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222 Y. F. Low and R. Martinez-Cancino
. MNE-Python: An open-source softwar e package for processing MEG and EEG
data in Python. It includes tools for preprocessing, source localization, and
statistical analysis, making it a versatile choice for researchers comfortable with
Python.
. OpenVibe: An open-source software platform designed for brain-computer
interface (BCI) applications, which includes functionalities for processing and analyzing EEG signals.
. BESA: A commercial software that is particularly well-suited for source analysis
and connect
ivity analysis of EEG and MEG data.
17.6 Concluding Remarks
We have explored several common preprocessing steps and shed light on the
essential considerations for developing an effective preprocessing pipeline. To
conclude, it is crucial to underscore that preprocessing techniques are not magical
solutions capable of transforming poor-quality data into high-quality data. Rather,
they serve to refine and enhance already good data, facilitating more accurate and
meaningful analysis. It is important to remember that the foundation of any successful EEG analysis lies in the quality of the raw signal—there is simply no substitute
for a clean, well-recorded signal. Therefore, while preprocessing is an indispensable
step, it should always be complemented by rigorous data acquisition practices to
ensure the best possible outcomes.
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Chapter 18
Introduction to EEG Oscillations
and Spectral Analysis
Ramon Martinez-Cancino and Yin Fen Low
Abstract Oscillatory brain activity reflects the dynam
processes. Understanding these oscillations requires methods beyon d the time
domain to reveal their frequency-domain information. This chapter introduces
spectral analysis as the foundation for studying EEG oscillations. Beginning with
the mathematical description of oscillatory processes, we build intuition around the
core concepts of amplitude, frequency, and phase. We then show how the dot
product provides the mathematical, geometric, and intuitive basis for Fourier analysis. The discrete Fourier transform (DFT) is presented as a practical tool for
decomposing EEG time series into the frequency domain, enabling precise characterization of oscillatory activity. Finally, we discuss the limitations of Fourier
analysis and outline ways to address them. Rather than thoroughly presenting
advanced methods in frequency analysis, this chapter aims to equip readers with
the conceptual and analytical framework needed to begin studying EEG time series
from a frequency-domain perspective.
Keywords Brain oscillations · Neural rhythms · Spectral analysis · Fourier
is · Dot product · Discrete Fourier transform · Power spectrum · Signal
analys
processing
ic coordination of neural
18.1 Introduction
As you may have discovered in earlier chapters, electroencephalography (EEG) is a
powerful tool for looking into the brain through its electrical activity. One of the
most distinctive features of EEG activity is its rhythmic natur e – an attribute that
extends beyond the human brain. In the 1870s, physician Richard Caton first noted
the presence of rhythmic electrical activity in the brains of animals (Caton,
few
decades later, Hans Berger observed similar phenomena in humans (Berger,
1875). A
R. Martinez-Cancino (*) · Y. F. Low
Brain Products GmbH, Gilching, Germany
e-mail:
ramon.martinez@brainproducts.com; yinfen.low@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_18
227
228 R. Martinez-Cancino and Y. F. Low
1929). Over time, researchers realized that these neural “rhythms” are not random
fluctuations but reflect structured patterns of neural activity. Many theories propose
that these oscillations help organize and synchronize neural activity across various
spatial and temporal scales (Varela et al.,
2001; Ward, 2003; Buzsaki & Draguhn,
2004). Such oscillatory coordination has been linked to processes ranging from
ion and cognition to consciousness (Ward,
percept
2003; Buzsaki & Draguhn, 2004).
Although a fascinating topic, this chapter does not deal with the historical origins
or biophysical mechanisms responsible for these rhythmic patterns. Instead, its goal
is to provide a theoretical base for interpreting the information they convey. Rhythms
have been our term of choice thus far, but moving forward, we will intentionally use
the term oscillations to emphasize the cyclic, repetitive nature of brain electrical
activity. Although brain rhythms and brain oscillations are often used interchangeably in neuroscience, a gradual conceptual evolution – fueled in part by interdisciplinary collaboration and the advent of refined analytical techniques – has prompted
a shift toward the more precise term oscillat ions.
This chapter explores the foundations of oscillation analysis: spectral analysis.
This
approach becomes more intuitive when we consider the brain’s electrical
signals to be inherently oscillatory. Given the strongly oscillatory nature of EEG
recordings, they have been characterized by the speed of these oscillations since their
early days. For instance, in his first observation of human EEG, Hans Berger
described what we know today as Alpha and Beta waves, which have frequency
ranges of 7–12 Hz and 12–30 Hz, respectively. Since that initial observation,
rhythmic activity in EEG has been widely reported and studied due to its ability to
characterize both normal and pathological brain states.
It is worth noting that embracing this o scillation-centric perspective is not just a
cal convenience. As Cohen (2014) points out, focusing on EEG oscillations
techni
means
acknowledging that it captures a dynamic, multidimensional domain of brain
activity. In this view, traditional event-related potentials (ERPs) represent only a
fraction of the information available in the EEG signal (Cohen,
oscillat
ion-based frequency analyses can reveal additional dimensions of brain
2014). Conversely,
dynamics that a purely time-domain approach might overlook (Herrmann et al.,
2014). This premise fundamentally shapes the content of the pages ahead.
In the
first part of the chapter, we introduce a mathematical description to
characterize oscillatory processes using a simple experimental setup. We discuss
the concepts of frequency, phase, and amplitude. Next, we build the intuition that
will allow us to justify the use of Fourier analysis, a key tool for frequency-domain
analysis of brain oscillations. Following, we introduce the dot product as a foundation for Fourier analysis. We then present the Discrete Fourier Transform (DFT) as a
practical implementation of this analysis. Throughout the chapter, the aim is to equip
you with the conceptual outlook and intuition needed to understand the spectral
analysis of EEG oscillations and unlock the rich information they contain in their
frequency domain. Finally, we present some of the DFT’s limitations and suggest
alternative approaches. A list of further readings is also provided at the end of the
chapter to complement the discussions.
18 Introduction to EEG Oscillations and Spectral Analysis 229
18.2 Character izing an Oscillatory Process
18.2.1 Fundamental Characteristics of an Oscillatory Process
Until now, we have been talking about frequency and oscillation without a proper
introduction. We will address these concepts now. Let us consider a mass – the bob –
suspended by a string from a fixed point, free to swing in a vertical plane. This setup
is known as a pendulum (see Fig. 18.1a). The bob’s position at any given time can be
describ
ed by its coordinates along the x and y axes. However, let us focus on its
motion along the x-axis.
From this perspective, as the pendulum swings, the projection (imagine its
shadow
) of the bob onto the x-axis traces a line between
travels from one extreme to the other with variable speed: it moves fastest at the
midpoint and slows to zero at the extremes, precisely at the instant it reverses
direction. We can model the bob’s position at time t, denoted x(t), with the following
sine-wave expression (Eq. 18.1):
Let us break this expression into smaller elements:
A and +A. The mass
ðx tð Þ¼ A sin 2πft þ θðÞ 18: 1Þ
. A represents
. sin is the sine function, which takes as
the maximum displacement of the motion around the midpoint.
its argument 2πft + θ.
. π (pi) is a fundamental constant approximately equal to 3.14159 , that is, the ratio
of a circle‘s circumference to its diameter (Wikipedia, 2025). In angle measurement,
π radians is equal to 180 degrees.
. f is the frequency, indicating how many
full swings the bob makes in a unit
of time.
Fig. 18.1 Pendulum motion and corresponding sinusoidal oscillation. (a) Pendulum oscillating in a
vertical plane, with its bob’s displacement on the x-axis restricted to the range A to A. (b) A
sinusoidal function describes the pendulum’s movement along the x-axis within this range
230 R. Martinez-Cancino and Y. F. Low
. θ is the phase (or phase angle offset), specifying where in its cycle the motion
starts at t 0.
Figure 18.1b shows the waveform generated by this function, and this is the
position
¼
of the bob over time. We have four important concepts to highlight here:
. Oscillation: Repetitive or cyclic variation in a system stat
pendulum example, a full swing of the bob – from one extreme to the other and
back – exemplifies an oscillation. This periodic motion can be described mathematically using sine or cosine functions.
. Amplitude: The maximum displacement from the mid
value, A, gives us an immediate idea of the “size” or “strength” of the oscillation.
This magnitude is related to the power or energy in each frequency – or frequency
band – through the squared ampl itude of the oscillation.
. Frequency: Indicates how many full oscillations
in hertz (Hz), where 1 Hz equals one complete cycle per second. In our pendulum
example, the frequency f captures the speed at which the pendulum swings back
and forth.
. Phase: Specifies the starting
in the cycle the pendulum begins its motion. A non-zero phase implies the bob
may already be going through a swing when we start observing it.
point of the oscillation at t ¼ 0. It determines where
occur in a unit of time, measured
e over time. In our
point of the oscillation. This
18.2.2 From Time to Frequency and Back
Consider now three sinusoids with different amplitudes and frequencies (Fig. 18.2a).
When added together, they produce a more complex waveform that still contains all
the original frequency components (Fig. 18.2b). This complex signal in the time
n can also be represented in the frequency domain by finding the signal’s
domai
components distribution of amplitudes over frequencies (Fig.
conveys critical information on the power contribution to a signal from its
tation
individual frequency components, and it is important to characterize the signal in this
domain.
A neat theoretical demonstration, you might say – but not exactly how things play
out
in real-world scenarios. In practice, we are usually presented with the opposite
challenge: Given a complex signal like the EEG, how can we identify and quantify
its underlying frequency components? This task is the core of spectral or frequency
analysis and is made possible by well-established signal processing methods like the
Fourier analysis.
18.2c). This represen-
18 Introduction to EEG Oscillations and Spectral Analysis 231
Fig. 18.2 Composite of sinusoids at 5, 10, and 15 Hz and corresponding frequency representation
(a). Sinusoids at three different frequencies: 5, 10, and 15 Hz. (b) Composite signal obtained by
summing the three sinusoids in panel a. (c) Power spectrum of the composite signal shown in panel
b, showing peaks at the corresponding frequencies
18.3 Foundation for Spectral Analysis: The Dot Product
The dot product is a mathematical operation on two vectors – think of two equallength sequences of numbers – that returns a single number. This operation is at the
core of the Fourier transform.
Computing the dot product is relatively simple; given the vectors A ¼ [a
] and B ¼ [b1, b2, ..., bn] where n is the number of elements of the vectors, to
a
n
, a2, ...,
1
compute the dot product between A and B, defined as A ∙ B , you multiply each
A vector element by their corresponding component of the vector B, and then sum up
all the values from each product (E q.
n
A ∙ B ¼
i¼1
18.2).
aibi ¼ a1b1þ a2b2 þ ⋯ þ anb
n
ð18:2Þ
The result of the dot product between two vectors can be interpreted in various
ways
depending on the context, although its definition remains consistent. For our
signal processing application, a particularly useful perspective is its geometric
interpretation. The dot product between the vectors A and B can also be defined by
Eq.
18.3.
A ∙ B ¼ A
cos θ ð18: 3Þ
j Bj j
j
where θ is the angle between the vectors A and B, and | | represents the length of the
vector. This relationship is depicted in Fig.
18.3a, where we have assumed that the
232 R. Martinez-Cancino and Y. F. Low
Fig. 18.3 Dot product between two vectors at different angles. Each panel illustrates the dot
product between vectors A (red) and B (blue). The green vector represents the projection of
A onto B, quantifying the component of A that lies in the direction of B. As the angle θ increases,
the projection shortens and can become negative, as seen in panel c. (a) θ ¼ 30 (π/6) (b)
¼
θ 90 (π /2) (c) θ 150 (5π/6)
¼
vectors in question can be represented in a bi-dimensional space, e.g., A ¼ [2, 1],
B ¼ [4, 0]. As the dimensions, or number of elements of the vectors grow, it may be
more complicated or even impossible to picture the graphical representation of the
vectors, but essentially, the interpretation of the dot product will remain as in our 2D
example. From this geometrical perspective, two common interpretations arise,
associated with aligning and projecting one vector onto the
other. From the projection perspective, the dot product is the length of the projection of A into the direction
of B when the two vectors are placed so that their origins coincide (Weisstein,
is, how much of A “lies along” the direction of B. From the alignment
That
1999).
perspective, the dot product may act as an alignment score between these two vectors
by indicating whether they point in similar directions or not. For example, see
Fig.
18.3a, if θ < π /2 the cosθ > 0 and A ∙ B > 0, here the two vectors point mainly
in
the same direction, and the dot product is a score of how much. In Fig. 18.3b
θ ¼ π /2 the cosθ ¼ 0 therefore A ∙ B ¼ 0, this indicates that these two vectors have
ortho
gonal directions, like a perfect cross. The last case, in Fig.
when
θ > π/2 the cosθ < 0 and A ∙ B < 0 which indicates that these two vectors have
18.3c, shows the case
somehow opposite directions. It may already be obvious that, despite the interpretation followed, the dot product indicates how similar the vectors involved in the
operation are.
18.4 Fourier Analysis
18.4.1 From Vectors to Sinusoids: The Fourier Connection
Nevertheless, why is there an insistence on discussing the dot product and its
geometrical interpretation? This is because it creates a powerful intuition that goes
beyond simple vectors. The same principle behind the dot product can be used to
decompose complex signals into sinusoids using the Fourier analysis. Here is a hint:
imagine a complex signal, a segment from an EEG recording, for example. This is
equivalent to a large vector with dimension equal to the number of sample points.
Then, imagine a pure sinusoid – a sinusoid with a single frequency. How would we