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.
EEG Analysis
The electroencephalogram can be analysed in several different ways. In diagnostic instances, a clinical specialist will visually review a recording, looking for visible changes in the signals, in order to identify bioelectrical abnormalities associated with disease. For example, the aberrant electrocortical activity associated with epilepsy, referred to as ictal activity, can be visually identified in an EEG (Beal et al., 2018; Krishnan, Chang, & Schomer, 2017; Spinosa, Liberalesso, Mehl, & Löhr Júnior, 2011).
Using information in the EEG that may not necessarily be amenable to visual analysis, certain disorders may also be distinguished from others. The electrocortical features that point to a disease, cognitive and/or emotional abnormalities are known as biomarkers (Nicotera et al., 2019; Smailovic et al., 2018; Snyder, Rugino, Hornig, & Stein, 2015). These features of an EEG signal are extracted using engineeringbased methodologies to undercover the information “hidden” from visual inspection (Delorme & Makeig, 2004; Oostenveld, Fries, Maris, & Schoffelen, 2011). Biomarkers can also be used to measure cognitive processes that unfold in realtime (Freeman, Mikulka, Prinzel, & Scerbo, 1999; McMahan, Parberry, & Parsons, 2015).
The discrete Fourier transform
EEG analysis rests on the foundations of digitalsignal processing methodologies (Delorme & Makeig, 2004; Klimesch, 1999; Oostenveld et al., 2011; Pfurtscheller & Lopes da Silva, 1999; Sanei & Chambers, 2007). The most important of all of signalprocessing methods is the Fourier transform. The Fourier transform, specifically the discrete Fourier transform (DFT), is used to transform a digitallysampled signal into a representation that allows one to assess the strength of the signal’s periodicities (rhythmic components) distributed as a function of frequency.
Analytically, this transform decomposes the timedomain signal into a sum of sinusoidal waves at discrete frequencies. From this sinusoidal representation, we can analyse the structure of the timedomain signal. Thus, we can talk about the spectral content of a signal or its spectrum. Alternatively, we can talk about transforming the signal into the frequencydomain (for an indepth discussion see: Oppenheim and Schafer, 2010, Ch 8 p. 648 – 741).
The equation below is the DFT for a finitelength signal of length :

(1) 
where . is complexvalued—it has a real and imaginary component. These components contain information about the combination of sinusoidal waves required to build the time domain signal . This information includes magnitude as well as phase information.
The inverse transform from frequencydomain back to the timedomain is given by:

(2) 
(Oppenheim and Schafer, 2010, p. 744).
The DFT is implemented in a number of different computational algorithms: Goertzel’s algorithm, the Winograd Fourier transform, and the fast Fourier transforms (FFT). As the name indicates, FFT algorithms exploit the essential properties of the Fourier transform in order to expeditiously compute .
Parceval’s theorem
According to Parceval’s theorem, the signal’s energy within the timedomain is conserved in the transformation into the frequency domain representation (Oppenheim & Schafer, 2010, p. 88). Thus, for a finitelength, discrete signal:

(3) 
where is the DFT of the signal .
The periodogram
The frequency domain representation, , is often expressed as a power spectral estimate: signal power varying as a discrete function of frequency. The power spectrum is an estimated quantity, representative of the true power spectrum of the underlying datagenerating process , sampled at discrete frequencies, where .
The power spectral estimate is also referred to as the periodogram, herein denoted as . Some important properties of are:
 nonnegative for realvalued signals;
 a realvalued function of frequency, and
 an even function of frequency (this property refers to the symmetric nature of the function, i.e. the function is mirrored at a midpoint of the estimate).
There is some fairly deep mathematical theory behind how the periodogram is derived, as well as its relationship to the true power spectrum of the datagenerating process, which we will not go into detail for simplicities sake. Herein, we focus on the instance of the DFT used in the context of EEG signalprocessing, providing the essential theory to enable EEG signalprocessing pipelines. Here we address the periodogram in general and lateron in the series focus on a specific application to EEG signals.
Windowing
In digitalsignal processing applications, an acquired signal, , will be windowed: prior to computing the DFT, the signal will be multiplied by a windowing (tapering) function, , the length of which is denoted . We denote the windowed signal :

(4) 
In this section, we assume that the window length is the same as the length of the signal, , and that is even.
Characteristically, EEG spectral analysis—estimating the periodogram and examining its properties—makes use of Gaussianshaped (cosinesum) functions, such as the Hamming window:

(5) 
(Oppenheim and Schafer, 2010, p. 565566. Note: for the purposes of spectral analysis, the window is computed at length and last sample point discarded, this is known as a periodic window (ibid. p. 566).
The window function, , effectively weightsdown (smooths) values at the start and end of the segment , less so in the middle—with the exception of a rectangular window, where all sample points are given equal weighting: . The periodogram estimated using a rectangular window is known simply as the periodogram, while an estimate using a nonrectangular window is known as the modified periodogram. Nonrectangular windows reduce spectral leakage, wherein discontinuities at the start and end of the timedomain signal result in signal power being attributed to frequencies outside true component of interest—this is also referred to as sidelobe leakage.
% generate a signal (random noise) x = 0 + 2.*randn(1,1024); N = length(x); L = N; % compute a periodic hamming window function w = zeros(); for n = 1:L+1 w(n) = 0.540.46*cos(2*pi*(n/L)); end w(end) = []; % window the signal v = x.*w;
Normalisation/Scaling
When using different window types (hamming vs. rectangular) in finite, smallsamples, one will observe a change in the average power of the spectral estimate owing to the different relative weightings applied to each sample point of the windowed signal. When the signal lengths approaches infinity, the estimates will be biased. Therefore, the power estimate will be scaled with a window normalisation constant, , multiplied by segment length . This ensures equivalent average power for PSD estimates under asymptotic assumptions, regardless of window type used, i.e. for all window types, the bias of the estimate will converge to zero as :

(6) 
(Oppenheim and Schafer, 2010, p. 865).
In the instance of a rectangular window, , otherwise window types yield a value where . Thus, assuming a modified periodogram, using a hamming window defined in Eq 5, the DFT of the windowed signal is:

(7) 
The periodogram is defined as:

(8) 
where the frequency index of the bin is given as, .
% compute the periodogram estimate Fs = 512; % sampling rate in Hertz V = fft(v); % compute the DFT via an FFT U = sum(w.^2)/L; % compute window normalisation constant I = (abs(V).^2)/(L*U); % periodogram F = (Fs/L)*(0:L1); % frequency index in Hertz
Properties of the Periodogram
The periodogram is an estimate of the underlying datagenerating process’s power spectrum within a range defined by , where is discrete. In the instance of the brain, the periodogram is an estimate of an electrocortical process’s true power spectral density under specific experimental conditions. The periodogram is known to be a biased estimate when estimated from finite length signals; however, a modified periodogram estimate, using appropriate scaling and normalisation, will be unbiased as N approaches infinity: it is an asymptotically unbiased estimate, therefore as , where the operator denotes the expected mean.
The periodogram, however, is an inconsistent estimate, i.e. the variance does not converge to zero as N approaches infinity. For large N, . In order, to address this, the periodogram routine introduces averaging over signal segments, which has the effect of decreasing the variance of the periodogram. The estimate becomes consistent, i.e. it converges to zero as the number of statistically independent segments approaches infinity: as . However, this decrease in variance tradesoff with a corresponding increase in estimation bias: a biasvariance tradeoff takes place.
Periodogram estimation and averaging in the context of EEG signals are covered in Part 2 of this series.
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