<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>.&nbsp;</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><p><br></p><p style="text-align: center;"><img src="@@PLUGINFILE@@/summation-two-signal.png" alt="The summation of two signals" width="561" height="450" class="img-responsive atto_image_button_text-bottom"><br></p><p style="text-align: center;">The summation of two signals<br></p><p style="text-align: center;"><br></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><p><br></p><p style="text-align: center;"><img src="@@PLUGINFILE@@/spectrogram-amplitude.png" alt="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" width="420" height="321" role="presentation" class="img-responsive atto_image_button_text-bottom"><br></p><p></p><p style="text-align: center;"><span style="font-size: 1rem;">How a perfect 10Hz (blue) and perfect 60Hz (red) signal </span><span style="font-size: 1rem;">would look&nbsp;<br></span><span style="font-size: 1rem;">in a spectrogram with Amplitude on the </span><span style="font-size: 1rem;">Y-axis and frequency on the X-axis</span></p><p style="text-align: center;"><span style="font-size: 1rem;"><br></span></p><p style="text-align: left;"><span style="font-size: 1rem;"></span></p><p>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.</p><p><br></p><p style="text-align: center;"><img src="@@PLUGINFILE@@/spectrogram-electrical.png" alt="Spectrogram showing diffuse electrical contamination at 16Hz" width="603" height="548" class="img-responsive atto_image_button_text-bottom"><br></p><p></p><p style="text-align: center;">Spectrogram showing diffuse electrical contamination at 16Hz<br></p><p style="text-align: center;"><br></p><p style="text-align: center;"><img src="@@PLUGINFILE@@/kurtosis-eeg.png" alt="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" width="604" height="553" class="img-responsive atto_image_button_text-bottom"><br></p><p style="text-align: center;">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<br></p><p style="text-align: center;"><br></p><p style="text-align: center;"><img src="@@PLUGINFILE@@/mathematical-eq.png" alt="The mathematical equation for frequency decomposition and the summation of frequencies " width="900" height="507" class="img-responsive atto_image_button_text-bottom"><br></p><p style="text-align: center;">The mathematical equation for frequency decomposition and the summation of frequencies<br></p>