Based on the formula Δ f=1/T=fs/N, it can be understood that in order to improve time or frequency resolution one has to either change signal properties or adjust window length. Then how can it be claimed that a particular signal processing technique offers better resolution than the other? For instance, it is well known that Stockwell transform has a better resolution as compared to Short Time Fourier Transform, BUT HOW?
Linear (and quadratic also) transforms are subject to the uncertainty principle, which states that you cannot improve time resolution AND frequency resolution at the same time. This means that you cannot improve both of them at the same time, and thus, if a transform is reported to have, say, "better" time resolution than another transform, it means that it will have "worst" frequency resolution, because the gain in one must be compensated by a lost of the other.
Create 2 sinusoidal signals, add noise, subject them to the 2 competing spectrum analysers. Check which one does separate the 2 frequencies better when their spectral distance approaches the inverse of the recording interval.
Dear Fernando, thank you for answering. Actually I need an answer from the perspective of characteristics of the transform used. Obviously, a better mathematical modelling or design consideration would lead to a more robust signal processing technique. But what characteristic, in particular, leads to resolution improvement? Is it adaptability of window being used in the transform or something else?
The main characteristic that leads to the improvement in spectral resolution is the choice of assumptions made about the behaviour of the signal outside the collection interval. For example, the Fourier-based spectrum analyser assumes periodicity while autoregressive-based spectrum analysers assume that the average value and a set of moments (derivatives) of the signals are maintained. This is akin to the Taylor series, where the future value depends on the previous value plus a weighted sum of first derivative, second derivative and so on.
Linear (and quadratic also) transforms are subject to the uncertainty principle, which states that you cannot improve time resolution AND frequency resolution at the same time. This means that you cannot improve both of them at the same time, and thus, if a transform is reported to have, say, "better" time resolution than another transform, it means that it will have "worst" frequency resolution, because the gain in one must be compensated by a lost of the other.
Dear Researchers, thank you for sharing your knowledge. Aparna, sorry I couldn't understand much from your answer. This thread became more interesting with the answer given by Leandro ! He has mentioned exactly what I wanted to discuss.
Let me draw your attention towards this interesting work: Synchroextracting transform (https://ieeexplore.ieee.org/abstract/document/7906573/)
Can we categorize this technique under linear/ quadratic transform (because it is a post-processed version of STFT)? Please share your views on the time-frequency resolution of this technique. It offers a frequency resolution as less as 1 Hz. And I think the time-frequency cell arrangement popular with the STFT and wavelet transforms would not be applicable here. This further creates a doubt regarding the adherence to the uncertainty principle. What do you all think?
Best Regards!
You should explore more on the theory behind Wigner-Ville distribution, its relation with STFT (you can obtain STFT from WV distribution), and its limitations.
To answer in short: the "transform" you are studying is not invertible, as it produces only an approximate reconstruction. It looses information. It will allow you to more precisely determine the frequency content of a monocomponent signal or one with very few simple, non crossing modes (that is why in all examples you see signals of those characteristics). I would like to see how it behaves in a more complex multimodal signal (like a speech signal, for example). I would not call it a "Transformation" in the sense of a complete representation of a signal, but more as an analysis technique allowing to determine the instantaneous frequencies for multicomponent signals where each mode does not interfer with each other. If that model is fulfilled, the method would allow to identify the instantaneous frequencies of each mode and its evolution, allowing for an approximate reconstruction by superposition of those modes.
There is a lot of bibliography on signal reconstructions from ridges (you can check the book "A wavelet tour on signal processing" by S. Mallat et al, in which signal reconstruction from ridges is presented for both STFT and wavelet transforms. In fact, I don't see why they have accepted to publish this paper as "new" material, on an already widely known method. As I mentioned, all the theory of that paper was already presented in section 4.4.1 of chapter 4 of Mallat's book (in my second edition version from 1998, in page 94). I recommend you the reading of that chapter, you will see that all equations in your referred paper are already written there.
Thank you Leandro ! Your comments are really insightful. I will definitely go through the literature that you have suggested.
In the endorsement and completion of the Leandro's hints, I suggest you other relevant books as "Wavelets and their Applications" by M. Misiti et al (esp. chapters 2 and 5) and "Wavelets for Sensing Technologies" by A.K. Chan and C. Peng (esp. chapters 2 and 3).
Dear Mahdi, thank you for your suggestions. I will go through these books.
Hello Utkarsh,
You might find answers to your questions by checking out the following references about synchrosqueezing for time-varying spectral analysis (inversible transform):
- Auger et al., " Time-Frequency Reassignment and Synchrosqueezing - An Overview ", 2013 - http://perso.ens-lyon.fr/patrick.flandrin/06633061.pdf
- Pham et al., " High-Order Synchrosqueezing Transform for Multicomponent Signals Analysis - With an Application to Gravitational-Wave Signal ", 2017 - http://www-ljk.imag.fr/membres/Sylvain.Meignen/recherche/High_order_sst.pdf
- Thakur et al., " The Synchrosqueezing algorithm for time-varying spectral analysis: Robustness properties and new paleoclimate applications ", 2013 - https://pdfs.semanticscholar.org/5d42/527fc87e42735c0d08d21afa528e3101b759.pdf
- Thakur et al., " Synchrosqueezing-based Recovery of Instantaneous frequency from Nonuniform Samples ", 2011 - https://pdfs.semanticscholar.org/2a94/f477426e10a1d00bd3add18eb25c3ad498af.pdf
Regards
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