This twopart series looks at the underlying mathematical theory and practical steps required to execute a spectral analysis of acquired electroencephalographic data. Spectral analysis, analysing how a signal’s power is distributed across a range of discrete frequencies, is fundamental to understanding an electroencephalogram as well as associated changes in psychological states. Solid, robust, replicable electrophysiology is built on the foundation of spectral analysis and one cannot emphasise enough the importance of the Fourier transform in this regard. The first part of this series deals with the general theory of the discrete Fourier transform (DFT), while the second part focuses on the mathematical/statistical properties of an EEG and how to apply a DFT in the context of biomedical signalprocessing.
Power spectral density of an electroencephalographic signal recorded over the occipital region during an eyesopen session. The shaded area around the spectrum is the 95% confidence interval around the mean of the estimate. The spectra has a number of prominent rhythmic components. The strong peak at 9Hz is the dominant alpha rhythm. One can also observe a strong beta component from 13Hz17Hz. Together, these spectral features indicate that the person is alert, awake, and aware of their environment.
The Electroencephalogram, Random processes and the Power Spectral Density
EEG signals are seen as arising from random or stochastic processes, which means the observed/acquired signal, , at any is determined by a unique underlying probability distribution and is related across time by joint probability distributions (Oppenheim & Schafer, 2010, p. 10671068). For a random process, the power spectral density (PSD) is theoretically derived by taking the DFT of the autocorrelation sequence of the signal; however, in practical signalprocessing, we can use the periodogram as an estimate of the PSD. Both methods are acceptable for purposes spectral estimation of random processes (ibid. Ch 10).
A signal generated by a random process is referred to as a realisation of that process. We may acquire multiple realisations of a given process during a single EEG recording. For analytical purposes, the acquired signal will be segmented into equal length segments—also referred to as epochs. For EEG signals, the segment length can be 0.5 – 2.5 seconds. Enforcing a fixed length onto signal segments will affect the lower bound of the analytical bandwidth, as at least one cycle of the lowest frequency would need to be represented in a segment to be amenable to analysis. Enforcing a segment length also ensures stationarity of the segment. In the context of random processes in general, and EEG signalprocessing specifically, widesense stationarity is a requirement for application of the DFT to a signal or segment. This is a statistical property of the signal, such that:
 the mean of the signal is timeinvariant;
 the autocovariance sequence is timeinvariant, and
 the variance of is finite across time points.
In EEG analysis, and in the analysis of most realvalued random processes, we often scale the power spectral estimate as the distribution of power in the frequency domain per unit frequency: Power/Hertz , this scaled spectrum is specifically referred to as a power spectral density (PSD). In essence, it is a periodogram estimate; however, the scaling and normalisation applied differ from Eq 8. This is explained further in the following section.
Truncating and Scaling the PSD
Practically speaking, we are only interested in the values within the range from 0Hz – Nyquist frequency, where the Nyquist frequency is defined as half the sampling rate of the acquired data and is expressed in Hertz. As mentioned previously, the periodogram will provide spectral estimates up to and including the sampling rate of the data. Thus, if the sampling rate was 128Hz, we would only be interested in spectral estimates up to and including 64Hz. The reason is that any sampled data above the Nyquist frequency is subject to aliasing. Essentially, rhythms in the data occurring above the Nyquist frequency are undersampled: one has not got enough datumpoints to represent the entire rhythmic cycle of such components. So, if we were to attempt to reconstructed the original signal from the sampled data, the rhythms above the Nyquist would be reconstructed at an incorrect frequency, i.e. they appear as something they are not, an alias. So, we truncate , removing all datumpoints above the Nyquist frequency and denoted the new length as :

(9) 
where the subscript T in denotes the truncated version of the DFT.
In order to conserve power correspondence between time and frequency domains—owing to the fact that spectral estimates above the Nyquist frequency were discarded, will be multiplied by 2 over the range , where is the DC component of the signal and is the Nyquist frequency. For this particular estimate of the PSD, we compute a new window normalisation constant :

(10) 
where ‘ denotes the vector transpose. In the case of a rectangular window W = LU = 1. In most practical instances, the scaled, normalised periodogram is taken as an estimate of the PSD:

(11) 
where the sampling interval ; the frequency index of each bin, , is given as, .
% compute power spectral density W = w*w'; % normalisation constant Vt = V(1:length(V)/2); % truncate DFT It = zeros(1,length(Vt)); % zeropadded buffer It(2:end1) = 2*(1/Fs)*(abs(Vt(2:end1)).^2/W); It(1) = 2*(1/Fs)*(abs(Vt(1)).^2/W); It(end) = 2*(1/Fs)*(abs(Vt(end)).^2/W);
Periodogram Averaging (Welch’s Periodogram)
Mentioned previously, for the purposes of spectral analysis using a DFT, will be segmented into equal length, widesense stationary segments of length —these maybe overlapping or not:

(12) 
where , and . The parameter, , is a positive integer value. When , the segments will overlap; when , the segments are nonoverlapping (contiguous). The subscript in , is the index of the segment, . is the total number of segments for which, . Thus, assuming then (Oppenheim and Schafer, 2010, p. 868).
A windowedDFT will be applied to each segment to yield an estimate, , computed as in Eq 11. The resultant local spectral estimates will be averaged to yield a final estimate:

(13) 
% generate a signal (random noise) clear rng default% reset random number generator % signal Fs = 512; % sampling rate duration = 120; % 120 second duration signal Q = Fs * duration; x = 0 + 2.*randn(1,Q); % zeromean random noise t = 0:1/Fs:duration1/Fs; % timeindex (2 seconds) x = cos(2*pi*100*t) + x; % sinusoidal at 100 Hz % segmentation parameters L = 1024; R = 1024; K = Q/L; F = (Fs/L)*(0:L1); % frequency index in Hertz % compute periodogram [w] = compHamming(L); % hamming window PSDs = zeros(K,L/2); for r = 0:K1 % window the signal startInd = (r*R)+1; endInd = (startInd+L)1; xr = x(1,startInd:endInd); v = xr.*w; [It] = computeScaledPSD(v,w,Fs); % PSD PSDs(r+1,:) = It; end % average across segments estPSD = mean(PSDs,1);
The PSD is represented in log or linear scale, or as a baseline corrected measure: measured relative to a baseline or reference period. The latter is often expressed as a percentage change in power, a zscore, or a difference score taken between logarithmicallyscaled spectral estimates. The choice is dependent upon the application (Delorme & Makeig, 2004; Gerhold, 2017; Klimesch, 1999; Pfurtscheller & Lopes da Silva, 1999). Baseline corrected EEG measurements are used extensively in consumer neuroscience applications
(see Vecchiato, Cherubino, Trettel & Bablioni, 2013 for detailed methodologies applied to television advertising assessments using many of the abovementioned baseline correction methodologies). The reader may also consult Gerhold (2017, Ch2, p. 1154), wherein the author(s) discuss the various tradeoffs involved in parametrisation of the DFT algorithm for analysis of eventrelated EEG data.
Measures of BandPower
As Parceval’s theorem holds (which is generally stated for a continuous case of the Fourier transform), integrating values between any two points on a frequency domain function will yield energy in a subband bounded by two points. In the case of a PSD, summing values bounded by two frequency points and yields total bandpower in that subband:

(14) 
Alternatively, the values including and between the upper and lower bound may be averaged to provide average bandpower, a ubiquitous measure of oscillatory power of an EEG signal:

(15) 
where is the total number of frequency bins summed across.
% compute band power measures
f1 = 7.5; f2 = 13;
f1 = find(F==f1); f2 = find(F==f2);
totalBandPower = sum(10*log10(estPSD(1,f1:f2)));
J = (f2f1)+1; avgBandPower = totalBandPower/J;
Through studying the power of the EEG signals in different subbands and how these bandpower measures covary with task demands, in terms of cognitive or affective processes, we can gain an understanding of how the human brain (specifically how the cerebral cortex) responds to and handles information and even make predictions about populations of participants and consumers. For pivotal methodological and theoretical literature, see Klimesch (1999) and Klimesch, Sauseng, & Hanslmayr (2007). For consumer neuroscience applications see Ramsøy, Skov, Christensen, & Stahlhut (2018), Shestyuk, Kasinathan, Karapoondinott, Knight, & Gurumoorthy (2019) and Vecchiato, Cherubino, Trettel, & Babiloni (2013).
Delorme, A., & Makeig, S. (2004). EEGLAB: an open source toolbox for analysis of singletrial EEG dynamics including independent component analysis. Journal of Neuroscience Methods, 134, 9–21.
Gerhold, M. (2017). A study of eventrelated electrocortical oscillatory dynamics associated with cued motorresponse inhibition during performance of the Go/NoGo task within a sample of prenatally alcoholexposed children and agematched controls. Cape Town: University of Cape Town.
Klimesch, W. (1999). EEG alpha and theta oscillations reflect cognitive and memory performance: a review and analysis. Brain Research Reviews, 29, 169–195.
Oppenheim, A. V., & Schafer, R. W. (2010). DiscreteTime Signal Processing (International Edition). Upper Saddle River: Pearson.
Pfurtscheller, G., & Lopes da Silva, F. H. (1999). Eventrelated EEG/MEG synchronization and desynchronization: basic principles. Clinical Neurophysiology, 110, 1842–1857.
Ramsøy, T. Z., Skov, M., Christensen, M. K., & Stahlhut, C. (2018). Frontal Brain Asymmetry and Willingness to Pay. Frontiers in Neuroscience, 12. https://doi.org/10.3389/fnins.2018.00138
Shestyuk, A. Y., Kasinathan, K., Karapoondinott, V., Knight, R. T., & Gurumoorthy, R. (2019). Individual EEG measures of attention, memory, and motivation predict population level TV viewership and Twitter engagement. PLOS ONE, 14, e0214507. https://doi.org/10.1371/journal.pone.0214507
Vecchiato, G., Cherubino, P., Trettel, A., & Babiloni, F. (2013). Neuroelectrical Brain Imaging Tools for the Study of the Efficacy of TV Advertising Stimuli and their Application to Neuromarketing. In Biosystems & Biorobotics. Retrieved from https://www.springer.com/gp/book/9783642380631
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