How to code the equation below?

I have an equation to code

To sigmaI, I is used, which is a 512*512 matrix. Also I need to calculate BER also. The equation is

The code that I wrote is,

clc;close all;clear all;
lambda = 532; 
k = 2*pi/lambda;
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z_diff = (diff(z)) ;
z_diff = [0 z_diff];
tilt_angle=10;
beta=(pi/180)*tilt_angle; 

Lx = 0.1;            
Ly = Lx;             
Nx = 512;           
Ny = Nx ;            
dx = Lx/Nx;          
dy = Ly/Ny;          

x = (-Nx/2:Nx/2-1)*dx;
y = (-Ny/2:Ny/2-1)*dy;
[X,Y] = meshgrid(x,y);
r = sqrt(X.^2 + Y.^2);


kx = meshgrid(-Nx/2:Nx/2-1)/(Nx.*dx);
ky = meshgrid(-Ny/2:Ny/2-1)/(Ny.*dy);
kz = ones(Nx);

Kx=cos(beta).*kx + sin(beta).*kz;
Ky=ky;
Kz=-sin(beta).*kx + cos(beta).*kz;

K2 = Kx.^2 + Ky.^2 + Kz.^2;

delfx = 1/(Lx);
delfy = delfx;
R = randn(Nx,Ny) ;     
R = R - mean(R(:,:));
R = R/std(R(:,:));   
Phi = 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w0=0.002;
F=2*(pi^2)*(k^2).*z_diff.*Phi;
x0 = linspace(-0.02,0.02,Nx);y0=x0;[X0,Y0] = meshgrid(x0,y0);
Uin=exp(-(X0.^2+Y0.^2)/w0^2);
x = x + z_diff(1)*tan(beta);
screen1 = exp(1i*((Kx.*x) + (Ky.*y))).*delfx.*delfy.*sqrt(F(1)).*R ; 
Uout = Uin.*exp(1i.*screen1);
c=[0.740664357812470	0.715162315217582	0.689780546528124	0.664556687838984	0.639501229779350	0.614618312979130	0.589909677254809	0.565375929219662	0.541017064806704	0.516832720551119	0.492822307973243	0.468985091286926	0.445320235117717	0.421826835235231	0.398503939126698	0.375350560217132	0.352365687964806	0.329548295192595	0.306897343515746	0.284411787427262	0.262090577416753	0.239932662380474	0.217936991503006	0.196102515739251	0.174428188990141	0.152912969040777	0.131555818312352	0.110355704466664	0.0893116008928944	0.0684224870996102	0.0476873490298864	0.0271051793136574	0.00667497746850089	0.0136042499421828	0.0337334891863653	0.0537137192700918	0.0735459118688980	0.0932310312800026	0.112770034392025	0.132163870669524	0.151413482150099	0.170519803452171	0.189483761791847	0.208306277007520	0.226988261591059	0.245530620724636	0.263934252322323	0.282200047075789	0.300328888503445	0.318321653002534	0.336179209903699	0.353902421527639	0.371492143243500	0.388949223528713	0.406274504030009	0.423468819625392	0.440532998486862	0.457467862143721	0.474274225546305	0.490952897130006	0.507504678879476	0.523930366392895	0.540230748946220	0.556406609557339	0.572458725050048	0.588387866117797	0.604194797387153	0.619880277480917	0.635445059080876	0.650889888990134	0.666215508194993	0.681422651926368	0.696512049720699	0.711484425480339	0.726340497533411	0.741080978693105	0.755706576316408	0.770217992362264	0.784615923449129	0.798901060911951	0.813074090858528	0.827135694225279	0.841086546832386	0.854927319438336	0.868658677793846	0.882281282695167	0.895795790036781	0.909202850863477	0.922503111421817	0.935697213210995	0.948785793033077	0.961769483042646	0.974648910795838	0.987424699298783	1.00009746705545	1.01266782811491	1.02513639211797	1.03750376434332	1.04977054575300	1.06193733303735	1.07400471865938	1.08597329089862	1.09784363389434	1.10961632768826	1.12129194826677	1.13287106760249	1.14435425369543	1.15574207061353	1.16703507853271	1.17823383377643	1.18933888885472	1.20035079250267	1.21127008971854	1.22209732180122	1.23283302638735	1.24347773748787	1.25403198552414	1.26449629736359	1.27487119635488	1.28515720236266	1.29535483180181	1.30546459767133	1.31548700958769	1.32542257381781	1.33527179331163	1.34503516773420	1.35471319349739	1.36430636379121	1.37381516861470	1.38324009480640	1.39258162607448	1.40184024302645	1.41101642319850	1.42011064108442	1.42912336816425	1.43805507293243	1.44690622092571	1.45567727475060	1.46436869411057	1.47298093583279	1.48151445389465	1.48996969944983	1.49834712085410	1.50664716369079	1.51487027079592	1.52301688228302	1.53108743556759	1.53908236539135	1.54700210384610	1.55484708039729	1.56261772190731	1.57031445265850	1.57793769437581	1.58548786624928	1.59296538495611	1.60037066468257	1.60770411714556	1.61496615161393	1.62215717492952	1.62927759152798	1.63632780345927	1.64330821040796	1.65021920971321	1.65706119638862	1.66383456314170	1.67053970039319	1.67717699629614	1.68374683675467	1.69024960544263	1.69668568382193	1.70305545116068	1.70935928455115	1.71559755892740	1.72177064708287	1.72787891968756	1.73392274530517	1.73990249040993	1.74581851940328	1.75167119463032	1.75746087639608	1.76318792298160	1.76885269065981	1.77445553371120	1.77999680443936	1.78547685318630	1.79089602834756	1.79625467638721	1.80155314185264	1.80679176738913	1.81197089375438	1.81709085983270	1.82215200264920	1.82715465738367	1.83209915738443	1.83698583418192	1.84181501750219	1.84658703528021	1.85130221367304	1.85596087707287	1.86056334811985	1.86510994771484	1.86960099503199	1.87403680753115	1.87841770097023	1.88274398941727	1.88701598526258	1.89123399923051	1.89539834039132	1.89950931617273	1.90356723237146	1.90757239316461	1.91152510112087	1.91542565721170	1.91927436082231	1.92307150976254	1.92681740027765	1.93051232705897	1.93415658325440	1.93775046047891	1.94129424882478	1.94478823687186	1.94823271169764	1.95162795888723	1.95497426254328	1.95827190529575	1.96152116831153	1.96472233130413	1.96787567254306	1.97098146886325	1.97403999567435	1.97705152696986	1.98001633533631	1.98293469196218	1.98580686664685	1.98863312780943	1.99141374249746	1.99414897639556	1.99683909383401	1.99948435779718	2.00208502993198	2.00464137055607	2.00715363866619	2.00962209194621	2.01204698677524	2.01442857823560	2.01676712012071	2.01906286494297	2.02131606394144	2.02352696708957	2.02569582310279	2.02782287944605	2.02990838234126	2.03195257677470	2.03395570650435	2.03591801406711	2.03783974078601	2.03972112677735	2.04156241095768	2.04336383105087	2.04512562359498	2.04684802394913	2.04853126630031	2.05017558367013	2.05178120792145	2.05334836976503	2.05487729876606	2.05636822335072	2.05782137081251	2.05923696731876	2.06061523791685	2.06195640654055	2.06326069601621	2.06452832806891	2.06575952332861	2.06695450133616	2.06811348054936	2.06923667834885	2.07032431104407	2.07137659387907	2.07239374103836	2.07337596565262	2.07432347980443	2.07523649453394	2.07611521984447	2.07695986470806	2.07777063707106	2.07854774385950	2.07929139098462	2.08000178334822	2.08067912484798	2.08132361838281	2.08193546585808	2.08251486819086	2.08306202531507	2.08357713618662	2.08406039878854	2.08451201013601	2.08493216628138	2.08532106231914	2.08567889239087	2.08600584969017	2.08630212646748	2.08656791403493	2.08680340277114	2.08700878212594	2.08718424062516	2.08732996587524	2.08744614456794	2.08753296248490	2.08759060450230	2.08761925459532	2.08761909584274	2.08759031043135	2.08753307966046	2.08744758394630	2.08733400282638	2.08719251496389	2.08702329815202	2.08682652931821	2.08660238452849	2.08635103899165	2.08607266706347	2.08576744225093	2.08543553721630	2.08507712378131	2.08469237293121	2.08428145481885	2.08384453876873	2.08338179328099	2.08289338603538	2.08237948389527	2.08184025291152	2.08127585832643	2.08068646457760	2.08007223530177	2.07943333333869	2.07876992073487	2.07808215874743	2.07737020784777	2.07663422772538	2.07587437729149	2.07509081468279	2.07428369726507	2.07345318163688	2.07259942363312	2.07172257832864	2.07082280004183	2.06990024233815	2.06895505803365	2.06798739919850	2.06699741716044	2.06598526250827	2.06495108509530	2.06389503404273	2.06281725774310	2.06171790386362	2.06059711934957	2.05945505042762	2.05829184260918	2.05710764069365	2.05590258877174	2.05467683022873	2.05343050774771	2.05216376331278	2.05087673821232	2.04956957304210	2.04824240770851	2.04689538143169	2.04552863274867	2.04414229951646	2.04273651891521	2.04131142745123	2.03986716096009	2.03840385460964	2.03692164290309	2.03542065968197	2.03390103812916	2.03236291077188	2.03080640948460	2.02923166549207	2.02763880937220	2.02602797105897	2.02439927984538	2.02275286438630	2.02108885270135	2.01940737217775	2.01770854957318	2.01599251101859	2.01425938202100	2.01250928746633	2.01074235162217	2.00895869814052	2.00715845006058	2.00534172981146	2.00350865921493	2.00165935948813	1.99979395124624	1.99791255450517	1.99601528868428	1.99410227260896	1.99217362451332	1.99022946204283	1.98826990225688	1.98629506163147	1.98430505606170	1.98230000086442	1.98028001078079	1.97824519997880	1.97619568205582	1.97413157004115	1.97205297639850	1.96996001302854	1.96785279127132	1.96573142190885	1.96359601516744	1.96144668072028	1.95928352768979	1.95710666465011	1.95491619962948	1.95271224011267	1.95049489304337	1.94826426482658	1.94602046133099	1.94376358789132	1.94149374931069	1.93921104986296	1.93691559329504	1.93460748282926	1.93228682116561	1.92995371048410	1.92760825244698	1.92525054820109	1.92288069838009	1.92049880310668	1.91810496199492	1.91569927415240	1.91328183818251	1.91085275218661	1.90841211376627	1.90596002002548	1.90349656757275	1.90102185252339	1.89853597050162	1.89603901664270	1.89353108559514	1.89101227152279	1.88848266810698	1.88594236854864	1.88339146557039	1.88083005141866	1.87825821786576	1.87567605621199	1.87308365728765	1.87048111145519	1.86786850861116	1.86524593818833	1.86261348915771	1.85997125003053	1.85731930886031	1.85465775324484	1.85198667032819	1.84930614680269	1.84661626891091	1.84391712244766	1.84120879276190	1.83849136475875	1.83576492290140	1.83302955121309	1.83028533327899	1.82753235224814	1.82477069083540	1.82200043132329	1.81922165556395	1.81643444498099	1.81363888057138	1.81083504290734	1.80802301213817	1.80520286799215	1.80237468977836	1.79953855638852	1.79669454629885	1.79384273757188	1.79098320785825	1.78811603439854	1.78524129402510	1.78235906316378	1.77946941783578	1.77657243365938	1.77366818585177	1.77075674923075	1.76783819821655	1.76491260683352	1.76198004871193	1.75904059708966	1.75609432481398	1.75314130434320	1.75018160774846	1.74721530671537	1.74424247254577	1.74126317615938	1.73827748809551	1.73528547851471	1.73228721720050	1.72928277356095	1.72627221663041	1.72325561507115	1.72023303717496	1.71720455086485	1.71417022369661	1.71113012286052	1.70808431518289	1.70503286712770	1.70197584479823	1.69891331393861	1.69584533993546	1.69277198781941	1.68969332226677	1.68660940760100	1.68352030779434	1.68042608646938	1.67732680690054	1.67422253201569	1.67111332439765	1.66799924628574	1.66488035957729	1.66175672582918	1.65862840625934	1.65549546174827	1.65235795284053	1.64921593974624	1.64606948234258	1.64291864017527	1.63976347246002	1.63660403808406	1.63344039560754	1.63027260326505	1.62710071896701	1.62392480030119	1.62074490453409	1.61756108861240	1.61437340916446	1.61118192250163	1.60798668461974	1.60478775120050	1.60158517761291	1.59837901891464	1.59516932985347	1.59195616486864	1.58873957809224	1.58551962335059	1.58229635416566	1.57906982375634	1.57584008503991	1.57260719063331	1.56937119285456	1.56613214372404	1.56289009496590	1.55964509800932	1.55639720398990	1.55314646375097	1.54989292784488	1.54663664653434	1.54337766979374	1.54011604731041	1.53685182848594	1.53358506243751	1.53031579799910	1.52704408372284	1.52376996788025	1.52049349846351	1.51721472318676	1.51393368948733	1.51065044452699	1.50736503519323	1.50407750810048	1.50078790959138	1.49749628573799	1.49420268234301	1.49090714494105	1.48760971879982	1.48431044892133	1.48100938004314	1.47770655663956	1.47440202292281	1.47109582284426	1.46778800009561	1.46447859811008	1.46116766006355	1.45785522887583	1.45454134721172	1.45122605748225	1.44790940184585	1.44459142220945	1.44127216022968	1.43795165731401	1.43462995462187	1.43130709306582	1.42798311331269	1.42465805578467	1.42133196066046	1.41800486787641	1.41467681712758	1.41134784786894	1.40801799931637	1.40468731044786	1.40135582000454	1.39802356649183	1.39469058818047	1.39135692310765	1.38802260907807	1.38468768366504	1.38135218421152	1.37801614783119	1.37467961140955	1.37134261160493	1.36800518484960	1.36466736735075	1.36132919509161	1.35799070383243	1.35465192911156	1.35131290624645	1.34797367033472	1.34463425625514	1.34129469866868	1.33795503201952	1.33461529053606	1.33127550823193	1.32793571890697	1.32459595614830	1.32125625333121	1.31791664362027	1.31457715997020	1.31123783512696	1.30789870162865	1.30455979180654	1.30122113778599	1.29788277148748	1.29454472462753	1.29120702871964	1.28786971507532	1.28453281480497	1.28119635881886	1.27786037782809	1.27452490234548	1.27118996268656	1.26785558897046	1.26452181112088	1.26118865886698	1.25785616174431	1.25452434909574	1.25119325007237	1.24786289363441	1.24453330855215	1.24120452340678	1.23787656659136	1.23454946631169	1.23122325058718	1.22789794725176	1.22457358395477	1.22125018816181	1.21792778715567	1.21460640803712	1.21128607772588	1.20796682296139	1.20464867030376	1.20133164613454	1.19801577665766	1.19470108790021	1.19138760571334	1.18807535577308	1.18476436358118	1.18145465446596	1.17814625358312	1.17483918591663	1.17153347627945	1.16822914931448	1.16492622949528	1.16162474112694	1.15832470834688	1.15502615512566	1.15172910526776	1.14843358241246	1.14513961003454	1.14184721144515	1.13855640979258	1.13526722806305	1.13197968908148	1.12869381551230	1.12540962986023	1.12212715447104	1.11884641153231	1.11556742307426	1.11229021097045	1.10901479693858	1.10574120254124	1.10246944918668	1.09919955812955	1.09593155047166	1.09266544716271	1.08940126900108	1.08613903663452	1.08287877056091	1.07962049112901	1.07636421853917	1.07310997284407	1.06985777394945	1.06660764161482	1.06335959545421	1.06011365493683	1.05686983938786	1.05362816798911	1.05038865977972	1.04715133365691	1.04391620837666	1.04068330255439	1.03745263466568	1.03422422304698	1.03099808589625	1.02777424127369	1.02455270710241	1.02133350116911	1.01811664112476	1.01490214448529	1.01169002863224	1.00848031081345	1.00527300814371	1.00206813760545	0.998865716049361	0.995665760195108	0.992468286631939	0.989273311819364	0.986080852087796	0.982890923639205	0.979703542547760	0.976518724760467	0.973336486097822	0.970156842254431	0.966979808799657	0.963805401178249	0.960633634710965	0.957464524595207	0.954298085905637	0.951134333594803	0.947973282493756	0.944814947312659	0.941659342641416	0.938506482950264	0.935356382590394	0.932209055794548	0.929064516677626	0.925922779237284	0.922783857354535	0.919647764794513	0.916514553007848	0.913626501586297	0.957181910437377	1.18507611517074];
Ri=z_diff./cos(beta);

screens = zeros(Nx,Nx,length(F)-1);
for i = 2:length(F)
        x = x + z_diff(i)*tan(beta);
        screens(:,:,i)= screens(:, :, i-1) + exp(1i*((Kx.*x)+(Ky.*y))).*delfx.*delfy.*sqrt(F(i)).*R;
        Hi = exp(1i*Kz*Ri(i) - c(i).*Ri(i)); 
        Ui = fftshift(fft2(Uout));     
        Ui = Hi.*Ui;                
        Ui = ifft2(ifftshift(Ui));  
        Uio = Ui.*exp(1i.*screens(:,:,i)); 
        I = Uio.*conj(Uio);
        Uout = Uio; 
end


% to calculate SigmaI
A1=0;A2=0;
for i1=1:Nx
    for i2=1:Nx
        I1 =   I(i1,i2)^2 ;
        A1 = A1 + I1    ; 
    end
end
A1

A1 = 3.2006e-257

for j1=1:Nx
    for j2=1:Nx
        I2 =  I(j1,j2)  ;
        A2 = A2 + I2;
    end
end
SI =  abs(((A1/Nx)/((A2/Nx)^2)) - 1) 

SI = 0.7497

%to calculate BER
syms u
PI = (1/(u*sqrt(2*pi*SI)))*exp(-((log(u) + SI )^2)/(2*SI)) ;
SNR0 = 1;
SNR = SNR0/sqrt(1+SI*SNR0) ;
fun = PI*erfc(real((SNR.*u)/(2*sqrt(2))));
ff = matlabFunction(fun,'Vars',u);
BER =  (0.5*integral(ff,0,Inf))

BER = 0.4032

I need to whether the codes for SigmaI and BER is correct.

Answer 1

[William Rose]

@Athira T Das,

I haven;t looked at the code, but I think the first equation is incorrect. You write

which appears to be missing a factor of 1/M in the fraction.