Blueprint Guide to Signal and System

BLUEPRINT GUIDE
TO SIGNAL AND
SYSTEM

NUR ALINA ZUREEN BINTI ROSLI

JABATAN KEJURUTERAAN ELEKTRIK
POLITEKNIK SEBERANG PERAI

PSP eBook | Blueprint Guide to Signal and System i

BLUEPRINT
GUIDE TO
SIGNAL AND

SYSTEM

NUR ALINA ZUREEN BT ROSLI

2021
JABATAN KEJURUTERAAN ELEKTRIK

POLITEKNIK SEBERANG PERAI

©All rights reserved. No part of this publication may be translated or reproduced in
any retrieval system, or transmitted in any form or by any means, electronic,
mechanical, recording, or otherwise, without prior permission in writing from
Politeknik Seberang Perai.

ii PSP eBook | Blueprint Guide to Signal and System

All rights reserved

No part of this publication may be translated or reproduced in any retrieval system,
or transmitted in any form or by any means, electronic, mechanical, recording, or

otherwise, without prior permission in writing from Politeknik Seberang Perai.

Published by

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Perpustakaan Negara Malaysia Cataloguing-in-Publication Data
Nur Alina Zureen Rosli, 1979-
BLUEPRINT GUIDE TO SIGNAL & SYSTEM / NUR ALINA ZUREEN ROSLI.
Mode of access: internet
eISBN 978-967-0783-80-2
1. Signal processing--Digital techniques.
2. Government publications.
3. Electronic books.
I. Title.
621.3822

PSP eBook | Blueprint Guide to Signal and System iii

Acknowledgement

Thank you so much to our e-book team, I appreciate and am truly grateful to them for giving
me this opportunity and experience in writing and compiling materials for Signal and System. I
sincerely and honestly hope that this e-book will be a guide to those beginners who are interested to
explore more on this topic.

iv PSP eBook | Blueprint Guide to Signal and System

Preface

This guide is designed to introduce the basic concepts of signal and system, subjecting
to the identification and classification inherent to the current digital signal processing (DSP)
found in current technology. More specifically shows the exposure to time-domain and
frequency domain representation of both continuous-time (CT) and discrete-time (DT)
systems. This outlook will provide readers with necessary engineering tools and techniques
to analyse Linear Time - Invariant (LTI) systems. Which allows making real-world connections
by introducing communications and control system applications.

Upon completion of this guide, the expected outcome is that the reader should be
able to develop a solid understanding of the concepts of time domain and frequency domain
with regard to signals and systems. Furthermore, a more comprehensive use of frequency
domain techniques to analyse continuous-time LTI systems being introduced. Readers are
also able to use frequency domain techniques to analyse given discrete-time LTI systems.
Understanding of the theoretical foundations of signal and system in the discretization
aspect. Discretization is the process through which we can transform continuous variables,
models or functions into a discrete form. This can be achieved or done by creating a set of
contiguous intervals also known as bins that go across the range of desired variable or model
or function. Meanwhile Continuous Data is measured, while Discrete Data is counted. Usage
or application of modern software tools such as Matlab enables easy sampling for system
analysis.

Learning the fundamental of signal and system fulfils the following Electrical
Engineering Program in all higher education institution. The main objective or outcome
expected from this guide is the reader will be able to define and diagnosed problems and
provide and implement electrical engineering solutions in an actual industrial environment.

In addition, the core outcome will be an ability to apply knowledge of mathematics,
science, and engineering to general electrical engineering and, in particular, to one or more
of the following areas such as, communications, computers, controls, power electronics, and
power systems. An ability to design a system, component, or process to meet desired needs.
Also developing an aptitude to identify, formulate, and solve electrical engineering problems.
Thus, these abilities will ensure the understanding of the handling of techniques, skills, and
modern engineering tools necessary for electrical engineering practice.

PSP eBook | Blueprint Guide to Signal and System v

Table of Content

CHAPTER PAGE

CHAPTER 1: INTRODUCTION TO SIGNAL AND SYSTEM 1
1.3 Overview 1
1.2 Classes of Signal Processing 3
CHAPTER 2: DEFINITION OF SIGNAL AND SYSTEM 5
2.1 Description of Signal 5
2.2 Description of System 6
2.3 Relationship between Signal and System 9
CHAPTER 3: CLASSIFICATION OF SIGNAL 11
3.1 Continuous-Time and Discrete-time Signal 12
3.2 Analog Signal and Digital Signal 16
3.3 Real Signal and Complex Signal 18
3.4 Deterministic Signal and Random Signal 19
3.5 Even Signal and Odd Signal 21
3.6 Periodic Signal and Non-Periodic Signal 23
CHAPTER 4: BASIC SIGNAL SEQUENCES AND FUNCTION OPERATION 25
4.1 Standard Signal Sequence 25
26
4.1.1 Unit Step Function 27
4.1.2 Unit Impulse Function 29
4.1.3 Complex Exponential Signals 31
4.1.3 Sinusoidal Signal 32
4.2 Signal Function Operation 32
4.2.1 Shifting 35
4.2.2 Scaling 37
4.2.3 Reflection 38
4.2.4 Addition 40
4.2.5 Subtraction 40
4.2.6 Multiplication 44
REFERENCES



PSP eBook | Blueprint Guide to Signal and System 1

CHAPTER 1: INTRODUCTION TO SIGNAL AND SYSTEM

1.1 Overview

Signals and systems can be seen applied in everyday life and also in various branches
of engineering and science. Signal properties for instance are determined through paradigms
such as periodicity, absolute tangibility, determinism and stochastic character. Stochastic
refers to the property of being well described by a random probability distribution.
Furthermore, in probability theory, the formal concept of a stochastic process is also referred
to as a random process.

A signal is defined as any physical quantity that changes with time, distance, speed,
position, pressure, temperature or some other quantity. A signal is a physical quantity that
consists of many sinusoidal of different amplitudes and frequencies. Examples as follows:

a) Continuous-Time Signal: x (t) = 10(t)
b) Discrete-Time signal: x[n] = 5x2+20x[n]

The special significance of signals is identified through the importance of the signal
behaviour in terms of the unit step, the unit impulse, the sinusoid, the complex exponential,
a particular specific time-limited signal and the most relevant sample is when the signal
condition is deemed to be in the form of continuous or discrete time signals otherwise it can
also be in the form of continuous and discrete amplitude signals.

System on the other hand is when the obtained signal whether it be for an input or an
output signal is generated through networks and combination of devices being used
throughout the circuit structure. The noticeable practice of electrical engineering in one or
more of the following areas indicates the usage of signal and system compound. These area
of practice can be found in communications, computers, controls, power electronics, and
power systems. In mathematics, in particular, functional analysis, convolution is a
mathematical operation on two different functions that produces a third function. That

2 PSP eBook | Blueprint Guide to Signal and System

expresses how the shape of one function, in this case signal, is modified by the other. The
term convolution itself refers to both the result function and to the process of computing it.
A System is a physical device that performs an operations or processing on a signal. Examples
of a system are filters and amplifiers.

Convolution is performed by sliding the essential signal over the image, commonly
starting at the top left corner, so as to move that particular signal through all the positions
where the said signal fits entirely within the boundaries of the image. Convolution also is
better known as a mathematical way of combining two signals to form a third signal. It is the
single most important technique in Digital Signal Processing. Convolution is important
because it relates the three signals of interest: the input signal, the output signal, and the
impulse response. Convolution needs to be visualized and then graphically presented, but
that is too qualitative. However, it is the graphical procedure which best describes the
concept and power of convolution.

In mathematical form, signal and system can be well represented by these two
algorithmic function which are Laplace and Fourier transform. The Laplace and Fourier theory
in addition with a trace of Z transforms and sampling of random signals is the core process of
signal and system. Essentially signal and system consist of computing through software. In the
field of Computer Science, it helps a lot in the algorithms and programming of signal and
systems, especially by using MATLAB to develop algorithm meant for Digital Signal Processing
(DSP) for efficient implementations. In addition, the applications to circuits and systems, at
least that makes the Laplace transform easy.

Digital Systems, in this particular case of signal and system has nothing to do with
gates and logic but more focused on real z-transform DSP. Followed by control and
communications, which is the next division on Laplace and Fourier.

PSP eBook | Blueprint Guide to Signal and System 3

1.2 Classes of Signal Processing

Analog signal Processing (ASP) is portrayed if the input signal given to the system is
analog then system does analog signal processing. For instances this condition is mostly
used when handling resistor, capacitor, inductor, or OP-AMP and the likes of these particular
components.

In the meantime, Digital signal Processing (DSP) is portrayed if the input signal given
to the system is digital then system does digital signal processing. As an example, it is
commonly found by means of Digital Computer, Digital Logic Circuits and such. The devices
called as Analog to Digital Converter (ADC) converts analog signal into digital and Digital to
Analog Converter (DAC) does vice-versa. The relationship of the Analog to Digital Converter
and Digital to Analog Converter can be seen in figure 1.1 below:

Analog ADC DIGITAL DAC Analog
Signal SYSTEM Signal

Figure 1.1: Signal Converters

Most of the signals generated are analog in nature. Hence, these signals are
converted to digital form by the analog to digital converter. Thus, Analog to Digital Converter
(ADC) generates an array of samples and gives it to the digital signal processor. This array of
samples or sequence of samples is the digital equivalent of input analog signal. The Digital
Signal Processing (DSP) performs signal processing operations like filtering, multiplication,
transformation or amplification etc. operations over this digital signal. The digital output
signal from the DSP is given to the Digital to Analog Converter (DAC).

4 PSP eBook | Blueprint Guide to Signal and System

1.3 Advantages and Disadvantages of Digital Signal Processing (DSP)
over Analog Signal Processing (ASP)

Focusing on the more advantageous aspect of this signal processing the physical size
of analog systems is quite large while digital processors are more compact and light in weight.
Analog systems are less accurate because of component tolerance ex R, L, C and active
components. Digital components are less sensitive to the environmental changes, noise and
disturbances. Digital systems are most flexible as software programs & control programs can
be easily modified. Digital signal can be stores on digital hard disk, floppy disk or magnetic
tapes. Hence becomes transportable. Thus easy and lasting storage capacity. Digital
processing can be done offline. Mathematical signal processing algorithm can be routinely
implemented on digital signal processing systems. Digital controllers are capable of
performing complex computation with constant accuracy at high speed. Digital signal
processing systems are upgradeable since that are software controlled. Possibility of sharing
DSP processor between several tasks. The cost of microprocessors, controllers and DSP
processors are continuously going down. Single chip microprocessors, controllers and DSP
processors are more versatile and powerful.

As for the disadvantages of Digital Signal Processing (DSP) over Analog Signal Processing
(ASP) designed for some complex control functions, it is not practically feasible to construct
analog controllers. Additional complexity in the converters (A/D & D/A Converters). Has a limit
in frequency. High speed AD converters are difficult to achieve in practice. In high frequency,
applications DSP are not preferred.

PSP eBook | Blueprint Guide to Signal and System 5

CHAPTER 2: DEFINITION OF SIGNAL AND SYSTEM

2.1 Description of Signal

Signals play a fundamentally important part in every type of electrical or electronic
system. An understanding i f the characteristics of signals and the way in which they interact
with systems is central to the ability of an engineer to understand electrical or electronic
systems. Signal is a function representing a physical quantity, and typically it contains
information about the behaviour or nature of the phenomenon.

Naturally when the word signal pops up in practice it is commonly symbolized in the
form of analog or digital. Approximately the analog signals are considered to be in the
continuous form whilst digital signals are discrete signals that have been quantizes.

Figure 2.1: Current flow in a simplified telecommunication circuit.

Electrical signals which are usually represented by voltage or the current value, is
shown in figure 2.1. To highlight, electrical signals are when voltages across the capacitor
currents flowing in the resistor. Typically these electrical signals are represented in the usage
of radio signals, TV signals, telephone signals, computer signals and such types of application.

Other than the typical electronic signals, signals can be in other forms or by using
different energy source, such as light, radio frequency signals, acoustic signals and
mechanical signals to name a few. From a communication point of view, a signal is any
function that carries some information or has a message to relay.

6 PSP eBook | Blueprint Guide to Signal and System

ELECTROMAGNETIC
WAVE

SOUND LIGHT
WAVE WAVE

Figure 2.2: Two types of signal waveform contained in the Electromagnetic Wave Spectrum.

Signals are generally shown in the form of electromagnetic waves for electrical signals.
Arbitrating to the electromagnetic spectrum, shown in figure 2.2, there are two main parts
that the electromagnet signals are able to function. These are through sound waves and light
waves. The easiest way to simulate these signals are by demonstrating it using mathematical
function.

2.2 Description of System

A system is defined as a physical device that generates a response or output signal for a
given input. System can also be described by the type of input and output it deals with. Since
here the focus is in the technical field, the signals that are dealt with depends heavily on in
what manner the system is being operated on. Therefore, in this case, the system being
applied would be in either any of these forms:

a) a mathematical model
b) a piece of code or software
c) a physical device
d) a black box
Wherein these forms whose input is a signal and it performs some processing on that
signal, and the output is a signal. The input is known as excitation and the output is known as
response.

Note: Meaning of input and output in a system.

PSP eBook | Blueprint Guide to Signal and System 7

Figure 2.3: Relationship of Signal & System
In the above figure 2.3, shows the relationship occurrence between signal and system.
Where signal is materialised with a system. The connection has been shown whose input and
output both are signals but the input is an analog signal and the output is a digital signal. It
means our system is actually a conversion system that converts analog signals to digital
signals.

Figure 2.4: A Simple Telephone Circuit System
Here, the above figure 2.4, indicates a common structure for a simple telephone
system. Let’s discuss the occurrence of signal in this austere system. As it is a simple
system, it is only possible for one-way communication. The circuit consisting of a MIC, an
inductor, a battery and a loudspeaker. Let’s take a look on how this simple telephone
works. A mic or microphone consisting of a cylindrical body in which its upper portion is

8 PSP eBook | Blueprint Guide to Signal and System

covered by a diaphragm and it is almost filled by carbon granules. At normal condition the
microphone offers a resistance and when we speak the diaphragm vibrates and the offered
resistance increase or decrease. Thus, sound waves are converted to variable resistance
and it will control the output voltage. This alternating voltage is fed to a loudspeaker after
the amplification of signal, the amplification is made possible by means of an inductor and
a battery, finally the amplified voltage is converted back to sound waves.

Other than the example given above, an audio amplifier, attenuator, TV set,
transmitter, receiver and the likes are all categorised as a system. Any type of machine or
engine are also identified as system, albeit a mechanical or mechatronic system. Examples
are shown in the figure 2.5 below.

Figure 2.5: Example of System Application

The main creation of system is to be able to process signals. Hence, systems process
signal to extract information in the sense of sequence analysis, enable transmission over
channels with limited capacity especially when using JPEG or MPEG coding, capable to
improve security over networks such as encountering encryption and watermarking and
another aspect is to support the formulation of diagnosis and treatment planning in the field
of medical imaging. System is fundamentally needed in order for a specific signal to manifest
through the required technology.

Figure 2.6: Transfer Function

PSP eBook | Blueprint Guide to Signal and System 9

Based on the concept shown in figure 2.6, the function linking to the output of the
system with the input signal is literally called a transfer function and it is typically indicated
with the symbol of h(*). Where ‘h’ is the impulse of the system.

2.3 Relationship between Signal and System

Systems customarily processes the information signal. Signals can be categorized as
either analog signals, in this case exemplified by continuous-time signal, or digital signals,
exemplified by discrete-time signal. As displayed in figure 2.7 below.

Figure 2.7: Simplified Relationship of Signal and System
There are very clear similarities and also some very significant variances to
differentiate between continuous-time signal and system and also discrete-time signal and
system. In other word continuous-time signal and discrete time signal carry the same
properties but only distinguished by either using the analog or digital system. The block
diagram in figure 2.8 and figure 2.9 shows these differences in the application of an analog
system and a digital system.

Figure 2.8: Example of an Analog System using a Continuous-Time (Analog) Signal

10 PSP eBook | Blueprint Guide to Signal and System

Figure 2.9: Example of a Digital System using a Discrete-Time (Digital) Signal
In discussing the approach of signal and system throughout this e-book for the most
part it is restricted to a specific type of classification that is mainly the linearity approach
which favours heavily on the time-invariant system. Extremely powerful tools and techniques
exist for both analysis and design of this class of system. In particular by discussing this class
of system a development of signal and system can be related and represented in both the
time domain and also the frequency domain.

Figure 2.10: Depiction of a System Processing the Desired Signal

Thus, the signals processed in linear time invariant relies on whether using the
continuous-time domain or the discrete-time domain. Usually it is done in one certain time
domain at a time but if both domains are done simultaneously then a sampler and quantizer
are need for a successful and smooth transmission.

PSP eBook | Blueprint Guide to Signal and System 11

CHAPTER 3: CLASSIFICATION OF SIGNAL

Mathematically, signals are represented as a function of one or more independent
variables. For instance, a black & white video signal intensity is dependent on , coordinates
and time , which is defined as ( , , ). In this e-book, we shall be exclusively concerned
with signals that are a function of a single variable in this case to highlight is Time. In the
domain of Time the signals are in the form of sinusoidal signal.

Figure 3.1: Representation of signals in two forms – (a) Graphical Representation
Form and (b) Functional Representation Form.

There are many aspects to consider when classifying a signal in a particular system. This
will make the procedure in the system run differently. For example, the signal can flow in a

12 PSP eBook | Blueprint Guide to Signal and System

single channel or multi-channel signals. The signal can also be in a single dimensional or Multi-
dimensional signal state. Normally for a continuous valued or discrete valued signal comes in
the form of symmetrical which are even signals or anti-symmetrical which are odd signals. But
mostly signals will be noted by the type of energy and power signal being used. The
classification that will be covered in this e-book is as such:

a. Continuous-Time and Discrete-time Signal
b. Analog and Digital Signals
c. Real and Complex Signals
d. Deterministic and Random Signals
e. Even and Odd Signals

Periodic and Non periodic Signals

Such classes are not disjoint, as an example there are digital signals that are periodic
of power type and others that are aperiodic (non-periodic) of power type, so for other
classification similar cases can happen. Any combination of single signal features from the
different classes stated above is absolutely possible by establishing the desired system.

3.1 Continuous-Time and Discrete-time Signal

Continuous-time (CT) signals are signals of which its amplitude varies continuously
with time. Common example is voltage and velocity. Denoted ℎ ℎ ( ), where the time
interval may be bounded which is finite or infinite. Amplitude value presents at all time during
the interval, depicted in figure 3.2 shown below.

PSP eBook | Blueprint Guide to Signal and System 13

Figure 3.2: Graphical Representation of Continuous-Time (CT) Signal
By the term continuous-time (CT) signal means a real or complex function of time x(t),
where the independent variable t is continuous. Continuous time signal is a signal that is
specified for every real value of the independent variable. The independent variable is
continuous, that is it takes any value on the real axis. The domain of the function representing
the signal has the cardinality of real numbers, identified as such:
a) Signal ↔ f=f(t)
b) Independent variable ↔ time (t), position (x)
c) For continuous-time signals: t ∈ R
Meanwhile the Discrete-time (DT) signal, [ ] which has the amplitude of the signal
varies at every discrete value nwhich is generally uniformly spaced. Common example
includes pixels, daily stock price (anything that a computer processes) Denote byx[n], here n
is an integer value that varies discretely.

14 PSP eBook | Blueprint Guide to Signal and System

Figure 3.3: Graphical Representation of Discrete-Time (DT) Signal

If “t” is a discrete variable, for example depiction is always done in the case of x(t),
x(t) is defined at discrete times, then the signal x(t) is a discrete-time signal. A discrete-time
signal is often identified as a sequence of numbers, denoted by x[n], where ‘n’ is an integer.
Digital signals in compose of discrete time and discrete amplitude are obtained by sampling
the analog signal at discrete instants of time, obtaining discrete time signals and then by
quantizing its values to a set of discrete values & thus generating discrete amplitude signals.
Sampling process takes place on x-axis at regular intervals and quantization process takes
place along y-axis. Quantization process is also called as rounding or truncating or
approximation process.

Discrete time signal: a signal that is specified only for discrete values of the
independent variable. It is usually generated by sampling so it will only have values at equally
spaced intervals along the time axis. The domain of the function representing the signal has
the cardinality of integer numbers, identified as such:

a) Signal ↔ f=f[n], also called sequence
b) Independent variable ↔ n
c) For discrete-time functions: t ∈ Z

Figure 3.4: Comparison between Continuous-Time and Discrete-Time Signal

PSP eBook | Blueprint Guide to Signal and System 15

Note: Comparison Table between Continuous-Time Signal and
Discrete-Time Signal

No Continuous-Time (CT) Discrete-time (DT)

1 This signal can be defined at any time This signal can be defined only at certain

instance & they can take all values in specific values of time. These time

the continuous interval (a, b) where a instance need not be equidistant but in

can be -∞ & b can be ∞. practice they are usually takes at equally

spaced intervals.

2 These are described by These are described by the difference

differential equations. equation.

3 This signal is denoted by x(t). These signals are denoted by x[n] or

notation x[nT] can also be used.

4 The speed control of a dc motor Microprocessors and computer based

using a techno-generator feedback or systems uses discrete time signals.

Sine orexponential waveforms.

5 Continuous Valued Discrete Valued

6 If a signal takes on all possible values If signal takes values from a finite set

on afinite or infinite range, it is said of possible values, it is said to be

to be continuous valued signal. discrete valued signal.

7 Continuous valued and continuous Discrete time signal with set of

time signals are basically analog signals. discrete amplitude are called digital

signal.

Table 3.1: Comparison Table between Continuous-Time Signal and Discrete-Time Signal

16 PSP eBook | Blueprint Guide to Signal and System

3.2 Analog Signal and Digital Signal

The definition of an Analog Signal is, it is a Continuous-time signal that can take on
any value in the continuous interval. In this case a Continuous time (CT) signals are signals of
which its amplitude varies continuously with time.

Figure 3.5: Analog Signal
If a continuous-time signal x(t) can take on any values in a continuous time interval,
then x(t) is called an analog signal with the signal shape as shown in figure 3.5. The analog
signal is a signal whose amplitude can take on any value in a continuous range. The amplitude
of the function f(t) (or f(x)) has the cardinality of real numbers. The difference between
analog and digital is similar to the difference between continuous-time and discrete-time. In
this case, however, the difference is with respect to the value of the function (y-axis). Analog
corresponds to a continuous y-axis, while digital corresponds to a discrete y-axis. An analog
signal can be both continuous time and discrete time.

Figure 3.6: Digital Signal
If a discrete-time signal can take on only a finite number of distinct values, x[n], then
the signal is called a digital signal. Digital signal is a signal which is one whose amplitude can

PSP eBook | Blueprint Guide to Signal and System 17

take on only a finite number of values, thus meaning it is quantized. The amplitude of the
function f[n] can take only a finite number of values. A digital signal whose amplitude can
take only M different values is said to be Mary Binary signals are a special case for M=2.

Figure 3.7: Analog and Digital signals in Continuous-Time and Discrete-Time Respectively

Note: Comparison Table between Analog Signal and Digital Signal

No Analog signal Digital signal
1 These are basically continuous These are basically discrete time signals and
discrete amplitude signals. These signals are
time and continuousamplitude signals. basically obtained by sampling and
quantization process.
2 ECG signals, Speech signal, Television All signal representation in computers and
signal and others signal alike. All the digital signal processors are digital.
signals generated from various sources
in nature are analog.

Table 3.2: Comparison Table between Analog Signal and Digital Signal

18 PSP eBook | Blueprint Guide to Signal and System

3.3 Real Signal and Complex Signal

A signal x( t ) is a real signal if its value is a real number, something that can be valued
or calculated without going through complicated algorithm. A real signal can be represented
by the following equation:

Formula

,

A signal x( t ) is a complex signal if its value is a complex number. A general complex
signal x( t ) is a function of the form. Here an equation has multiple value with a combination
of real numbers and imaginary numbers represented by ‘j ‘. Real numbers are numbers
obtained immediately whilst imaginary numbers are values that has to be analyzed and
calculated through an evaluated algorithm, seen in the following equation:

Formula

,

Where are real signals and a complex signal represented with ‘j’,

here (t) signifies either an it is a continuous variable or a discrete variable.

Figure 3.8: Signal Representation of Real Signal and Complex Signal Respectively

PSP eBook | Blueprint Guide to Signal and System 19

Note: Comparison Table between Real Signal and Complex Signal

SIGNAL AMPLITUDE IN REAL COMPLEX / INTEGER
TIME OR SPACE

REAL ANALOG CONTINOUS-TIME DIGITAL CONTINOUS-TIME

COMPLEX / INTEGER ANALOG DISCRETE-TIME DIGITAL DISCRETE -TIME

Table 3.3: Comparison Table between Real Signal and Complex Signal

3.4 Deterministic Signal and Random Signal

Deterministic signals are those signals whose values are completely specified for any
given time. The values of signal are completely specified for any given time an example of this
classification is the Radio Frequency (RF) signal.

Figure 3.9: Deterministic Signal

The deterministic signal is easily describe as a signal whose physical description in
known completely. A deterministic signal is a signal in which each value of the signal is fixed
and can be determined by a mathematical expression, rule, or table. Because of this the future
values of the signal can be calculated from past values with complete confidence. There is no
uncertainty about it is amplitude values, Examples: signals defined through a mathematical
function or graph.

20 PSP eBook | Blueprint Guide to Signal and System

Random signals are called in a few ways, one such is known as Non-Deterministic
Signal and the other, which also can be called as a Probabilistic Signal. All three, Random,
Non-Deterministic and Probabilistic Signal carry the same meaning in its signal
representation. Random signals are those signals that take random values at any given times.
Taking random values at any given time or sequence that cannot be calculated
mathematically such as noise.

Figure 3.9: Random Signal

Random or probabilistic signals is when the amplitude values cannot be predicted
precisely but are known only in terms of probabilistic descriptors. The future values of a
random signal cannot be accurately predicted and can usually only be guessed based on the
averages of sets of signals. They are realization of a stochastic process for which a model could
be available. Signal examples that have random outcome are EEG, evocated potentials, noise
in CCD capture devices for digital cameras and many more.

Note: Comparison Table between Deterministic Signal and Random
Signal

No Deterministic signals Random signals

1 Deterministic signals can be Random signals that cannot be
represented ordescribed by a represented or described by a
mathematical equation or lookup mathematical equation or lookup table.
table.
Not Preferable. The random signals can
2 Deterministic signals are preferable be described with the help of their
because for analysis and processing statistical properties.
of signals we can usemathematical
model of the signal.

PSP eBook | Blueprint Guide to Signal and System 21

3 The value of the deterministic The value of the random signal cannot
signal can be evaluated at time be evaluated at any instant of time.
(past, present or future) without
certainty. Example Noise signal or Speech signal

4 Example Sine or exponential
waveforms.

Table 3.4: Comparison Table between Deterministic Signal and Random Signal

3.5 Even Signal and Odd Signal

An even signal is a signal that is symmetric about the vertical axis. An even signal is
any signal ‘f’ such that f (t) = f (-t). Even signals can be easily spotted as they are symmetric
around the vertical axis. In this case it is a mirrored image.

(a) Even Signal in Continuous –Time

(b) Even Signal in Discrete –Time
Figure 3.10: Representation of Even Signals

Note: A signal ( ) or [ ] is referred to as an even signal if:-

22 PSP eBook | Blueprint Guide to Signal and System

An odd signal is asymmetric about the vertical axis. An odd signal, on the other hand,
is a signal ‘f’ such that f (t) = - [f (-t)]. An asymmetric signal is when the equivalent signal is in
the opposite direction.

(a) Odd Signal in Continuous –Time

(b) Odd Signal in Discrete –Time
Figure 3.11: Representation of Odd Signals

Note: A signal ( ) or [ ] is referred to as an odd signal if:-

Some properties of even and odd functions
EVEN function x ODD function = ODD function
ODD function x ODD function = EVEN function
EVEN function x EVEN function = EVEN function
Table 3.5: Representation of Even and Odd Signals

PSP eBook | Blueprint Guide to Signal and System 23

3.6 Periodic Signal and Non-Periodic Signal

Signal can be classified based on its periodicity, in terms of whether or not it is a
periodic or aperiodic signal. For a signal to be considered at its periodic state when a periodic
signal will have a definite pattern that repeats again and again over a certain period of time.
Usually the signal flows continuously. Therefore, a signal that satisfies the condition below is
called a periodic signal.

Formula: x(t + nT0) = x(t), T0 ≠ 0

With,

t = time
T0 = fundamental period
n = 1, 2, 3

Where the smallest value of T0 that satisfies the definition is called the periodic signal.

Figure 3.12: Sample of Periodic Signal

24 PSP eBook | Blueprint Guide to Signal and System

Discrete time signal is periodic if its frequency can be expressed as a ratio of two
integers. f= k/N where k is integerconstant. A signal x(t) is a periodic signal if x(t) = x(t +
nT0), where T0 is called the period and the integer n > 0.

When the angular frequency is given by:

Formula ; with:

So if no value of T satisfies the condition mentioned then it is called non-periodic. A
signal that does not satisfy the above condition is automatically called a non-periodic or also
known as aperiodic signal. Which is a signal that does not repeats its pattern over a certain
period length. A non-periodic signal is measured at one cycle of the signal. Signal is at a certain
time. If the signal does not satisfy above property called as Non-Periodic signals.

Figure 3.12: Sample of Non-Periodic Signal

The signal x(n) is said to be periodic if x(n+N) = x(n) for all n where N is the
fundamental period of the signal. If x(t) ≠ x(t + T0) for all t and any T0, then x(t) is a non-
periodic or a periodic signal.

PSP eBook | Blueprint Guide to Signal and System 25

CHAPTER 4: BASIC SIGNAL SEQUENCES AND FUNCTION
OPERATION

4.1 Standard Signal Sequence

Signal sequences is needed to ascertain the flow of signal in the system being
exercised especially in the Digital Signal Processing (DSP) field. In spite of continuing advances
in speed, cost, capability, and size of digital hardware software development continues to be
a challenge. In addition, the need for conversion of continuous-time signals to and from
discrete-time sequences adds complexity and cost in the form of analog-to-digital and digital-
to-analog converters and associated analog filters.

As a point, applications requiring very high speed processing still may be beyond the
capabilities of digital hardware. Thus, signals and sequences are essential to be manipulated
in the particular system. In order to observe many similar characteristics in signals and
sequences and also some differences, these basic signal function sequence will be observed:

a) Unit Step Function
b) Unit Impulse Function
c) Complex Exponential Signals
d) Sinusoidal Signals

It is now ready to consider in detail continuous-time signals that occur in
analog or continuous-time (CT) systems and discrete-time or digital sequences in discrete-
time (DT) systems. These basic signal types that are mentioned above are frequently
encountered in Digital Signal Processing (DSP). In the next segment both their continuous-
time and discrete-time versions is explained. Note that the analog or continuous-time
versions of several of these signals are artificial constructs and they violate some of the
conditions stated above for real-world signals and cannot actually be realized.

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4.1.1 Unit Step Function

The unit step function u(t), also known as the Heaviside unit function, is defined as the
“unit step”, also often referred to as a Heaviside function, which the signal is literally taking a
step. Signal moves a step in the desired direction. As an example, signal has 0 value until time
0, at which point, it abruptly switches to 1.0. The unit step represents events that change
state, for instance the switching on of a system, or of another signal. It is usually represented
as u(t) in continuous time and u[n] in discrete time, as seen below:

Figure 4.1: Unit Step Function in Continuous-Time

Figure 4.2: Unit Step Function in Discrete-Time
The unit step sequence or function u[n] is defined as which is shown in Figure 4.2.
Note that the value of u[n] at n=0 is defined, unlike the continuous-time step function u(t) at
t=0 and equals unity. Similarly, the shifted unit step function sequence in discrete-time, which
is u[n-k] is defined as shown in Figure 4.2. In practical terms, the step function represents the

PSP eBook | Blueprint Guide to Signal and System 27

switching on or off of a device or a signal. The continuous-time step function u(t) violates the
criterion we have stated above of smoothness, since it changes instantaneously from 0 to 1
at t=0. Practical realizations are only approximations, which can change value extremely
quickly, but never instantaneously.

4.1.2 Unit Impulse Function

The Continuous-Time (CT) unit impulse function is commonly denoted by the given
expression:

Formula

These equations say that the impulse is zero everywhere accept at the origin and the
area under the unit impulse is unity of 1. The unit impulse function it is also known as Dirac
delta function, ( ). The continuous-time or analog version briefly, but somewhat non-
rigorously of the delta function is identified as Dirac delta as illustrated in Figure 4.1 and
defined by:

An alternative definition of is ( ):

The above definition states that the impulse function is zero everywhere except at
t=0, and the area under the function is 1.0. Technically, the impulse function is not a true
function at all, and the above definition is imprecise. If one must be precise, the impulse
function is defined only through its integral, and its properties. Specifically, in addition to the

28 PSP eBook | Blueprint Guide to Signal and System

property ∫ ( ) = 0 shows when∫ ( ) = 1, it has the property that for any function of
( ) which is bounded in value at = 0, makes the function into ∫ ( ) ( ) = (0).
The impulse function signal can be exemplified as such:

Figure 4.1: Unit Impulse Function in Continuous-Time
Note that the Dirac delta function itself is not smooth and is unbounded in amplitude.
The discrete-time or a more digitalized version of the delta function is also known as the
Kronecker Delta Function. It is precisely defined as the definition of unit impulse or also
known as unit sample is in a sequence of [ ], and similarly, the shifted unit impulse or
otherwise sample is in a sequence of [ − ], both are defined as follows:

Figure 4.2: Unit Impulse Function in Discrete-Time

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The impulse function, also known as the delta function, is possibly the most important
signal to know about in signal processing. It is important enough to require a special section
of its own in these notes. Technically, it is a signal of unit energy, which takes non-zero
values at exactly one instant of time, and is zero everywhere else.

4.1.3 Complex Exponential Signals

The exponential signal literally represents an exponentially increasing or falling series,
represented in the terms:

Continuous time: ( ) =
A continuous time exponential signal can be represented by the following expression;

( ) =
Where and are in general complex numbers. For real exponential signal, and are reals.
As for the complex exponential are represented as such;

A - Amplitude
ω, Ω - Angular frequencies
Ǿ - Phase angle
The complex exponential signal can be illustrated as such:

Figure 4.1: Complex Exponential Signal in Continuous-Time

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The case > represents exponential growth. Some signals in unstable systems
exhibit exponential growth. The case < represents exponential decay. Some signals in
stable systems exhibit exponential decay. . The exponential signal models the behaviour of
many phenomena, such as the decay of electrical signals across a capacitor or inductor. The
exponential signal violates bounded-ness, since it is infinite in value either at = −∞ for the
value of < or at = ∞ for the value of > .

Formula: For a discrete-time the complex exponential sequence is of the form:

[ ] = Ω0
Again, using Euler’s formula, [ ]can be expressed as:

[ ] = Ω0 = cos Ω0 + Ω0
Thus [ ] is a complex sequence whose real part is cos Ω0 and imaginary part is Ω0 .

Figure 4.2: Complex Exponential Signal in Discrete-Time

In discrete time the definition of the exponential signal takes a slightly different form:
Discrete time: [ ] = n

Note that although superficially similar to the continuous version, the effect of on
the behaviour of the signal is slightly different. The sign of does not affect the rising or falling
of the signal, instead it affects its oscillatory behaviour. Negative alphas cause the signal to
alternate between negative and positive values. The magnitude of affects the rising and
falling behaviour stated as such:

Note: | | < 1 results in falling signals, whereas | | > 1 results in rising signals.

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4.1.4 Sinusoidal Signal

The sinusoid is a smooth signal and has finite power and violates none of the criteria
for real-world signals. The sinusoid is a familiar signal. A sinusoidal sequence can be expressed
as such:

Continuous-Time: (t) = cos(ωt + )
Discrete-Time: [ ] = cos(Ω + )

Given if is dimensionless, then both Ω and have unit of radians. The sinusoid is one
of the most important signals in signal processing. We will encounter it repeatedly. The
continuous-time version of it shown above is a perfectly periodic signal. Note that the above
equations are defined in terms of cosines. We can similarly define the sinusoid in terms of
sine’s as sin(ωt + ) and cos(Ω + ). Cosines and sine’s are of course related through a
phase shift of π/2 will have a function of cos( ) = sin( - π/2).

Figure 4.3: Sinusoidal Signal for CT & DT

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4.2 Signal Function Operation

Operations on discrete time signals can be composed by manipulating and combining
other signals. There are two variable parameters in general:

a) Amplitude
b) Time
Consider these manipulations briefly. The following operation can be performed with either
amplitude or time:
a) Shifting
b) Scaling
c) Reflection
d) Adding
e) Subtraction
f) Multiplication

4.2.7 Shifting

Shifting is when signal x (n) can be shifted in time. By using this method it is possible to
delay the sequence or advance the sequence. This is done by replacing integer n by n-k where
k is integer. If k is positive signal, means that the signal is delayed in time by k samples and
the arrow gets shifted on left hand side. But if k is a negative signal then the signal is advanced
or anticipated in time k samples and the arrow gets shifted on right hand side.

The shifting of signal can be depicted by the movement of the following function given
bellow:
X(n) = { 1, -1 , 0 , 4 , -2 , 4 , 0 ,……}

n=0

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Delayed by 2 samples : X(n-2)= { 1, -1 , 0 , 4 , -2 , 4 , 0 ,……}

n=0

Anticipated by 2 samples : X(n+2) = { 1, -1 , 0 , 4 , -2 , 4 , 0 ,……}

n=0

From the given sequence, the delayed or anticipated movement can be seen in the
figure 4.5, as shown below:

Figure 4.5: Sample of Shifted Signal

Whereas the signal is displaced along the intendant axis by or N for discrete time. If
is positive, the signal is delayed and if is negative the signal is advanced. The represented
function is derived as follows:

CoCnotnintiunouuosutsimtime:e: ( () =) = ( (− − ) )
DiDscisrcerteetteimtime:e: [ [ ] =] = [ [ − − ] ]

34 PSP eBook | Blueprint Guide to Signal and System

a) Shifting in Continuous-Time

b) Shifting in Discrete-Time
Figure 4.5: Depiction of a shifted (a) CT and (b) DT signal

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4.2.8 Scaling

Amplitude scaling can be done by multiplying signal with some constant. Suppose
original signal is x(n). Then output signal is A x (n). Thus, simply scaling a signal up or down by
a gain term.

Continuous time: ( ) = ( )
Discrete time: [ ] = [ ]

→ , ℎ ℎ > 1
Given if the signal needs to multiply by times term then the signal will be in a compress
state but if the signal is divided by times term then the signal will be in a expand state, where
both of these condition does not affect the amplitude of the function. The shifting method is
illustrated in figure 4.6 below:

Figure 4.6: Sample of Scaling Signal

36 PSP eBook | Blueprint Guide to Signal and System

a) Scaling in Continuous-Time

a) Scaling in Discrete-Time
Figure 4.7: Depiction of a scaling (a) CT and (b) DT signal

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A more vivid demonstration of signal scaling can be seen in the sample in figure 4.8.
Where x(t) is an amplitude scaled version of x(t) whose amplitude is scaled by any given factor
that is appropriate for used signal.

Figure 4.8: Sample of scaling for CT and DT signal

4.2.9 Reflection

Folding or reflection, literally means the mirrored image of the signal in use. It is
folding of signal about time origin at n=0. In this case replace n by –n. the function obtained
will be as the sequence shown below:
Original signal:
X (n) = { 1, -1 , 0 , 4 , -2 , 4 , 0}

n=0
Folded signal:
X (-n) = { 0 , 4 , -2 , 4 , 0 , -1 , 1}

n=0

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This folding method can also be called as the inversion method or a time reversal
approach, where the mirror image of f(t) is at the vertical axis. Where it can be a real or
imaginary, positive or negative. Say when f(-t) indicates that the function is negative, the
signal is flipped across the y-axis. Reflection sample can be seen as in figure 4.9, with x(-t) is
the time reversal of the signal x(t).

Figure 4.9: Sample of Reflected Signal

4.2.10Addition

An addition operation where given signals are x1(n) and x2(n), which produces output
y(n) where y(n) = x1(n)+ x2(n). An adder generates the output sequence which is the sum of
input sequences. Addition of two signals is nothing but addition of their corresponding
amplitudes. This can be best explained by using the following example:

Continuous-time:
( ) = 1( ) + 2( )

Discrete-time:
[ ] = 1[ ] + 2[ ]

PSP eBook | Blueprint Guide to Signal and System 39

a) Addition in Continuous- Time

b) Addition in Discrete- Time
Figure 4.10: Depiction of addition signal in (a) CT and (b) DT

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4.2.5 Subtraction

Subtraction of two signals is nothing but subtraction of their corresponding
amplitudes. This can be best explained by the following example:

Figure 4.11: Sample of Subtraction in Signal
As seen from figure 4.11. above the following condition will take place:
-10 < t < -3 amplitude of z (t) = x1(t) - x2(t) = 0 - 2 = -2
-3 < t < 3 amplitude of z (t) = x1(t) - x2(t) = 1 - 2 = -1
3 < t < 10 amplitude of z (t) = x1(t) + x2(t) = 0 - 2 = -2

4.2.6 Multiplication

Multiplication is the product of two signals is defined as y(n) = x1(n) * x2(n). The concept
of multiplication can be depicted as follows:

Continuous-time:
( ) = 1( ) 2( )

Discrete-time:
[ ] = 1[ ] 2[ ]

PSP eBook | Blueprint Guide to Signal and System 41

a) Multiplication in Continuous-Time

b) Multiplication in Discrete-Time
Figure 4.12: Sample of Multiplication in (a) & (b) of Signal

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With 1[ ] and 2[ ], can themselves be obtained by manipulating other signals. For
example below we have a truncated exponential begins at t=0. This signal can be obtained by

multiplying:

When 1[ ] =
And 2[ ] = ( )
Where ( ) = ( )
for < 0

Same is true for discrete time signals. In genral one-sided signals can be obtained by
multiplying by u[n] or shifted or time-reversed versions of u[n] or u(t). In general one-sided
signals can be obtained by multiplying by u[n] or shifted or time-reversed versions of u[n] or
u(t). Multiplication of two signals is nothing but multiplication of their corresponding
amplitudes. This can be best explained by the following example:

Figure 4.13: Sample of Multiplication in Signal

As seen from the diagram above,
-10 < t < -3 amplitude of z (t) = x1(t) ×x2(t) = 0 ×2 = 0
-3 < t < 3 amplitude of z (t) = x1(t) ×x2(t) = 1 ×2 = 2
3 < t < 10 amplitude of z (t) = x1(t) × x2(t) = 0 × 2 = 0

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Useful signal operations are when these methods of shifting, scaling or inversion are
being introduced to the function and are done combined operations as such f(t) → f(at-b).
Then two possible sequences of operations are performed:

1) Time shift f(t) by to obtain f(t-b). Now time scale the shifted signal f(t-b) by a to
obtain f(at-b).

2) Time scale f(t) by a to obtain f(at). Now time shift f(at) by b/a to obtain f(at-b).

Note that you have to replace t by (t-b/a) to obtain f(at-b) from f(at) when replacing t by
the translated argument namely t(t-b/a).