The Fourier Transform
Baron Jean Baptiste Joseph Fourier was a French mathematician who in his Theorie analytique de la chaleur (Analytical Theory of Heat), developed the technique known as Fourier Analysis. This technique has proven to have application in many other unrelated disciplines including (in our case) the analysis of electromagnetic signals. Fourier's Theorem essentially states that the frequency content of any signal can be described as the sum of a specific set of sine waves. The sine wave is the only pure frequency and any distortion of this shape represents harmonics of some fundamental frequency. Thus any wave, no matter how oddly shaped, can be broken down into its component sine waves.
For example: fig.a shows a fundamental frequency with its 3rd harmonic. Combining them, we get a composite wave form that looks something like fig.b. If we combine enough harmonics of increasing frequency and decreasing amplitude, we end up with a near perfect "square" wave (fig.c).
There are two types of Fourier Analysis. The first is Fourier Series analysis for periodic signals
(i.e. signals that have exactly the same pattern for each cycle, such as quasars, 60Hz wall current
noise or regular heart beats also referred to as “time invariant” signals). The second form of analysis
is called the Fourier transform which deals with non periodic signals such as the human EEG which varies
continuously over time. Thus the Fourier transform of a non periodic signal produces a continuous
transform. When the input signal is periodic (repeats itself exactly with each cycle) its frequency
content can be represented by a discrete set of numbers called the Fourier series coefficients. They
signify the "weight" given each frequency that is required to reconstruct the original signal. Furthermore,
the frequencies that correspond to these different coefficients are harmonically related ; each being
an integer multiple of some fundamental frequency. This paper will confine itself primarily to exploring
the background and application of the Fourier transform since this is the signal processing technique
utilized in Neurofeedback (the EEG being intensely nonperiodic).
The Fourier transform is one of the most commonly used methods of signal analysis. It is simply a mathematical transformation that changes a signal from a time domain representation to a frequency domain representation thereby allowing one to observe and analyze its frequency content. Plotting a Fourier transform gives us a visual representation of the relative proportion of different frequencies in an input signal. Some examples: the FT of single sine wave would appear as a single spike (see below), indicating that only one frequency was present. Similarly, the noise from florescent lights appears as a prominent peak at 60 Hz. Where as the processing of a periodic signal with the Fourier series produces a discrete set of coefficients, a nonperiodic signal requires a mathematical tool that will produce a continuous transform (i.e. one that changes continuously over time instead of the “photolike” transform of the periodic signal) . The Fourier transform therefore, requires something a bit more powerful. When moving from the time domain x(t) into the frequency domain X(f), the time function x(t) must be evaluated for all values of t (time). The following is the actual Fourier transform which illustrates the mathematical relationship between the time domain x(t) and the frequency domain X(f).
The product of the time function x(t) and a [complex trigonometric expression] yield the frequency function X(f) when integrated over time t for a specific frequency f (Integration involves finding the area defined by a specific function over small increments, in this case dt.) Go over that last sentence slowly three more times and then please note: Although mathematical formulas will appear in this paper they are more for historical significance than anything else. The emphasis will be on developing an intuitive understanding of the concepts discussed.
Here are some examples of signals in the time domain and their corresponding Fourier transform:
1.)
2.)
transforms to:
3.)
4.)
Regarding examples 3 and 4: Signals that are narrow in the time domain (e.g. an impulse) are wide in the frequency domain (flat line of same amplitude). Conversely, signals that are wide in the time domain (e.g. DC signal flat line) are narrow in the frequency domain (impulse w/ frequency =0).
Also note: For our purposes in the field of EEG biofeedback, the magnitude of each frequency is often all that is shown on a spectral display. However for the complete reconstruction of a signal, phase information is necessary as well.
The following is an example of a fast Fourier transform performed on a wave form similar to those used in EEG biofeedback. Note that a "fast" Fourier transform (or FFT) is simply a computationally efficient algorithm designed to speedily transform the signal for real time observation.
The dominant frequency of this signal is approximately 3.5 Hz. The instrument samples the input signal
at a rate of 200 times per second and performs FFTs at a rate of 5 per second (a little on the slow
side as FFTs go.) Notice that the sampling rate is not the same thing as the FFT rate! The sampling
rate is the frequency with which the data points are gathered while the FFT rate is an indication of
how often the mathematical operation is performed on the sampled points. Sampling rate does, however,
play a role in the ultimate resolution of the displayed signal. The width of the FFT window in this
case is 1 second (a window is a defined period of time within which a mathematical operation is performed).
Since the FFT rate is 5 /second, a new FFT is calculated every .2 seconds. Having a window of 1 second
implies that .8 seconds of old data is being averaged in with the .2 seconds of new data. This process
is known as "smoothing" in that the response is a bit more sluggish to the eye of the patient. The
smaller the ratio of new data to old, the slower the response of the feedback. Conversely, if we were
to narrow the FFT window to .2 seconds (matching the FFT rate in this case and eliminating old data
from the average), we would have zero smoothing leading to a highly reactive and what many would consider,
agitating form of feedback. In this case an FFT window of less than .2 seconds would, of course, leave
gaps in the sampled data.
Filters Remember that the Fourier transform shows the frequency content of a signal. Filtering is the process of removing certain portions of the input signal in order to create a new signal. A familiar example would be the bass and treble controls on a CD player or electric guitar. There are four basic filter types:
 The Low pass filter which removes all frequencies above the cut off frequency. (Typically used
for noise removal and data smoothing.)
 The High pass filter removes all frequencies below the c/o frequency. (Used for DC or low frequency
drift.)
 Band pass filter removes all frequencies outside f 1  f 2. (Used often in EEG measurements.)
 Notch filter removes all frequencies between f 1 and F 2. (60Hz noise removal).
So how does a filter actually “remove” unwanted frequencies? This is accomplished by sending the input signal through a system function H(s) which determines the degree of amplification for each frequency in the signal. The desired frequencies are boosted by the instrument gain while the unwanted frequencies are boosted by a gain of zero. The following is the general form of a system function:
H(s) = g(s) / x(s)
where "s" represents a complex frequency. (Note that the term "complex" refers to a combination of real and imaginary numbers  specifically the square root of (1) which, technically, does not exist. Hence the name "imaginary" number. The concept of complex numbers is just a mathematical trick to keep our equations happy and running smoothly and will not be discussed further.)
The frequencies to be eliminated are those for which H(s) = 0. More specifically, they are the roots of the function g(s) (i.e. those values of s which make g(s) = 0). The solutions are referred to as “zeros.” Conversely, the roots of the function in the denominator, x(s), are those that cause H(s) to go to infinity (maximum gain). Solutions to x(s) are referred to as "poles." For example:
This implies that s =2 is a zero; and s =1 and s = 3 are poles. VOILA'!! We have just created a simple filter.
Designing a filter involves nothing more than designing a system function that has the desired frequency response (i.e. placing the poles and zeros in the correct configuration in the frequency plane to obtain the desired waveform). This response is known as a "Bode plot" named after (who else?) Dr Bode. A computer is generally used to determine the roots (pole/zero locations) of the appropriately chosen polynomials. These can be found in most filter design or operational amplifier textbooks.
Obviously there is no such thing as an "ideal" filter. The frequency response of most filters look something like the following low pass filter.
Terms: Pass band: the frequency range you are interested in looking at.
Transition band: the band over which the frequency response transitions from the pass band to the stop band.
Attenuation: this refers to the degree of difference between the pass band and stop band.
Attenuation rate: refers to the slope or roll off rate. Usually measured in dBs (decibels) per octave. Note that an octave is a doubling of a frequency (e.g. the difference in pitch between C1 and C2 on a piano).
Cutoff frequency: the edge of the pass band; also called the corner frequency.
Gain: the amount of amplification of the signal in the pass band.
Order: the number of poles in the system function H(s) (equal to the highest degree of the polynomial in the denominator). The higher the order, the steeper the rolloff (see below). Higher order filters are more complicated to build.
Phase Response As with the section on the FFT, the emphasis has been on the magnitude response of a filter. For a complete discussion on signal processing, the issue of phase must be included. Ideally, a filter should have a ”linear” phase response. This means that there is a constant time delay difference from the input for all input frequencies. If the phase response is not linear, then different frequencies would be delayed by different amounts. For example: when opera music is put through a filter, a cymbal crash might be heard over the singers voice instead of after.
Sampling
Sampling is the process of taking a continuous time signal and representing it by a series of discrete data points. Any (band limited) signal can be represented in this way as long as the samples are equally spaced and are close enough together in time. Sampling theory makes our life easier by efficiently converting a signal from the analog world (A) to the digital world (D) and back again with virtually no loss of information provided that the above requirements are met. The overall process is as follows:
So why all the bother? Why not simply leave everything in the analog domain and be happy? The reason is that signal processing in the digital domain, with all of its complexity, gives us much greater flexibility. For instance; changing filter characteristics involves reprogramming a few numbers rather than pulling out and replacing resistors and capacitors.
Aliasing
We know that any band limited continuous time signal can be reconstructed with a specific number of samples. However this process breaks down when the samples are spaced too far apart. An “alias” is a spurious signal that is obtained when the digital sampling of an analog signal occurs at too slow a rate. When a high frequency signal is sampled too slowly the high frequency signal will alias (look a whole lot like) low frequency activity upon analog reconstruction.
Example: a high frequency signal that is sampled too slowly.
The laws of signal processing dictate that the lowest frequency sine wave (the dotted trace) that will fit the sampled data points will be the frequency that gets integrated into the reconstructed analog signal. In the above example, the sampled points are simply too sparse for an accurate reconstruction of the high frequency signal (solid trace). In this case, aliasing will tend to corrupt the low end of the frequency spectrum making it appear that there is more low frequency activity than may actually occurring.
The frequency at which a signal aliases f(a) is equal to the difference between the sampling frequency f(s) and the maximum frequency f(m) in the band being measured.
or f(a) = f(s)  f(m)
In all cases, the value of f(s) must always be at least twice that of f(m). But just for fun , let's assume otherwise. Let's say:
f(s) < 2f(m)
If this is so then f(s)  f(m) < f(m) which implies that f(a) < f(m). What this means is that the frequency at which we experience aliasing is less than the maximum frequency of the signal in the band. Bad news! Here is an example involving actual numbers:
Let's say f(s) = 30Hz and f(m) = 18Hz. Then f(a) = 30 Hz  18Hz = 12Hz.
This means that we will begin to experience signal corruption beginning at 12 Hz, well within the band we are interested in measuring.
Thus, it is vital that the sampling rate always be greater than twice the maximum frequency being measured:
f(s) > 2f(m).
This rule is known as The Nyquist Sampling Theorem.
To ensure that aliasing does not occur in the sampling process, the signal to be sampled is passed through a lowpass filter to impose the appropriate band limitations on the signal. This filter is known as an antialiasing filter.
Another method of ensuring the integrity of signal reconstruction is called oversampling. The Nyquist theorem states that it is possible to completely reconstruct a signal as long as the sampling rate is at least twice the maximum frequency in the signal. The closer to the Nyquist threshold the signal, the sharper (higher order) the analog low pass filter required to band limit the signal (i.e. the roll off must be very steep). Rather than constructing one of these monster filters, the technique of oversampling is used. In this way the sampled points are close enough together so a more realistic lowpass filter can be used. As an example: CD players are advertised as having 4x or 8x oversampling. The human auditory system cannot hear frequencies higher than 20KHz. The typical CD contains audio information sampled at a rate of 44KHz. Typical sampling rates for EEG biofeedback is 150 samples per second or greater. This means that the EEG biofeedback process generally involves frequencies that are below 75 Hz.
Amplifiers and Voltage Laws
Basic Terms and Quantities
Charge: The quantity of electricity. Unit symbol: C for Coulomb which is 6.242 x 10 to the 18th power number of electrons (or protons).
Current: Represented by the symbol " i ". Unit measure is the ampere (A). A flow of charge equivalent to 1 Coulomb per second.
Voltage: Symbol: V or E. Unit is the Volt (V). The amount of electromotive force pushing current between two points.
Resistance: Symbol: R. Unit is the ohm. The ratio of voltage divided by current for a given conductor under a given set of conditions. A form of Impedance (see below) related to DC circuits ( i.e. circuits with currents that flow in one direction at a relatively constant rate).
Reactance: Symbol: (usually X). Unit is the ohm. The part of impedance of an AC circuit ( i.e. a circuit with currents that reverse (alternate) direction due to capacitance and/ or inductance).
Impedance: Symbol: Z. Unit is the ohm. The vector combination of resistance and reactance. Impedance is an apparent force that is in opposition to the flow of current in a circuit.
Ohms Law
Voltage is equal to the product of the current and the resistance: V = I R
If provided with any 2 quantities, the third can be easily calculated: I = V / R or R = V / I
Ohms Law also applies to impedance (AC circuits): V = I Z I = V / Z and Z = V / I
Kirchhoff's Voltage Law Kirchhoff's law states that the sum of the voltage drops (resistance) and rises around a closed loop within a circuit is equal to zero.
Kirchhoff's law is a result of the fact that the output of a voltage generator (Vg) is equal to the sum of the voltage drops across the resistive elements within the circuit: Vg = v1 + v2 + v3
which implies that:  (Vg) + v1 + v2 + v3 = 0
Note that (Vg) is actually a negative voltage drop since it is the output of the generator (in other words it represents a voltage rise). This law will prove very important regarding the subject of electrode vs. amplifier input impedance which we will now consider.
Impedance
When current flows through a closed loop, it is analogous to water flowing through a hose. In other words the amount of water (current) that enters the hose (loop) is the same amount that comes out the other end. And like a hose, if there is a difference in the amount of current flowing past any two different points in the circuit, then there is a current drain somewhere in the circuit. (This is the principle on which the “ground fault interrupter” works in biofeedback instrumentation to insure against an electrical shock to the patient. When the GFI detects a current drop between the “live” wire and a neutral location in the circuit, a breaker is triggered that shuts down the system.) In the following example we will examine how this feature of current and Kirchhoff's Voltage Law explain one of the reasons behind the importance of low electrode impedance and high input instrument impedance when performing EEG biofeedback. For simplicity and for reasons that will become clear later in this paper, we will only be dealing with the resistance component of impedance.
Remember that the current ( i ) that flows through each element (r1, r2 and r3) is exactly the same. (Note that r2 is the input impedance of the measuring instrument). In order to have maximum accuracy in measuring the voltage being generated (Vg), the voltage drop across the measuring instrument (V2) must be very close to Vg. Thus we want the voltage drop (V1 andV3) across r1 and r3 to be minimal. (r1 and r3 can represent the contact resistance of the biofeedback sensors). Since, according to Kirchhoff's Law, Vg = V1 + V2 + V3 , we want V1 and V3 to be negligible in order for V2 to be as close to Vg as possible. Since the current is the same across each element, the only way we can obtain this outcome is by controlling the impedance (i.e. very LOW contact resistance at r1 and r3 with very HIGH instrument input impedance at r2) . This is why it is common to see instruments with whopping impedances of 1 million megohms (that's a million million ohms folks !). When compared to the voltage drop at the contact sensors (due to 5000 to 10,000 ohm contact resistance) we can see that the voltage loss is fairly insignificant. However there is a much more crucial issue concerning the importance of low contact resistances, as we shall soon see.
Differential Amplifiers An amplifier is simply a device that increases the magnitude of voltage, current or power. In our case, for neurofeedback purposes, we are interested in the differential amplifier which amplifies voltage differences between 2 points. This is different from , say, a power amplifier that is used for a public address system. Since this sort of amplifier is non differential, it does not discriminate and thus everything in the line gets amplified. In the case of a differential amplifier, the difference between the voltages V+ (the positive terminal) and V ( the negative terminal) are amplified by the gain (A) giving us the resultant voltage Vo (voltage out). The reason this is necessary is because we are dealing (in the case of the EEG) with extremely small microchanges in potential and would not be able to see them if not amplified massively.
Vo = A [( V+)  (V)]
A differential amplifier with a very high gain and extremely high input impedance is called an operational amplifier or simply an "opamp." The inputs have extremely high impedances and the output node has near zero resistance allowing it to behave like an ideal voltage source supplying as much current as is necessary.
In order to get a sense of how a differential amplifier works, think of 2 microphones at the extreme ends of a choir with sopranos placed at one end and bassos at the other. In the middle (approximately equidistant from the 2 mics) place the midrange voices altos, contraltos, tenors, etc. If this were a non differential amplifier (several mics on separate channels placed around the choir) all of the voices would be amplified to the same degree. However a differential amplifier would only amplify the sopranos and bassos. Why? Because the sound from the midrange voices would arrive at both microphones simultaneously with the same intensity and frequency causing the input at both mics to be exactly the same thus cancelling each other out. This quality, unique to differential amplifiers, is referred to as common mode rejection. The signals that are common to both inputs cancel themselves out while the signals that are unique to each input are preserved for amplification.
In the field of EEG biofeedback this particular feature of the differential amplifier becomes critically important. The reason for this is that we tend to work in an extremely polluted environment from an electromagnetic standpoint. Anytime electric current is run through a circuit, an electromagnetic field forms around the device and extends out into space anywhere from a few feet to a few miles. As such, the feature of common mode rejection becomes one of the technicians greatest ally in attaining clean measurements.
So what are some of the sources of common mode input? Basically, it is any power source that is far enough away to influence each electrode to nearly the same degree. These can include: computers, power outlets, radios, motors, ignition switches, generators, florescent lights and toaster ovens.
Electric Fields Given all of the preceding information, it is clear that the measuring instrument itself needs to be taken into consideration as a source of signal corruption in biofeedback or any other form of biomedical measurement. When a person is placed in an electromagnetic field, there will be a flow of current through the body across the lines of force of the field. (The lines of force represent different strengths of the magnetic force located within the field. Think of the concentric circles that form around a bar magnet when one pours iron filings in the vicinity of the magnet . The strength of the field remains constant along an individual line of force but varies across the lines). We live in an electromagnetically "enriched" environment where the build up of electrical potential differs from one place to another. Since the human body is fairly extensive, different extremities can exist in different parts of an electric field. Depending on the degree of potential difference (voltage) between any two locations on the body, there can be either a large or small current flow through the body, which, if substantial enough, can present a significant degree of interference. If the electric field of the instrument and the client are very much alike, there will be less current flow between them. (This is why putting the biofeedback instrument on someone's lap can sometimes alleviate the problem. This is generally not recommended in EEG biofeedback although touching the instrument, thereby putting both instrument and client at equal potential, can sometimes be helpful ).
In addition, it is almost inevitable that an imbalance will exist between the electrode skin resistance of the two electrodes which, if large enough, can completely defeat the common mode rejection feature of the differential input system. A significant difference in the electrode impedances can change a common mode signal (which is subject to complete cancellation) to a differential mode signal resulting in corruption of the measurement. This is why low electrode impedances (contact resistance) are important for accurate readings.
For example:
Let's say that an electromagnetic field induces a current flow of 0.2 nA (nano amperes) on the above system. A nano amp is equal to 0.000,000,001 amps. The above simplified schematic represents two electrodes with skin resistances of 15 K ohms and 5 K ohms leading into the differential amplifier. Using Ohms Law would give us a voltage of :
0.2 nA x 15 K ohms = 3.0 uV at the top electrode skin interface.
and
0.2 nA x 5 K ohms = 1.0 uV at the bottom electrode skin interface.
This would lead to a differential voltage of: 3.0 uV  1.0 uV = 2.0 uV. Thus Vo would include this added 2.0 uV in its output to the computer for analog reconstruction. Since we are concerned with the difference between the electrodeskin resistances, we could actually have resistances as high as we wanted so long as the difference between them remained small (correspondence from Siegfried Othmer). However it is difficult to have high electrode skin resistances that would be close enough to each other in value to sufficiently cancel the extraneous voltage. Keeping both of them under 5,000 ohms will give us a reasonable margin given the extremely small common mode currents we deal with in performing EEG biofeedback.
Impedance vs. Resistance Revisited
The reason we use electrode skin resistance instead of impedance is because at the low frequencies (those we deal with in performing EEG biofeedback), the impedance is almost entirely resistive. (This may be a good time to revisit the term definitions at the beginning of the paper.) The reason this is so is due to the fact that in an AC circuit, voltage and current share a phase relationship. In a DC circuit, which is purely resistive, there is no phasing between voltage and current since the current does not change direction. An AC circuit is considered “DC equivalent” if the phasing between i and v is 0 degrees (i.e. v and i are in phase). At higher frequencies this phasing will widen as one will tend to lag the other. (Think of yourself trying to mirror the movements of someone else. While they are moving slowly it is quite easy to move in time with them. However as they begin to move faster and faster, your response time introduces a time lag or “phase delay”). This is similar to what happens with a voltage driven current source. As V reverses polarity at an ever increasing rate, the reversal in direction of the current flow becomes slightly delayed. Thus the phasing increases as a function of higher frequency. This phase delay is due to the inherent capacitance and inductance qualities of the circuit. Capacitance inhibits instantaneous voltage changes; inductance inhibits instantaneous current changes.
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