What is MTW (Multi Time Window)?

Created by Jake Bedard, Modified on Tue, 18 Jun at 2:07 PM by Jake Bedard

What is MTW (Multi Time Window)?


Multi Time Window is Smaart's way of maximizing the frequency range that contains a useful number of data points. Historically, Fourier analysis made use of one FFT per measurement, which used a single size FFT, forcing the user to choose between options and weigh pros and cons.



Potential Pitfalls of Single-Sized FFTs


The key trade-off of this method is the inverse relationship between time resolution and frequency resolution. They are both a function of the "time constant", the time it takes to record enough samples of a DFT of a given size while at a given sampling rate (aka the "time window"). As such, longer time windows provide more detailed frequency resolution (smaller gaps between data points, often more than we want at high frequencies) but at the expense of less-detailed time resolution (less responsive measurement).




You can calculate the time constant for an FFT (in seconds) by dividing the sampling rate used to record the time-domain signal by the FFT size in samples. For example, a 16K FFT recorded at 48000 samples/second has a time constant of 0.341 seconds (16384/48000) or 341 milliseconds.



Since, by definition, low frequencies have longer cycle times than higher frequencies, they take longer to resolve. In fact, the lowest frequency that an FFT (or any other kind of DFT) can clearly "see" is 1/T, where T is the FFT time constant in seconds. Using the example of a 16K FFT at a 48k sample rate, frequency resolution in that case works out to 2.93 Hz (1/0.341).




The frequency resolution of an FFT is equal to the frequency of a sinewave that cycles exactly once within the FFT time window. All other frequency bins are at harmonics of that fundamental frequency, so knowing the time constraint also tells you how far apart the frequency bins are.



Each doubling of FFT size (in samples) doubles the FFT frequency resolution and extends its frequency range an octave lower.



This large single size FFT perfectly fine when taking a long-term average or a steady-state measurement using pink noise or a similar signal. However, it isn’t particularly useful when analyzing a dynamic signal such as speech or music, where you may need to see features of the signal that are very closely spaced in time as separate events. For example, if two drum beats occur within the time constant of a single FFT, the resulting spectrum in the frequency domain includes the energy from both hits as a single figure at each frequency. 


If you needed to see each beat as a separate event you would need to shorten the time window, which would result in more widely-spaced frequency bins and loss of resolution at lower frequencies. As a longer FFT takes more time to resolve, so do lower frequencies, which is where we get this loss of low frequency content when the FFT is smaller. 


For instance, in terms of speech analysis, typical speaking rates for native English speakers range from about 140-180 words per minute or about 200-300 syllables per minute, so a 16K FFT can get you words but not syllables. Dropping the FFT size to 8k would double the time resolution to a minimum of about 352 frames per minute (enough to keep up with insanely fast music or distinguish individual syllables in speech) but does so at the expense of some loss of detail at low frequencies.



Why Use MTW?


Here is where Smaart’s Multi Time Window (MTW) is helpful: it attempts to let you have your high-resolution, low-frequency cake and eat it in real-time, too! We are able to sidestep these issues by using a series of small FFTs at progressively lower sampling rates to deliver approximately 1 Hz resolution at low frequencies without incurring excessively high resolution in the upper octaves. (Smoothing the transfer function also helps to clean up excess resolution at high frequencies and works for both MTW and single-FFT size measurements.)


Having only about 800 frequency data points or fewer (depending on the measurement type and base sample rate selections) also helps with readability of MTW measurements when compared to single-size FFT measurements, while maintaining comparable low-frequency resolution. The use of shorter time windows at higher frequencies makes the coherence function a much more useful tool for detecting time mismatches between reference and measurement signals, as well.



MTW+


An in-depth overview of MTW+, a similar Smaart feature, can be found here.


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