Digital comm. systems Project: Simplified digital have...

Project:Simplifieddigitalcommunica5onsystems

WatcharapanSuwansan5suk

EIE/ENE324KingMongkut’sUniversityofTechnologyThonburi

Digitalcomm.systemshavethesemajorcomponents

Sampler

message(inputwaveform)

Quan5zer Sourceencoder

Channelencoder Modulator

Interpolator Tablelookup

Sourcedecoder

Channeldecoder

message(outputwaveform)

Channel

transmiRer

receiver

Demodulator

2

Afocusofthisproject

Theprojectcoversthesimplestcase:thecaseofnochannelencoding

•  isaguessof

3

Channelencoder Modulator

Channeldecoder

GaussianChannel

transmiRer

receiver

Demodulator

X1, X2, X3, . . .

X̂1, X̂2, X̂3, . . .

Xi 2 {�1, 1}where

independentandiden5callydistributed(iid)

X̂i Xi

Thisblockcanbesimplified(seenextslide)

Byusingorthornormalsignalstomodulate,wecansimplifythediagram

•  Randomvariablesappearattheinput/outputofeachblock•  Givenindependence,thedecoderdecoderseachseparately

4

Channelencoder

Channeldecoder

transmiRer

receiver

X1, X2, X3, . . .

X̂1, X̂2, X̂3, . . .

Xi 2 {�1, 1}whereiid

+

X1, X2, X3, . . .

iidGaussianN (0,�2)

Z1, Z2, Z3, . . .

Y1, Y2, Y3, . . .

Yi = Xi + Ziwhere

(given)

(given)

(consequence:Yi’sareiid)

Yi

Youwillsimulatethesimplestformofadigitalcommunica5onsystem

noise

+

transmiRedbit receivedsymbol

X 2 {�1, 1}

XY

Z

Discreterandomvariable(rv)(Bernoullirvwith)

Con5nuousrv,independentofX(Gaussianornormaldistribu5on)Mean=0,variance=

Y = X + Z

�2

Con5nuousrv

Decoder

receivedbitX̂

Discreterv

X̂ 2 {�1, 1}P {X = 1} = p

5

A^erthisproject,youwillbeableto•  Understand the simplest structure of the digitalcommunica5onsystem

•  Generate realiza5ons (random samples) of discrete andcon5nuousrandomvariablesusingMatlab

•  Compare thehistogramof the realiza5ons to theprobabilitydistribu5onoftherandomvariable

•  Designadecisionruleatareceiver•  Simulatethebiterrorrateasafunc5onofthesignal-to-noisera5o

6

Thisexampleshowsrealiza5onsofrvs

•  Supposethedecisionrule(thedecoder)isalwaysoutput

noise

+

transmiRedbit receivedsymbol

Y = X + ZDecoder

decodedbit

X = 1

Z = �0.1

= 0.9

X̂ = 1

X̂ = 1

7

Thisprojectconsistsof4parts

noise

+

transmiRedbit receivedsymbol

XY

Z

Decoder

decodedbitX̂

Part1:TransmiRer

Part2:Channel

Part3:Receivedsymbol Part4:Decodedbit

8

YouwillmodifysixMatlabfilesthatareprovidedtoyouinthecoursewebsite

•  Someofthefilesareusedinmul5plepaths

getBernoulli.m commsys_1_transmit.m

commsys_2_channel.mgetNormal.m

commsys_3_rec_symb.m

commsys_4_detect.m

isusedinPart1

Part2

Part3

Part4

9

PART1:THETRANSMITTER

10

Youwillgeneraterealiza5ons(observedvalues)ofa{-1,+1}-Bernoullirv

•  ThepmfofXis

•  Arealiza5on(observedvalue)ofXhasthevalueofeither+1or-1

•  Inalongrunifyougenerateonerealiza5ona^eranother,thepropor5onof+1thatyouobserveisp

P {X = +1} = p

P {X = �1} = 1� p

11

Usethesevaluesofpinyourexperiment

Group p

5 0.55

6 0.60

7 0.65

8 0.70

12

•  Thevalueofpwillbeassignedtoyouinclass

Group p

1 0.30

2 0.35

3 0.40

4 0.45

Yourtaskisthefollowing1.  FillinthecodeoffilegetBernoulli.m2.  Fill in the code of func5on getRela5veFreq of file

commsys_1_transmit.m3.  Modify the value of p in the first line of func5on

plotBernoulliHistoffilecommsys_1_transmit.m4.  Runcommsys_1_transmit.m,andmakesurethehistogram

and the pmf matches (otherwise your code in 1-3 arewrong)

5.  S u b m i t M a t l a b c o d e g e t B e r n o u l l i . m a n dcommsys_1_transmit.m(so^copy/email)

6.  Submitthehistogram(so^copy/email)

13

Matlabexample:generaterealiza5onsfromauniformdiscreterv

•  Suppose randomvariableW is equally likely to take a valuefromtheset{1,2,3,...,10}

•  Thatis,thepmfofWis

•  Matlabfunc5onunidrnd(10)givesyourealiza5onsofW>>unidrnd(10,1,3)%1-by-3matrixans=9102

P {W = k} =

(110 , k = 1, 2, . . . , 10

0, otherwise

14

TheseareMatlabfunc5onsthatgeneraterealiza5onofcommondiscretervs

Distribu<on Matlabfunc<on

Binomial binornd

Geometric geornd

Poisson poissrnd

Uniform unidrnd

15

Rela5vefrequencyisapropor5onthatasymboloccursinagivensequence

•  Supposethe5realiza5onsofXare+1,-1,-1,+1,-1

•  Therela5vefrequencyof+1is

•  Therela5vefrequencyof-1is

16

# of times that 1 occurs

5

=

2

5

# of times that �1 occurs

5

=

3

5

PART2:THECHANNEL

17

Youwillgeneraterealiza5onsofazero-meanGaussian(normal)rv

•  GaussianrvZhasameanofzeroandthevarianceof•  ThepdfofZis

•  Arealiza5onofZisarealnumber•  Inalongrunifyougenerateonerealiza5ona^eranother,thehistogramoftheserealiza5onslookslikethepdfofY

�2

fZ(z) =1p2⇡�2

e�z2/2, �1 < z < 1

18

Yourtaskisthefollowing1.  Fill in the code of func5on getNormalPDF in the file

commsys_2_channel.m2.  Fillinthecodeoftheonlyfunc5oninfilegetNormal.m3.  Changethevalueofsiginline11ofcommsys_2_channel.m

tobeanyvalueofyourchoice(sig= =standarddevia5onofnoiseZ)

4.  Run commsys_2_channel.m, and make sure that thehistogramand thepdfmatch (otherwise, your code in1-3arewrong)

5.  S u bm i t M a t l a b c o d e g e t N o rm a l P D F .m a n dcommsys_2_channel.m(hardcopy)

6.  Submitthehistogram(hardcopy)

�

19

Matlabexample:generaterealiza5onsfromauniformcon5nuousrv

•  SupposerandomvariableUisuniformontheinterval(0,1)•  Thatis,thepdfofUis

•  Matlabfunc5onrandgivesyourealiza5onsofU>>rand(1,5)%1-by-5matrixans=0.39220.65550.17120.70600.0318

fU (u) =

(1, 0 < u < 1

0, otherwise

20

TheseareMatlabfunc5onsthatgeneraterealiza5onofcommoncon5nuousrvs

Distribu<on Matlabfunc<on

Exponen5al exprnd

NormalorGaussian normrnd

Uniform rand

21

PART3:THERECEIVER(RECEIVEDSYMBOL)

22

YouwillderivethepdfofY•  Therela5onshipiswhereisa{-1,1}-Bernoullirvandisanormalrv,independentof

•  Arealiza5onofYisgeneratedforyouinthetemplate,callingyourfunc5onsgetBernoulli.mandgetNormal.m

•  Realiza5onofY=realiza5onofX+realiza5onofZ

Y = X + Z

X Z N (0,�2)

X

23

Yourtaskisthefollowing1.  Fillinthevalueofsiginline10ofcommsys_3_rec_symb.m,

usingthesamevalueofsiginpart22.  DerivethepdfofYasa func5onofsig.Usethevalueofp

thatwasassignedtoyourgroupforthederiva5on3.  Fillinfunc5ongetPdfYoffilecommsys_3_rec_symb.m4.  Runcommsys_3_rec_symb.m,andmakesurethehistogram

andthepdfmatches(otherwise,yoursteps1-3arewrong)5.  Submitthederiva5onofthepdfinstep2(hardcopy)6.  Submit theMatlab code commsys_3_rec_symb.m and the

histogram(hardcopy)

24

Hereisaderiva5onforthecase.

•  ThecdfofYis

p = 1/2

P {Y y} = P {Y y and X = 1}+ P {Y y and X = �1}

= P {Y y | X = 1}P {X = 1}+ P {Y y | X = �1}P {X = �1}

=1/2 =1/2Y=X+Z

=1

2P {X + Z y | X = 1}+ 1

2P {X + Z y | X = �1}

=1

2P {1 + Z y | X = 1}+ 1

2P {�1 + Z y | X = �1}

independence independence

25

(con5nued)deriva5onofthepdfofY

•  Differen5atethecdfwithrespecttoyandgetthepdf:

P {Y y} =1

2P {1 + Z y}+ 1

2P {�1 + Z y}

=1

2P {Z y � 1}+ 1

2P {Z y + 1}

fY (y) =1

2fZ(y � 1) · d(y � 1)

dy+

1

2fZ(y + 1) · d(y + 1)

dy

fY (y) =1

2fZ(y � 1) +

1

2fZ(y + 1)

=1 =1

26

PART4:THERECEIVER(DECODEDBIT)

27

YouwilldesignadecisionruleandsimulatethecorrespondingBER

•  The decoder “guesses” whether the transmiRed bit is -1 or+1,usingtheinputvalueY

•  isthevalueoftheguess•  Bit error rate (BER) or the bit error probability (BEP) is theprobabilityofmakinganerrorinthedecision:

receivedsymbol(realnumber)

YDecoder

receivedbit

X̂ 2 {�1, 1}

BER = Pn

X 6= X̂o

X̂

28

Yourtaskisthefollowing1.  Set p in line 10 of commsys_4_detect.m to be the value

assignedtoyourgroup2.  Fill-infunc5onfromSNRdB,whichconvertsfromtheSNRin

dBtothestandarddevia5onsigofnoise.Therela5onshipis

3.  Designyourowndecisionruleandfill-inthefunc5ondecide4.  Run commsys_4_detect.m to simulate the BER using the

decisionruleyoudesignedinstep35.  ModifythedecisionrulesothatthesimulatedBERisclose

tothebaseline(themaximumlikelihoodes5mateorMLE)6.  SubmitMatlab code commsys_4_detect.m and the plot of

BER(hardcopies)

SNR (dB) = 10 log10

E�X2

2E {Z2} = 10 log10p

�2

29

Anexampleofanaivedecisionruleistoalwaysguess.

•  ThentheMatlabcodeforfunc5ondecidewillbe% Decode the received symbols% Input:% y - a vector of received symbols (real numbers)% p - the probability that a bit +1 is sent at the % transmitter% sig - the standard deviation of Gaussian noise% Output:% xhat - a vector of +1's and -1's, of the same % size of 'y’. xhat(i) is the decoded bit for % the received symbol y(i)function xhat = decide( y, p, sig ) xhat = ones( size(y) ); end

X̂ = 1

30

ThentheBERwillbeatthedots

31

−10 −5 0 510−4

10−3

10−2

10−1

100

SNR (dB)

BER

Part 4: Bit error rate (BER) at different signal to noise ratios (SNRs)

Simulation (my decision rule)Baseline