Wavelet transform Wavelet transform is a relatively new concept (about 10 more years old) First of all, why do we need a transform, or what is a transform anyway? We have answered in Fourier Transform. TIME-DOMAIN FREQUENCY-DOMAIN Signal is stationary

A non-stationary signal. Interval 0 to 300 ms: 100 Hz, Interval 300 to 600 ms: 50 Hz, Interval 600 to 800 ms: 25 Hz, Interval 800 to 1000 ms: 10 Hz sinusoid. FT cannot answer it! At what times (or time intervals), do these frequency components occur?

FT is not a suitable for non-stationary signal, with one exception: FT can be used for non-stationary signals, if we are only interested in what spectral components exist in the signal, but not interested where they occur. However, if this information is needed, i.e., if we want to know, what spectral component occur at what time (interval) , then Fourier transform is not the right transform to use. FT gives what frequency components exist in the signal. Nothing more, nothing less! THE ULTIMATE SOLUTION:THE WAVELET TRANSFORM Wavelet transform is capable of providing the time and frequency information simultaneously, hence giving a time-frequency representation of the signal.

Example: Suppose we have a signal which has frequencies up to 1000 Hz Using filter, we can break the signal into different bands, say, 0-125 Hz, 125-250 Hz,250-500 Hz, and 500-1000 Hz. We can continue this until we have decomposed the signal to a pre-defined certain level. We can plot them on a 3-D graph, time in one axis, frequency in the second, and amplitude in the third axis. This will show us which frequencies exist at which time.

Before introducing WT, let’s look at short time Fourier Transform (STFT) There is only a minor difference between STFT and FT. In STFT, the signal is divided into small enough segments, where these segments of the signal can be assumed to be stationary. How to do it? Using window as a filter. Do FT on series of The red one shows the window located at t=t1', the blue shows t=t2', and the green one shows the window located at t=t3'. These will correspond to three different FTs at three different times. Therefore, we will obtain a true time-frequency representation of the signal.

What if frequency changes continues? we can only know what Interval 0-250 s: 300 Hz, 250-600s: 200 Hz, 600-800s: 100 Hz, 800-1000s: 50 Hz Isn’t that great? 60 30 20 10 5 0 250 600 800 1000 X10 What if frequency changes continues? we can only know what frequency bands exist at what time intervals.

Resolution Problem The uncertainty principle, Heisenberg states that the momentum and the position of a moving particle cannot be known simultaneously. Same thing here, We cannot know what spectral component exists at any given time instant. The best we can do is to investigate what spectral components exist at any given interval of time. This is a problem of resolution Higher frequencies are better resolved in time, and Lower frequencies are better resolved in frequency.

Let’s look at the problem using window If a window has an infinite length, it gets back to FT, which gives perfect frequency resolution, but loses time information. If a window is of finite length, it obtains time information, but it covers only a portion of the signal, which causes the frequency resolution to get poorer.

Narrow window ===>good time resolution, poor frequency resolution. Wide window ===>good frequency resolution, poor time resolution. Examples window

Four peaks are well separated from each other in time. But in frequency domain, every peak covers a range of frequencies, instead of a single frequency. 60 30 20 10 5 0 250 600 800 1000 X10 Peaks are not well separated from each other in time, but, in frequency domain the resolution is much better. 0 250 600 800 1000 60 30 20 10 5 X10

An excellent frequency resolution, but a terrible time resolution 0 250 600 800 1000 60 30 20 10 5 X10 What kind of a window to use? Narrow windows give good time resolution, but poor frequency resolution. This is the main reason why researchers have switched to WT from STFT. STFT gives a fixed resolution at all times, whereas WT gives a variable resolution The main differences between the STFT and the WT The width of the window is changed as the transform is computed for every single spectral component in the WT, but is fixed in the STFT

Continuous Wavelet Transform Fourier analysis Wavelet analysis

A wavelet transform is defined as *: complex conjugate The parameter scale s in the wavelet analysis is similar to the scale used in maps. As in the case of maps, large scales correspond to a non-detailed global view, and small scales correspond to a detailed view. Similarly, in terms of frequency, low frequencies (large scales) correspond to a global information of a signal (that usually spans the entire signal), whereas high frequencies (small scales) correspond to a detailed information of a signal (that usually lasts a relatively short time). Large scale  Low frequency Small scale  High frequency

Fourier analysis Wavelet analysis

Morlet wavelet Small scale compressed high frequency Large scale dilated low frequency

Translation

Commonly used mother wavelet other than Morlet wavelet Haar Mexican Hat Meyer

How to do wavelet transform? Reconstruction

Wavelet transform is much easier and faster in the Fourier domain

Example Morlet wavelet Time domain Fourier domain Wavelets Time domain Fourier domain wavelet transform

Example Mexican hat wavelet Time domain Fourier domain Wavelets Time domain Fourier domain wavelet transform

How to Connect Scale to Frequency? Pseudo-frequency corresponding to a scale. is a scale. is the sampling period. is the center frequency of a wavelet in Hz. is the pseudo-frequency corresponding to the scale a, in Hz.