The Fourier transform does not give information about the local time–frequency
characteristics of the signal which are especially important in the context of nonstationary
signal analysis. Various other mathematical tools like the Wavelet transform,
Gabor transform,Wigner–Ville distribution, etc. have been used to analyze such
kind of signals. The Fractional Fourier transform (FrFT) which is a generalization
of the integer order Fourier transform can also be used in this context
Its probable easiest to illustrate the difference with examples. Try to visualise your time signal as being the x-axis of a graph, and the Fourier domain (frequency domain) as being the Y-axis of a graph. This is exactly how you would represent a STFT. The Fourier transform operation is a mapping that "rotates" your data from the time axis to the frequency one.
As a side note, you may find that the STFT does what you seem to require in that is allows you to look at the changing frequency content as a function of time.
Now ask yourself what happens if you don't fully rotate by 90 degrees from time to frequency - you end up somewhere in between. This is what a fractional Fourier transform does; it takes you to a domain that is neither frequency or time but between the two
Note that the concept of frequency applies when your basis functions are (co)sinusoids. However we could use a different set of basis functions, that rather than expanding and contracting by changing thier frequency, are stretched by a scaling factor. You could use any family of pulses from the wide variety availableand decompose your signals into pulses of different scales and amplitudes; this is a wavelet transform. Clearly when looking to decompose against a wavelet base, frequency is now an irrelevant concept
Hope that helps
A wavelet is a mathematical function used to divide a given function or continuous-time signal into different scale components. Usually one can assign a frequency range to each scale component. ... A wavelet transform is the representation of a function by wavelets.
Hi
Please check the following work. It describes the matters in details. I believe it will help you.
https://www.researchgate.net/publication/3342219_On_the_relationship_between_the_Fourier_and_fractional_Fourier_transforms
Regards
The question asked is related to the Fractional Fourier Transform (FrFT) and the Wavelet Transform (DWT).
DWT is known to be superior to ordinary FT since it performs a multi-resolution analysis.
However, FrFT is far superior to DWT in terms of spatial analysis capability since DWT can analyze an image in only 3 directions: horizontal, diagonal and vertical.
In very simple terms, the fractional Fourier transform (FrFT) is a generalization of the ordinary Fourier transform. Specifically, the FrFT implements the so-called order parameter a which acts on the ordinary Fourier transform operator. In other words, the alpha-th order fractional Fourier transform represents the alpha-th power of the ordinary Fourier transform operator. When alpha=pi/2, we obtain the Fourier transform, while for alpha=0, we obtain the signal itself. Any intermediate value of alpha (0
in the mathematics in the area of harmonic analysis,the fractional Fourier transform is a family of linear transformations generalizing the Fourier transform.it can be thought of as the Fourier transform to the nth power,where n need not to be an integer thus,it can transform a function to any intermediate domain between time and frequency
Dear ITTAPALLI
When you are not an expert in the field, you should avoid answering a question by copying and pasting from WIKIPEDIA.
Such an attitude is not beneficial to the scientific community, nor to our Ph.D students.
Regards