Physics124:Lecture8

OddsandEndsBinary/Hex/ASCII

Memory&PointersinCDecibels&dBScalesCoherentDetection

AdaptedfromTomMurphy’slectures

Binary,HexadecimalNumbers• Computersstoreinformationinbinary– 1or0,correspondingtoVCC and0volts,typically– theCCsubscriptoriginatesfrom“collector”oftransistor

• Becomefamiliarwithbinarycountingsequence

2Phys124:Lecture8

binary decimal hexadecimal

00000000 0 0x00

0000 0001 1 0x01

00000010 2 0x02

00000011 2+1=3 0x03

00000100 4 0x04

00000101 4+1 =5 0x05

etc.

1111 1100 128+64+32+16+8+4 =252 0xfc

11111101 128+64+32+16+8+4+1=253 0xfd

11111110 128+64+32+16+8+4+2=254 0xfe

11111111 128+64+32+16+8+4+2+1=255 0xff

BinarytoHex:easy!• Noteseparationofprevious8-bit(one-byte)numbersintotwo4-bitpieces(nibbles)– makesexpressioninhex(base-16;4-bits)natural

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binary hexadecimal decimal

0000 0 0

0001 1 1

0010 2 2

0011 3 3

0100 4 4

0101 5 5

0110 6 6

0111 7 7

1000 8 8

1001 9 9

1010 A(lowercasefine) 10

1011 B 11

1100 C 12

1101 D 13

1110 E 14

1111 F 15

0 1 2 3 4 5 6 7

0 NUL ^@ null (\0) DLE ^P SP space 0 @ P ` p

1 SOH ^A startofhdr DC1 ^Q ! 1 A Q a q

2 STX ^B starttext DC2 ^R “ 2 B R b r

3 ETX ^C endtext DC3 ^S # 3 C S c s

4 EOT ^D endtrans DC4 ^T $ 4 D T d t

5 ENQ ^E NAK ^U % 5 E U e u

6 ACK ^F acknowledge SYN ^V & 6 F V f v

7 BEL ^G bell ETB ^W ‘ 7 G W g w

8 BS ^H backspace CAN ^X ( 8 H X h x

9 HT ^I horiz.tab(\t) EM ^Y ) 9 I Y i y

A LF ^J linefeed (\r) SUB ^Z * : J Z j z

B VT ^K verticaltab ESC escape + ; K [ k {

C FF ^L formfeed FS , < L \ l |

D CR ^M carriage ret(\n) GS - = M ] m }

E SO ^N RS . > N ^ n ~

F SI ^O US / ? O _ o DELPhys124:Lecture8 4

firsthexdigitsecond

hexdigit

ASCIITable inHex

ASCIIinHex

• NotethepatternsandconveniencesintheASCIItable– 0thru9ishex0x30to0x39(justadd0x30)– A-Zparallelsa-z;justadd0x20

• startsat0x41and0x61,soHis8th letter, is0x48,etc.

– thefirst32charactersarecontrolcharacters,oftenrepresentedasCtrl-C,denoted̂ C,forinstance• associatedcontrolcharactersmirror0x40to0x5F• putcommoncontrolcharacters inred;useful toknowinsomeprimitiveenvironments

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Two’sComplement• Unsignedaredirectbinaryrepresentation• Signedintegersusuallyfollow“two’scomplement”

– rule:togetneg.number, flipallbitsandaddone• example:-2:00000010à 11111101+1=11111110• Alsocalled“bias”,2" =256forsingleprecision,2$% fordoubleprecision

– addingpos.&neg.à 00000000(ignoreoverflow bit)Phys124:Lecture8 6

binary hex unsigned 2’scomplement

00000000 0x00 0 0

00000001 0x01 1 1

00000010 0x02 2 2

01111111 0x7F 127 127

10000000 0x80 128 -128

1000 0001 0x81 129 -127

11111110 0xFE 254 -2

11111111 0xFF 255 -1

FloatingPointNumbers• MoststandardisIEEEformat

– http://en.wikipedia.org/wiki/IEEE_754-1985– https://en.wikipedia.org/wiki/IEEE_floating_point#IEEE_754-2008

𝟎. 𝟏𝟓𝟔𝟐𝟓$, = 𝟎. 𝟎𝟎𝟏𝟎𝟏- =2./ + 2.1 = +𝟏.𝟎𝟏-×2./• Threeparts:sign, mantissa, exponent

– sign:0ispositive,1isnegative;mantissa,1isimplied– exponent:biasby127,i.e.,127-3-124 =0b01111100– single-precision(float)has32bits(1,8,23,resp.)

• 7digits,10±38: log(10)/log(2)=3.32,so223 ≈107;±127/3.32≈38– doubleprecision(double)has64bits(1,11,52,resp.)

• 16digits,10±308

• Theactualconvention isnotcriticalforustounderstand,asmuchas:– limitationstofiniterepresentation– spaceallocationinmemory:just32or64bitsof1’s&0’s

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Phys124:Lecture8 8

Arrays&StorageinC

• Wecanholdmorethanjustonevalueinavariable– buttheprogramneeds toknowhowmanyplacestosavein

memory• Examples:

– wecaneither sayhowmanyelements toallowandleavethemunset;sayhowmanyelements andinitialize allelements tozero; leaveoutthenumberofelements andspecify explicitly; specifynumberofelements andcontents

– characterarraysarestrings– stringsmustendin‘\0’ tosignaltheend– mustallow room:char name[4]=“Bob”

• fourthelementis‘\0’ bydefault

int i[8], j[8]={0}, k[]={9,8,6,5,4,3,2,1,0};double x[10], y[10000]={0.0}, z[2]={1.0,3.0};char name[20], state[]=“California”;

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IndexingArraysint i,j[8]={0},k[]={2,4,6,8,1,3,5,7};double x[8]={0.0},y[2]={1.0,3.0},z[8];char name[20],state[]="California";

for (i=0; i<8; i++){z[i] = 0.0;printf(”j[%d] = %d, k[%d] = %d\n",i,j[i],i,k[i]);

}name[0]='T';name[1]='o';name[2]='m';name[3] = '\0';printf("%s starts with %c and lives in %s\n",name,name[0],state);

• Indexarrayintegers, startingwithzero• Sometimes initializeinloop(z[] above)• Stringassignmentawkwardoutsideofdeclaration line

– #include <string.h> provides“useful”stringroutines• doneautomatically inArduino,butalsoStringtypemakesmanythingseasier

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MemoryAllocationinArrays• state[]=“California”;→

• name[11]=“Bob”;→

– empty spacesattheendcouldcontainanyrandomgarbage

• int i[] = {9,8,7,6,5,4,3,2};→

– indexingi[8] isoutofbounds,andwilleither causeasegmentation fault(ifwriting),orreturngarbage(ifreading)

C a l i f o r n i a \0

B o b \0

9 8 7 6 5 4 3 2

eachblockis8-bitchar

eachblockis16or32-bitint

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Multi-DimensionalArrays

• Cisarow-majorlanguage:thefirstindexdescribeswhichrow(notcolumn),andarrangedinmemoryrow-by-row– memoryis,afterall, arrangedone-dimensionally

• Havetheoptionoftreatinga2-Darrayas1-D– arr[5] = arr[1][1] = 3

• Canhavearraysof2,3,4,…dimensions

int i,j,arr[2][4];

for (i=0; i<2; i++){for (j=0; j<4; j++){arr[i][j] = 4+j-2*i;

}} 4 5 6 7 2 3 4 5

4 5 6 7

2 3 4 5

0

1

0 1 2 3

i

j

inmemoryspace:

Phys124:Lecture8 12

Arraysandfunctions• Howtopassarraysintoandoutoffunctions?• AnarrayinCisactuallyhandledasa“pointer”

– apointer isadirectiontoaplaceinmemory• Apointertoavariable’saddressisgivenbythe&symbol

– youmayremember thisfromscanf functions• Foranarray,thenameisalreadyanaddress

– because it’sablockofmemory, thenamebyitselfdoesn’t containauniquevalue

– instead, thenamereturns theaddressofthefirstelement– ifwehaveint arr[i][j];

• arr and&arr[0] and&arr[0][0] meanthesamething:theaddressofthefirstelement

• Bypassinganaddresstoafunction,itcanmanipulatethecontentsofmemorydirectly,withouthavingtopassbulkyobjectsbackandforthexplicitly

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Example:3x3matrixmultiplication

void mm3x3(double a[], double b[], double c[])

// Takes two (3x3 matrix) pointers, a, b, stored in 1-d arrays nine// elements long (row major, such that elements 0,1,2 go across a// row, and 0,3,6 go down a column), and multiplies a*b = c.// Dimensionality and encoding of the arguments are assumed.{

double *cptr; // define a pointer variable to doubleint i,j;

cptr = c; // without *, it’s address; point to addr. for c

for (i=0; i<3; i++){for (j=0; j<3; j++){

*cptr++ = a[3*i]*b[j] + a[3*i+1]*b[j+3] + a[3*i+2]*b[j+6];// calc value to stick in current cptr location, then// increment the value for cptr to point to next element// * gets at contents

}}

}

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mm3x3,expanded

• Thefunctionisbasicallydoingthefollowing:

– whichyoucouldconfirmisthepropersetofoperationsformultiplyingout3×3matrices

*cptr++ = a[0]*b[0] + a[1]*b[3] + a[2]*b[6];*cptr++ = a[0]*b[1] + a[1]*b[4] + a[2]*b[7];*cptr++ = a[0]*b[2] + a[1]*b[5] + a[2]*b[8];

*cptr++ = a[3]*b[0] + a[4]*b[3] + a[5]*b[6];*cptr++ = a[3]*b[1] + a[4]*b[4] + a[5]*b[7];*cptr++ = a[3]*b[2] + a[4]*b[5] + a[5]*b[8];

*cptr++ = a[6]*b[0] + a[7]*b[3] + a[8]*b[6];*cptr++ = a[6]*b[1] + a[7]*b[4] + a[8]*b[7];*cptr++ = a[6]*b[2] + a[7]*b[5] + a[8]*b[8];

Phys124:Lecture8 15

Notesonmm3x3• Thefunctionis constructedtodealwith1-dinsteadof2-darrays– 9elementsinsteadof3×3– itcouldhavebeendoneeitherway

• Thereisapointer,*cptrbeingused– byspecifyingcptr asadoublepointer,andassigningitsaddress(justcptr)toc,wecanstockthememorybyusing“pointermath”

– cptr istheaddress;*cptr isthevalueatthataddress– justlike&x_val isanaddress,whilex_val containsthevalue

– cptr++ bumpstheaddressbytheamountappropriatetothatparticulardatatype(doubleprec.here),called“pointermath”

– *cptr++ = value; assignsvalueto*cptr,thenadvancesthecptr count

Phys124:Lecture8 16

Usingmm3x3

• passingjustthenames(addresses)ofthearrays– fillingouta andb, butjustmaking spaceforc– notefunctiondeclarationbeforemain

#include <stdio.h>

void mm3x3(double a[], double b[], double c[]);

int main(){

double a[]={1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0};double b[]={1.0, 2.0, 3.0, 4.0, 5.0, 4.0, 3.0, 2.0, 1.0};double c[9];

mm3x3(a,b,c);

printf("c = %f %f %f\n",c[0],c[1],c[2]);printf(" %f %f %f\n",c[3],c[4],c[5]);printf(" %f %f %f\n",c[6],c[7],c[8]);

return 0;}

Phys124:Lecture8 17

double a[3][3]={{1.0, 2.0, 3.0},{4.0, 5.0, 6.0},{7.0, 8.0, 9.0}};

double b[3][3]={{1.0, 2.0, 3.0},{4.0, 5.0, 4.0},{3.0, 2.0, 1.0}};

double c[3][3];

mm3x3(a,b,c);

Anotherwaytoskinthecat

• Here,wedefinethearraysas2-d,knowingthatinmemorytheywillstillbe1-d– wewillgetcompilerwarnings,butthethingwillstillwork– notarecommended approach,justpresentedherefor

educational purposes– Notethatwecouldreplacea with&a[0][0] inthe

functioncall,andthesame fortheothers,andgetnocompiler errors

Decibels

• Soundismeasuredindecibels,ordB– asaremanyradio-frequency(RF)applications

• Logarithmicscale– commonfeatureisthatevery10dBisafactorof10inpower/intensity

– otherhandymetrics• 3dBis2ו 7dBis5ו obviouslypiling2× and5× is10×,whichis10dB=3dB+7dB

– decibelsthuscombinelikelogarithms:additionrepresentsmultiplicativefactors

Phys124:Lecture8 18

Phys124:Lecture8 19

SoundIntensity

• Soundrequiresenergy(pushingatoms/moleculesthroughadistance),andthereforeapower

• Soundischaracterizedindecibels(dB),accordingto:– soundlevel =10×log(I/I0)=20×log(P/P0)dB– I0 =10−12 W/m2 isthethresholdpowerintensity (0dB)– P0 =2×10−5 N/m2 isthethresholdpressure (0dB)

• atmosphericpressureisabout105 N/m2

• 20 outfrontaccountsforintensitygoinglikeP2

• Examples:– 60dB(conversation) means log(I/I0)=6,soI =10−6W/m2

• andlog(P/P0)=3,soP =2×10−2 N/m2 =0.0000002atmosphere!!– 120dB(painthreshold) means log(I/I0)=12,soI =1W/m2

• andlog(P/P0)=6,soP =20N/m2 =0.0002atmosphere– 10dB(barelydetectable) means log(I/I0)=1,soI =10−11W/m2

• andlog(P/P0)=0.5,soP ≈ 6×10−5 N/m2

Phys124:Lecture8 20

Soundhittingyoureardrum• Pressurevariationsdisplacemembrane(eardrum,microphone)whichcanbeusedtomeasuresound– myspeakingvoiceismovingyoureardrumbyamere1.5×10-4 mm=150nm=1/4wavelengthofvisiblelight!

– thresholdofhearingdetects5×10-8 mmmotion,one-halfthediameterofasingleatom!!!

– painthresholdcorrespondsto0.05mmdisplacement• Earignoreschangesslowerthan20Hz– sothoughpressurechangesevenasyouclimbstairs,itistooslowtoperceiveassound

• Eardrumcan’tbewiggledfasterthanabout20kHz– justliketryingtowiggleresonantsystemtoofastproducesnosignificantmotion

A-weighting:accountforrelativeloudness

Phys124:Lecture8 21

• Relatedtomeasurementofsoundpressurelevel• DefinedinstandardIEC61672:2003,forenvironmentaland

industrialnoise• Accountsforlowsensitivityofhumanearatlowfrequencies,

reducesimportanceoflowfreq.inhumanimpactassessmentsOtherweightings available:B,C,D

Phys124:Lecture8 22http://www.dot.ca.gov/dist2/projects/sixer/loud.pdf

SoundscalesnotforhumansNoiseCriterion(NC)

Phys124:Lecture8 23

dBScales• Intheradio-frequency(RF)world,dBisusedseveralways

– dBisarelative scale:aratio:oftencharacterizingagainorloss• +3dBmeansafactoroftwomore• −17dBmeansafactorof50loss,or2%throughput

– dBm isanabsolute scale,inmilliwatts: 10×log(P/1mW)• 0dBm is1mW,a30dBm signalis1 W• 36dBm is4W(note6dBistwo3dB,eachafactorof2à 4×)• −27dBm is2µW

– dBc issignalstrengthrelative tothecarrier• oftencharacterizesdistortionfromsinusoid• −85dBc meansanydistortionsarealmostnineorders-of-magnitudeweakerthanthemainsinusoidal“carrier”

• Voltageistheequivalentofsoundpressure,anamplitude:10×log 89:;<=

89:;<>= 20×log ?9@ABC;=

?9@ABC;>dB

Phys124:Lecture8 24

CoherentDetection

• Sometimesfightingtodiscernsignalagainstbackgroundnoise– photogate inbrightsetting,forinstance

• Oneapproachiscoherentdetection– modulatesignalatknownphase,inON/OFFpatternat50%dutycycle

– accumulate(add)in-phaseparts,whilesubtractingout-of-phaseparts

– haveintegratorperformaccumulation,ortryinsoftware• butifbackgroundisnoisyinadditiontohigh,integrationbetter

– basicallybackgroundsubtraction– gainmorethegreaterthenumberofcyclesintegrated

Phys124:Lecture8 25

RawSignal,Background,andNoise

Phys124:Lecture8 26

ModulatedSignal;stillhardtodiscern

Phys124:Lecture8 27

Integration,subtracting“OFF”portions

Phys124:Lecture8 28

ExpressedinElectronics

Phys124:Lecture8 29

firstop-ampjustinverting;secondsumstwoinputs,onlyoneonatatimehaseffectofaddingpartswhenRef=+1,subtractingwhereRef=-1

clears“memory”ontimescale ofτint =RintCcouldalsoconceiveofperformingmathinsoftware

Lock-indetection

Phys124:Lecture8 30

https://www.zhinst.com/applications/principles-of-lock-in-detection

Announcements

• ProjectProposalsdueintwoweeksFriday,Nov.3• Lab3duenextweekMon/Tue(10/23,10/24)• Lab4duein2+weeksTue/Wed(11/6,11/7)• MidtermWed.11/8

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