# FFT with High Time and Frequency Resolution

Can anyone describe in a bit more detail or provide references to the techniques for simultaneous enhancements to time and frequency resolution described in a high level in HEAD acoustics International - Advanced FFT Analysis HSA - frequency and time resolution as you want video? Can anyone think of any downsides?

The time "overlap" is mostly straightforward, but I don't quite follow how different window sizes for the same time step would be aggregated together.

• All high resolution methods make assumptions about the signal, i.e. they assume a certain signal model. The downsides are that anything that doesn't match the model is lost or misrepresented. This particular example apparently uses a sparse sinusoidal model. Look for relevant patents if you want to know the details, but be aware that what you've been watching is a marketing video. Commented Apr 25, 2022 at 7:12
– Royi
Commented May 6, 2022 at 6:01
• This is simple minded but is true. The sample rate, $f_\mathrm{s}$ alone sets the degree of time-resolution. The time resolution is $\frac{1}{f_\mathrm{s}}$. The frequency resolution is set by the sample rate, $f_\mathrm{s}$, and the FFT size, $N$, and is $\frac{f_\mathrm{s}}{N}$. The choice of windows does help smooth things out, which is why I recommend the Gaussian window for time/frequency analysis. Commented May 30, 2023 at 20:47

As Jazzmaniac said in his comments: this is a marketing video which is heavy on hype and light on technical details.

There is no way around the basic limitation of frequency and time resolution. The techniques used in HSA appear to be standard interpolation methods: you can adjust hop size and window length independently. This is all fine and well: You just need to be aware that this interpolation of the original data and NOT new data. You don't learn anything new, you just display it differently.

There is certainly nothing wrong in optimizing the display of your data to the properties of your signal or the requirements of your application. It's just not a "one size fits all" approach that is fundamentally better (or worse) than any other approach.

Specially this works really well in the video, because the frequencies of the signal only change very slowly with time. If you would use the same analysis parameters for a signal with rapid pitch changes (such as someone singing or speech), you would see that the time transitions are very smeared out and it would be quite difficult to see when the pitch change actually happens.

So it all boils down to: use the right tool and parameters for the task at hand. In some cases that may be the HSA as advertised but in many cases it will not.

• This is incorrect. You mean given no information about the signal you are limited to the resolution by the uncertainty principal applied on the time interval. But are you dealing with signals which you have no information on? I'd guess that rarely.
– Royi
Commented Apr 25, 2022 at 8:32
• Sorry I don't understand the comment. If you use the Fourier Transform you are ALWAYS limited by it's uncertainly principal. It doesn't matter what you know or don't know about the signal Commented Apr 25, 2022 at 11:12
• We talking here in the context of resolution of frequency of the DFT. The ability to see different frequency components within a limited time interval. The resolution you imply is from the limitation of the time window. This can be beaten and I have shown it multiple times on this community.
– Royi
Commented Apr 25, 2022 at 12:20
• @Royi there's nothing surprising about the fact that you can fit model parameters to beat time-frequency uncertainty, but you lose generality and don't have a linear decomposition of the signal. Time-frequency uncertainty is a mathematical fact for linear decompositions. So what's your point? That prior knowledge can improve your estimate? Sure, but most of the time it's not prior knowledge that is being applied but mere speculation that leads to the assumption of the validity of a certain model. And the outcome doesn't tell you if the model assumptions were valid. Commented Apr 25, 2022 at 12:28
• @Jazzmaniac, It is highly surprising that so many DSP engineers don't under stand on which cases the assertion of frequency resolution holds. They all think always while the whole Fourier Analysis (In DSP context) is based on the fact that I don't know anything about the signal but its bandwidth. Yet in most DSP applications you know much more than the bandwidth. You're asking the questions like most of the people here know about priors and modeling.
– Royi
Commented Apr 25, 2022 at 13:35

As others stated: without extra information you cannot increase the time resolution and the frequency resolution at the same time. But imagine there is only one sinusoid and you know it, then you can calculate the real frequency and amplitude perfectly. 3 samples of a sine wave are already enough to do it. So if you know your signal consists of some sine waves then you can do such tricks. The FFT is perfect, it does not add or remove any information.

I disagree with the following:

• A) Heisenberg applies no matter what
• B) Heisenberg can only be broken with assumptions

Synchrosqueezing proves this. I'll avoid going into detail, but B is true in the general case, and A is simply false:

• "B not always true": if the signals are nice enough, we can localize them in time and in frequency almost perfectly, over a subset of the observation interval - e.g. if our data is 10 secs, we may perfectly know 5 secs of it. This pertains to information scarcity near boundaries.
• "A isn't true": statistical priors fill the missing information that the limitation assumes. Heisenberg's principle itself is an assumption that we lack certain knowledge, to maximize generality.

Incidentally, I'll also comment that I was onto something here.

Time frequency resolution is a long debate in the DSP communities.
But modern models has proved that the resolution isn't limited by the DFT because usually we know more about the signal but its bandwidth.

Remark: Usually the counter argument is the Uncertainty Principle. In this context we should understand that it holds given all the knowledge we have is limited to the bandwidth of the signals.

Resolution, or the ability to see different signals as different components, is a function of the knowledge of the signal in question.
If you know nothing, then you probably have zero resolution as you can not infer anything form what you see.

If all you know is something about the signal bandwidth, then you're limited by the properties of the DFT once you apply it.

The magic happens when you add more information to the problem. Information either on the signal, the other signals or the properties of the system itself which is the case in the video.

By skimming through the video I'd describe their method as following:

1. Apply DFT.
2. Since the time interval (Time window) is known and modeled as multiplication with the original data one could infer the data in the DFT is a result of a convolution of the signal of interest and the DFT transform of the window.
3. The above model leaves us with a Deconvolution problem. Which in many cases can be solved.
4. Solving the deconvolution model using iterative method.
5. Mitigating the noise amplification due to the deconvolution.

The nice thing about this model is that it almost, at least not explicitly, assumes nothing about the data. But if it almost assumes nothing, how can it beat the known limitations.
Well, there are assumptions which are buried in the deconvolution process. To name few:

1. The SNR is quite high.
2. The signal of interest is following the model of the deconvolution. Usually it means it is LPF, In many other cases smooth, etc...
3. The artifacts by other surrounding signals are limited (Namely they don't match the "high pass pattern" of the deconvolution process).
• I'd be happy for a feedback from the one who -1.
– Royi
Commented May 30, 2023 at 12:35