The document provides an introduction to signals, covering their types, transformations, and classifications. It explains mathematical models such as Fourier and Laplace transforms, as well as filter types like passive and active filters. Applications in audio, communications, and signal processing are highlighted, detailing how modulated signals, filtering techniques, and frequency analysis are essential for effective signal design and analysis.

1. Introduction to
Signals
Signals are the fundamental building blocks of communication, carrying
information through various mediums like sound, light, and electricity.
Understanding the mathematical models and classifications of signals is
crucial for designing efficient communication systems.

2. Mathematical Model of Signals
1
Continuous-Time Signals
Signals that are defined for all points in
time, such as analog audio or video
waveforms.
2
Discrete-Time Signals
Signals that are defined only at specific,
equally-spaced time instants, such as
digital audio samples.
3
Signal Transformations
Applying mathematical operations like
Fourier transforms to analyze and
manipulate signals in the frequency
domain.

3. Signal Classification
Periodic vs. Aperiodic
Periodic signals repeat
themselves after a fixed
time interval, while
aperiodic signals do not.
Deterministic vs.
Random
Deterministic signals have
a known, predictable
behavior, while random
signals have
unpredictable, statistical
properties.
Energy vs. Power
Signals
Energy signals have a
finite amount of total
energy, while power
signals have a continuous
energy flow over time.

4. Elementary Signals in
Continuous Time
Continuous-time signals are functions that vary with time, like sound
waves or radio signals. Understanding the fundamental properties of
these signals is crucial for analyzing and processing them in engineering
applications.

5. Sinusoidal Signals
1 Amplitude
The maximum value reached by the signal.
2 Frequency
The number of cycles per unit of time.
3 Phase
The initial position of the signal in its cycle.
Sinusoidal signals are fundamental building blocks for many physical phenomena and
engineering applications. Their periodic nature and easily-defined properties make them
invaluable for signal analysis and system design.

6. Exponential Signals
Growth
Exponential signals can
model processes that
increase rapidly over time,
like population growth or
the spread of a disease.
Decay
They can also represent
phenomena that diminish
over time, such as the
discharge of a capacitor or
the absorption of light in a
medium.
Oscillation
Combining exponential
growth and decay creates
oscillating signals, which
are fundamental to many
electronic circuits and
communication systems.
Exponential signals are another important class of continuous-time functions, exhibiting either
growth or decay over time. They have wide-ranging applications in science, engineering, and
beyond.

7. Periodic Signals in
Continuous Time
Periodic signals in continuous time are mathematical functions that
repeat themselves at regular intervals. These signals are essential in
various fields, such as electronics, communications, and signal
processing, where they are used to transmit and process information.
by ECP-Ege

8. Fourier Series Representation
Trigonometric Form
Periodic signals can be
represented as a sum of
sine and cosine waves
with different amplitudes
and frequencies, known as
the Fourier series in
trigonometric form.
Harmonic Form
The Fourier series can
also be expressed in
harmonic form, where the
signal is represented as a
sum of harmonically
related sinusoids with
integer multiples of the
fundamental frequency.
Complex Form
The Fourier series can be
represented in complex
form, using complex
exponentials, which
provides a more concise
mathematical expression
and simplifies certain
calculations.

9. Amplitude, Phase, and Power Spectral
Distributions
1 Amplitude
Spectrum
The amplitude
spectrum of a
periodic signal
reveals the relative
magnitudes of the
frequency
components that
make up the signal.
2 Phase Spectrum
The phase spectrum
provides information
about the relative
timing or phase shift
of the frequency
components in the
signal.
3 Power Spectral
Distribution
The power spectral
distribution
represents the
distribution of power
across the frequency
components of the
signal, which is
important for signal
analysis and
processing.

10. Nonperiodic Signals
in Continuous Time
Nonperiodic signals in continuous time are not repeating patterns over
time. These signals do not have a well-defined period and can have a
wide range of irregular shapes and frequencies. Understanding the
characteristics of nonperiodic signals is crucial for analyzing and
processing real-world data.

11. The Fourier Transform
1
Definitions
The Fourier transform is a mathematical
tool that decomposes a signal into its
frequency components, allowing for
analysis in the frequency domain.
2
Conditions of Existence
For the Fourier transform to exist, the
signal must be absolutely integrable and
satisfy certain continuity and
boundedness conditions.
3
Amplitude and Phase Spectra
The Fourier transform provides the
amplitude and phase information of the
signal's frequency components, known
as the amplitude and phase spectra.

12. Spectral Energy Distribution
1 Aperiodic Signals
The spectral energy distribution of
an aperiodic signal is continuous
and often irregular, lacking the
distinct peaks associated with
periodic signals.
2 Energy Concentration
The spectral energy distribution
reveals how the signal's energy is
distributed across the frequency
spectrum, providing insights into the
signal's characteristics.
3 Frequency Analysis
Analyzing the spectral energy distribution can help identify dominant frequency
components and understand the underlying dynamics of the aperiodic signal.

13. Introduction to
Laplace Transform
The Laplace transform is a powerful mathematical tool used to analyze
and solve linear differential equations. It converts a function of time into a
function of a complex variable, enabling simpler analysis and solutions.

14. Definitions and Conditions of Existence
1 Definition
The Laplace
transform of a
function f(t) is
denoted as F(s),
where s is a complex
variable.
2 Conditions of
Existence
For the Laplace
transform to exist, the
function f(t) must be
piecewise continuous
and of exponential
order.
3 Exponential Order
This means that f(t)
must be bounded by
an exponential
function as t
approaches infinity.

15. Unilateral and Bilateral Laplace
Transforms
Unilateral Laplace
Transform
The unilateral Laplace
transform is defined for t ≥
0, and is commonly used
to analyze causal
systems.
Bilateral Laplace
Transform
The bilateral Laplace
transform is defined for all
real values of t, and is
used to analyze non-
causal systems.
Applications
Unilateral transforms are
used in control systems,
circuit analysis, and signal
processing, while bilateral
transforms are used in
quantum mechanics and
other fields.

16. Modulated Signals:
Amplitude Modulation
(AM)
Amplitude Modulation (AM) is a fundamental technique used in radio
communication to encode information onto a carrier wave. In an AM
signal, the amplitude of the carrier wave is varied in proportion to the
instantaneous value of the modulating signal.

17. Modulation Coefficients,
Spectral Content, and Useful
Band
The modulation coefficient determines the depth of amplitude modulation.
The AM signal's spectral content consists of the carrier frequency and two
sidebands, one above and one below the carrier. The useful band for an
AM signal is the total bandwidth occupied by the carrier and both
sidebands.

18. Effective Value of an AM Signal
1 Carrier Power
The effective value of
an AM signal
depends on the
carrier power, which
is reduced by the
modulation process.
2 Modulation Depth
The effective value
also depends on the
modulation depth,
which determines
how much the
amplitude fluctuates
around the carrier.
3 Sideband Power
The sidebands carry
the modulated
information and their
power contributes to
the overall effective
value.

19. Introduction to
Frequency Modulated
Signals
Frequency modulation (FM) is a method of encoding information in a
radio wave by varying the frequency of the carrier signal. This allows for
clearer audio quality and reduced interference compared to amplitude
modulation (AM) signals.

20. Principles of Frequency Modulation
1 Carrier Wave
The constant radio frequency that is modulated to carry the signal.
2 Modulating Signal
The audio or data that is encoded onto the carrier wave.
3 Frequency Deviation
The maximum change in frequency from the carrier wave caused by the
modulating signal.

21. Applications of FM Signals
Radio Broadcasting
FM signals are commonly
used for high-quality audio
broadcasts due to their
resistance to interference
and static.
Telecommunications
FM is used in two-way
radio systems, such as
those used by emergency
services and military
communications.
Telemetry
FM signals are employed
in telemetry applications to
transmit data from remote
sensors and instruments.

22. Passive Electric
Filters
Passive electric filters are electronic circuits that use only passive
components like resistors, capacitors, and inductors to selectively filter or
block certain frequencies while allowing others to pass. These filters find
applications in audio systems, power supplies, and various electronic
devices.

23. Constant K Filters
Design
Constant K filters are a
type of passive filter that
use a specific design
formula to determine the
component values. This
ensures a well-defined
frequency response and
makes them useful for
applications that require
precise filtering.
Applications
Constant K filters are
commonly used in
telephone systems, radio
frequency circuits, and
audio equipment to
remove unwanted signals
or frequencies.
Advantages
Their simple design,
predictable performance,
and relatively low cost
make constant K filters a
popular choice for many
filtering needs.

24. General Analysis of Constant K Filters
1 Frequency Response
Constant K filters exhibit a well-defined frequency response, with a sharp
cutoff at the desired cutoff frequency.
2 Impedance Matching
The filter design ensures that the input and output impedances are matched,
reducing signal reflections and maximizing power transfer.
3 Bandwidth Control
The filter parameters can be adjusted to control the bandwidth, allowing for
selective filtering of signals.

25. Constant K Filters: An
Introduction
Constant K filters are a fundamental type of analog electronic filter used
to control the frequency response of circuits. They offer a simple and
reliable way to create low-pass, high-pass, band-pass, and stop-band
filters for various applications.

26. Practical Filter Structures
Low-Pass
Low-pass filters allow low
frequencies to pass
through while attenuating
high frequencies. This is
useful for removing
unwanted high-frequency
noise.
High-Pass
High-pass filters do the
opposite, allowing high
frequencies to pass
through while blocking low
frequencies. This can be
used to remove unwanted
low-frequency
interference.
Band-Pass
Band-pass filters only
allow a specific range of
frequencies to pass
through, blocking both low
and high frequencies
outside this band.

27. Design and Implementation
Considerations
1 Component Selection
Choosing the right resistors,
capacitors, and inductors is crucial for
achieving the desired filter
characteristics.
2 Impedance Matching
Ensuring proper impedance matching
between filter stages is important for
minimizing reflections and signal loss.
3 Stability and Tolerances
Filter performance can be affected by
component tolerances and
environmental factors, requiring
careful design and testing.
4 Applications
Constant K filters find use in audio
processing, instrumentation,
communications, and many other
areas requiring frequency selectivity.

28. Active Filters:
Generalities
Active filters are electronic circuits that selectively allow or block certain
frequency components of a signal. They are widely used in audio,
communications, and control systems to shape the frequency response of
a system.

29. Voltage Transfer Functions
The voltage transfer function describes the relationship between the input
and output voltages of a filter. It is a mathematical expression that
captures the frequency-dependent behavior of the filter, allowing
engineers to analyze and design active filters with desired frequency
characteristics.

30. Simple Second-Order Active Filters
General Design
Second-order active filters
are constructed using op-
amps, resistors, and
capacitors. They provide a
more flexible frequency
response compared to
passive filters, allowing for
adjustable cutoff
frequencies and filter
slopes.
Key Parameters
Important design
parameters include the
cutoff frequency, filter type
(low-pass, high-pass,
band-pass, etc.), and
quality factor, which
determines the sharpness
of the frequency response.
Applications
Simple second-order
active filters find
widespread use in audio
processing, signal
conditioning, and control
systems, where precise
frequency shaping is
required.

31. Introduction to Active Filters
How Active Filters
Work
Active filters use amplifiers
to shape the frequency
response of a signal,
providing improved
performance compared to
passive filters.
Advantages of Active
Filters
Active filters offer
advantages like higher gain,
lower noise, and more
flexible design options
compared to passive filters.
Designing Active
Filters
Engineers can carefully
design active filters to
shape the frequency
response of a signal and
achieve the desired
performance.

32. Types of Active Filters
Low-Pass Filter
Allows low frequencies to
pass through while reducing
high frequencies, useful for
smoothing signals.
High-Pass Filter
Allows high frequencies to
pass through while blocking
low frequencies, useful for
removing DC offsets.
Band-Pass Filter
Allows a specific range of
frequencies to pass through
while blocking others, useful
for isolating signals.

33. Design Considerations and
Applications
1 Cutoff Frequency
The frequency at
which the filter's
response drops by
3dB, a key design
parameter.
2 Component
Selection
Choosing appropriate
resistors, capacitors,
and amplifiers to
achieve the desired
filter characteristics.
3 Applications
Active filters are used
in audio processing,
instrumentation,
control systems, and
more.

34. Active Filters with
Feedback Loops
Active filters are electronic circuits that use feedback loops to selectively
allow or block certain frequency ranges. These filters play a crucial role in
audio, communication, and signal processing applications, enabling
precise control over the frequency response of a system.

35. Feedback Loops in Active Filters
1
Input Signal
The input signal enters the filter circuit
and is processed through various
stages.
2
Amplification
The signal is amplified to the desired
level, ensuring sufficient gain for the
subsequent filtering stages.
3
Feedback Loop
The filtered output is fed back to the
input, creating a feedback loop that
enhances the filter's selectivity and
stability.

36. Low-Pass Active Filters
Concept
Low-pass active filters
allow low-frequency
signals to pass through
while blocking high-
frequency signals, creating
a smooth, filtered output.
Applications
These filters are
commonly used in audio
systems to remove
unwanted high-frequency
noise and in power
supplies to smooth out
ripple signals.
Structures
Common low-pass active
filter structures include the
Sallen-Key and Multiple
Feedback (MFB)
topologies.

37. High-Pass Active Filters
1 Concept
High-pass active
filters allow high-
frequency signals to
pass through while
blocking low-
frequency signals,
creating a sharp,
filtered output.
2 Applications
These filters are used
in audio systems to
remove unwanted
low-frequency noise,
such as rumble or
hum, and in
communication
systems to remove
DC offsets.
3 Structures
Common high-pass
active filter structures
include the Sallen-
Key and Multiple
Feedback (MFB)
topologies, similar to
low-pass filters.

38. Band-Pass Active Filters
Concept
Band-pass active filters allow a specific
range of frequencies to pass through
while blocking both low and high
frequencies, creating a selective
output.
Applications
These filters are used in audio systems
to isolate specific frequency bands,
such as for tone control, and in
communication systems to extract
narrowband signals from a wider
frequency range.
Structures
Common band-pass active filter structures include the State Variable and Multiple
Feedback (MFB) topologies, which provide independent control over the center
frequency and bandwidth.

39. Band-Stop Active Filters
Narrow Bandwidth
Band-stop active filters
have a narrow frequency
range that is blocked,
allowing all other
frequencies to pass
through.
Selective Rejection
These filters are used to
remove specific unwanted
frequency components,
such as power line hum or
other narrowband
interference.
Flexible Design
Band-stop active filters can
be designed using various
topologies, including the
State Variable and Multiple
Feedback (MFB)
configurations.

40. Ege Alp Kuleli
Erasmus Student
Electrical Electronics Engineering
Oradea Univertisty