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18 Introduction to EEG Oscillations and Spectral Analysis 233
determine how much of that sinusoid is present in our EEG signal? Of course, it can
be done by taking a dot product!
Conceptually, the Fourier analysis operates by projecting the signal onto a whole
set of sinusoidal “basis vectors” with different frequencies, and each time asking:
How well does my signal match this specific sine wave? By doing this for all relevant
basis vectors, we decompose the signal into several sinusoidal components that,
when weighted and summed, can describe the original signal exactly. In this sense,
the dot product serves as the engine driving Fourier decom position; therefore,
developing a strong intuition for this operation is highly relevant. In the following
sections, we will dive more into the Fourier analys is.
18.4.2 The Fourier Family
While many individuals have contributed to the development of Fourier analysis, the
name itself honors Jean-Baptiste Joseph Fourier (1768–1830), a French mathematician and physicist, for his mathematical insights into the practical applications of
these techniques. The origins of Fourier analysis can be traced back to his first paper
on the Theory of Heat published in 1807 and expanded in 1822 (Fourier,
where
he discussed the expansion of functions in trigonometric series.
Fourier proposed that any continuous and smooth function could be expressed in
of trigonometric or exponential functions. Following his work, many others
terms
expanded on this field, making the study of Fourier analysis seem daunting today
due to the variety of names and classifications used.
Signals, for example, can be classified as either continuous, where values are
de
fined at every instant in time, or discrete, where values are only defined at specific
time points. Additionally, signals can be periodic, featuring repeating patterns, or
aperiodic, which do not show such patterns. The combination of these two characteristics creates four distinct categories of signals, each requiring different
approaches and terminology within Fourier analysis (Smith,
context of EEG analysis, some assumptions can be made to help us decide
In our
which approach is more appropriate. For example, our signals are recorded digitally
over limited periods of time. This means we do not have infinite recordings and do
not measure signals continuously. Instead, we sample them at regular time intervals,
resulting in a finite number of data points. However, no such Fourier Transform is
designed to deal with finite-length signals. Furthermore, sines and cosines are
defined as infinite signals, from minus to plus infinity, and they cannot be used to
build up a finite signal. The trick is imagining the signals in question have infinite
samples on the left and right of the actual points with value zero. Then, the resultant
signal looks discrete and aperiodic, and the Discrete Fourier Transform (DFT) can be
applied (Smith,
1997).
1997).
1822),
234 R. Martinez-Cancino and Y. F. Low
18.4.3 Discrete Fourier Transform
We now know that the DFT fits the type of data we deal with when analyzing the
EEG in digital signal processing (DSP) settings: sampled at fixed intervals and
limited in duration. Now let us get into the detai ls: imagine we have a signal sampled
regularly at time intervals Δt during the recording process into N discrete points x[n],
where n ¼ 0, 1, 2, ⋯, N
duration NΔt, and a sampling frequency f
this discrete-time signal into a discrete-frequency representation using the transformation in Eq.
18.4.
1. This means our signal has a finite length, with a time
¼ 1/Δt. The DFT allows us to transform
s
N 1
X k
n¼0
x n e
i2πkn=N
, k ¼ 0, 1, 2, ⋯, N 1 ð18: 4Þ
Here, X[k] is the – complex – Fourier coefficient of the time series x at frequency
k
. In this expression, you can also identify the main terms within the sum operator;
one is the signal to decompose, x[n], and the other one is an exponential function
i2πkn/N
e
with a complex argument. Here, we say complex because the term i ¼
1p is present, as in the coefficient X[k]. This may look intimidating at first, but it is
another way to represent trigonometric functions in an exponential notation through
Euler’s formula (Eq. 18.5).
iθ
¼ cos θ þ i sinθ ð18:5Þ
e
Introducing the Euler notation into Eq. 18.4 lead us to Eq. 18.6, a more familiar
expres
sion in line with our previous discussions (remember our dot product!).
N 1
X k
x n cos 2πkn=N i x n sin 2πkn=NðÞ, k
n¼0
¼ 0, 1, 2, ⋯, N 1 ð18 :6Þ
This clearly illustrates
how the DFT quantifies the alignment of our original
signal with the corresponding cosine and sine waves at specific frequencies k.
A valid question at this point might be: How is this set of frequencies
k
determined? To answer this, we need to understand that the highest frequency
that can be extracted from a time series is given by its Nyquist frequency, which is
half its sampling frequency (Eq.
18.7).
f
s
¼
f
N
2
ð18:7Þ
18 Introduction to EEG Oscillations and Spectral Analysis 235
And that the number of unique frequencies that can be extracted from an N-length
time series – known as the frequency resolution of the signal – equals one-half of the
number of data points plus the zero frequency, N/2 + 1 (Cohen,
frequencies in Eq. 18.6 correspond then to linearly spaced N/2 + 1 frequencies
k
between
zero and f
. Here, we have assumed N to be even, which is the most
N
2014). The
common case in practice.
It’s important to mention that the DFT delivers an exact reconstruction of the
signa
l in its frequency domain and is fully reversible without any loss of information.
This is done by the inverse Fourier transform (Eq.
N 1
x½n
1
N
k¼0
X½k e
i2πkn=N
, n ¼ 0, 1, 2, ⋯, N 1 ð18:8Þ
18.8).
This inverse reconstruction can be particularly useful, for instance, when a signal
has
been transformed into its frequency domain; some frequency components are
rejected, and then the signal is reconstructed. This exemplifies a typical filtering
use case.
It is particularly important to know that the DFT, as presented here, only serves
ic purposes. Yes, it delivers exact results that allow the representation of a
didact
signal in the frequency space and its reconstruction back to the time domain with
zero loss. However, it is hugely inefficient computationally for practical use. There
are at least three methods used to implement the DFT for practical applications. First,
the problem can be approached as a set of simultaneous equations. The second
method, DFT by correlation, is based on detecting a known signal within another –
think of correlating the signal with sinusoidal basis functions, and the last, and
probably the most popular, is the Fast Fourier Transform (FFT). The FFT is an
ingenious algorithm that decomposes a DFT with N points into N DFTs, each with a
single point (Smith,
the
square of the number of sample points in the signal, N
1997). Computationally, the speed of the DFT is proportional to
2
, but in the FFT case it is
N log N. This could be the difference between 1 min and 1000 min (17 h) (Cohen,
2014)!
18.4.4 Power Spectrum
After performing the Fourier analysis, the next step is to present and interpret its
results. As mentioned previously, the Fourier coefficients are complex magnitudes
with real and imaginary parts; therefore, they can be represented as in Eq.
X k X
From here,
in Eq.
we can easily compute the magnitude of each frequency component as
18.10; the power is then obtained by taking the square of this value.
k i X
Re
k 18: 9Þ
Im
18.9.
236 R. Martinez-Cancino and Y. F. Low
Fig. 18.4 Representation of the Fourier analysis of sinusoids. (a) A 3 Hz sine wave (left) and its
spectrum (right), showing a single peak at 3 Hz. (b) An 8 Hz sine wave (left) and its spectrum
(right), showing a single peak at 8 Hz. (c) The sum of the 3 Hz and 8 Hz sine waves (left). The
corresponding spectrum (right) displays two peaks, reflecting the presence of both frequency
components
X k
¼ X
j j
Re
k 2þ X
2
k
Im
ð18: 10Þ
A common way of visualizing the results from the Fourier analysis is through a
plot showing the values of Eq. 18.10: frequency on the x-axis and power or
2-D
amplit
ude on the y-axis (Cohen,
2014) as in Fig. 18.4.
18.4.5 Limitations of Fourier Analysis and Advanced
Methods
Fourier analysis is widely regarded as one of the most powerful tools for performing
and understanding frequency and time-frequency analysis in signal processing.
Nevertheless, it does have its limitations. In this section, we will briefly discuss
these limitations and suggest alternative methods that can help mitigate their drawbacks. To complement the discussions in this chapter, we provide a list of further
readings to enhance your understanding of the concepts covered.
18 Introduction to EEG Oscillations and Spectral Analysis 237
Fourier analysis assumes that the signal is stationary throughout the analysis
window. This means that the statistical properties of the data remain constant over
time, and the power representation from the Fourier perspective does not change
over time. For EEG signals, this limitation is critical when we analyze processes that
change over time, such as the response to external stimuli or due to endogenous
processes like the development of an epileptic seizure. This limitation has two
practical consequences for our results: first, it reduces the “resolution” of the Fourier
analysis, since to represent a more complex frequency structure on a non-stationary
signal, more closely spaced frequency components need to be added. Second, we
implicitly give away all the temporal information in our signal; there is no description of how the spectrum evolves over time. Of course, this can be overcome if we
analyze shorter data segments where we can assume the signal is stationary, and by
making this a moving window, we can achieve a dynamic description of the
frequency domain over time. This is indeed one of the main rationales behind the
short-time Fourier transform (Oppenhei m,
with
the trade-off between time and frequency resolution. A shorter window yields
1999). However, here we are presented
better time localization but poor frequency resolution, while a longer window
improves frequency discrimination at the cost of smearing timing information
(Daubechies, 1992; Mallat, 1999; Samar et al., 1999, Rosenblatt et al., 2014).
The limitations discussed can be addressed by using wavelet transforms (Oppenheim, 1999), which inherently handle non-stationary signals by analyzing EEG data
both time and frequency domains. Unlike the Discrete Fourier Transform (DFT),
in
wavelet analysis does not assume that signals are stationary, allowing the spectral
content to change over time. Another alternative approach to the DFT is the
Multitaper Analysis (Thomson, 2000), which improves spectral estimates by reducing variance, making it particularly useful for analyzing short or noisy time series.
Further Readings
Addison, P. S. (2002). The illustrated wavelet transform handbook. CRC Press.
Cohen, M. X. (2014). Analyzing neural time series data: Theory and practice. MIT Press.
Cohen, M. X. (2014). Fundamentals of time-frequency analyses in Matlab/Octave. MIT Press.
Smith, S. W. (1997). The scientist and engineer’s guide to digital signal processing. California
Technical.
van Drongelen, W. (2007). Signal processing for neuroscientists. Introduction to the analysis of
physiological
signals. Academic.
References
Berger, H. (1929). Über das elektroenkephalogramm des menschen. Archiv für Psychiatrie und
Nervenkrankheiten, 87(1), 527–570.
238 R. Martinez-Cancino and
Y. F. Low
Buzsaki, G., & Draguhn, A. (2004). Neuronal oscillations in cortical networks. Science, 304(5679),
1926
–1929.
Caton, R. (1875). Electrical currents of the brain. The Journal of Nervous and
Mental Disease,
2(4), 610.
Cohen, M. X. (2014). Analyzing neural time series data: Theory and practice. MIT Press.
Daubechies, I. (1992). Ten lectures on wavelets. SIAM.
Fourier, J. (1822). Theorie analytique de la chaleur. par M. Fourier, Chez Firmin Didot, père et fils.
Herrmann, C. S., Rach, S., Vosskuhl, J., & Strüber, D. (2014). Time–frequency analysis of event-
potentials: A brief tutorial. Brain Topography, 27, 438–450.
related
Mallat, S. (1999). A wavelet tour of signal processing. Elsevier.
Oppenheim, A. V. (1999). Discrete-time signal processing. Pearson Education India.
Rosenblatt, M., Figliola, A., Paccosi, G., Serrano, E., & Rosso, O. A. (2014). A quantitative
analysis
of an EEG epileptic record based on multiresolution wavelet coefficients. Entropy,
16(11), 5976–6005.
Samar, V. J., Bopardikar, A., Rao, R., & Swartz, K. (1999). Wavelet analysis of neuroelectric
waveforms:
A conceptual tutorial. Brain and Language, 66(1), 7–60.
Smith, S. W. (1997). The scientist and engineer’s guide to digital signal processing. California
Technical
Pub.
Thomson, D. J. (2000). Multitaper analysis of nonstationary and nonlinear time series data. In
W.
Fitzgerald, R. Smith, A. Walden, & P. Young (Eds.), Nonlinear and nonstationary signal
processing (pp. 317– 394). Cambridge University Press.
Varela, F., Lachaux, J.-P., Rodriguez, E., & Martinerie, J. (2001). The brainweb: Phase synchro-
and large-scale integration. Nature Reviews Neuroscience, 2(4), 229–239.
nization
Ward, L. M. (2003). Synchronous neural oscillations and cognitive processes. Trends in Cognitive
Sciences,
7(12), 553–559.
Weisstein, E. W. Dot product. MathWorld—A Wolfram Web Resource. https://mathworld.
wolfram.com/DotProduct.html
Wikipedia contributors. (2025, June 21). Pi. Wikipedia. Retrieved June 27, 2025, from https://en.
wikipedia.org/w/index.php?title 1296692125
¼Pi&oldid¼
Chapter 19
Event-Related Potentials
Michael Hoppstädter
Abstract Event-related potentials (ERPs) are one of the most common applications
EEG. Many ERPs are linked to different sensory or cognitive processes, and they
for
are investigated whenever researchers are interested in brain responses time-locked
to an event.
This chapter will enable you to create your own ERPs from pre-processed EEG
and provide you with some background information on the basic principles
data
behind ERPs and how to interpret them. The characteristic shape of the ERP
waveform will be discussed, and some typical examples of ERP components will
be covered in brief. The chapter will close with a discussion of typical ERP features
that can be extracted for statistical analysis.
Keywords ERP · Evoked response ·
Sensory component · Cognitive component ·
Epoching · Baseline correction
19.1 Introduction
EEG is a well-established measure and has been around for more than 100 years.
While ERPs were also investigated early in the 1930s (Luck, 2014), their rise to
inence only started in the 1960s with publications on the contingent negative
prom
variation or CNV (Walter et al.,
What Is an ERP The explanation is already in the name—Event-Related Potential.
the focus of interest is an electrical potential (since the recording technique is
Thus,
EEG) that is related to an event. The term event usually refers to stimulation of any
sensory modality, for instance, visual, auditory, haptic, olfactory, or even pain.
However, the responses evoked by these stimuli are so small that they do not
stand out from the ongoing oscillatory activity, and it would be hard to detect and
analyze them. Overcoming this issue is rather simple: the response to either the same
M. Hoppstädter (✉)
Brain Products GmbH, Gilching, Germany
e-mail:
michael.hoppstaedter@brainproducts.com
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2026
T.
Warbrick (ed.), The EEG Handbook,
https://doi.org/10.1007/978-3-032-20450-9_19
1964) and the P300 (Sutton et al., 1965).
239
240 M. Hoppstädter
Fig. 19.1 Schematic of the ERP logic. Each trial contains ongoing task-unrelated background
activity and evoked activity that is due to the stimulation. Averaging over many repetitions reduces
the ongoing activity and amplifies the evoked activity in the final ERP
or similar stimuli is not only measured once but many times, and then all responses
are averaged to yield the mean activity related to that type of event—the ERP.
The evoked response to the stimulus is very similar across repetitions as the EEG
oscillations are roughly showing the same deflections at the same time. They are time
and phase-locked. Therefore, averaging these time and phase-locked responses
amplifies the evoked activity. The ongoing oscillatory background activity, however, will be random across trials as it is neither time nor phase-locked. Such random
activity will mostly cancel out through averaging. Hence, the stimulus-unrelated
background noise will be attenuated (see Fig.
19.1). From a signal detection per-
spective, this means that the signal-to-noise ratio (SNR) increases by adding more
repetitions, as the noise (i.e., task-unrelated variability) is continuously reduced
(Luck,
2014). Whether the ERPs are generated by many evoked responses that
sum up or through a phase resetting of ongoing background oscillations (Sauseng
et al.,
2007) is a matter of debate.
19.2 How to Get from EEG to ERPs
A Simple ERP Experiment Since ERPs have been investigated for decades, many
classical paradigms are known in the EEG community. An easy and widely used
example is the oddball paradigm which has been investigated in many variations in
thousands of studies. In its simplest form, you have 2 stimuli, they can be for
example different shapes (like X’s and O’s) or different tones. Imagine the same
19 Event-Related Potentials 241
tone is repeated many times and the participant only needs to listen. Sometimes the
sequence will be interrupted with another tone of a different pitch. This second tone
occurs more rarely and it cannot be predicted. The data segments around the tones
are then averaged separately for frequent and rare tones to get the ERPs for the two
conditions. Finally, the two averages are contrasted to reveal any differences in the
ERP waveform between conditions. This very easy paradig
explaining the logic of ERP experiments, and I will come back to it at different
points of this chapter.
How to Measure the Right Responses Two factors are key to this process: precise
timing and correct labeling. Timing is important because the brain reacts extremely
quickly to external stimuli. To accurately detect the neuronal responses that occur
within a few hundred milliseconds, you need to know exactly when a stimulus is
presented, or when a response is given. This is achieved by sending triggers to the
EEG amplifier (see Chap.
must be avoided, and therefore it is essential to properly set up and test your
jitter)
recording pipeline (see Chap.
been presented. Coming back to the oddball example, you need to know which of the
two tones was presented when. For most experiments, there will be more than two
stimuli, thus it is important to know to which category, or condition, each stimulus
belongs. Otherwise, it would be impossible to average all stimuli of the same
condition. Thus, appropriate labeling of the triggers is the second key factor.
14: Triggers). Any time delays (especially unsystematic
10: Pilot Testing). It is also crucial to know what has
m is a great example for
19.2.1 How to Process Your ERP Data
19.2.1.1 Pre-processing
It is necessary to treat the raw, continuous EEG data with adequate pre-processing
techniques to clean and prepare it for ERP analysis. There is no general standard
pre-processing pipeline for ERPs, but some steps are commonly used across many
ERP studies. I want to mention a few key factors that play a big role in
pre-processing and should be considered when planning the analysis:
Sampling Rate Consider before recording which sampling rate you need to accomplish
the planned analyses. It is not necessary to keep a very high rate for ERPs, but
do not down-sample too far.
Filtering You need to
ERPs. Consider especially the low cutoff which removes slow frequencies from the
data. Also think about whether you need a notch filter (to remove mains noise). See
also Chap.
ERP
Optimal Ref
can always re-reference offline. The final reference for the analysis should be chosen
22: Quantifying EEG and ERP Data Quality for a detailed discussion of
components and appropriate filter parameters.
erence While the online reference is often preset by the hardware, you
determine the right filter band that does not distort your
242 M. Hoppstädter
based on the spatial distribution of the ERP effect of interest. Consider the choice of
reference in publications using similar paradigms to ensure comparability of your
findings.
Artifact Handling In order to avoid contamination of the ERP waveform, it is often
necessary to attenuate or reject large artifacts. You should focus on ocular, muscle,
and cardiac artifacts and large head or body movements.
Generally, pre-processing should always be adapted to your experiment and
is goals. See Chap. 17: EEG Pre-processing and Artifact Handling for a
analys
detailed discussion of this topic.
19.2.1.2 Trial Selection
Once the continuous EEG data has been cleaned, it is time to select the time intervals
that
you want to further investigate. A few different names are used to refer to the
intervals around experimental stimuli: trials, epochs, and segments are commonly
used but they mean the same thing. The procedure is often called epoching or
segmentation. For simplicity, I will stick to the term trial. If the recording contains
markers (as a result of sending triggers during the experiment) that tag the time point
and the condition of each stimulation, these can be used to segment the data into
smaller chunks, only keeping the experimental trials.
In addition to identifying the right data portions in your recording, you also need
to
decide how long the trials should be. Generally, the trials should be long enough
to capture the full ERP response. In most cases, you will keep a section of data
around the stimulation marker, so that you have a pre-stimulus and a post-stimulus
period. The pre-stimulus period is usually used for baseline correction (see Sect.
19.2.1.3) while the post-stimul us period is the interval in which you observe the
ion to the stimulus, i.e., the ERPs. Note that there are exceptions: some ERPs
react
occur prior to the stimulation while others are locked to the time of a response to the
stimulus (see Sect.
the
same for all trials that will be averaged together, and also across all conditions
19.4.3 for examples). Importantly, the timeline of a trial should be
that you will compare against each other. The frequency of the stimulus presentation
should also be considered when you design the experiment. The inter-stimulus
interval (ISI) should be long enough so that successive trials will not overlap.
However, this is not always possible, for example, in paradigms with fast
stimulation.
19.2.1.3 Baseline Correction
As EEG
is constantly fluctuating due to ongoing oscillations, the voltage level is
changing over the course of the recording. Hence, the mean voltage between trials
can vary considerably. As soon as you want to compare the amplitudes between
trials, this becomes a problem. Thus, removing this offset in amplitude level between