1. 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

2. 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?

3. 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.

4. 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, Hz, Hz, and 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.

5. 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.

6. What if frequency changes continues? we can only know what
Interval s: 300 Hz,
s: 200 Hz,
s: 100 Hz,
s: 50 Hz
Isn’t that great?
X10
What if frequency changes continues? we can only know what
frequency bands exist at what time intervals.

7. 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.

8. 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.

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

10. 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.
X10
Peaks are not well separated
from each other in time, but,
in frequency domain the
resolution is much better.
X10

11. An excellent frequency resolution, but a terrible time resolution
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

12. Continuous Wavelet Transform
Fourier analysis
Wavelet analysis

13. 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

14. Fourier analysis
Wavelet analysis

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

16. Translation

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

18. How to do wavelet transform?
Reconstruction

19. Wavelet transform is much easier and faster in the Fourier domain

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

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

23. 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.