Fundamentals of
Measurement Technology
(1)
Prof,Wang Boxiong
1.Introduction
1.1 Measurement technology,background
and development
1.2 Essence of measurement and its basic
prerequisites
1.3 Standards and their units
Major points to study
1,What is measurement technology all
about?
2,Significance of measurement technology
in science,production and social
progress
3,Major components of a measuring
system and their respective functions
4,What to study
1.Introduction
Historical background
Knowledge acquisition often begins with
measurement,
Measurement science (metrology) was
generated and developed with the
development of human society.
Early measurements:
Length (distance),time,area and weight,
etc,
1.1 Measurement technology,background and development
The field of measurement was enlarged
continuously,and the means and methods
for measurements became more
sophisticated:
Elle,created by the Egyptians in about 3000 B.C.
A unified standard of length.
Defined as the length of the forearm of the Pharao
measured from his elbow to the tip of his stretched
middle finger plus the width of a palm twig in his hand.
Realized with a black granite as the primary standard.
Emperor Qin Shi established a unified system of
weights and measures and gave severe
punishments to those who violated the system.
1.1 Measurement technology,background and development
Measurement technology plays an important role in
promoting production development and social
progress.
The rapid development of science and technology
has invigorated measurement technology,and the
development of modern electronics,especially the
information technology has pushed forward further
its development,
Every subject can find its trace in measurement
technology,
The development of measurement technology has,
in turn,promoted the development of other
subjects,
1.1 Measurement technology,background and development
The extensiveness,essence and scientific
significance of measurement technology:
“Whatever exists,exists in some amount.”
“I often say that when you can measure what you
are speaking about and express it in numbers,you
know something about it,and when you cannot
measure it,when you cannot express it in numbers,
your knowledge is of meager and unsatisfactory
kind,It may be the beginning of knowledge,but
you have scarcely,in your thought,advanced to
the stage,whatever the matter be.”
1.1 Measurement technology,background and development
-- Lord Kelvin (William Thompson)
Classification of physical quantities (in terms
of time-dependent properties):
Static quantities and dynamic quantities
Static quantity,does not vary or varies
slowly with time.
Dynamic quantity,varies rapidly with time.
Static measurement:the measurement of a
static quantity,
Dynamic measurement:the measurement of
a dynamic quantity.
1.1 Measurement technology,background and development
Applications of dynamic measurement:
Machine part,vibration measurement;
Flight of an aircraft,using dials and instruments
to indicate its course,speed,acceleration and
mileage etc,
The process of steel rolling,detecting thickness
and width of the rolled strip steel.
1.1 Measurement technology,background and development
Whether a measurement is static or dynamic
is totally relative,sometimes they can be
changed with each other.
The kind of the measurement to be
performed depends on:
what is measured;
what are the requirements on the measurement:
accuracy,stability,etc.
1.1 Measurement technology,background and development
Composition of a measurement
system:
1.1 Measurement technology,background and development
Fig,1.1 General configuration of a measuring system
Sensors and
transducers
Signal
conditioning
and
conversion
Data display
& recording
Measured
object Observer
First stage,sensors or sensing elements
Obtaining information from the measured object
and converting them into signals or variables
suitable for the measurement.
Examples:
Spring scale,the spring is a sensor or a sensing
element,converting force into an elastic deformation –
the displacement,
Temperature measurement,liquid-in-glass
thermometer is a sensor,converting change in
temperature into the displacement of the liquid in
capillary tube,
1.1 Measurement technology,background and development
For a given measurement task,the first step is
to effectively get the information from the
measurand pertinent to the measurement.
Sensors or transducers play an important role in
the whole measurement system,
1.1 Measurement technology,background and development
The second stage,signal conditioning and
conversion
Further processing of the signals from the
sensor:
Signal conversion,amplification,filtering,data
storage and replay,and some special processings.
The third stage,signal displaying and
recording
Display or record the signal from the signal
conditioning stage by means of the methods
and equipment suitable for observation and
further analysis,
1.1 Measurement technology,background and development
All functions of the three stages are realized
through sensors and various measuring
instruments and equipment,which constitute
the core of a measurement system.
The measured object and the observer must
be also considered as the constituents of the
system.
The measurand,influence the sensor through
connections or couplings when a sensor is
employed to pick up signals from measurand.
The observer,influence the characteristics of a
system through his/her own behaviors,
1.1 Measurement technology,background and development
Distortionless or accurate measurement,
the system’s output will truly,duplicate” or
reflect the input.
How to realize an accurate or a
distortionless measurement becomes the
focus of study in our text.
1.1 Measurement technology,background and development
Purpose of the course:
Grasp the basic theories and methods of
measurement technology,the methods and
means for measuring and analyzing parameters
of a mechanical system.
Lay a foundation for further studying and solving
engineering problems of measurement
technology,
1.1 Measurement technology,background and development
Major contents of the course:
1,Basic signal theories and the theories and
methods for signal processing:
signal representations in time and frequency
domains;
signal spectrum and the methods for spectrum
analysis;
signal convolution and correlation;
basic theories and methods for signal processing.
1.1 Measurement technology,background and development
2,Parameters of measuring systems and their
evaluations:
representations in time and frequency domains for
the transfer characteristics of measuring systems;
impulse response and frequency response
functions;
descriptions of the dynamic performances of first-
order and second-order systems;
measurement of the parameters,as well as the
conditions for an accurate measurement.
1.1 Measurement technology,background and development
3,Theories of transducers:
operating principles for various commonly-used
transducers;
configurations and characteristic performances;
typical applications.
4,Principles and methods of signal conditioning:
bridges;
signal modulation and demodulation;
signal filtering;
A/D and D/A conversions;
typical applications.
5,Operating principles and configurations of
displaying and recording instruments,their
dynamic performances and applications.
1.1 Measurement technology,background and development
Chapter 2 Signal Analysis and
Processing in Measurement
Prof,Wang Boxiong
2.1 Signal and measuring system
2.2 Signal representation
2.3 Digital signal processing
Chap,2 Signal Analysis and Processing in Measurement
Major points to study
1,Significance of signal and measuring
system analyses
2,Methods for representing signals in time-
domain and frequency-domain
Frequency-domain representation of periodic
signals,their discrete frequency spectrums
Frequency-domain representation of
nonperiodic signals,their continuous
frequency spectrum
Fourier transform,major properties and
applications
Fourier transforms of typical signals and their
applications
Chap,2 Signal Analysis and Processing in Measurement
3,Time-domain and frequency-domain
analyses for system’s transfer
characteristics:
Time-domain,unit impulse response function,
convolution integration
Frequency-domain,frequency response
function and transfer function,multiplication
Depicting system’s characteristics by use of its
response functions
Responses of first-order and second-order
systems to typical excitations (impulse,step,
ramp,sinusoidal signal)
Major points to study
Chap,2 Signal Analysis and Processing in Measurement
4,Requirements on measuring systems for
accurate or distortionless measurements
5,Significance of digital signal processing
(DSP) and its characters
6,Elements of DFT and FFT,their properties
and formulae
7,Basic theories for DSP,sampling theorem,
windowing,signal truncation,etc.
8,Typical algorithms for FFT
Major points to study
Chap,2 Signal Analysis and Processing in Measurement
A measuring system is composed of three
stages,
– signal sensing;
– signal conditioning;
– signal recording and displaying,
Measuring systems for different
measurands may have different
configurations and operating principles,
According to its operating principle,a
measuring system can be a mechanical,an
electrical or a hydraulic one,etc,
2.1 Signal and measuring system
2.1 Signal and measuring system
Fig.2.1 Block diagram of a measuring system’s structure
To deal with systems of different characteristics,
we always
– neglect the concrete physical meanings of systems;
– abstract them with an ideal physical model for obtaining
the common laws governing the behaviors of systems of
the same kind,
Signals,all the physical quantities or variables of
systems,such as
– force,displacement,acceleration,voltage,current and
light intensity,etc.
It is the objective to disclose the system’s transfer
characteristics for signals through the study of laws
of signals in a measuring system,
2.1 Signal and measuring system
Signals and systems are closely related,
For a measuring system,
– input signal,the excitation to the system,
– output signal,the response to the system,
The signal theories include:
– signal analysis,
– signal processing,
– signal synthesis,
2.1 Signal and measuring system
The word signal comes from sign
(signum in Latin),which denotes an
object,a mark,a language element,or
an agreed-upon symbol that serves as
an information vector,
2.2 Signal representation
Signal,the physical representation of the
information that it carries from its source to
its destination,
Noise,any phenomenon (interference,
random distortion,et cetera) disturbing signal
perception or interpretation,after the
acoustical effects bearing the same name,
A noise is also a signal,
2.2.1 Signal definition
Signal-to-noise ratio (S/N or SNR),a
measure of the extent of signal
contamination by noise,expressed as
or
2.2.1 Signal definition
ns PP /
(2.1)
10lo g10?dB (2.2)
The difference between signal and noise is
purely artificial and depends only on the
user’s criteria,
What makes the difference between signal
and noise is only the observer’s focus of
interest,
Example,gear noise
Signal theory must encompass noise theory,
2.2.1 Signal definition
Experimental constraints
– An experimental signal is the image of a
physical process and must be physically
achievable,
– Constraints for a signal:
Its energy must be finite.
Its amplitude is necessarily bounded,
This amplitude is a continuous function because
the source system inertia prohibits any
discontinuity,
The signal spectrum is also necessarily bounded
and must tend towards zero when the frequency
tends towards infinity,
2.2.2 Signal classification
Classification methods
– Phenomenological classification based on
the evolution type of the signals,its predefined
character,or its random behavior
– Energy classification defining two classes of
signals,those with a finite energy and those
with a finite mean power but an infinite energy
– Morphological classification based on the
continuous or discrete character of the
amplitude or of the independent variable of the
signal
2.2.2 Signal classification
– Dimensional classification,which is based
on the number of independent variables,in the
signal model
– Spectral classification,which is based on
the shape of the frequency distribution of the
signal spectrum
2.2.2 Signal classification
1,Deterministic or random signals
Definitions
– Deterministic signals,have an evolution that
is perfectly predictable with an appropriate
mathematical model
– Random signals,have unpredictable
behavior and can generally be described only
through statistical observations
– Deterministic signals can be easily depicted
mathematically with handy formulae,usually
used in laboratories,also encountered in
production,
E.g,electrical energy by electromechanical
generators driven by a turbine,
2.2.2 Signal classification
Subdivisions of deterministic signals,
– periodic signals,complying with the relation,
i.e.,obeying a regular cyclical repetition law with
a period T,
sinusoidal signals
periodic compound signals
pseudorandom signals
– nonperiodic signals,do not enjoy this property,
quasiperiodic signals
transient signals
2.2.2 Signal classification
)()( kTtxtx
(2.3)
Sinusoidal signals:
2.2.2 Signal classification
)](2s in [)2s in ()( tTAtTAtx (2.4)
Fig,2.2 Sinusoidal signal
Pseudorandom signals form a particular category of
periodic signals with quasirandom behavior,
2.2.2 Signal classification
Fig,2.3 Pseudorandom signal
quasiperiodic signals,resulting from the sum of sine
waves with uncommensurable periods
2.2.2 Signal classification
Fig,2.4 Quasiperiodic signal
transient signals:short-lived
2.2.2 Signal classification
Fig,2.5 Transient signals,x(t)=rectangular pulse; y(t)=decaying
exponential pulse; z(t)=sinusoidal pulse,
Subdivisions of random signals
– Stationary random signals,the statistical
characteristics of which are invariant in time,
Ergodic,the statistical average values,or moments,
of a stationary signal are equal to its time average
values.
– Nonstationary random signals,do not have
this property,
A random signal with a transient behavior is
nonstationary,
2.2.2 Signal classification
2.2.2 Signal classification
Fig,2.6 Stationary random signal,x(t)=wideband signal (with noise);
y(t)=lowpass filtered signal,
2.2.2 Signal classification
Fig,2.7 Nonstationary random signal
s i n u s o i d a l p e r i o d i c
c o m p o u n d
p s e u d o -
r a n d o m
p e r i o d i c
t r a n s i e n t
non-
p e r i o d i c
deterministic
e r g o d ic q u a s i-
p e r io d ic
non-
e r g o d ic
s t a t io n a r y
s p e c ia l
c la s s if ic a t io n
non-
s t a t io n a r y
rand om
S ig n a ls
2.2.2 Signal classification
2,Energy classification
Finite energy signals
– include all the transient deterministic or
random signals.
Finite average power signals
– include nearly all the periodic or
quasiperiodic signals and the permanent
random signals.
2.2.2 Signal classification
Applying a voltage to a unit resistor R(R=1Ω )
to get the instantaneous power
When x(t) complies with the following condition,
x(t)is a finite energy signal,also called energy
signal,
Examples,rectangular pulses,exponentially
decaying signals,etc.
)()()( 22 txR txtp (2.5)
dttxdttptW )()()( 2(2.6)
dttx 2)( (2.7)
2.2.2 Signal classification
When a signal complies with the condition:
i.e.,the signal has a finite (nonzero) average
power,it is called the finite average power
signal,or power signal,
Example,undamped
vibration system
2.2.2 Signal classification
2/ 2/ 2)(1lim0 T TT dttxT (2.8)
Fig,2.8 One-degree-of-freedom vibration system
Observations
The function x2(t) corresponds to a
time distribution of the signal energy,
The average power Px(t) is the energy
mean distribution over the chosen
interval,
2.2.2 Signal classification
3,Continuous signals and discrete
signals
Continuous signals and discrete signals,
depengding on whether the signal’s amplitude
is a continuous or discrete variable and the
argument t (here considered as time) is itself
continuous or discrete.
Signals with continuous amplitude and
continuous time are called analog signals,
Signals with discrete amplitude and continuous
time are called quantized signals,
2.2.2 Signal classification
2.2.2 Signal classification
Fig,2.9 Continuous-time signals
To make the definition of a function become
complete,we stipulate,the value of a function f(t),
if it has a discontinuity point at t=0,is equal to the
value halfway between the left-hand limit f(t0-) and
the right-hand limit f(t0+),
where
The unit-step function is then defined as:
Digital signals,signals with discrete amplitude and
discrete time,
2.2.2 Signal classification
)()(21)( 000 tftftf (2.10)
),(lim)( 000 tftf d e f )(lim)( 000 tftf d e f
0,1
0,2/1
0,0
)(
t
t
t
t
d e f
(2.11)
2.2.2 Signal classification
Fig,2.10 Morphological classification of signals
Processing system classification:
– Analog systems,amplifiers,classical filters,
multipliers,signal modulators,etc.
– Sampled data systems,charge coupled
devices,switched capacitor filters,etc.
– Digital systems,digital filters,correlators,
Fourier transformers,and other specialized
processors,
2.2.2 Signal classification
Two methods for representing a signal:
1,representation in time domain
2,representation in frequency domain
Time representation,
– reflects mainly the features of signal’s
amplitude in variation with time,
– For analysis,
Classical differential or difference equations are
utilized.
Concepts such as unit-impulse response and unit
sequence response are also employed with the
help of convolution integral,
2.2.3 Signal representations in time domain and in frequency domain
Frequency representation:
– Convert a time variable function or sequence of
a signal into its corresponding function of a
certain variable in frequency domain to disclose
the frequency features of the signal and the
system,
– For continuous signals and systems,Fourier
transform and Laplace transform are employed,
whereas for discrete signals and systems Z-
transform is used,
2.2.3 Signal representations in time domain and in frequency domain
– Complex signals are often decomposed into a
sum of some basic signals of certain type,which
comply with certain mathematical conditions and
are easy to realize and analyze,
basic signals,sinusoidal or harmonic signals,complex
exponential signals,step signals,impulse signals,etc,
– With signal’s frequency representation a time
signal is converted into a frequency signal and
expressed with a sum of basic signals in
frequency representation.
– The signal’s frequency composition and the
relationship between the amplitude and the phase
of all frequency components of the signal are
analyzed,
2.2.3 Signal representations in time domain and in frequency domain
Note:
It should be noted,though,that whichever
of the representations is adopted,identical
signals contain the same information and
the information can not be increased or
reduced with different representations.
2.2.3 Signal representations in time domain and in frequency domain
Measurement Technology
(1)
Prof,Wang Boxiong
1.Introduction
1.1 Measurement technology,background
and development
1.2 Essence of measurement and its basic
prerequisites
1.3 Standards and their units
Major points to study
1,What is measurement technology all
about?
2,Significance of measurement technology
in science,production and social
progress
3,Major components of a measuring
system and their respective functions
4,What to study
1.Introduction
Historical background
Knowledge acquisition often begins with
measurement,
Measurement science (metrology) was
generated and developed with the
development of human society.
Early measurements:
Length (distance),time,area and weight,
etc,
1.1 Measurement technology,background and development
The field of measurement was enlarged
continuously,and the means and methods
for measurements became more
sophisticated:
Elle,created by the Egyptians in about 3000 B.C.
A unified standard of length.
Defined as the length of the forearm of the Pharao
measured from his elbow to the tip of his stretched
middle finger plus the width of a palm twig in his hand.
Realized with a black granite as the primary standard.
Emperor Qin Shi established a unified system of
weights and measures and gave severe
punishments to those who violated the system.
1.1 Measurement technology,background and development
Measurement technology plays an important role in
promoting production development and social
progress.
The rapid development of science and technology
has invigorated measurement technology,and the
development of modern electronics,especially the
information technology has pushed forward further
its development,
Every subject can find its trace in measurement
technology,
The development of measurement technology has,
in turn,promoted the development of other
subjects,
1.1 Measurement technology,background and development
The extensiveness,essence and scientific
significance of measurement technology:
“Whatever exists,exists in some amount.”
“I often say that when you can measure what you
are speaking about and express it in numbers,you
know something about it,and when you cannot
measure it,when you cannot express it in numbers,
your knowledge is of meager and unsatisfactory
kind,It may be the beginning of knowledge,but
you have scarcely,in your thought,advanced to
the stage,whatever the matter be.”
1.1 Measurement technology,background and development
-- Lord Kelvin (William Thompson)
Classification of physical quantities (in terms
of time-dependent properties):
Static quantities and dynamic quantities
Static quantity,does not vary or varies
slowly with time.
Dynamic quantity,varies rapidly with time.
Static measurement:the measurement of a
static quantity,
Dynamic measurement:the measurement of
a dynamic quantity.
1.1 Measurement technology,background and development
Applications of dynamic measurement:
Machine part,vibration measurement;
Flight of an aircraft,using dials and instruments
to indicate its course,speed,acceleration and
mileage etc,
The process of steel rolling,detecting thickness
and width of the rolled strip steel.
1.1 Measurement technology,background and development
Whether a measurement is static or dynamic
is totally relative,sometimes they can be
changed with each other.
The kind of the measurement to be
performed depends on:
what is measured;
what are the requirements on the measurement:
accuracy,stability,etc.
1.1 Measurement technology,background and development
Composition of a measurement
system:
1.1 Measurement technology,background and development
Fig,1.1 General configuration of a measuring system
Sensors and
transducers
Signal
conditioning
and
conversion
Data display
& recording
Measured
object Observer
First stage,sensors or sensing elements
Obtaining information from the measured object
and converting them into signals or variables
suitable for the measurement.
Examples:
Spring scale,the spring is a sensor or a sensing
element,converting force into an elastic deformation –
the displacement,
Temperature measurement,liquid-in-glass
thermometer is a sensor,converting change in
temperature into the displacement of the liquid in
capillary tube,
1.1 Measurement technology,background and development
For a given measurement task,the first step is
to effectively get the information from the
measurand pertinent to the measurement.
Sensors or transducers play an important role in
the whole measurement system,
1.1 Measurement technology,background and development
The second stage,signal conditioning and
conversion
Further processing of the signals from the
sensor:
Signal conversion,amplification,filtering,data
storage and replay,and some special processings.
The third stage,signal displaying and
recording
Display or record the signal from the signal
conditioning stage by means of the methods
and equipment suitable for observation and
further analysis,
1.1 Measurement technology,background and development
All functions of the three stages are realized
through sensors and various measuring
instruments and equipment,which constitute
the core of a measurement system.
The measured object and the observer must
be also considered as the constituents of the
system.
The measurand,influence the sensor through
connections or couplings when a sensor is
employed to pick up signals from measurand.
The observer,influence the characteristics of a
system through his/her own behaviors,
1.1 Measurement technology,background and development
Distortionless or accurate measurement,
the system’s output will truly,duplicate” or
reflect the input.
How to realize an accurate or a
distortionless measurement becomes the
focus of study in our text.
1.1 Measurement technology,background and development
Purpose of the course:
Grasp the basic theories and methods of
measurement technology,the methods and
means for measuring and analyzing parameters
of a mechanical system.
Lay a foundation for further studying and solving
engineering problems of measurement
technology,
1.1 Measurement technology,background and development
Major contents of the course:
1,Basic signal theories and the theories and
methods for signal processing:
signal representations in time and frequency
domains;
signal spectrum and the methods for spectrum
analysis;
signal convolution and correlation;
basic theories and methods for signal processing.
1.1 Measurement technology,background and development
2,Parameters of measuring systems and their
evaluations:
representations in time and frequency domains for
the transfer characteristics of measuring systems;
impulse response and frequency response
functions;
descriptions of the dynamic performances of first-
order and second-order systems;
measurement of the parameters,as well as the
conditions for an accurate measurement.
1.1 Measurement technology,background and development
3,Theories of transducers:
operating principles for various commonly-used
transducers;
configurations and characteristic performances;
typical applications.
4,Principles and methods of signal conditioning:
bridges;
signal modulation and demodulation;
signal filtering;
A/D and D/A conversions;
typical applications.
5,Operating principles and configurations of
displaying and recording instruments,their
dynamic performances and applications.
1.1 Measurement technology,background and development
Chapter 2 Signal Analysis and
Processing in Measurement
Prof,Wang Boxiong
2.1 Signal and measuring system
2.2 Signal representation
2.3 Digital signal processing
Chap,2 Signal Analysis and Processing in Measurement
Major points to study
1,Significance of signal and measuring
system analyses
2,Methods for representing signals in time-
domain and frequency-domain
Frequency-domain representation of periodic
signals,their discrete frequency spectrums
Frequency-domain representation of
nonperiodic signals,their continuous
frequency spectrum
Fourier transform,major properties and
applications
Fourier transforms of typical signals and their
applications
Chap,2 Signal Analysis and Processing in Measurement
3,Time-domain and frequency-domain
analyses for system’s transfer
characteristics:
Time-domain,unit impulse response function,
convolution integration
Frequency-domain,frequency response
function and transfer function,multiplication
Depicting system’s characteristics by use of its
response functions
Responses of first-order and second-order
systems to typical excitations (impulse,step,
ramp,sinusoidal signal)
Major points to study
Chap,2 Signal Analysis and Processing in Measurement
4,Requirements on measuring systems for
accurate or distortionless measurements
5,Significance of digital signal processing
(DSP) and its characters
6,Elements of DFT and FFT,their properties
and formulae
7,Basic theories for DSP,sampling theorem,
windowing,signal truncation,etc.
8,Typical algorithms for FFT
Major points to study
Chap,2 Signal Analysis and Processing in Measurement
A measuring system is composed of three
stages,
– signal sensing;
– signal conditioning;
– signal recording and displaying,
Measuring systems for different
measurands may have different
configurations and operating principles,
According to its operating principle,a
measuring system can be a mechanical,an
electrical or a hydraulic one,etc,
2.1 Signal and measuring system
2.1 Signal and measuring system
Fig.2.1 Block diagram of a measuring system’s structure
To deal with systems of different characteristics,
we always
– neglect the concrete physical meanings of systems;
– abstract them with an ideal physical model for obtaining
the common laws governing the behaviors of systems of
the same kind,
Signals,all the physical quantities or variables of
systems,such as
– force,displacement,acceleration,voltage,current and
light intensity,etc.
It is the objective to disclose the system’s transfer
characteristics for signals through the study of laws
of signals in a measuring system,
2.1 Signal and measuring system
Signals and systems are closely related,
For a measuring system,
– input signal,the excitation to the system,
– output signal,the response to the system,
The signal theories include:
– signal analysis,
– signal processing,
– signal synthesis,
2.1 Signal and measuring system
The word signal comes from sign
(signum in Latin),which denotes an
object,a mark,a language element,or
an agreed-upon symbol that serves as
an information vector,
2.2 Signal representation
Signal,the physical representation of the
information that it carries from its source to
its destination,
Noise,any phenomenon (interference,
random distortion,et cetera) disturbing signal
perception or interpretation,after the
acoustical effects bearing the same name,
A noise is also a signal,
2.2.1 Signal definition
Signal-to-noise ratio (S/N or SNR),a
measure of the extent of signal
contamination by noise,expressed as
or
2.2.1 Signal definition
ns PP /
(2.1)
10lo g10?dB (2.2)
The difference between signal and noise is
purely artificial and depends only on the
user’s criteria,
What makes the difference between signal
and noise is only the observer’s focus of
interest,
Example,gear noise
Signal theory must encompass noise theory,
2.2.1 Signal definition
Experimental constraints
– An experimental signal is the image of a
physical process and must be physically
achievable,
– Constraints for a signal:
Its energy must be finite.
Its amplitude is necessarily bounded,
This amplitude is a continuous function because
the source system inertia prohibits any
discontinuity,
The signal spectrum is also necessarily bounded
and must tend towards zero when the frequency
tends towards infinity,
2.2.2 Signal classification
Classification methods
– Phenomenological classification based on
the evolution type of the signals,its predefined
character,or its random behavior
– Energy classification defining two classes of
signals,those with a finite energy and those
with a finite mean power but an infinite energy
– Morphological classification based on the
continuous or discrete character of the
amplitude or of the independent variable of the
signal
2.2.2 Signal classification
– Dimensional classification,which is based
on the number of independent variables,in the
signal model
– Spectral classification,which is based on
the shape of the frequency distribution of the
signal spectrum
2.2.2 Signal classification
1,Deterministic or random signals
Definitions
– Deterministic signals,have an evolution that
is perfectly predictable with an appropriate
mathematical model
– Random signals,have unpredictable
behavior and can generally be described only
through statistical observations
– Deterministic signals can be easily depicted
mathematically with handy formulae,usually
used in laboratories,also encountered in
production,
E.g,electrical energy by electromechanical
generators driven by a turbine,
2.2.2 Signal classification
Subdivisions of deterministic signals,
– periodic signals,complying with the relation,
i.e.,obeying a regular cyclical repetition law with
a period T,
sinusoidal signals
periodic compound signals
pseudorandom signals
– nonperiodic signals,do not enjoy this property,
quasiperiodic signals
transient signals
2.2.2 Signal classification
)()( kTtxtx
(2.3)
Sinusoidal signals:
2.2.2 Signal classification
)](2s in [)2s in ()( tTAtTAtx (2.4)
Fig,2.2 Sinusoidal signal
Pseudorandom signals form a particular category of
periodic signals with quasirandom behavior,
2.2.2 Signal classification
Fig,2.3 Pseudorandom signal
quasiperiodic signals,resulting from the sum of sine
waves with uncommensurable periods
2.2.2 Signal classification
Fig,2.4 Quasiperiodic signal
transient signals:short-lived
2.2.2 Signal classification
Fig,2.5 Transient signals,x(t)=rectangular pulse; y(t)=decaying
exponential pulse; z(t)=sinusoidal pulse,
Subdivisions of random signals
– Stationary random signals,the statistical
characteristics of which are invariant in time,
Ergodic,the statistical average values,or moments,
of a stationary signal are equal to its time average
values.
– Nonstationary random signals,do not have
this property,
A random signal with a transient behavior is
nonstationary,
2.2.2 Signal classification
2.2.2 Signal classification
Fig,2.6 Stationary random signal,x(t)=wideband signal (with noise);
y(t)=lowpass filtered signal,
2.2.2 Signal classification
Fig,2.7 Nonstationary random signal
s i n u s o i d a l p e r i o d i c
c o m p o u n d
p s e u d o -
r a n d o m
p e r i o d i c
t r a n s i e n t
non-
p e r i o d i c
deterministic
e r g o d ic q u a s i-
p e r io d ic
non-
e r g o d ic
s t a t io n a r y
s p e c ia l
c la s s if ic a t io n
non-
s t a t io n a r y
rand om
S ig n a ls
2.2.2 Signal classification
2,Energy classification
Finite energy signals
– include all the transient deterministic or
random signals.
Finite average power signals
– include nearly all the periodic or
quasiperiodic signals and the permanent
random signals.
2.2.2 Signal classification
Applying a voltage to a unit resistor R(R=1Ω )
to get the instantaneous power
When x(t) complies with the following condition,
x(t)is a finite energy signal,also called energy
signal,
Examples,rectangular pulses,exponentially
decaying signals,etc.
)()()( 22 txR txtp (2.5)
dttxdttptW )()()( 2(2.6)
dttx 2)( (2.7)
2.2.2 Signal classification
When a signal complies with the condition:
i.e.,the signal has a finite (nonzero) average
power,it is called the finite average power
signal,or power signal,
Example,undamped
vibration system
2.2.2 Signal classification
2/ 2/ 2)(1lim0 T TT dttxT (2.8)
Fig,2.8 One-degree-of-freedom vibration system
Observations
The function x2(t) corresponds to a
time distribution of the signal energy,
The average power Px(t) is the energy
mean distribution over the chosen
interval,
2.2.2 Signal classification
3,Continuous signals and discrete
signals
Continuous signals and discrete signals,
depengding on whether the signal’s amplitude
is a continuous or discrete variable and the
argument t (here considered as time) is itself
continuous or discrete.
Signals with continuous amplitude and
continuous time are called analog signals,
Signals with discrete amplitude and continuous
time are called quantized signals,
2.2.2 Signal classification
2.2.2 Signal classification
Fig,2.9 Continuous-time signals
To make the definition of a function become
complete,we stipulate,the value of a function f(t),
if it has a discontinuity point at t=0,is equal to the
value halfway between the left-hand limit f(t0-) and
the right-hand limit f(t0+),
where
The unit-step function is then defined as:
Digital signals,signals with discrete amplitude and
discrete time,
2.2.2 Signal classification
)()(21)( 000 tftftf (2.10)
),(lim)( 000 tftf d e f )(lim)( 000 tftf d e f
0,1
0,2/1
0,0
)(
t
t
t
t
d e f
(2.11)
2.2.2 Signal classification
Fig,2.10 Morphological classification of signals
Processing system classification:
– Analog systems,amplifiers,classical filters,
multipliers,signal modulators,etc.
– Sampled data systems,charge coupled
devices,switched capacitor filters,etc.
– Digital systems,digital filters,correlators,
Fourier transformers,and other specialized
processors,
2.2.2 Signal classification
Two methods for representing a signal:
1,representation in time domain
2,representation in frequency domain
Time representation,
– reflects mainly the features of signal’s
amplitude in variation with time,
– For analysis,
Classical differential or difference equations are
utilized.
Concepts such as unit-impulse response and unit
sequence response are also employed with the
help of convolution integral,
2.2.3 Signal representations in time domain and in frequency domain
Frequency representation:
– Convert a time variable function or sequence of
a signal into its corresponding function of a
certain variable in frequency domain to disclose
the frequency features of the signal and the
system,
– For continuous signals and systems,Fourier
transform and Laplace transform are employed,
whereas for discrete signals and systems Z-
transform is used,
2.2.3 Signal representations in time domain and in frequency domain
– Complex signals are often decomposed into a
sum of some basic signals of certain type,which
comply with certain mathematical conditions and
are easy to realize and analyze,
basic signals,sinusoidal or harmonic signals,complex
exponential signals,step signals,impulse signals,etc,
– With signal’s frequency representation a time
signal is converted into a frequency signal and
expressed with a sum of basic signals in
frequency representation.
– The signal’s frequency composition and the
relationship between the amplitude and the phase
of all frequency components of the signal are
analyzed,
2.2.3 Signal representations in time domain and in frequency domain
Note:
It should be noted,though,that whichever
of the representations is adopted,identical
signals contain the same information and
the information can not be increased or
reduced with different representations.
2.2.3 Signal representations in time domain and in frequency domain
