME-DOMAIN signals in their raw format. It means that measured signal is a function of time. Why do we need the frequency information? In many cases, the most distinguished information is hidden in the frequency content of the signal. Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 2 1
Stationary signal Frequency content of stationary signals do not change in time. All frequency components exist at all times 20Hz x( t) = cos(2π f1t) + cos(2πf 2t) +... + cos(2πf nt) 80Hz 120Hz Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 3 Fourier Transformation In 19 th century, the French mathematician J. Fourier, showed that any periodic function can be expressed as an infinite sum of periodic complex exponential functions. X(f) = + x(t) e -2πjft Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 4 2
X(f) Fourier Transformation = + x(t) e -2πjft Raw Signal (time domain) cos( 2π ft) + j.sin(2πft) x(t) cos(2πft) 1 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8-1 0 0.2 0.4 0.6 0.8 1 5Hz 1 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8-1 0 0.2 0.4 0.6 0.8 1 10Hz 5 1 Hz Σ x(t).*cos(2πft) = -8.8e-15 Amplitude 1 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8-1 0 0.2 0.4 0.6 0.8 1 time Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 6 3
2 Hz Σ x(t).*cos(2πft) = -5.7e-15 Amplitude 1 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8-1 0 0.2 0.4 0.6 0.8 1 time Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 7 3 Hz Σ x(t).*cos(2πft) = -4.6e-14 Amplitude 1 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8-1 0 0.2 0.4 0.6 0.8 1 time Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 8 4
4 Hz Σ x(t).*cos(2πft) = -2.2e-14 Amplitude 1 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8-1 0 0.2 0.4 0.6 0.8 1 time Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 9 4.8 Hz Σ x(t).*cos(2πft) = 74.5 Amplitude 1 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8-1 0 0.2 0.4 0.6 0.8 1 time Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 10 5
5 Hz Σ x(t).*cos(2πft) = 100 Amplitude 1 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8-1 0 0.2 0.4 0.6 0.8 1 time Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 11 5.2 Hz Σ x(t).*cos(2πft) = 77.5 Amplitude 1 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8-1 0 0.2 0.4 0.6 0.8 1 time Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 12 6
6 Hz Σ x(t).*cos(2πft) = 1.0e-14 Amplitude 1 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8-1 0 0.2 0.4 0.6 0.8 1 time Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 13 20, 80, 120 Hz FT (X) Amplitude Frequency 14 7
Non-Stationary signal Frequency content of stationary signals change in time. Magnitude 20 Hz 80 Hz 120 Hz Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 15 FT (X) Amplitude Frequency 16 8
Conclusion So, how come the spectrums of two entirely different signals look very much alike? Recall that the FT gives the spectral content of the signal, but it gives no information regarding where in time those spectral components appear. Once again please note that, the FT gives what frequency components (spectral components) exist in the signal. Nothing more, nothing less. Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 17 Conclusion Almost all biological signals are non-stationary. Some of the most famous ones are ECG (electrical activity of the heart, electrocardiograph), EEG (electrical activity of the brain, electroencephalogram), and EMG (electrical activity of the muscles, electromyogram). ECG EEG EMG Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 18 9
Short-Time Fourier Transformation Can we assume that, some portion of a non-stationary signal is stationary?? The answer is yes. In STFT, the signal is divided into small enough segments, where these segments (portions) of the signal can be assumed to be stationary. For this purpose, a window function "w" is chosen. Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 19 Short-Time Fourier Transformation FT X Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 20 10
Short-Time Fourier Transformation FT X Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 21 Short-Time Fourier Transformation FT X Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 22 11
Short-Time Fourier Transformation FT X Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 23 Short-Time Fourier Transformation FT X Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 24 12
Short-Time Fourier Transformation FT X Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 25 Short-Time Fourier Transformation FT X Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 26 13
Short-Time Fourier Transformation FT X Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 27 Short-Time Fourier Transformation FT X Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 28 14
Short-Time Fourier Transformation FT X Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 29 Short-Time Fourier Transformation FT X Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 30 15
Window width = 0.05 Time step = 100 milisec time-frequency representation (TFR) Amplitude Frequency Time step Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 31 FT X(f) = + 2 x(t) e - πjft dt STFT X(t, f) = + 2 [x(t) ω(t - t')] e - πjft dt ω ω ω 32 16
Window width = 0.02 Time step = 10 milisec Amplitude Amplitude Frequency Time step Time step Amplitude Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 33 Frequency Narrow windows give good time resolution, but poor frequency resolution. Window width = 0.1 Time step = 10 milisec Amplitude Amplitude Frequency Time step Time step Amplitude Wide windows give good frequency resolution, but poor time resolution; Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 34 Frequency 17
Conclusion What kind of a window to use`? The answer, of course, is application dependent: If the frequency components are well separated from each other in the original signal, than we may sacrifice (loss) some frequency resolution and go for good time resolution, since the spectral components are already well separated from each other. The Wavelet transform (WT) solves the dilemma of resolution to a certain extent, as we will see. Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 35 Multi Resolution Analysis MRA, as implied by its name, analyzes the signal at different frequencies with different resolutions. Every spectral component is not resolved equally as was the case in the STFT. MRA is designed to give good time resolution and poor frequency resolution at high frequencies and good frequency resolution and poor time resolution at low frequencies. This approach makes sense especially when the signal at hand has high frequency components for short durations and low frequency components for long durations. Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 36 18
Continuous Wavelet Transformation The continuous wavelet transform was developed as an alternative approach to the short time Fourier transform to overcome the resolution problem. The wavelet analysis is done in a similar way to the STFT analysis, in the sense that the signal is multiplied with a function, wavelet, wavelet similar to the window function in the STFT, and the transform is computed separately for different segments of the time domain signal. Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 37 Mexican hat Morlet 2 t 2 1 2 t 2σ ω( t) = e 3 2 2πσ σ t 2σ ω = e iat e ( t 1) Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 38 2 19
t = 0 Scale = 1 Ψ(s,t) Inner product x(t) X Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 39 t = 50 Scale = 1 Ψ(s,t) Inner product x(t) X Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 40 20
t = 100 Scale = 1 Ψ(s,t) Inner product x(t) X Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 41 t = 150 Scale = 1 Ψ(s,t) Inner product x(t) X Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 42 21
t = 200 Scale = 1 Ψ(s,t) Inner product x(t) X Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 43 t = 200 Scale = 1 0 Ψ(s,t) Inner product x(t) X Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 44 22
t = 0 Scale = 10 Ψ(s,t) Inner product x(t) X Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 45 t = 50 Scale = 10 Ψ(s,t) Inner product x(t) X Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 46 23
t = 100 Scale = 10 Ψ(s,t) Inner product x(t) X Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 47 t = 150 Scale = 10 Ψ(s,t) Inner product x(t) X Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 48 24
t = 200 Scale = 10 Ψ(s,t) Inner product x(t) X Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 49 Scale = 10 0 Ψ(s,t) Inner product x(t) X Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 50 25
Scale = 20 Ψ(s,t) Inner product x(t) X Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 51 Scale = 30 Ψ(s,t) Inner product x(t) X Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 52 26
Scale = 40 Ψ(s,t) Inner product x(t) X Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 53 Scale = 50 Ψ(s,t) Inner product x(t) X Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 54 27
Continuous Wavelet Transformation CWT ψ x ( τ,s) = 1 s x(t) ψ t τ ( )dt s As seen in the above equation, the transformed signal is a function of two variables, τ and s, the translation and scale parameters, respectively. Ψ(t) is the transforming function, and it is called the mother wavelet. If the signal has a spectral component that corresponds to the value of s, the product of the wavelet with the signal at the location where this spectral component exists gives a relatively large value. Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 55 Magnitude 20 Hz 50 Hz 120 Hz Translation increment=50 milisecond Scale inc.=0.5 56 28
10 Hz 20 Hz 60 Hz 120 Hz Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 57 1. Introduction Signal Processing. Extract information from a signal. Time domain is not always the best choice. Frequency domain: Fourier Transform: Easily implemented in computers: FFT. Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 58 29
1. Introduction. Fourier Analysis. Can not provide simultaneously time and frequency information: Time information is lost. Small changes in frequency domain cause changes everywhere in the time domain. For stationary signals: the frequency content does not change in time, i.e.: Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 59 1. Introduction. Fourier Analysis. Basis Functions are set: sinusoids. Many coefficients needed to describe an edge or discontinuity. If the signal is a sinusoid, it is better localized in the frequency domain. If the signal is a square pulse, it is better localized in the time domain. Short Time Fourier Transform Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 60 30
1. Introduction. Short Time Fourier Analysis. Time-Frequency Analysis. Windowed Fourier Transform (STFT). Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 61 1. Introduction. Short Time Fourier Analysis. Discrete version very difficult to find. No fast transform. Fixed window size (fixed resolution): Large windows for low frequencies. Small windows for high frequencies. Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 62 31
1. Introduction. Wavelet Transform. Small wave. Energy concentrated in time: analysis of nonstationary, transient or time-varying phenomena. Allows simultaneous time and frequency analysis. Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 63 1. Introduction. Wavelet Transform. Fourier: localized in frequency but not in time. Wavelet: local in both frequency/scale (via dilations) and in time (via translations). Many functions are represented in a more compact way. For example: functions with discontinuities and/or sharp spikes. Fourier transform: O(nlog2(n)). Wavelet transform: O(n). Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 64 32
1. Introduction Wavelet Transform. t b WT ( a, b) = < f, ψ, b >= f ( t) ψ ( a a ) dt Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 65 1. Introduction Wavelet Transform: The effect of the Wavelet Transform is a convolution to measure the similarity between a translated and scaled version of the Wavelet and the signal under analysis. Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 66 33
2. Definitions Mathematical Background. Topic of pure mathematics, but great applicability in many fields. Linear decomposition of a function: i Integer index for the finite or infinite sum c i Expansion Coefficients ψ i Expansion Set Decomposition Unique: { ψ i ( t) } Basis for f ( t) Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 67 2. Definitions Mathematical Background. Basis orthogonal Then the coefficients can be calculated by : Since: Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 68 34
2. Definitions Mathematical Background. Linear decomposition of a vector: {(1,0,0),(0,1,0),(0,0,1)} (3, 2,1) Basis in R 3 Generator system In R 3 Independent Orthogonal Normal Basis in R 3 Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 69 2. Definitions Mathematical Background Otherwise : Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 70 35
2. Definitions Mathematical Background. Example of linear decomposition of a function. Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 71 2. Definitions Mathematical Background. Example of linear decomposition of a function: Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 72 36
2. Definitions Example: Four dimensional space (four non-null coefficients). Synthesis Formula: Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 73 2. Definitions Example: Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 74 37
2. Definitions Example: Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 75 2. Definitions Example: If f columns are orthonormal. Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 76 38
3. Multiresolution Analysis Multiresolution Formulation. If a set of signals can be represented by a weighted sum of ϕ(tk), a larger set (including the original), can be represented by a weighted sum of ϕ(2t-k). Signal analysis at different frequencies with different resolutions. Good time resolution and poor frequency resolution at high frequencies. Good frequency resolution and poor time resolution at low frequencies. Used to define and construct orthonormal wavelet basis for L 2. Two functions needed: the Wavelet and a scaling function. Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 77 3. Multiresolution Analysis. Multiresolution Formulation. Effect of changing the scale. Scaling function: ϕ(t) To decompose a signal into finer and finer details. Subspace: Increase the size of the subspace changing the time scale of the scaling functions: Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 78 39
3. Multiresolution Analysis. Multiresolution Formulation Subspace: The spanned spaces are nested: Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 79 3. Multiresolution Analysis. Multiresolution Formulation. Increasing j, the size of the subspace spanned by the scaling function, is also increased. Wavelets span the differences between spaces w i. More suitable to describe signals. Wavelets and scaling functions should be orthogonal: simple calculation of coefficients. Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 80 40
3. Multiresolution Analysis. Multiresolution Formulation. scaling function coefficients Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 81 3. Multiresolution Analysis Example: Multiresolution Formulation Haar Wavelet and Scaling Function: Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 82 41
3. Multiresolution Analysis Multiresolution Formulation Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 83 4. Discrete Wavelet Transform. Wavelet Transform: Building blocks to represent a signal. Two dimensional expansion. Wavelet set: Discrete Wavelet Trannsform of f(t) Time-Frequency localization of the signal. The calculation of the coefficients can be done efficiently. Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 84 42
4. Discrete Wavelet Transform. Wavelet Transform: Wavelet Systems are generated from a mother Wavelet by scaling and translation. Multiresolution conditions satisfied. Filter bank: the lower resolution coefficients can be calculated from the higher resolution coefficients by a tree-structured algorithm. Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 85 4. Discrete Wavelet Transform. Wavelet Transform Calculation: 1. Choose a mother wavelet. 2. Given two values of j and k, calculate the coefficient according to: 3. Translate the Wavelet and repeat step 2 until the whole signal is analyzed. 4. Scale the Wavelet and repeat steps 2 and 3. Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 86 43
4. Discrete Wavelet Transform. Discrete Wavelet Transform Calculation: Follow steps 1-4 defined previously. Change the expressions: Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 87 4. Discrete Wavelet Transform. Discrete Wavelet Transform Calculation: Using Multiresolution Analysis: Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 88 44
4. Discrete Wavelet Transform. Discrete Wavelet Transform Calculation: Using Multiresolution Analysis Dyadic grid Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 89 4. Discrete Wavelet Transform Discrete Wavelet Transform Calculation: Calculation to be implemented in a computer. Is there any method, such as FFT, for DWT efficient calculation? Pyramid Algorithm Basic Idea: DWT (direct and inverse) can be thought of as a filtering process. Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 90 45
4. Discrete Wavelet Transform Pyramid Algorithm: Filtering: π Signal is passed through a halfband lowpass filter Sampling Frequency:2 Bandwidth: π π Signal is passed through a halfband highpass filter Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 91 4. Discrete Wavelet Transform Pyramid Algorithm: After filtering, half of the samples can be eliminated: subsample the signal by two. Subsampling: Scale is doubled. Filtering: Resolution is halved. Decomposition is obtained by successive highpass and lowpass filtering. Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 92 46
4. Discrete Wavelet Transform Pyramid Algorithm: Filters h and g are not independent. Relation between filters Quadrature Mirror Filters. Signal length must be a power of 2 (due to down sampling by 2). Maximum decomposition level depends on signal length. Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 93 4. Discrete Wavelet Transform Pyramid Algorithm: Wavelet Decomposition Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 94 47
4. Discrete Wavelet Transform Pyramid Algorithm: Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 95 4. Discrete Wavelet Transform Wavelet Decomposition Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 96 48
4. Discrete Wavelet Transform Wavelet Decomposition (book notation) Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 97 4. Discrete Wavelet Transform Signal Analysis and Synthesis (book notation) Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 98 49
4. Discrete Wavelet Transform- wavelet packet Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 99 4. Discrete Wavelet Transform Pyramid Algorithm: Reconstruction Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 100 50
4. Discrete Wavelet Transform Wavelet Reconstruction Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 101 4. 2D Wavelet Transform- Decomposition Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 102 51
4. 2D Wavelet Transform- Synthesis Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 103 Examples: Wavelet decomposition Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 104 52
Examples: full wavelet packet decomposition Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 105 Examples: Optimal wavelet packet decomposition Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 106 53
4. Discrete Wavelet Transform. Wavelet Transform: Wavelet Transform representation: Time-Scale Plane: Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 107 4. Discrete Wavelet Transform. Effect of choosing different Mother Wavelets: The Wavelet transform in essence performs a correlation analysis, therefore the output is expected to be maximal when the input signal most resembles the mother wavelet. Depending on the application, mother wavelet should be as similar as possible to the signal or to the signal portion under analysis. For example, the Mexican Hat Wavelet is the optimal detector to find a Gaussian. Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 108 54
4. Discrete Wavelet Transform. Effect of choosing different Mother Wavelets: Adaptive approach Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 109 5.1- Application : Denoising Finite Length Signal with Additive Noise Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 110 55
5.1- Denoising General Method: 1) Choose a wavelet and number of levels. 2) Threshold the detail coefficients. For example, hard thresholding, (setting to zero the elements whose absolute values are lower than threshold.). 3) Perform a wavelet reconstruction based on the original approximation and modified detail coefficients. Questions: Which Thresholding Method? Which Threshold? Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 111 5.1- Denoising Thresholding Method (Diagonal Linear Projection): Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 112 56
Example: Denoising Symlets wavelet, level 2 Threshold = 94.9 Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 113 5.2.- Abrupt Change Detection Low level details contain information of high frequency components. Small support wavelets are better suited to detect such changes. Noise makes it difficult. Application: QRS complex detection. Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 114 57
5.3.- Long-Term Evolution. Foundations: Slow changes which take place in the signal, i.e., low frequency variations. In the frequency domain: the higher the approximation level, the narrower the low pass filter is. Problem: depending on the baseline (trend) frequency, the approximation level may change. The baseline is considered as additive noise. Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 115 5.3.- Long-Term Evolution. Finite Length Signal with baseline variation: Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 116 58
5.3.- Long-Term Evolution. With the Wavelet Approximation of certain level, we estimate the baseline. It can be used as extra information about the signal or to reduce the baseline oscillation. Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 117 5.3.- Long-Term Evolution. Application: Electrocardiogram baseline reduction. Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 118 59
5.4.- Compression Methods: Quantization in the Wavelet domain. The set of possible coefficients values is reduced: Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 119 5.4.Compression Implementation: 1) Perform wavelet transform on the signal or image (Basis choice is very important.). 2) Discard those coefficients considered insignificant : Thresholding. 3) Perform inverse wavelet transform on the modified coefficients. Coefficients Histogram: Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 120 60
5.4.Compression Application: ECG compression. Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 121 6- Other Topics Feature Extraction. Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 122 61