<p></p><h3><a name="_Toc29929037"></a><a name="_Toc3535252">JTFA (Joint Time Frequency Analysis)</a></h3>

<p>Joint-Time Frequency-Analysis
(JTFA)is used for real-time EEG biofeedback and is used to compute statistical
Z scores in real-time. JTFA relies on complex demodulation, which is an
analytic technique that multiples a time series by a sine wave and a cosine
wave and then applies a low pass filter (Granger et al, 1964; Otnes and
Enochson, 1977). JTFA computes instantaneous power, instantaneous phase
differences, and instantanous coherences. Unlike the Fourier transform, &nbsp;which depends on windowing and integration
over an interval of time, JTFA complex demodulation computes and creates an
instantaneous Z score within subject variance of instantaneous electrical
activity as well as the between subject variance for subjects of a given age.
The summation of instantaneous Z scores is Gaussian distributed and has high
crossvalidation (Thatcher et al, 1987; 2007), however, the individual time
point by time point Z score is always smaller than the summation due to within
subject variance.</p><p>INSERT
VIDEO OF JTFA POWRE ANALYSIS</p><p></p><h3><a name="_Toc29929038"></a><a name="_Toc3535253">FFT (Fast Fourier Transform)</a></h3>

<p>Jean Baptiste Fourier demonstrated
that any repetitive signal, over the course of time, could be broken down into
a series of sine waves by boiling it down into its component parts, in integral
numbers. His Fast Fourier Transform is a mathematical calculation that
transforms time-related EEG into a frequency domain. So, in Fast Fourier
decomposition, the raw EEG is broken down into disparate frequencies, divided
into bandwidths: delta (0.5 – 4HZ), theta (4 – 8Hz), alpha (8-12 Hz), Beta 1
(12-15Hz), Beta 2 (15 – 19Hz), Beta 3 (19-25 Hz) and High Beta (25 – 30Hz), as
well as individual hertz bins. Many normative databases also break down gamma
into individual Hz bins, thought not all give normative data for gamma
frequencies.</p>

<p>Essentially, the Fast Fourier
Transform takes a signal and breaks it down into sine waves of different amplitudes
and frequencies. Fourier’s theorem states that any waveform within a time
domain can be represented by the <b><u>weighted sum of sines and cosines</u></b>.
</p><p></p><p>In the below image, we are first
seeing two sine waves, where one is three times as fast as the other.&nbsp; When you add them, you can see you get a
different signal. This is the sum of the two frequencies. A Fast Fourier
transform will allow us to see them as separate frequencies.</p><h3>The summation of two signals</h3><p></p><br><p></p><p style="text-align: center;"><img src="@@PLUGINFILE@@/Summation%20of%20Two%20Signals.png" alt="Summation of Two Signals" width="561" height="450" class="img-responsive atto_image_button_text-bottom" style="font-size: 1rem;"></p><p>In Fisch, it is written: “The FFT
function is based on the fact that any signal can be described as a combination
of sine and cosine waves of various places, frequencies and amplitudes (Fisch,
1999, p. 125).&nbsp; Although the etiology of
EEG does not neatly compartmentalize into individual frequency bins, FFT is one
way to better understand the role of various frequencies. When using a
decomposition method, such as FFT, you lose the morphology of the raw EEG,
which is essential for understanding the fundamentals of the person’s data.</p><p>If we were to take a signal
generator and create a perfect 10 Hz wave, but receive electrical contamination
at 60Hz, we would see a sharp peak at 10Hz and at 60Hz in a spectrogram. In the
below picture, 10HZ is represented by the blur arrow and 60Hz is represented by
the red arrow.</p><h3>How a perfect 10Hz (blue) and perfect 60Hz (red) signal would
look in a spectrogram with Amplitude on the Y-axis and frequency on the X-axis</h3><p style="text-align: center;"><span style="font-size: 1rem;">&nbsp;</span><img src="@@PLUGINFILE@@/Noise%20and%20Sine%20Wave.png" alt="Noise and Sine Wave" width="600" height="321" class="atto_image_button_middle" style="font-size: 1rem; text-align: center;"></p><p></p><p style="text-align: center; "><br></p><p style="text-align: center; "><br></p><p style="text-align: center; "><br></p><p></p><p style="text-align: left;">One way to tell real EEG from
artifact generated by electricity, is the spectrogram will show an abnormal
peak, signifying one frequency. Subharmonics of electricity in the United
States and western nations are at 16Hz and at 28Hz. The peak seen at 16Hz in
the following image is the result of a saltbridge that occurred at the ground
electrode, contaminating every other electrode, as a consequence. A real human
EEG signal will have “kurtosis,” which means “curved” or “arching” in Greek and
is a measure of tailedness within the probability distribution of EEG
frequency. We expect to see kurtosis in all dominant frequencies of the EEG.<span lang="EN"></span></p><h2 style="text-align: left;">Spectrogram showing diffuse electrical contamination at 16Hz</h2><p style="text-align: center;"><span style="font-size: 1rem;">&nbsp;</span><img src="@@PLUGINFILE@@/Spectrogram.png" alt="Spectrogram" width="603" height="548" class="img-responsive atto_image_button_middle" style="font-size: 1rem; text-align: center;"><span style="font-size: 1rem;">&nbsp;</span></p><p></p><p style="text-align: center; "><br></p><p></p><h3 style="text-align: left;">A peak showing sides, or kurtosis in real human EEG, indicating
that this client not only produces 10Hz, but the also produce the frequencies
surrounding 10Hz, as one would in a normal distribution</h3><p style="text-align: center;"><span style="font-size: 1rem;">&nbsp;</span><img src="@@PLUGINFILE@@/alpha%20peak.png" alt="alpha peak" width="604" height="553" class="img-responsive atto_image_button_middle" style="font-size: 1rem; text-align: center;"></p><p></p><p style="text-align: center; "><br></p><p style="text-align: center; "><br></p><p style="text-align: center; "><br></p><p></p><h3 style="text-align: center; ">The mathematical equation for frequency decomposition and
the summation of frequencies </h3><p style="text-align: center;"><img src="@@PLUGINFILE@@/FFT.png" alt="FFT" width="1209" height="682" class="img-responsive atto_image_button_text-bottom" style="font-size: 1rem;"></p><p></p><p></p><p style="text-align: center;">The&nbsp;mathematical&nbsp;details&nbsp;of&nbsp;complex&nbsp;demodulation&nbsp;used&nbsp;to&nbsp;compute&nbsp;the&nbsp;instantaneous&nbsp;Z&nbsp;</p>

<p style="text-align: center;">scores&nbsp;as&nbsp;contained&nbsp;in&nbsp;the&nbsp;Applied&nbsp;Neuroscience,&nbsp;Inc.&nbsp;“dll”&nbsp;are&nbsp;published&nbsp;in&nbsp;Otnes&nbsp;and&nbsp;</p>

<p style="text-align: center;">Enochson,&nbsp;(1977);Granger&nbsp;and&nbsp;Hatanaka,&nbsp;(1964);&nbsp;Bloomfield,&nbsp;2000;&nbsp;Thatcher&nbsp;et&nbsp;al,&nbsp;</p>

<p style="text-align: center;">(2007).&nbsp;Complex&nbsp;demodulation&nbsp;is&nbsp;a&nbsp;time&nbsp;domain&nbsp;digital&nbsp;method&nbsp;of&nbsp;spectral&nbsp;analysis&nbsp;</p>

<p style="text-align: center;">whereas&nbsp;the&nbsp;fast&nbsp;Fourier&nbsp;transform&nbsp;(FFT)&nbsp;is&nbsp;a&nbsp;frequency&nbsp;domain&nbsp;method.&nbsp;&nbsp;These&nbsp;two&nbsp;</p>

<p style="text-align: center;">methods&nbsp;are&nbsp;related&nbsp;by&nbsp;the&nbsp;fact&nbsp;they&nbsp;both&nbsp;involve&nbsp;sines&nbsp;and&nbsp;cosines&nbsp;and&nbsp;both&nbsp;operate&nbsp;in&nbsp;</p>

<p style="text-align: center;">the&nbsp;complex&nbsp;domain&nbsp;and&nbsp;in&nbsp;this&nbsp;way&nbsp;represent&nbsp;the&nbsp;same&nbsp;mathematical&nbsp;descriptions&nbsp;of&nbsp;the&nbsp;</p>

<p style="text-align: center;">power&nbsp;spectrum.&nbsp;&nbsp;The&nbsp;advantage&nbsp;of&nbsp;complex&nbsp;demodulation&nbsp;is&nbsp;that&nbsp;it&nbsp;is&nbsp;a&nbsp;time&nbsp;domain&nbsp;</p>

<p style="text-align: center;">method&nbsp;and&nbsp;less&nbsp;sensitive&nbsp;to&nbsp;artifact&nbsp;and&nbsp;it&nbsp;does&nbsp;not&nbsp;require&nbsp;windowing&nbsp;nor&nbsp;even&nbsp;integers&nbsp;</p>

<p style="text-align: center;">of&nbsp;the&nbsp;power&nbsp;of&nbsp;2&nbsp;as&nbsp;does&nbsp;the&nbsp;FFT.&nbsp;&nbsp;&nbsp;The&nbsp;FFT&nbsp;integrates&nbsp;power&nbsp;in&nbsp;a&nbsp;frequency&nbsp;band&nbsp;over&nbsp;</p>

<p style="text-align: center;">the&nbsp;entire&nbsp;epoch&nbsp;length&nbsp;and&nbsp;requires&nbsp;windowing&nbsp;functions&nbsp;which&nbsp;can&nbsp;dramatically&nbsp;affect&nbsp;</p>

<p style="text-align: center;">the&nbsp;power&nbsp;values&nbsp;whereas,&nbsp;as&nbsp;mentioned&nbsp;previously,&nbsp;complex&nbsp;demodulation&nbsp;does&nbsp;not
require&nbsp;windowing&nbsp;(Otnes&nbsp;and&nbsp;Enochson,&nbsp;1972).&nbsp;&nbsp;&nbsp;Complex&nbsp;demodulation&nbsp;was&nbsp;computed&nbsp;</p>

<p style="text-align: center;">for&nbsp;the&nbsp;linked&nbsp;ears&nbsp;and&nbsp;eyes&nbsp;open&nbsp;and&nbsp;eyes&nbsp;closed&nbsp;conditions&nbsp;for&nbsp;all&nbsp;625&nbsp;subjects&nbsp;in&nbsp;the normative database.</p><p style="text-align: center;"><span style="font-size: 1rem;">&nbsp;</span><img src="@@PLUGINFILE@@/JTFA%20Imaging.png" alt="JTFA feedback for Z-scores" width="586" height="592" class="img-responsive atto_image_button_middle" style="font-size: 1rem; text-align: center;"></p><p style="text-align: center;"><br></p><p></p>