It has been challenged that FFT and JTFA give "completely" different results when applied to the same data, and that it is therefore not acceptable to use FFT-based norms when computing z-scores from an instantaneous JTFA (Quadrature Filter) calculation. This is not correct, and the equivalence of the two outputs is demonstrated below. Anyone can repeat this exercise with the BrainMaster software. The BrainAvatar software is used for this demonstration because it automatically produces an Excel file output with 8/second data, showing the instantaneous results. These results show clearly that the two computations, FFT (Fast Fourier Transform) and JTFA (Joint Time-Frequency Analysis), produce equivalent results when the outputs are compared on an "apples to apples" comparison. They further show that the results are stable over extended periods, as shown from the two examples below, taken using 1-minute and 5-minute samples.
In order to produce equivalent results, it is necessary to take into account that the FFT normally produces a "power" measure in "microvolts squared," and that the JTFA produces a magnitude in "microvolts." In addition, the windowing used in the FFT epochs is an additional form of attenuation, so that it needs to be compensated for when comparing results. The BrainMaster software is configured so that when we combine the FFT bins (on 1 Hz boundaries) to produce a spectral estimate, it produces a compensated value in "microvolts" that can be compared directly with the JTFA output.
Demonstration that FFT and JTFA (Quadrature Filter) results converge for both short-term and long-term statistics
It is straightforward to show that the results of either an FFT or a JTFA calculation yield the same results, both in short-term statistics, and in long-term averages. This can be shown by using the following procedure:
Run the BrainAvatar software
Create new folder, e.g. “FFTJTFA Test”
Create and use default settings
Enter the following equations into the Event Wizard and make the events "enabled"
Event 1: x = Bnd(1, 8, 12); damping factor 0 (this computes the sum of the FFT bins for 8 to 12 Hz)
Event 2: x = C1AA/2; damping factor 10 (this accesses the digital filter amplitude for alpha filter 8-12).
Use settings and close
Now do either:
File / Open EEG file / Playback
Run a live EEG
You can watch the event values in a text stats panel you create, or using Event Trends.
After 1, 2, or 5 minutes, or any length of time you desire
Go to C:\ProgramData\BrainMaster\Studies\FFTJTFA Test
Delete the first two rows of labels
Select columns 2 and 4 in Excel, e.g.
Select column 2
<CTRL> Select column 4
Insert / Scatter / OK
You will then see a scatter plot like the following. Each point represents the sample for a time location of 1/8 second. In other words, the system writes the results 8 times per second. The FFT results are for a sliding 1-second epoch, while the JTFA results are exponentially damped using the damping factor applied. These plots were made by using a playback of real EEG, so they reflect the real world in addition to the match which is theoretically guaranteed. The difference seen in the regression (less than 3%, evidenced by the R=.9784) ) is explained by the minute differences in the exact time-response of the processing. If the signals were steady-state, then this difference would disappear and FFT and JTFA would give absolutely identical results. However, even in the presence of time-varying signals, both methods definitely converge on identical "targets," both in the short-term and in the long-term. The 8% difference in the regression (evidenced by the 1.0756 term) is due to the fact that the filters have not been meticulously tuned at the passband edges for this particular demonstration. This could be brought into a perfect match b y adjusting the coefficients on the FFT band sum to match the quadrature filters exactly. This is a consistent ratio, and could also simply be incorporated into the software, so that the FFT and JTFA results would match with a ratio of 1.00. In the produce, we do match each band using real data, so that the data BrainMaster provides to each DLL has been tuned for optimal accuracy.
The existence of the R=.9784 is the important demonstration that the average variation between the two methods (JTFA and FFT) is less than 3%. It is important to note that the observed scale factor of 7.56% is not an error, but a systematic and constant scale factor that reflects the difference in the band edges between the FFT and the JFTA. It should be noted tht this is not technically an "error" but rather simply a difference between two estimation methods. It is arguable which is most representative, the epoch-based FFT result, or the more instantaneous JTFA result. Furthermore, this difference can be easily compensated for in the software, so that the net agreement between the two methods is at an average (RMS error) of less than 3%. The actual difference therefore consists of no more than plus or minus 1.5%. When applied to z-scores, a difference of 1.5% is insignificant, reflecting well below 0.1 standard deviations of difference. For example, for a 10 microvolt alpha level, the maximum possible difference error is only plus or minus 0.15 microvolts. Moreover, the difference is statistically random, so that when doing combined z-score training, it is essentially undetectable.
The following plot shows the agreement (over 97%) between FFT and JTFA over 1 minute. This includes 480 samples, being 8/second
The following plot shows the agreement (over 97%) between FFT and JTFA over 5 minutes. This includes 2400 samples, being 8/second.
Conclusion: Because the expected 1:1 correspondence between JTFA and FFT measures has been demonstrated, it is justified to provide JTFA (quadrature) data to a database constructed using FFT's, and expect correct results.