Sampling Impr

8/12/2019 Sampling Impr

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Basic sampling theory


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Overview

Introduction/motivation

Sampling basics

A/D conversion

Nyquist frequency

Binary numbers and quantiation

!achine representation

Bitwise operations in " Summary


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#ou may have heard of$$$

%oltage and current &Ohm's law( )%*( )"*+

Operational amplifiers

*inear dynamic circuits

Sinusoidal signals

Analog filters

All of these concepts aredefined in continuous time


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,lectronic systems

In the -old days. people constructed all electronic systemsusing purely analog devices

*ower verticaldeflection plate

"athode

"ontrol electronicslass tube&vacuum+

0pper verticaldeflection plate0pper verticaldeflection plate


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,lectronic systems

,lectronic systems are very often used together with non1electronic devices2 motors( loudspea3ers( antennas( *,Ddisplays( etc$ etc$

4achometer

%oltage proportional to speed

5egulator

!otor current

D"1motor


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!odern electronic systems

5oughly since the 6789's( analogsystems have gradually beenreplaced with digital designs(because2

4hey are cheaper

4hey are easier to design

4hey are (re-)programmable


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A general system framewor3

System/reality !otor :uman ear Atmosphere Supertan3er

Actuator ;ower supply :eadphones Antenna 5udder( engine( propeller

Sensor 4achometer !icrophone Antenna ;S receiver

Computer Speed controller !;< encoding =iltering 5oute planning

Analog/ContinuousDigital/Discrete


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"ontinuous signals

4he signals and physical phenomena we observe around us arenormally continuous:

"urrents and voltages in the power grid

>ater flow in a district heating system

5otation speed of an engine

,tc$ etc$$$


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,lectrocardiogram


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,lectromyogram


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Signals vs$ components

"omponent/subsystem

"omponent/subsystem

"omponent/subsystem

"omponent/subsystem

SignalSignal

SignalSignal


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Discretiation of signals

"omputers only -understand. -numbers. :ence( continuous signals must be discretizedin order for

computers to be able to process them

4his is 3nown assampling

Discrete time"ontinuous time

Am

plitude

Am

plitude


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Sampling

"ontinuous time2 x x&t+( x R! t R

Discrete time2 x x&t+( x R! t "#! # R+! " Z

After sampling( we obtain ase$uence o% discrete &alues2

x" x&"#+

# ? s @ is calledsampling time

%s 6/4 ? : @ is called thesampling %re$uency

&sometimes also given in ?rad/s@+


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Sampling 1 schematics

4imerAlgorithm

A/D converter

Actuator

Sensor

D/A converter


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A/D "onversion the result

"ontinuous1timesignal

Sampled signal

Sampling process


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Information in sampled signals

Say we want to sample the following sinusoidal signalC how dowe choose an appropriate sampling frequency


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Sampling with E9 :


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Sampling with F9 :


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Sampling with 699 :


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Sampling with 6EG :


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Sampling with E99 :


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Sampling with


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Sampling with G99 :


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Nyquist frequency

4he Nyquist1Shannon sampling theorem2

'% a %unctionx(t)contains no %re$uencies higher than hertz! itis completely determined by gi&ing its ordinates at a series o%points spaced / &*+seconds apart+

"onversely( it isimpossibleto reconstruct a continuous1timesinusoidal signal containing frequencies higher than half thesampling frequencyH 4his frequency is called the,y$uist%re$uency

In practice( it is difficult to detect frequencies of more than about of the sampling frequency


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Sinusoidal signal models

Loseph =ourier discovered that anyrepetiti&e signal may be written asa &potenitially infinite+sum o%sines/cosines

and.idthrefers to the frequencieswhere the amplitude of thesine/cosine components have non1negligible amplitude$

Loseph =ourier( 68M16


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=requency spectrum

A frequency spectrum is a graph that shows how much 'power'each sinusoidal component contributes with

9

A

%

E$< :

6$7 sin&9$ t+

9$E sin&E$< t+

9$7 sin&$< t+

9$ : $< :


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Aliasing

In old movies you could sometimes see wagon wheels rotating-bac3wards. 1 this is an optical illusion caused by a limitedsampling frequency in the film cameras used$

li i


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Aliasing

Image E Image 6Image hen a frequency in a continuous1time signal eJceeds theNyquist1frequency( but is still below the actual samplingfrequency( we see a -reflection. of that frequency in the sampledsignalC this is 3nown as aliasing

Increasing frequency content in an


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Increasing frequency content in ananalog signal( fiJed%

s

"omp ting the freq enc of the


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"omputing the frequency of thealiased signal

4he frequency of the aliased signal(%a( is found via the

eJpression

%a&,+ P%, %

sP

%sis the sampling frequency

%is the actual signal frequency , is an integer


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=or eJample$$$

=ind%a if%s is 6EG : and% is 699 :

4est different values of, in the eJpression%a&,+ P%, %

sP

, 6 2 f = 125 100 = 25

N= 2 : f = |125 200| = 75N= 3 : f = |125 300| = 175

We can thus expect to see an aliased signal component at25 H!

Sampling a 699 : signal with


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p g g%

s 6EG :

4wo and a half wave in9$6 s

Q -oscillation. at EG :


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:ow to avoid aliasing

;re1filter the signal before sampling

*owpass1filter A/D "onverter

Analog signal affectedby unwantedfrequencies

Analog signal containingonly relevant &low+frequencies

Digital signal


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Bac3 to A/D "onversion

"ontinuous1timesignal

Sampled signal

Sampling process

ObsHinaryrepresentation of

sampled signal


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Binary numbers

Binary numbers with ndigits can representdifferent values

,$g$ for n


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Ruantiation

Numbers represented by binary digits are countable thismeans that not all real numbers can be represented on acomputer

But we can get arbitrarily close by increasing the number of

digits


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Ruantiation error

,ach bit corresponds to avoltage interval

:ence( quantiation leads to a

$uantization errorqof at

most the same sie as thisinterval &+

Best1case &5ounding off+2

q" #


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Summary

A/D conversion is a way of representing continuous signals asdiscrete values

)eywords2 discretization and $uantization

Sampled &aluesare represented in a computer using binarydigits

4ime

Amplitude

4ime

Amplitude

S


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Summary

It is not possible to reconstruct sinusoidal signals containingfrequencies higher than half the sampling frequency the,y$uist %re$uency

Normally( a sensible choice of sampling frequency is G to E9

times the highest frequency -of interest. to the application(called the band.idth

>hen sampling signals with lots of frequencies( you need to beaware of aliasing

On the slides after this( you will find some reference materialthat you might find of relevance later


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A/D " i t b i


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A/D "onversion comparator basics

A/D "onversion conversion to logic


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gvalues

A/D "onversion several


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comparators in parallel

A continuous voltage signal iscompared with differentreference voltages and areconverted into either % or 9 %

4hese values are associated withlogic values either 9 or 6

Bit i ti i "


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Bitwise operations in "

AND 6 if both bits are 6( 9 otherwise AND and assignment

P OR 6 if either bits are 6( 9 if they both are 9

P OR and assignment

T XOR 6 if ON, of the bits is 6 and the other is 9

T XORand assignment

U one's complement =lips all bits

KK Shift Left Shifts all bits to the left( inserting 9's

KK Shift Leftand assignment

VV Shift Right Shifts all bits to the right

VV Shift Right and assignment


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Introdu3tion til digitale filtre


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Introdu3tion til digitale filtre

*et us say that we have a sequence of samples

that we wish to process in some way

4ypically( thex'es are used as input to some digital filter

The general formula for a digital filter is

where the a's og b's are filter coefficientesand they's arethe output of the filter at the indicated sample numbers

{x"}"=9,

i=9

naaiy"i=i=9

nbbix "i

Introduction to digital filters


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Introduction to digital filters

>ritten out and rearranged2

naand n

bare usually small integersC furthermore( it it is common

to scale such that a9 6$ b

9is often &but not always+ 9$

=or eJample( a so1called second order filter2

y "=b6x"6+bEx"Ea6y "6aEy "E

a9y"=b9x"+b6x"6++b/x"nba6y"6ana y "na

Simple filters without dynamics


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Simple filters without dynamics

0nit gain

Simple gain

Signal offset

y"=0x "

y "=x"c

y "=x"

4ime delay and difference


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4ime delay and difference

4ime delay

Simple difference filter corresponds to di%%erentiation

y "=x"x"6

y "=x"6

Averaging filters


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Averaging filters

Second order averaging filter

4hird order averaging filter

"entral difference filter

y "=6

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Sampling Impr