Hello ppl,
The idea of wavelet transform has its base idea
from Fourier transform. So, it is mandatory to understand the physical
significance of everything that is mathematical. This post explains the interpretation
of Fourier transform that gives the frequency response of the signal.
It is easy to tell the frequency of a periodic
signal. In case of periodic signal, the frequency can be explained as the number
of times the signal repeats in a unit time (1 second). Given an Aperiodic signal, how can someone
tell the frequencies present in that??
A simple solution can be representing the Aperiodic signal by sum of periodic signals of varying frequencies.
So the next question arises à which periodic signal to use?
At this
point, a very important finding of Fourier, the Fourier series comes handy. The
idea of Fourier series is that, any periodic signal can be represented as a sum
of sine and cosine waves of varying frequencies and amplitude. The mathematical representation is,
There are various forms of writing the same,
which we don’t need as of now.
But, this idea is for periodic signals. What
about Aperiodic signal?
We can represent the Aperiodic signal to be a periodic signal with a period of infinity.
The mathematical derivations for Fourier series and Fourier transform are available in multiple sources. The mathematical representation of Fourier transform aka frequency response of the signal can be given as,
This actually gives the value of the particular frequency present in that signal. But how can we understand this? The Fourier transform here has two dot products. One is the dot product of signal with sine and other with cosine. Here, for all the interpretations, the usage of periodic signals is for better understanding. This concept applies for Aperiodic signals as well.
Interpretation of dot product
The dot product gives the measure of
similarity. We can see that using an example. Here we use sine and cosine
waves.
The below image is the point-wise product of signals. Here the dot product of two sine waves gives a
value around 24 and the dot product of sine and cosine waves give 0. This can be seen in the image below,
So it is easy to see that the dot product gives
the measure of similarity of the two signals.
The Fourier transform, in general gives two
responses of a signal: Magnitude and phase response. We can interpret both the results.
Interpretation of Magnitude response
We are generating a signal which is a sum of sine
and cosine wave of 1 Hz and 2 Hz
Now we are getting the value of the two dot product terms in the Fourier transform for varying frequencies. The dot product with sine waves of varying frequency is given below:
We can see the spike in value only for the
particular frequency. So the magnitude response is the norm of these 2 values.
Any signal can be represented as a sum of sine and cosine waves. So this gives
the magnitude of the particular frequency present in that signal.
Interpretation of Phase response
The complex number, in general can be plotted in a
graph below,
Here, it is easy to get the magnitude of a particular complex value. We are about to try the same here. We are generating sine signal with varying phase angle. We are generating sine signal with varying phase angle. This can be represented by,
sin (t+phase_angle)
The phase angle is varied from 0 to pi. The sine signals with varying phase is shown below,
After getting the Fourier dot products of the signal, we are calculating the phase using the formula,
where the Fourier Transform is of the form, a+ i b. The 'a' is the dot product of signal with cosine waves and 'b' is the negative of dot product of signal with sine waves.
The plot above shows that the phase calculated from FT
is linearly varying with respect to the actual phase change.
The code behind all the images is available in this link.
The concept of dot product is the base of everything.
In the next part of the post, we can discuss the evolution of STFT and wavelets.
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