Fish and spehlmann eeg primer rapidshare

The BSR is calculated as the sum of intervals of suppression each at least 0. The random character of the EEG dictates that QEEG parameters extracted will exhibit a moment-to-moment variation without discernible change in the patient's state. Thus, output parameters are often smoothed by a moving average before display. Because of the particularly variable non-stationary nature of burst suppression, the BSR should be averaged over at least 15 epochs 60 s. An important alternative approach to time domain analysis examines signal activity as a function of frequency.

So-called frequency domain analysis has evolved from the study of simple sine waves. A simple sine or cosine wave is a function of time, t, that can be completely described by Equation 2with three parameters: amplitude, frequency, and phase angle.

Amplitude is one half the peak-to-peak voltage; frequency is number of complete cycles per second; and phase angle is the way to describe the starting point of the waveform. The phase angle can be understood by considering a sine wave as the vertical displacement of a spoke on a spinning wheel, as illustrated in Figure 8.

The convention is that the spoke starts in the horizontal position 0 [degree sign] , starting the sine at the midpoint of its vertical excursion. If upright, the spoke would be considered at 90 [degree sign] phase. In this model of a sine wave generator, the radius of the wheel is the sine amplitude, and the number of revolutions per second is the sine wave frequency. Figure 8. A rotating vector or spoke describes a sinusoid over time. In this example, image a wheel, rotating counterclockwise, with a light source in its rim adjacent to the marked spoke.

As the wheel turns, a graph of the vertical position of the light versus time will produce the indicated sine wave. The rotational speed of the wheel determines the frequency of the resulting sine wave. The initial angle of the spoke is the phase angle of the sine wave as used in Equation 2. In this illustration, the wheel starts at three different phase angles.

Note that the sine frequency is independent of the phase angle. Returning to EEG signals, it can be demonstrated Fourier's theorem that any arbitrarily complicated time-varying waveshape, x t , can be decomposed into the sum of simple sine or cosine waves. Formally this is true as long as the original waveform was repetitive e. Informally, nonrepetitive signals are commonly decomposed with little error if the following caveats are noted.

In practice, waveforms are finite and may not be truly repetitive. Nonperiodic signals can be treated as though the period of repetition has become infinitely long [Greek small letter tau][arrow right][infinity].

The fundamental frequency then becomes infinitely small as does the separation of spectral components; thus the spectrum of nonperiodic signal is a continuous function. The Fourier Equation Equation 3 states that a periodic signal, x t , consists of discrete components: a DC zero frequency amplitude, A 0 , plus sinusoids at the fundamental frequency, f 0. This rule applies whether the signal is in an analog or digital form. Figure 9 demonstrates this process with the synthesis of a square wave from sinusoidal harmonics.

The series of coefficients, A m and B m , comprise the amplitude spectrum of the original signal, x t , and the [Greek small letter theta] m coefficients form the phase spectrum. Figure 9. An example of the Fourier theorem that any repetitive waveform can be constructed from, or deconstructed to, a series of simple sine wave harmonics. Harmonics are sine waves whose frequencies are integer multiples of the slowest component wave.

In this case, a square wave is constructed by sequentially adding odd harmonics. As more harmonics are added, the reconstructed wave becomes closer to an ideal square wave. The final square wave, x t , can be written as:Equation 11where f 0 is the fundamental frequency of the square wave and a 0 is its amplitude. In the case of a square wave, all the harmonic phase angles are zero.

A Fourier analysis generates a frequency spectrum, which is simply a histogram of amplitudes or phase angles as a function of frequency. The concept is well illustrated by the effect of passing a white light through a glass prism, creating a rainbow or spectrum. Each color of light represents a unique frequency photon, and the relative brightness among the colors is a measure of the energy amplitude at each frequency. Any measured signal transformed by the Fourier technique into the frequency domain will have both an amplitude and a phase component for each harmonic frequency.

Mathematically it is usually expedient to describe both components with a single complex number. To review, regular real numbers can be considered as describing a point on a number line.

A complex number, z, is a vector describing a point on the complex plane. The vector has two components, x, which is the projection of the vector on the real number axis, and jy, which is the projection on the imaginary axis j is the right angle vector for an imaginary number, and it is defined as the square root of An operation frequently performed in signal processing is complex conjugation where the sign of the imaginary component is inverted, i.

The complex number vector can also be described as a magnitude i. The Fourier equation Equation 3 is thus written using the compact notation of complex exponentials:Equation 4. In Equation 4, m, again is the harmonic number, and c m is a complex Fourier coefficient that contains amplitude and phase information for each harmonic.

The conversion of a time domain waveform, x t , into its sine was frequency components, X f , is known as a Fourier transformation. This transformation, under ideal conditions, does not alter or reduce the information content within the waveform, and an inverse Fourier transformation will reconstitute the original waveform i.

Performing the Fourier transformation is then just computing the series of amplitude coefficients, A m and B m , and the harmonic phase angles, [Greek small letter theta] m , from Equation 3, or the series of complex coefficients, c m per Equation 4. The complex number version of the Fourier equation Equation 4 defines a time domain signal to be a sum of complex harmonics. One may easily separate the amplitude components of the spectrum by taking the magnitude of each of the complex magnitudes, [vertical bar] c m [vertical bar], which provides a set of values that comprise the amplitude spectrum.

Squaring the values of amplitude spectrum creates the power spectrum. The phase spectrum is the set of [Greek small letter phi] m coefficients. Recall that m is the harmonic, a function of frequency expressed as multiple of the fundamental frequency. The conversion of an analog signal to a sampled digital signal further modifies the signal spectrum. Equation 5 is the digital version of the Fourier transform and uses the signal processing convention that time domain signals are written in lower case, x t or x kT , and the equivalent complex frequency domain spectrum in upper case, X f.

Increasing the duration of the sampling epoch increases the number of digital samples of time domain data, which in turn, increases the number of resulting frequency domain spectral components. The frequency resolution of the DFT is the reciprocal of the epoch duration, e.

The arbitrary slicing of a continuous signal stream into finite epochs an epoch consists of a sequential set of samples can introduce contamination in the form of artifactual frequencies created by the abrupt transitions at the ends of the epoch. For example, a digitized epoch of a sine wave starting and ending at a phase angle of 0 [degree sign] would have zero-valued samples at either end of the epoch, and the computed spectrum would be an accurate representation of the true frequency content of the signal, in this case, a single non-zero value at the frequency of the sine wave.

In the case of a sampled sine wave starting and ending at 90 [degree sign] phase, the sampled values abruptly start and end at a non-zero value.

The non-zero start and finish points cause the frequency domain to artifactually add high frequency components to the spectrum to match the abrupt, step-like transitions in the time domain signal. This type of distortion is minimized by multiplying each time domain amplitude point within the epoch against the corresponding value of a window function.

A window function is a numerical series containing the same number of elements as the number of signal samples in the epoch. Window element values tend toward zero at both ends and toward unity in the middle. A variety of window functions have been described, including the rectangle, the triangle, the Hanning, and the Blackman functions. Until recently, commercial EEG analyzers did not perform windowing on sampled data.

Figure A An illustration of time domain-based EEG processing. The top waveform is the original signal after analog anti-alias filtering with a bandpass of 0. The middle tracing demonstrates the effect of windowing on the original signal. Windowing is a technique used to reduce distortion from epoch end artifacts in subsequent frequency domain processing.

A window consists of a set of digital values with the same number of members as the data epoch. In this case, a Blackman window was used. The Blackman window is defined as:Equation 12where i is the sample number and n is the number of samples in the epoch. The window operation multiplies each data sample value against its corresponding window value, i. The bottom tracing is the autocorrelation function of this epoch of EEG.

The autocorrelation provides much of the same information as a frequency spectrum because it can identify rhythmicities in the data.

B Continuing with the same epoch of digitized EEG, the top two tracings are the real and imaginary component spectra, respectively, resulting from the Fourier transform. The middle trace is the phase spectrum, which is usually discarded because of the present lack of known clinically useful correlation. The bottom tracing is the power spectrum. It is calculated as the sum of the squared real and imaginary components at each frequency i.

Recall that power equals squared voltage. Note that the power spectrum, by reflecting only spectral magnitude, has explicitly removed whatever phase versus frequency information present in the original complex spectrum. From the power spectrum, the QEEG and relative band powers are calculated as described in the text and in Table 1. The original integral-based approach to computing a Fourier transform is computationally laborious, even for a computer.

In , Cooley and Tukey published an algorithm for efficient computation of Fourier series from digitized data. More information about the implementation of FFT algorithms may be found in the text by Brigham.

In clinical monitoring applications, the results of a EEG Fourier transform are graphically displayed as a power versus frequency histogram, and the phase spectrum has been traditionally discarded as uninteresting. Whereas the frequency spectrum is relatively independent of the start point of an epoch relative to the waveforms contained , the Fourier phase spectrum is highly dependent on the start point of sampling and thus very variable.

Spectral array data from sequential epochs are plotted together in stack like pancakes so that changes in frequency distribution over time are readily apparent. Raw EEG waveforms, because they are stochastic, cannot be usefully stacked together because the results would be a random superposition of waves.

However, the EEG's quasistationarity creates spectral data that are relatively consistent from epoch to epoch, allowing enormous visual compression of spectral data by stacking and thus simplified recognition of time-related changes in the EEG.

There are two types of spectral array displays available in commercial instruments: the compressed spectral array CSA and the density spectral array DSA. The CSA presents the array of power versus frequency versus time data as a pseudo-three-dimensional topographic perspective plot Figure 11 , and the DSA presents the same data as a gray scale-shaded or colored two-dimensional contour plot. Although both convey the same information, the DSA is more compact, whereas the CSA permits better resolution of the power or amplitude data.

The creation of a spectral array display involves the transformation of time domain raw EEG signal into the frequency domain via the fast Fourier transform.

The resulting spectral histograms are smoothed and plotted in perspective with hidden-line suppression for CSA displays on left or by converting each histogram value into a gray value for the creation of a DSA display on right.

The effort to glean useful information from the EEG has led from first order mean and variance of the amplitude of the signal waveform to second order power spectrum, or its time domain analog, autocorrelation statistics and now to higher order statistics.

Higher order statistics include the bispectrum and trispectrum third and fourth order statistics, respectively. Little work has been published to date on trispectral applications in biology, but there have been more than papers and abstracts to date related to bispectral analysis of the EEG.

Whereas the phase spectrum produced by Fourier analysis measures the phase of component frequencies relative to the start of the epoch, the bispectrum measures the correlation of phase between different frequency components as described below. What exactly these phase relationships mean physiologically is uncertain; one simplistic model holds that strong phase relationships relate inversely to the number of independent EEG pacemaker elements.

Bispectral analysis has several additional characteristics that may be advantageous for processing EEG signals: gaussian sources of noise are suppressed, thus enhancing the signal-to-noise ratio for the non-gaussian EEG, and bispectral analysis can identify non-linearities, which may be important in the signal generation process.

A complete treatment of higher order spectra may be found in the text by Proakis et al. As noted previously, the bispectrum quantifies the relationship among the underlying sinusoidal components of the EEG. For each triplet, the bispectrum, B f 1 , f 2 , a quantity incorporating both phase and power information, can be calculated as described below. The bispectrum can be decomposed to separate out the phase information as the bicoherence, BIC f 1 , f 2 , and the joint magnitude of the members of the triplet, as the real triple product, RTP f 1 , f 2.

The defining equations for bispectral analysis are described in detail below. An example of such phase relationships and the bispectrum is illustrated in Figure The bispectrum is calculated in a two- dimensional space of frequency 1 versus frequency 2 as represented by the coarsely cross-hatched area. Because of the symmetric redundancy noted in the text and the limit imposed by the sampling rate, the bispectrum need only be calculated for the limited subset of frequency combinations illustrated by the darkly shaded triangular wedge.

In panel A, three waves having no phase relationship are mixed together producing the waveform shown in the upper right. The bispectrum of this signal is everywhere equal to zero. In panel B, two independent waves at 3 and 10 Hz are combined in a non-linear fashion, creating a new waveform that contains the sum of the originals plus a wave at 13 Hz, which is phase-locked to the 3- and Hz components.

Calculation of the bispectrum of a digitized epoch, x i , begins with an FFT to generate complex spectral values, X f. This multiplication is the heart of the bispectral determination: if at each frequency in the triplet, there is a large spectral amplitude i.

Finally, the complex bispectrum is converted to a real number by computing the magnitude of the complex product. Equation 6. If one starts by sampling EEG at Hz into 4-s epochs, then the resulting Fourier spectrum will extend from 0 to 64 Hz at 0. If all triplets were to be calculated, there would be 65, [middle dot] points.

Fortunately, it is unnecessary to calculate the bispectrum for all possible frequency combinations. The unique set of frequency combinations to calculate a bispectrum can be visualized as a wedge of frequency versus frequency space Figure The combinations outside this wedge need not be calculated because of symmetry i. Still, because this calculation must be performed, using complex number arithmetic, for at least several thousand triplets, it is easy to see that it is a major computational burden.

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