lecture 6 - signals and systems basics.ppt

8/10/2019 Lecture 6 - Signals and Systems Basics.ppt

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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/


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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

lecture 6 - signals and systems basics.ppt

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