Methods to increase FFT time resolution without lowering FFT size

Question

It is known that in FFT time and frequency resolution are inversely related if FFT resolution is increased then time resolution will decrease. However is it possible to overcome issues in time resolution by methods such as overlap FFT etc. and also how much is theoretically possible given all other parameters such as sampling frequency are fixed?

Edit: Visualization of the problem shown below:

STFT of the above signal is to be performed and the start of the pulse is to be estimated to the highest possible accuracy. Also, when significant noise is added the signal above, the resulting signal in time domain looks like below:

Now the presence of the signal can be determined by taking an FFT when the overlap window starts sliding left beyond the 200th sample, but is it still possible to determine where exactly the signal started even though it is buried in noise?

Answer 1

Yes. Windowing will decrease the frequency resolution and therefore increase time resolution. The resolution bandwidth as the inverse of time resolution is well tabulated in the classic paper by fred harris "On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform". Further you can compute the resolution bandwidth for any window directly with the following formula:

$$B = N \frac{\sum(w[n]^2)}{(\sum w[n])^2}$$

Where $B$ is the equivalent noise bandwidth in bins, and $w[n]$ are the window values, equal in length to the FFT.

The OP has clarified that the objective is to determine the starting edge of a pulse in time. I assume the pulse is bandwidth constrained in the presence of other significant noise or interference in other frequencies and hence the motivation to use the FFT as an approach to band filtering the signal of interest. I would suggest alternatively time domain techniques rather than frequency domain for purpose of determining time domain features. Specifically consider using a time-domain band pass filter to select the frequency range of interest and then from the filtered result in the time domain the time location for the start of the pulse can be easily determined with threshold detection on the pulse amplitude.

Further details, considerations and suggestions with that approach are as follows:

Use a linear phase filter so that the introduced delay of the filter can be easily removed / compensated for and will not cause any signal distortion (for a linear phase filter the delay is simply half the number of coefficients).

The bandwidth of the filter will effect the rise time of the resulting pulse, so there will be a trade of time accuracy and noise. For example, if the pulse itself was rectangular, this would require infinite bandwidth, which would also introduce the maximum amount of noise to the point where we may not be able to detect any signal in the noise. As we reduce the bandwidth, rise time will increase and noise will decrease, but the slower rise time will give us more uncertainty in the actual time event of what we refer to as "start of pulse" in the presence of noise. As a guideline, the relationship between the 10% to 90% rise and fall time to bandwidth is:

$$t_r = \frac{0.35}{BW}$$

Where $t_r$ is the 10% to 90% rise time (or 90% to 10% fall time) in seconds, and $BR$ is the bandwidth in Hz. This is specifically for a first order system, but will approximately hold very well for higher order systems as well.

Answer 2

is it possible to overcome issues in time resolution by methods such as overlap FFT

Yes and no. You can certainly increase time resolution by having more overlap, but this results only in interpolation but not additional information

Answer 3

The answer depends on your assumptions and facts about the signal, and what you really require when using the word "resolution". If the signal is stationary with isolated spectral peaks surrounded by a significantly lower noise floor, then it is possible to increase spectral peak estimation resolution if those assumptions are met. But you will get bogus results if those requirements are not met (e.g. won't work if you are trying to resolve closely spaced spectral peaks, highly varying-in-time spectrum, or spectrum of interest buried in noise).

If the signal meets the above requirements, then you can interpolate spectral peaks between FFT result bins from a single FFT using Sinc (or window-transform) interpolation. Or you can use phase vocoder analysis or spectrum reassignment algorithms with an overlapped set of FFT results to increase spectral peak estimation precision (such as for a higher resolution spectrogram plot). The magnitude and phase differences between overlapped FFT frames provides the information gain (assuming spectral stationarity across those overlaps).

For more time resolution from a given FFT length, you can always use a narrower window, either a window the same length with a narrower main lobe, or a shorter window with zero-padding on one or both sizes. But with narrower windows, if you don't want to lose information or miss events between FFT frames, you may need more frame overlap.

If you can't overlap frames more in your original capture, but still need more time resolution, you could always IFFT each frame, make copies, and then FFT each copy using different sub-offsets of narrower windows.

As for how much is theoretically possible, for a single pure sinusoid in zero noise you can estimate all its parameters using only 3 or 4 non-aliased samples, with the resolution error bounded by arithmetic and sampling quantization.

Answer 4

Detecting the onset of a constant frequency sine with additive noise can probably be done by using a matched filter (use the sine as a correlator template) of the longest possible length (either the acceptable latency or the minimum expected tone duration), followed by some fiddling to avoid duplicate registrations at +T, +2T etc due to sine periodicity.

I fail to see how you can do anything better than that.