Simulation of Spatially Correlated Large-Scale Parameters and Obtaining Model Parameters from Measurements

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Gwendolyn Casey
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1 Simulation of Spatially Coelated Lage-Scale Paamete and Obtaining Model Paamete fom PER ZETTERBERG Stockholm Septembe 8 TRITA EE 8:49

2 Simulation of Spatially Coelated Lage-Scale Paamete and Obtaining Model Paamete fom Pe Zettebeg Electical Engineeing Royal Intitute of Technology Intoduction In the WINNER deliveable D54 we intoduced the concept of lage cale vaiable [] Thee ae vaiable which decibe aveage popetie of the channel in local aea uch angle-pead, hadow-fading and delay-pead They ae alway metic of the entie channel and neve pe-clute model A local aea i an aea of -3wavelength whee the lage cale paamete can be aumed contant The eaon fo calling them lage-cale paamete i that they change at appoimately the ame ate a the hadow-fading which i ometime called lage-cale fading in contat to mall-cale fading The mall-cale fading (often called fat-fading, Dopple fading o Rayleigh fading i vaying on a ub-wavelength cale Figue illutate how a mobile tavel on a tajectoy and whee the tajectoy i divided into local aea The lage-cale ae almot contant within one local-aea and change between local aea The coelation between the lage cale paamete between two poition (,y and (,y hould be a deceaing function of the Euclidian ditance between the two point Thi i a fom of patial coelation ince (,y and (,y ae epaated in pace Howeve, we note that the tem i uually ued fo mall-cale coelation between the fading of two (o moe antenna on a node (bae-tation/elay/mobile-tation,y,y 3,y3 4,y4 Figue : Illutation of a tajectoy with local aea A the tem indicate, lage-cale paamete i a vecto of andom vaiable Thee vaiable may (in the geneal cae have diffeent ditibution and be coelated (ie the coelation mati at zeo lag i not a diagonal To achieve thee objective the vecto of lage-cale (, paamete i modeled a (, = g, (* ( (,

3 [ T (, = R, y + ( μ ζ whee (, (, K m (, i a vecto of poibly patially coelated N(, independent Gauian pocee in two dimenion, R i a coelation mati, μ i vecto = ] of mean and g [ g (,K, g ( ] = ( m m Q geneal g ( i given by g ( = ( f ( whee f ( of (, y and the Q function i given by Q i a vecto of tanfomation function In t π ( = ep dt i the ditibution function It i common to model the patial coelation of the log-nomal (hadow fading by an eponential function of the Euclidean ditance Baed on thi obevation, we intoduce eponential autocoelation in the element of ie ( + ( y y { ( ( } ( E = =, y, y, y y ep λ The autocoelation of an individual lage-cale paamete (, will then no longe be eponentially (in geneal ditibuted fo two eaon Fit the miing mati R make the autocoelation of each element of the tanfomed vecto of lage cale paamete (, to become a um of eponential, and finally the tanfomation alo change the coelation Howeve, in pactice the eulting autocoelation ae geneally quite cloe to a ingle eponential In Section 5, below we how a method of how to obtain the paamete fom meauement data The meauement ae fom uban maco-cellula meauement [] Geneation of (, uing filteing Below we decibe how to geneate (, uing a filteing appoach Since the element of (, ae independent it uffice to decibe how to geneate one element of (, ay eg element ie (, Thi andom vaiable i Gauian, ha mean zeo and an autocoelation function given by ( + ( y y { ( ( } ( E = =, y, y, y y ep λ The deivation below ae baed on the theoy in [] fo two-dimenional image pocee

4 3 Baic appoach We notice that if we tat with a Gauian andom vaiable η (, which i patially white ie { (, y η( y } = δ ( δ ( y y E η,, and then pa it though a two-dimenional filte h (,, the output of that filte υ (, i given by υ (, = h(, (, y y d d η y and it autocoelation by { υ (, y (, y } (, y y h(, h(, y y y d d υ = ν = + + y E Thu by appopiately chooing h (, we can obtain (, by filteing the white map η (, In ode to elect h (, we note that the powe pectal denity of ν ie R ν ( f, f y i elated to the Fouie tanfom of h (, though Thu by etting R ( f f R ( f, f the Fouie tanfom of tanfom it ie ( f f H ( f, f R = ν, y y ν, y = y we can numeically obtain ( (, h, by fit calculating, and then taking the quae oot of the eult and invee h { } ( I R ( f, f, y = The following matlab-code obtain the filte h (, (called hy in the code in a gid of 3636 point paced λ apat whee λ i mete lambda= %% De-coelation ditance of i ampling_inteval=*lambda %% R: Two-dimenion auto-coelation function %% ize(r = (Nauto,Nauto Nauto=; %% Mut be even =(:Nauto; =(-mean(*ampling_inteval; y=(:nauto'; y=(y-mean(*ampling_inteval; % i ize (Nauto,Nauto % (i,j i the ditance fom, in a gid of point =ab(epmat(,nauto, +j*epmat(y,,nauto; R=ep(-/lambda; F=fft(R; F=(ab(F^*ep(j*angle(F; %% F i baically eal-valued But ince R(, i not oigo it will have a phae coeponding to the offet of oigo to R(, Thi i an offet we want to keep in F theefoe we copy the phaed of F h=eal(ifft(f; N=36;

5 i=((nauto/-(n/+:((nauto/+(n/; %% Select the tap with ma powe P=um(um(ab( h(i,i^; %% Powe of the elected tap Ptot=um(um(ab( h^; %% Powe of all tap (-P/Ptot i a meaue of the tuncation eo hy=h(i,i*inv(qt(p; %% Save eult in hy, compenate fo lo of powe Figue 3: Code fo geneation of the impule epone ( h, Aume now that we want to imulate an aea of ize 3λ 3λ fo a ingle ite We may then ue the impule epone h (, a decibed in the code of below, to obtain (, in a gid of point with paced λ To obtain (, at poition between the gid point, intepolation can be ued Npoint_out=3; Npoint_in=Npoint_out+*ize(hy,; i_gid=conv(hy,andn(npoint_in,npoint_in; %% Two-dimenional convolution %% Cop to emove tanient i_gid=i_gid(ize(hy,+(:npoint_out,ize(hy,+(:npoint_out; Figue 3: Code fo geneating (, in a gid of point fo a ingle ite 4 Tiling The appoach above ha the diadvantage of equiing lage memoy pace fo toing the map (ie the gid of ealization of (, To wok aound thi poblem we utilize cyclic convolution when geneating the gid, ay with length N ie we geneate (, in the point = Δ c, yk = Δ k, whee c, k {, K, N }, and Δ i the gid-pacing uing the cyclic convolution between h(, and a N by N ized mati of independent Gauian andom vaiable η, y The cyclic convolution i defined a ( c k N N h η c, k = c, k = k = ( (, c, k {, N } cc mod N, k k mod N We poition the impule epone in h c, k o that h c, k i non-zeo only fo c, k < N h uch a wa done in the code of Figue 3 fo N h = 36 (ie we have alo tuncated the impule epone Fo the inne point of the mati c, k, ie c, k { N h, K, N N h } (whee N h i the length of the impule epone it i clea that c,k i identical to a nomal linea (ie non-cyclic convolution between h and η Fo non inne point, the cicula convolution i identical to a linea convolution between and η, whee η i a cyclically epeated veion of η Fom thi it follow that a cyclic epetition of t c, k will have the ame autocoelation popetie a the c, k fo offet c c < N N h, < N N h Baed on thi we may epeat o tile t c, k to obtain a gid coveing any ize of an aea With pope pogamming we till only need to toe only one N by N ized mati The only diadvantage with the popoed appoach i that the ame patten will be epeated, albeit typically at ditance fa away The cyclic convolution can advantageouly be implemented with a two-dimenional dicete Fouie tanfom a illutated by the code in Figue 4 Ndft=^8; i_tilde_gid=ifft(fft(hy,ndft,ndft*fft(andn(ndft,ndft;

Figue 4: Illutation of computation of one tile to be epeated ove a modeled evice aea 5 Multiple Site When we have multiple bae-tation ite we in pinciple need multiple ealization of (, which again tat

6 Figue 4: Illutation of computation of one tile to be epeated ove a modeled evice aea 5 Multiple Site When we have multiple bae-tation ite we in pinciple need multiple ealization of (, which again tat to tain ou memoy eouce To e-olve thi poblem we may define a ite-pecific tile fo each ite a n c, k = ( ccn mod N,( k kn mod N, c, k whee c n, k n ae ite pecific offet If the n n of diffeent bae-tation ae paced ufficiently fa apat, the pocedue will enue that the ealization of diffeent baetation in a given point will be independent a deied Again we note that by pope pogamming we till only have to toe one by N ized mati A common appoach when imulating cellula ytem i to tat with a finite evice aea a illutated in Figue 5 and the fold the edge of the imulation aea to obtain an edgele evice aea The eaon fo thi i that mobile and bae-tation nea the edge will othewie eceive le intefeence than thoe in the cente In Figue 5 we would fold bode A againt bode B and bode C againt bode D o a to ceate a Donut haped planet With thi pocedue, the uppe-left bae-tation in Figue 5 would become a neighbo bae-tation to the uppe-ight bae-tation of the ame plot The cicula convolution and tiling concept fit thi model pefect a the ealization on the edge D will be coelated with thoe on the edge C N Figue 5: Illutation of tiling and infinite imulation aea The mak the poition of the bae-tation and the quae mak the tile

7 6 Obtaining Model Paamete fom Meauement Data Below we decibe the tep of obtaining the equied model paamete fom meauement data The data wa collected fom an uban maco-cell at GHz [] The data only allow u to etimate AoD angle-pead, AoA angle-pead, and hadowfading Thu ou ( i in thi cae eponential 6 Chaacteize the ditibution of the individual element of ( In Figue 6 the ditibution of the tee component of ( 3 Ditibution of lage-cale vecto:aod pead Ditibution of lage-cale vecto:aoa pead 45 Ditibution of lage-cale vecto:shadow lo Fequency of occuance 5 Fequency of occuance Fequency of occuance Degee Degee Figue 6: Ditibution of the lage-cale paamete 5 5 Lo facto 6 Find a tanfomation g ( uch that the tanfomed lage cale vecto ( = g( ( i a vecto of Gauian andom vaiable Alo find the invee of ( g The following tanfomation have been identified ( ( (ep( if ep( ( = ep( = =, g log = g < 3 ( ( = =, 3 g 3 3 log 3 othewie Nomal pobability plot of the obtained etimate ae hown in Figue 6 Nomal pobability plot tanfomed lage-cale vecto:aod pead Nomal pobability plot tanfomed lage-cale vecto:aoa pead Nomal pobability plot tanfomed lage-cale vecto:shadow lo Pobability Pobability Pobability Data Data Data Figue 6: Nomal pobability plot of the tanfomed lage-cale paamete The invee of the tanfom ae given by ( = =, g

8 ( ( ln( < = g ( ln 3 = othewie ( 3 = = 3 g Etimate the paamete λ, K,λm Plot the etimated theoetical auto-coelation coefficient function obtained fom the model (* in the plot of analyi item pat 3 The paamete ae tuned by hand The fitted lope ae given by 7m, m and 7m, epectively The fitted cuve ae plotted in Figue 63 Coelation function of tanfomed SL paamete: numbe and numbe Coelation function of tanfomed SL paamete: numbe and numbe Coelation function of tanfomed SL paamete: numbe and numbe Coelation function of tanfomed SL paamete: numbe and numbe Coelation function of tanfomed SL paamete: numbe and numbe Coelation function of tanfomed SL paamete: numbe 3 and numbe Figue 63: Meaued auto-coelation function (geen and fitted eponential cuve (blue 64 Analyi item pat 5: Geneate data and check ome ditibution and co-coelation The following figue how a compaion of the actually meaued and the imulated angle-pead The imulated angle-pead can become negative Thi can be olved with a imple thehold function Ditibution of lage-cale vecto:aoa pead Simulated DoA angle-pead Fequency of occuance Fequency of occuance Degee Angle-pead Figue 64: Meaued and imulated angle-pead in left and ight ubfigue epectively The mati off meaued coelation coefficient fo the untanfomed data i given by

9 c= c= c=3 = = 38 4 = The mati of imulated coelation coefficient of the tanfomed data i given by c= c= c=3 = 4 48 = 4 36 = Refeence [] William K Patt, Digital Image Poceing, John Wiley and Son, ISBN

Histogram Processing

Histogram Processing Hitogam Poceing Lectue 4 (Chapte 3) Hitogam Poceing The hitogam of a digital image with gay level fom to L- i a dicete function h( )=n, whee: i the th gay level n i the numbe of pixel in the image with