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Basics of Signal
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Basic definitions A signal is an abstraction of any measurable quantity that
is a function of one or more independent variables such astime or space Voltages and currents are common electrical signals
Signals can be continuous or discrete A continuous time signal is one that is present for all
instants in time and space example is voltage on a wire A discrete time signal is only present at discrete times
Often discrete time signals are samples of a continuoustime signal
A system is an abstraction of anything that takes an input
signal, operates on it and produces an output signal Signals and systems theory is the framework for mostengineering knowledge
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Basic Periodic Signal Terminology
Periodic repeating
Mathematical model from trigonometry
2
)sin(
Period phase
f requency
Amplitude A
t A y
Frequency is defined in radians/second where radians = 1 cycle or 360 degrees 2
)()( t vT t v
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Example from Sim ple Sine Wave inTim e and Freq Do m ain.VI
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Period or WaveLength(one cycle)
Am
pl i t u d e
Basic Periodic Signal Terminology
Frequency =1/Period (cycles/second)
or
rads/sec period 2
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Periodic Signal Terminology
Frequency (f)= 1/(time to perform one cycle) This yields a value in hertz or cycles/per second Often we talk in radians per second There are radians in a 360 degree cycle
So x f = frequency (rads/sec) =
2
2
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sin(x) = cos(x - 90 degrees) = cos( x - )sin(x) lags cos(x)cos(x) leads sin (x)
2
Phase difference
Phase Terminology
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Fourier Series
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Fourier Series
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Fourier square wave.vi
N=20
N=4
N=3
N=2
N=1
N=0
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Fourier triangle.vi
N=0 N=1
N=2 N=4
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N=20
Fourier rectangular sawtooth wave.vi
N=0
N=1
N=2
N=3
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Time and Frequency Domains
Previous examples have shown signals varying as a function
of time. These are said to be representations in the timedomain.
Signals can also be represented in the frequency domain In the frequency domain they are expressed as functions of
frequency A typical way to look at a signal in the frequency domain iswith a power spectral density (PSD) plot
A PSD shows the distribution of power in the variousfrequencies of a signal
As we can see from Fourier Series, a signal may in fact becomposed of many different signals When we look at a composite signal as components of different
frequencies, we are working in the frequency domain
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Power Spectral Density Example
Example from M ul tiple Sin e Waves in Time and Freq Domain.VI
Plot of sin(x) + sin(2x) + 2 sin(6x) + 3cos(9x)
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Noise Noise is undesired signal or contamination of a signal we want to
measure Average White Gaussian Noise (AWGN) equal power at all frequencies Frequency Specific Noise
Power at a specific frequency Alternating current (AC) power in house wiring in India is a periodic
waveform at 50 hertz It is not uncommon to find 50 hertz noise in electrical systems due to
electromagnetic interference from wiring systems The amount of signal present versus the noise present is expressed in the
Signal to Noise Ratio (SNR) It is usually expressed in decibels
Much of the work of instrumentation engineers involves extractingsignals and rejecting the noise
SNR is thus an important figure of merit to instrumentation engineering
Example using Signals and Noise.VI
power
power
voltage
voltage Noise
Signal
Noise
Signal SNR log10log20
Amplitude
freq
AWGN
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Sources of Noise
CONDUCTIVECAPACITIVEINDUCTIVE
RADIATIVECOUPLING CHANNEL
NOISESOURCE
RECEIVER(SIGNALCIRCUIT)
AC POWER LINESCOMPUTERS
DIGITAL LINES
TRANSDUCERSIGNAL CABLES
MEASUREMENT CIRCUIT
From Improved Signal Quality Via Conditioning by Lauren Sjoboen atwww.ni.com
http://www.ni.com/http://www.ni.com/http://www.ni.com/8/10/2019 Lecture 6 - Signals and Systems Basics.ppt
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Filters
Instrumentation engineers use filters to rejectunwanted signals (noise) and leave only the desiredsignals
Filters are classified by the frequencies they acceptor reject
Filters are a key part of signal conditioning in anyinstrumentation and data acquisition system Here we just want to understand the idea that we
can filter signals to remove signal content ofundesired frequencies
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Types of FiltersType Ideal Transfer as a
function of
frequency (|H(f)|)
Description Example Use
Lowpass Removes all signalswith frequency > fc
Noise removal, datainterpolation, smoothing
Highpass Removes all signalswith frequency < fc
Removing DC or lowfrequency drift, edgedetection orenhancement
Bandpass Removes all signals
outside of the rangeof f1 to f2
Tuning in a frequency on
a radio receiver,separating a subcarrier
Band Rejector Notch
Removing all signalat a particularfrequency range f1
to f2
Removing a particularnoise like power linehum at 60 hz
0 fc
1
0 fc
1
0 f1
1
f2
0 f1
1
f2
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Example from Extr act the Sine Wave.VI
Noisecontaminatedsignal
SignalafterLowPassFilter
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Real filter from Signals and Systems Made Ridiculously Simple
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Steady State and TransientResponse
Most systems have two types of response to an input The dynamic or transient response short lived
response driven by an imbalance of forces The steady state response a balanced unchanging state
This is not only for electrical systems but also for structuralsystems (mass spring damper), thermal systems andchemical systems
The study of dynamic response is a critical part ofengineering that is based on the use of differential
equations and the Laplace transform.
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Why is steady state and transientresponse important for understanding
instrumentation ?
We have to be able to characterize and separate theresponse of sensors to a changing input from the responseof the system to changing conditions If a sensor is bouncing around in response to an input, it
will not provide a good measurement Measurement errors result when the transient or steady
state response of a sensor is not perfect (non-ideal) Most measurement time histories are a combination of
transient and steady state response We need to be able to use the terminology properly to
describe what we are measuring
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Types of ideal inputs
Type of Input Time DomainRepresentation
Description Example
Unit Impulse (DiracDelta Function)
Instantaneousapplication andremoval ofinput
A hammer strikeon a structure, ahigh speedelectrical signal
Unit Step Instantaneousapplication ofsignal whichremains
Power on ofequipment.Application ofweight on astructure
Unit Ramp Continuouslyincreasinginput
Fluid level of atank
Time A m p l
i t u d e
11
Time A m p
l i t u d e
11
Time A m p l
i t u d e
11
For this lecture we are going to concentrate on unit step response
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First Order Dynamics If the quantity under measurement is x(t) and the sensor output is v(t)
then a sensor with first order dynamics can be represented by the
ordinary differential equation
zeroat timeof valuetheiswhere
exp1)1()(
solutionformclosedahasthis,)()()(
systemtheof constanttimetheiswhere1
bysides bothMultiply
sensor theof frequencynaturala w here()()(
0
00
x X
t K X et v
t x K t vt v
a
t) Kxt avt v
at a K X
x
x x
x x
If this is the case, the response is 63 % of the steady state in secsWithin 5% of the steady state value forAnd within 2% of the response within 2% of the steady state for
This should be recognizable as the time response of a simple R-C circuit from thelast lecture where
Many thermal sensors are first order sensors
2t 3t
RC
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First order system response to astep input
from Northrop Introduction to Instrumentation andMeasurements
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Types of Second Order Dynamics Second order sensor
dynamics fall into one of
three categories,depending on the locationof the roots of thecharacteristic equation ofthe differential equationthat describes the sensor
These categories are underdamped (complex
conjugate roots) critically damped (two
equal roots) Overdamped (unequalreal roots)
These three cases arerepresented by the
corresponding ordinarydifferential equation
)()(
)()2(
)()2(
2
2
t Kxabvbavv
t Kxavavv
t Kxvvv
x x x
x x x
n xn x x
sec)/( _
)( _
rad frequencynatural
zeta factor damping
n
kA
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Equations for Second Order Response Solving the differential equations for the step response
leads to the following results
),d(overdampe )(1
1)(
damped)y(criticall 1)(
1tan
ed)(underdamp 1sin1
11)(
0
20
21
2
220
abaebeabab KX
t v
ateea
KX t v
where
t e KX
t v
bt at x
at at x
nt
n x
n
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2nd Order StepResponse from
NorthropIntroduction toInstrumentationandMeasurements
This is a timedomainrepresentation ofthe response to astep input
Underdamped
Critically damped
Overdamped