SUMMER INTERNSHIP 2015

Multi User MIMO channel Submitted by

AMIT KUMAR

Under the guidance of

DR. PRASANT KUMAR SAHU

Department

Of

  School of Electrical Sciences

Indian Institute of Technology, Bhubaneswar

DEPARTMENT OF ELECTRICAL SCIENCES

IIT BHUBANESWAR

ACCEPTANCE CERTIFICATE

The Summer Internship Project entitled “MULTI user MIMO” by Amit Kumar for the purpose of Internship Certificate.

Dated: 22nd June 2015

Signature ______________

Name: Dr. Prasant Kumar Sahu

DECLARATION

I am aware of the Institute’s policy on plagiarism and I certify that this piece of work is my own with all sources fully acknowledged.

______________

Name: Amit Kumar

Acknowledgment

Myself AMIT KUMAR summer intern of Multiuser MIMO is over whelmed in all humbleness and gratefulness to acknowledge my sincere thanks to all those who have helped me to put my ideas and assigned work, well above the level of simplicity and into something concrete.

I thank whole heartedly DR. PRASANT KUMAR SAHU for selecting me a part of his valuable project, constantly motivating for doing better and showing complete confidence in my work.

I am very thankful to my mentors Dr . Bijayananda Patnaik and Mr. Himansu Shekhar Pradhan for their valuable help. They were always there to show me the right track when I was in need. With help of their valuable suggestions, guidance and encouragement, I was able to complete my tasks properly and with satisfaction. Also in the process, I learnt a lot other technical and non-technical things from them and I consider myself to be very fortunate to have such mentors.

ABSTRACT The channel modelling is an important part of mobile radio communication. The channel simulators help us to study how real world mobile communication systems behave in effective and efficient manner.

In this project I have discussed in detail about MIMO channel, MIMO channel modelling and I have implemented MIMO concept on optical fibre.My first part of project is mainly focused on modelling of MIMO channel, where statistical properties of a MIMO channel is discussed mainly using one ring geometrical model and PAS models. In one ring geometrical model isotropic scattering condition is assumed [4], high accuracy of the simulation model is demonstrated by comparing its statistical properties with those of the underlying reference model.Since in MIMO system signal propagates over multipath so at the receiver side delayed version of same signal is received therefore study of spatial correlation between signals is very important. In fact, a mathematical analysis for spatial correlation requires a distribution of PAS for the real environments channel [2] as indoor, outdoor, microcell, macro cell. Therefore, various PAS models are discussed and their results are simulated using MATLAB.In the second part of this project MIMO channel implemented on optical fibre using wavelength division multiplexing technique and to increase Q value of output signal at 180km distance, equalizer is used .Here back propagation algorithm is used to realise equalizer.

INTRODUCTION

MIMO is sending and receiving multiple signals simultaneously by using array of antennas at both transmitter and receiver sides.2×2MIMO means system can support 2 channel and 2 antennas it have. By using spatial diversity technique MIMO systems increase data rate in limited available bandwidth as well increases channel capacity.

MIMO system typically consists of M T transmit and M R receive antennas by using same channel, every antenna receives not only direct component intended for it but also indirect component intended for other antennas. The direct component from antenna 1 to antenna 1 is specified with h 1 1 and indirect connection from antenna 1 to 2 is defined as cross component h 2 1.

The figure below shows the fundamental concept of MIMO system [1].

Output of the system can be expressed as [3]

y(τ) = H(τ) ∗ s(τ) + n(τ)

Where H (τ) is the channel impulse response and s (τ) is the transmitted signal y (τ) is the received signal and n ( τ) is the additive white Gaussian noise ∗ denotes convolution.If bandwidth is narrow so that channel can be treated as approximately constant over frequency (flat

channel) then corresponding input-output relationship simplifies to [3] y = Hs + n

In many cases, the elements of narrowband MIMO channel matrix are assumed [3] to be independent and identically distributed (IID) when studying the MIMO channel capacity. In reality, however, due to insufficient spacing between antenna elements and limited scattering in the environment, the fading is not always independent causing a lower MIMO channel capacity compared to the ideal, IID case. Therefore the proposed MIMO channel models should take this effect into account.

The data to be transmitted is divided into independent data streams M. The number of data elements is always less than or equal to number of antennas.

When the individual data streams are assigned to various users then it is called multi user MIMO.

CHANNEL MODELLING

Modelling a channel is calculating all physical processing which are affecting a signal from transmitter to the receiver. Initially some basic faded channel modelling is described later on MIMO channel modelling is studied.

Mainly, most of the channel model simulators are implemented based on the sum-of-sinusoids principle which is first introduced by Rice and then developed later. Starting from the idea that each signal can be expressed under sum-of-sinusoids, we assumed that any model type is able to be implemented following the sum-of-sinusoids principle. In mobile channel modelling, we begin from fundamental channel models which are Rayleigh and Rice models. Thus, these two latter are the result of the sum of two Gaussian processes. Here, the Gaussian process is structured as follow: for reference model N is infinite. [6]

For simulation model N is finite number of harmonic functions [6]

When using the Rice method, we assume that numbers of sinusoids are infinite. But, in real-world we consider N as small as possible aiming to let the implementation of the simulation realizable, LPNM calculation method gives N as 25 most suitable for simulation model [1]

We compare the simulation model with the reference model. Thus, we try to minimize the error modelling as much as possible to achieve best match. These relationships will accompany for all

channel models. The simulation models are based on the principle of deterministic channel modelling. The principle of deterministic channel modelling consists of the following steps to proceed [4]Step1: Starting point reference model. Based on one or several Gaussian processes, each with prescribe autocorrelation function.Step2: Derive a stochastic simulation model from the reference model by replacing the Gaussian process by sum-of sinusoid with fixed gain, fixed frequency, and random phases.Step3: Determine a deterministic simulation model by fixing all model parameters of the stochastic simulation model including the phases.Step4: Compute the model parameters of the simulation model by fitting the relevant statistical properties of the deterministic (or stochastic) simulation model to those of reference model.Step5: Perform the simulation of one (or some few) sample functions

Pictorially steps can be shown as [7]

MIMO channel modelling

simulation modelDeterministic SOS

3.

sample functionSimulation of

5.

simulation model

Stochastic SOS 2.

Reference model 1.

ComputationParameter

4.

One sample function Infinite no of samples Infinite no of samples ≈10) iNFinite complexity ( ≈10) iN∞) Finite complexity ( iNInfinite complexity (

>.< Time avarage E{.} E{.}

Fixed parameters

Channel modelling is process and method of designing MIMO system to address the issues, problems relevant to the system, and provide analysis to enhance and develop the system by simulating.Multipath is vital characteristic of data transmission in wireless communication systems. Wireless channel affects the signal to travel in multipath between transmitter and receiver. It causes different impairment to transmitted signal. Receiver gets the reflection of same symbols in delay versions. Delays which is also called as fading occurs due to reflection, refractions, diffractions, shadowing etc. due to the buildings, trees, aircrafts, humidity, temperature etc. Delay or fading could be in form of changing phase or magnitude of signals. Multipath affects and delayed symbols reduce the channel efficiency, throughput and causes corrupted information at receiver.

Typical mobile radio propagation scenario [7]

One of the techniques deployed to counter act the multipath fading is MIMO to retrieve the strongest signal from the channel.

In order to model MIMO following factors are taken into consideration to get maximum required results [8]

• Free space loss and path loss • Trees, building which cause Shadowing • For mobile environment Doppler shift and delay spread due to

multi path

• Joint correlation of antenna at sending and receive end • Channel matrix singular value distribution

Modelling of MIMO channel can be done by using generalized principles of deterministic channel modelling in which.

>>starting point is a geometrical model with an infinite number of scatterers e.g., one ring model >>from geometrical model reference model is derived where number of scatterers are infinite>>by taking finite number of scatterers and fixing all model parameters of stochastic simulation model, deterministic simulation model is obtained

GEOMETRICAL ONE RING MODEL

Assumptions: [4]1 .All scatters are located on the same ring around the Mobile station.2. Base station is elevated.3. No line of sight is there between mobile station and base station.

For convenience, the BS and the MS play the roles of the transmitter and receiver, respectively.

FIGURE: geometrical one ring model [4]

Where;

δbs: Antenna element spacing at base stationδms : Antenna element spacing at mobile stationαbs : Multi-element antenna tilt angle at base stationαms : Multi-element antenna tilt angle at mobile stationαv : Angle of motionnb s : Angle of departure at base stationnms : Angle of arrival at mobile stationD: The distance between the base station and the mobile stationR: Radius of the ring of scatters around the mobile station

If the number of scatterers N approaches to ∞ then discrete AOA nms becomes continuous random variables with a given distribution p(ms),e.g. The uniform distribution, the Von Mises distribution, the Laplacian distribution.Since at base station and at mobile station 2 element antenna is used therefore channel matrix can be given as [1]

H(t)=[h i j (t)]= [h 11(t ) h12(t )h 21(t) h22(t )]

The matrix elements are the channel gains for every antenna element at the base station towards every antenna element at the mobile station and mathematically they can be written as [1]

h11 (t) = limN→∞

1√ N ∑

n=1

N

an bn ej (2∏ f n t +θn)

h12 (t) = limN→∞

1√ N ∑

n=1

N

an¿ bnej(2∏ f n t+θn)

h21 (t) = limN→∞

1√ N ∑

n=1

N

an bn¿ej(2∏ f n t+θn)

h22 (t) = limN→∞

1√ N ∑

n=1

N

an¿ bn¿ej(2∏ f n t+θn)

Where phases θn are i.i.d Rvs.

A measure for the correlation between the channel components h11 (t) and h22 (t) is the so-called space-time CCF, which is defined as [1]

ρ11 ,22(δBS , δ MS , τ )=E {h11( t )h22¿ ( t +τ )};

= limN →∞

1N ∑

n=1

N

an2 bn

2 e− j2 Πf nτ

=∫−π

π

an2 (δ BS )bn

2( δMS )e− j2 πf n τ

p(ϕ MS)dφMS

PAS model

PAS is an important factor in determining the spatial correlation between antenna elements.For different channel environment there are various PAS model [2]-1. n t h power of cosine pas model- here PAS is expressed in n t h power of cosine function,2.uniform PAS model is useful in rich scattering environment such as indoor , 3 .The Truncated Laplacian PAS model is commonly employed for macro cell or microcell environments.

MATLAB CODE & SIMULATION RESULTS1. PERFORMANCE EVALUATION-In this section, we assess the accuracy of the proposed stochastic simulation model by comparing its statistical properties with those of the reference model.1. Comparison of time ACFThe transmit correlation function of the reference model and the corresponding correlation function of the stochastic simulation model are presented in first simulation result. Notice that the presented results not only confirm the theory but they visualize as well that the approximationTime ACF (simulation) ≈ Time ACF (reference) is excellent in the range of 0 to tau (max) = 0.08

MATLAB code:1. Comparison of Time ACF of reference model and simulation modelN=25;f_max=91;k=10;phi_0=pi/ (2*N);alphav=pi/12;tau=linspace(0,0.1,200);phiMS=linspace(-pi,pi,100);von_mises=exp(k*cos(phiMS-phi_0))/(2*pi*besseli(0,k));ACF_reference= zeros(200);ACF_simulation= zeros(200);phi=zeros(N,1);for n=1:Nphi(n)=(1/N)*2*pi*n-0.5*pi/N;endfor k=1:length(tau)ACF_reference(k)=trapz(phiMS,exp(-sqrt(-1)*2*pi*f_max*cos(phiMS-….alphav).*tau(k)).*von_mises);ACF_simulation(k)=sum(exp(-sqrt(-1).*2.*pi.*f_max.*cos(phi-…alphav).*tau(k)))/length(phi);endplot(tau,ACF_reference, 'r ',tau,ACF_simulation, '--')legend ( 'ACF reference' , 'ACF simulation')ylabel( 'time autocorrelation function' )xlabel( 'time seperation')

2.2D SPACE CCF of reference modelN=25;k=10;mu_0=pi/(2*N);f_max=91;t=1;tau=linspace(0,N/182,30);mu_0=pi/(2*N);phiBS_max=pi/90;alphaBS=pi/2;alphaMS=pi/2;deltaBS=linspace(0,30,30);deltaMS=linspace(0,3,30);phiMS=linspace(-pi,pi,25);phi_NMS=zeros(N,1);von_mises=exp(k*cos(alphaMS-mu_0))/(2*pi*besseli(0,k));

CCF_reference=zeros(length(deltaBS),length(deltaMS));for n=1:Nphi_NMS(n)=(1/N)*2*pi*n-0.5*pi/N;endfor q=1:length(deltaBS)for g=1:length(deltaMS)a=exp(sqrt(-1).*pi.*deltaBS(q).*(cos(alphaBS)+phiBS_max.*sin(alphaBS).*sin(phi_NMS)));b=exp(sqrt(-1).*pi.*deltaMS(g).*cos(phi_NMS-alphaMS));CCF_reference(q,g)=trapz(phiMS,a.^2.*b.^2.*von_mises);endendsurf(real(CCF_reference));set(gca, 'YDir', 'reverse');title( '2D space CCF OF reference model' )ylabel( 'deltaMS/lambda')xlabel( 'deltaBS/lambda')zlabel( '2D space CCF')

3.2D SPACE CCF of Simulation modelN=25;k=10;f_max=91;t=1;tau=linspace(0,N/(2*f_max),30);mu_0=pi/(2*N);phiBS_max=pi/90;alphaBS=pi/2;alphaMS=pi/2;deltaBS=linspace(0,25,30);deltaMS=linspace(0,3,30);phiMS=linspace(-pi,pi,25);phi_NMS=zeros(N,1);von_mises=exp(k*cos(alphaMS-mu_0))/(2*pi*besseli(0,k));for n=1:Nphi_NMS(n)=(1/N)*2*pi*n-0.5*pi/N;endCCF_simulation=zeros(length(deltaBS),length(deltaMS));for q=1:length(deltaBS)for g=1:length(deltaMS)

aa=exp(sqrt(-1).*pi.*deltaBS(q).*(cos(alphaBS)+phiBS_max.*sin(alphaBS).*sin(phi_NMS)));bb=exp(sqrt(-1).*pi.*deltaMS(g).*cos(phi_NMS-alphaMS));CCF_simulation(q,g)=sum(aa.^2.*bb.^2)/length(phi_NMS);endendsurf(real(CCF_simulation));set(gca,'YDir','reverse');title('2D space CCF OF simulation model')ylabel('deltaMS/lambda')xlabel('deltaBS/lambda')zlabel('2D space CCF')

4.Simulation of mimo channels :channel capacityN=200;SNR=10^(17/10);alphaBS=pi/2;alphaMS=pi/2;f_max=91;alphaV=pi/12;phiBS_max=pi/90;tau=linspace(0,0.5,200);theta= 0 + (2*pi-0).*rand(length(tau),1);phi_MS=zeros(N,1);a=zeros(N,1);b=zeros(N,1);f=zeros(N,1);h11=zeros(length(tau),1);h12=zeros(length(tau),1);h21=zeros(length(tau),1);h22=zeros(length(tau),1);X(n)=zeros(length(N),1);Y(n)=zeros(length(N),1);Z(n)=zeros(length(N),1);V(n)=zeros(length(N),1);cap11=zeros(N,1);for n=1:Nphi_MS(n)=(1/N)*2*pi*n-0.5*pi/N;endfor n=1:Na(n)=exp(sqrt(-1)*pi*0.5*(cos(alphaBS)+phiBS_max*sin(alphaBS)*sin(phi_MS(n))));

b(n)=exp(sqrt(-1)*pi*0.5*cos(phi_MS(n)-alphaMS));f(n)=f_max*cos(phi_MS(n)-alphaV);endfor n=1:N X(n)=a(n).*b(n); Y(n)=a(n)'.*b(n); Z(n)=a(n).*b(n)'; V(n)=a(n)'.*b(n)';endfor t=1:length(tau)h11(t)=sum(X*exp(sqrt(-1)*(2*pi.*f*t+theta)))/sqrt(N);h12(t)=sum(Y*exp(sqrt(-1)*(2*pi.*f*t+theta)))/sqrt(N);h21(t)=sum(Z*exp(sqrt(-1)*(2*pi.*f*t+theta)))/sqrt(N);h22(t)=sum(V*exp(sqrt(-1)*(2*pi.*f*t+theta)))/sqrt(N);endI=eye(2);H=[h11 h12;h21 h22];for i=1:Ncap11(i)=log(det(I+SNR/2*(H(i)')*(H(i))))/(2*log(2));endplot(tau,cap11)

PAS MODELS5.N-th power cosine function PAS modelphi=linspace(-100,100,100);n1=2;n2=4;n3=8;f1 = @(x) (cos(x)).^n1;Q1 = integral(f1,-pi/2,pi/2);f2 = @(x) (cos(x)).^n2;Q2 = integral(f2,-pi/2,pi/2);f3 = @(x) (cos(x)).^n3;Q3= integral(f3,-pi/2,pi/2);Y=degtorad(phi);m1=((cos(Y)).^n1)./Q3;m2=((cos(Y)).^n2)./Q3;m3=((cos(Y)).^n3)./Q3;plot(phi,m1,phi,m2,'--',phi,m3,':')legend('n=2','n=4','n=8')title('nth power of cosine function for pas model')ylabel('power azimuth spectrum');xlabel('phi')

6.UNIFORM PAS modelPASphi=linspace(-200 ,200,100);y1= degtorad(10);y2= degtorad(30);q1=1/(2*sqrt(3)*y1);pas1=q1*rectangularPulse(-25,25,phi);q2=1/(2*sqrt(3)*y2);pas2=q2*rectangularPulse(-50,50,phi);plot(phi,pas1,phi,pas2,'--')grid onlegend('as=10','as=30')title('Uniform PAS, AoA=0[degree]')xlabel('phi [degree]')ylabel('P(phi)')

7.UNIFORM PAS modelspatial correlation coefficient

x=linspace(0,6,250);phi_0=degtorad(0);as1=degtorad(10);as2=degtorad(30);Q1=1/(2*sqrt(3)*as1);Q2=1/(2*sqrt(3)*as2);delta_phi1=sqrt(3)*as1;delta_phi2=sqrt(3)*as2;sum1=zeros(1,250);sum2=zeros(1,250);for m=1:250 sum1=sum1+(besselj(2*m,2*pi*x))*(cos(2*m*phi_0))*(sin(2*m*delta_phi1)/(2*m));end for m=1:250 sum2=sum2+(besselj(2*m,2*pi*x))*(cos(2*m*phi_0))*(sin(2*m*delta_phi2)/(2*m));endRxx1=besselj(0,2*pi*x)+4*Q1.*sum1;Rxx2=besselj(0,2*pi*x)+4*Q2.*sum2;

plot(x,abs(Rxx1),x,abs(Rxx2),'--')grid onlegend('AS=10(degree)','AS=30(degree)')title('Uniform PAS model- spatial correlation coeff.')xlabel('d/lambda')ylabel('|Rxx|')

8.Trucated Gaussian PASPAS

deltaphi=180;phi1=linspace(-200,200,400);phi=degtorad(phi1)phi_0=degtorad(0); sigma1=degtorad(30); p1= ((1/erf(deltaphi/(sqrt(2)*sigma1)))/(sqrt(2*pi)*sigma1))*exp(-(phi-phi_0).^2/(2*sigma1.^2)); sigma2=degtorad(10); p2= ((1/erf(deltaphi/(sqrt(2)*sigma2)))/(sqrt(2*pi)*sigma2))*exp(-(phi-phi_0).^2/(2*sigma2.^2)); plot(phi1,p2,phi1,p1,'--') legend('AS=10[DEGREE]','AS=30[degree]') xlabel('phi(degree)'); ylabel('power azimuth spectral') title('Truncated gaussian PAS')

9.Trucated Gaussian PAS modelSpatial correlation coefficient

clearclfx=linspace(0,6,250);delta_phi=pi;phi_0=degtorad(0);sum1=zeros(1,250);sum2=zeros(1,250);sigma1=degtorad(10);sigma2=degtorad(30);q1=1/erf(pi/(sqrt(2)*sigma1));q2=1/erf(pi/(sqrt(2)*sigma2));

for m=1:250 sum1=sum1+(besselj(2*m,2*pi.*x)).*(exp(2*sigma1^2*m.^2))*(cos(2*m*phi_0))*(erf(pi/(sigma1*sqrt(2)))-erf(-pi/(sigma1*sqrt(2)))); endrxx1=(besselj(0,2*pi.*x))+(q1.*sum1);for m=1:250 sum2=sum2+(besselj(2*m,2*pi.*x)).*(exp(2*sigma2^2*m.^2))*(cos(2*m*phi_0))*(erf(pi/(sigma2*sqrt(2)))-erf(-pi/(sigma2*sqrt(2)))); endrxx2=(besselj(0,2*pi.*x))+(q2.*sum2);plot(x,abs(rxx1),x,abs(rxx2),'--')grid onlegend('AS=10(degree)','AS=30(degree)')title('Truncated Gaussian PAS model spatial correlation coeff.')xlabel('d/lambda')ylabel('|Rxx|')

10.truncated laplacian modelpas

deltaphi=degtorad(180);phi1=linspace(-200,200,400);phi=degtorad(phi1);phi_0=degtorad(0); sigma1=degtorad(10); p1=((1/(1-exp(-sqrt(2)*deltaphi/sigma1)))/(sqrt(2)*sigma1))*(exp(-(sqrt(2)*abs(phi-phi_0))/sigma1)); sigma2=degtorad(30); p2=((1/(1-exp(-sqrt(2)*deltaphi/sigma2)))/(sqrt(2)*sigma2))*(exp(-(sqrt(2)*abs(phi-phi_0))/sigma2)); plot(phi1,p1,phi1,p2,'--') grid on legend('AS=10[DEGREE]','AS=30[degree]') xlabel('phi(degree)'); ylabel('power azimuth spectral') title('Truncated laplacian PAS')

11.truncated laplacian model-spatial correlation coefficient.clearclfx=linspace(0,6,250);delta_phi=pi;phi_0=degtorad(0);sum1=zeros(1,250);sum2=zeros(1,250);sigma1=degtorad(10);sigma2=degtorad(30);q1=1/erf(pi/(sqrt(2)*sigma1));q2=1/erf(pi/(sqrt(2)*sigma2));for m=1:250 sum1=sum1+besselj(2*m,2*pi.*x)*cos(2*m*phi_0)*((sqrt(2)/sigma1+(exp(-(sqrt(2)*delta_phi)/sigma1))*(2*m*sin(2*m*delta_phi)sqrt(2)*cos(2*m*delta_phi)/sigma1))/(sqrt(2)*sigma1*((sqrt(2)/sigma1)^2+4*m^2))); endrxx1=(besselj(0,2*pi.*x))+(4*q1.*sum1);for m=1:250 sum2=sum2+besselj(2*m,2*pi.*x)*cos(2*m*phi_0)*((sqrt(2)/sigma2+(exp(-(sqrt(2)*delta_phi)/sigma2))*(2*m*sin(2*m*delta_phi)-sqrt(2)*cos(2*m*delta_phi)/sigma2))/(sqrt(2)*sigma2*((sqrt(2)/sigma2)^2+4*m^2))); endrxx2=(besselj(0,2*pi.*x))+(4*q2.*sum2);plot(x,abs(rxx1),x,abs(rxx2),'--')grid onlegend('AS=10(degree)','AS=30(degree)')title('Truncated Laplacian PAS model spatial correlation coeff.')xlabel('d/lambda')ylabel('|Rxx|')

SIMULATION RESULTS OF MATLAB

1.comparision of Time ACF of reference model and simulation model

2.2D space CCF of reference model

3.2D space CCF of simulation model

4. CAPACITY of MIMO channel

5.POWER AZIMUTH SPECTRUM of nth power of cosine pas model

6. POWER AZIMUTH SPECTRUM of uniform pas model

7.spatial correlation coefficient of uniform pas model

8. POWER AZIMUTH SPECTRUM of truncated gaussian pas model

9. Truncated Gaussian PAS model spatial correlation coeff.

10. POWER AZIMUTH SPECTRUM of truncated laplacian pas model

11.spatial correlation coefficient of truncated laplacian pas model.

2 BACK PROPAGATION ALGORITHM:

In simple terms back propagation is like learning from mistakes. Here, for some given inputs, we know (label) the desired/expected output .    

Initially all the edge weights are randomly assigned. For every input in the training dataset, the ANN is activated and its output is observed. This output is compared with the desired output that we already know, and the error is "propagated" back to the previous layer. This error is noted and the weights are "adjusted" accordingly. This process is repeated until the output error is below a predetermined threshold.

ALGORITHM STEPS : Here neuron A is a hidden layer neuron and neuron B is an output neuron and has the weight WAB.

1. First apply the inputs to the network and work out the output – remember this initial output could be anything, as the initial weights were random numbers.

2. Next work out the error for neuron B. The error is What you want – What you actually get, in other words: ErrorB = OutputB (1-OutputB)(TargetB – OutputB) The “Output(1-Output)” term is necessary in the equation because of the Sigmoid Function – if we were only using a threshold neuron it would just be (Target – Output).

3. Change the weight. Let W + AB be the new (trained) weight and WAB be the initial weight. W + AB = WAB + (ErrorB x OutputA) Notice that it is the output of the connecting neuron (neuron A) we use (not B). We update all the weights in the output layer in this way.

4. Calculate the Errors for the hidden layer neurons. Unlike the output layer we can’t calculate these directly (because we don’t have a Target), so we Back Propagate them from the output layer (hence the name of the algorithm). This is done by taking the Errors from the output neurons and running them back through the weights to get the hidden layer errors. For example if neuron A is connected as shown to B and C then we take the errors from B and C to generate an error for A. ErrorA = Output A (1 - Output A)(ErrorB WAB + ErrorC WAC) Again, the factor “Output (1 - Output )” is present because of the sigmoid squashing function.

5. Having obtained the Error for the hidden layer neurons now proceed as in stage 3 to change the hidden layer weights. By repeating this method we can train a network of any number of layers .since threshold error should be as low as possible therefore in my code I have taken threshold error as 0.07;

MATLAB code used in SINGLE CHANNEL and for f irst four cases.OutputPort1=InputPort1;inp=InputPort1.Sampled.Signal;target=InputPort2.Sampled.Signal;inp=10*inp;target=10*target; inp1=real(inp);inp2=imag(inp);targ1=real(target);targ2=imag(target);len=length(inp1);for i=1:len if targ1(i)<0 s ign1(i)=-1; else s ign1(i)=+1; endendfor i=1:len if targ2(i)<0 s ign2(i)=-1; else s ign2(i)=+1; endendinp1=abs(inp1);inp2=abs(inp2);targ1=abs(targ1);targ2=abs(targ2);finout=zeros(1, len);finout1=zeros(1, len);eta=0.98;wij=[0 0 0];wjk=[0 0 0];bias=[0.051 0.052 0.053];c=1;coun=1;for i=1:len while c if inp1(i)~=targ1(i) %output calculation inpij=(inp1(i)*wij)+bias; for l=1:3 outj( l)=(1/(1+exp(-inpij(l)))) ; end

inpjk=outj*wjk' ; outk=(1/(1+exp(-inpjk))); %error calculat ion ek=targ1(i)-outk; delk=outk*(1-outk)*(outk-targ1(i)) ; for l=1:3 delj(l )=outj( l)*(1-outj( l))*delk*wjk(l); end %weight updation for l=1:3 delwjk(l)=(-eta)*delk*outj( l) ; wjk(l)=wjk(l)+delwjk(l) ; end for l=1:3 delwij(l)=(-eta)*delj( l)*inp1(i); wij( l)=wij(l)+delwij(l) ; end for l=1:3 delbias(l)=(-eta)*delj(l) ; bias(l)=bias(l)+delbias(l) ; end coun=coun+1; ek=abs(ek); if coun==5000 wij=[1.5 2.5 3.5];wjk=[1.55 2.55 3.55]; bias=[1.1 1.2 1.3]; end if coun==50000; wij=[10.5 11.5 12.5];wjk=[15.55 20.55 25.55];bias=[5.1 5.2 5.3]; end %display(ek); if ek>=0 && ek<=0.0040 | | coun>=1000000 c=0; finout(i)=outk; end else c=0; finout(i)=targ1(i) ; %display(finout); end end coun=1; c=1;endwij=[0 0 0];wjk=[0 0 0];bias=[0.051 0.052 0.053];c=1;coun=1; for i=1:len while c if inp2(i)~=targ2(i)

%output calculation inpij=inp2(i)*wij ; for l=1:3 outj( l)=(1/(1+exp(-inpij(l)))) ; end inpjk=outj*wjk' ; outk=(1/(1+exp(-inpjk))); %error calculat ion ek=targ2(i)-outk; delk=outk*(1-outk)*(outk-targ2(i)) ; for l=1:3 delj(l )=outj( l)*(1-outj( l))*delk*wjk(l); end %weight updation for l=1:3 delwjk(l)=(-eta)*delk*outj( l) ; wjk(l)=wjk(l)+delwjk(l) ; end for l=1:3 delwij(l)=(-eta)*delj( l)*inp2(i); wij( l)=wij(l)+delwij(l) ; end for l=1:3 delbias(l)=(-eta)*delj(l) ; bias(l)=bias(l)+delbias(l) ; end coun=coun+1; ek=abs(ek); if coun==5000 wij=[1.5 2.5 3.5]; wjk=[1.55 2.55 3.55]; bias=[1.1 1.2 1.3]; end if coun==50000; wij=[10.5 11.5 12.5];wjk=[10.55 11.55 12.55]; bias=[5.1 5.2 5.3]; end if ek>=0 && ek<=0.004 | | coun>=1000000 c=0; finout1(i)=outk; end else c=0; finout1(i)=targ1(i) ; %display(finout1); end end coun=1; c=1;

endfinout=finout.*sign1;finout1=finout1.*sign2;finout1=(0+1i)*finout1;finout=finout+finout1;finout=0.1*finout;OutputPort1.Sampled.Signal=finout;Simulation ResultsCASE1: single channel with & without equalizer for fiber length 180km.

LAYOUT DIAGRAM

EYE DIAGRAM without equalizer

EYE DIAGRAM with equalizer

CASE2: MIMO without equalizer & without amplifier for 180km length of fiberLAYOUT

In this case for all frequency components q factor is coming as zero, so I am showing here for 193.1 THz signal only, rest of all three have same output.

CASE3: MIMO without equalizer & with pre amplification

F=193.1THz

193.2 THz

193.3 THz

193.4 THz

CASE3A: MIMO without equalizer and inline amplification

193.1 THz

193.2 THz

193.3 THz

193.4 THz

CASE4: MIMO with equalizer and amplifier with inline amplification

193.1 THz

193.2 THz

193.3 THz

193.4 THz

MATLAB code for FIFTH CASE.

OutputPort1=InputPort1;OutputPort2=InputPort2;OutputPort3=InputPort3;OutputPort4=InputPort4;inp1=InputPort1.Sampled.Signal;inp2=InputPort2.Sampled.Signal;inp3=InputPort3.Sampled.Signal;inp4=InputPort4.Sampled.Signal; target1=InputPort5.Sampled.Signal;target2=InputPort6.Sampled.Signal;target3=InputPort7.Sampled.Signal;target4=InputPort8.Sampled.Signal;inp1=10*inp1;inp2=10*inp2;inp3=10*inp3;inp4=10*inp4;target1=10*target1;target2=10*target2;target3=10*target3;target4=10*target4; inp5=real(inp1);inp6=imag(inp1);inp7=real(inp2);inp8=imag(inp2);inp9=real(inp3);inp10=imag(inp3);inp11=real(inp4);inp12=imag(inp4);targ5=real(target1);targ6=imag(target1);targ7=real(target2);targ8=imag(target2);targ9=real(target3);targ10=imag(target3);targ11=real(target4);targ12=imag(target4);len1=length(inp1);len2=length(inp2);len3=length(inp3);len4=length(inp4);if f==193.1for i=1:len1 if targ5(i)<0 sign1(i)=-1; else sign1(i)=+1; endendfor i=1:len1 if targ6(i)<0 sign2(i)=-1; else sign2(i)=+1; endendendif f==193.2for i=1:len2 if targ7(i)<0 sign3(i)=-1; else sign3(i)=+1; endendfor i=1:len2 if targ8(i)<0 sign4(i)=-1; else sign4(i)=+1; endendendif f ==193.3for i=1:len3 if targ9(i)<0 sign5(i)=-1; else sign5(i)=+1; endendfor i=1:len3 if targ10(i)<0 sign6(i)=-1; else sign6(i)=+1; endendendif f==193.4for i=1:len4

if targ11(i)<0 sign7(i)=-1; else sign7(i)=+1; endendfor i=1:len4 if targ12(i)<0 sign8(i)=-1; else sign8(i)=+1; endendendinp5=abs(inp5);inp6=abs(inp6);inp7=abs(inp7);inp8=abs(inp8);inp9=abs(inp9);inp10=abs(inp10);inp11=abs(inp11);inp12=abs(inp12);targ5=abs(targ5);targ6=abs(targ6);targ7=abs(targ7);targ8=abs(targ8);targ9=abs(targ9);targ10=abs(targ10);targ11=abs(targ11);targ12=abs(targ12);finout=zeros(1,len1);finout1=zeros(1,len1);finout2=zeros(1,len2);finout3=zeros(1,len2);finout4=zeros(1,len3);finout5=zeros(1,len3);finout6=zeros(1,len4);finout7=zeros(1,len4);eta=0.98;wij=[0 0 0];wjk=[0 0 0];bias=[0.051 0.052 0.053];c=1;coun=1;if f==193.1for i=1:len1 while c if inp5(i)~=targ5(i) inpij=(inp5(i)*wij)+bias; for l=1:3 outj(l)=(1/(1+exp(-inpij(l)))); end inpjk=outj*wjk'; outk=(1/(1+exp(-inpjk))); ek=targ5(i)-outk;delk=outk*(1-outk)*(outk-targ5(i)); for l=1:3 delj(l)=outj(l)*(1-outj(l))*delk*wjk(l); end for l=1:3 delwjk(l)=(-eta)*delk*outj(l);wjk(l)=wjk(l)+delwjk(l); end for l=1:3 delwij(l)=(-eta)*delj(l)*inp5(i); wij(l)=wij(l)+delwij(l); end for l=1:3 delbias(l)=(-eta)*delj(l);bias(l)=bias(l)+delbias(l); end coun=coun+1; ek=abs(ek); if coun==5000 wij=[1.5 2.5 3.5]; wjk=[1.55 2.55 3.55];bias=[1.1 1.2 1.3]; end if coun==50000; wij=[10.5 11.5 12.5]; wjk=[15.55 20.55 25.55];bias=[5.1 5.2 5.3]; end if ek>=0 && ek<=0.0040 || coun>=1000000 c=0;finout(i)=outk; end else c=0;finout(i)=targ5(i); end end coun=1; c=1;endwij=[0 0 0];wjk=[0 0 0];bias=[0.051 0.052 0.053];c=1;coun=1; for i=1:len1 while c

if inp6(i)~=targ6(i) inpij=inp6(i)*wij; for l=1:3 outj(l)=(1/(1+exp(-inpij(l)))); end inpjk=outj*wjk; outk=(1/(1+exp(-inpjk))); ek=targ6(i)-outk;delk=outk*(1-outk)*(outk-targ6(i)); for l=1:3 delj(l)=outj(l)*(1-outj(l))*delk*wjk(l); end for l=1:3 delwjk(l)=(-eta)*delk*outj(l);wjk(l)=wjk(l)+delwjk(l); end for l=1:3 delwij(l)=(-eta)*delj(l)*inp6(i); wij(l)=wij(l)+delwij(l); end for l=1:3 delbias(l)=(-eta)*delj(l);bias(l)=bias(l)+delbias(l); end coun=coun+1;ek=abs(ek); if coun==5000 wij=[1.5 2.5 3.5]; wjk=[1.55 2.55 3.55];bias=[1.1 1.2 1.3]; end if coun==50000; wij=[10.5 11.5 12.5];wjk=[10.55 11.55 12.55]; bias=[5.1 5.2 5.3]; end if ek>=0 && ek<=0.004 || coun>=1000000 c=0;finout1(i)=outk; end else c=0; finout1(i)=targ6(i); end end coun=1; c=1;endendif f==193.2for i=1:len2 while c if inp7(i)~=targ7(i) inpij=(inp7(i)*wij)+bias; for l=1:3 outj(l)=(1/(1+exp(-inpij(l)))); end inpjk=outj*wjk';outk=(1/(1+exp(-inpjk))); ek=targ7(i)-outk; delk=outk*(1-outk)*(outk-targ7(i)); for l=1:3 delj(l)=outj(l)*(1-outj(l))*delk*wjk(l); end for l=1:3 delwjk(l)=(-eta)*delk*outj(l); wjk(l)=wjk(l)+delwjk(l); end for l=1:3 delwij(l)=(-eta)*delj(l)*inp7(i); wij(l)=wij(l)+delwij(l); end for l=1:3 delbias(l)=(-eta)*delj(l);bias(l)=bias(l)+delbias(l); end coun=coun+1;ek=abs(ek);

if coun==5000 wij=[1.5 2.5 3.5]; wjk=[1.55 2.55 3.55];bias=[1.1 1.2 1.3]; end if coun==50000; wij=[10.5 11.5 12.5]; wjk=[15.55 20.55 2]; bias=[5.1 5.2 5.3]; end if ek>=0 && ek<=0.040 || coun>=1000000 c=0;finout2(i)=outk; end else c=0; finout2(i)=targ7(i); end end coun=1; c=1;endwij=[0 0 0];wjk=[0 0 0];bias=[0.051 0.052 0.053];c=1;coun=1;for i=1:len1 while c if inp8(i)~=targ8(i) inpij=inp8(i)*wij; for l=1:3 outj(l)=(1/(1+exp(-inpij(l)))); end inpjk=outj*wjk'; outk=(1/(1+exp(-inpjk)));ek=targ8(i)-outk;delk=outk*(1-outk)*(outk-targ8(i)); for l=1:3 delj(l)=outj(l)*(1-outj(l))*delk*wjk(l); end for l=1:3 delwjk(l)=(-eta)*delk*outj(l);wjk(l)=wjk(l)+delwjk(l); end for l=1:3 delwij(l)=(-eta)*delj(l)*inp8(i);wij(l)=wij(l)+delwij(l); end for l=1:3 delbias(l)=(-eta)*delj(l); bias(l)=bias(l)+delbias(l); end coun=coun+1; ek=abs(ek); if coun==5000 wij=[1.5 2.5 3.5]; wjk=[1.55 2.55 3.55];bias=[1.1 1.2 1.3] end if coun==50000; wij=[10.5 11.5 12.5];wjk=[10.55 11.55 12.55];bias=[5.1 5.2 5.3]; end if ek>=0 && ek<=0.004 || coun>=1000000 c=0;finout3(i)=outk; end else c=0; finout3(i)=targ8(i);end end coun=1; c=1;endendif f==193.3for i=1:len3 while c if inp9(i)~=targ9(i) inpij=(inp9(i)*wij)+bias; for l=1:3 outj(l)=(1/(1+exp(-inpij(l))));

end inpjk=outj*wjk'; outk=(1/(1+exp(-inpjk))); ek=targ9(i)-outk; delk=outk*(1-outk)*(outk-targ9(i)); for l=1:3 delj(l)=outj(l)*(1-outj(l))*delk*wjk(l); end for l=1:3 delwjk(l)=(-eta)*delk*outj(l); wjk(l)=wjk(l)+delwjk(l); end for l=1:3 delwij(l)=(-eta)*delj(l)*inp9(i); wij(l)=wij(l)+delwij(l); end for l=1:3 delbias(l)=(-eta)*delj(l); bias(l)=bias(l)+delbias(l);end coun=coun+1; ek=abs(ek); if coun==5000 wij=[1.5 2.5 3.5];wjk=[1.55 2.55 3.55];bias=[1.1 1.2 1.3]; end if coun==50000; wij=[10.5 11.5 12.5];wjk=[15.55 20.55 25.55]; bias=[5.1 5.2 5.3]; end if ek>=0 && ek<=0.0040 || coun>=1000000 c=0;finout4(i)=outk; end else c=0; finout4(i)=targ9(i); end end coun=1; c=1;endwij=[0 0 0];wjk=[0 0 0];bias=[0.051 0.052 0.053];c=1;coun=1; for i=1:len3 while c if inp10(i)~=targ10(i) inpij=inp10(i)*wij; for l=1:3 outj(l)=(1/(1+exp(-inpij(l)))); end inpjk=outj*wjk'; outk=(1/(1+exp(-inpjk)));ek=targ10(i)-outk;delk=outk*(1-outk)*(outk-targ10(i)); for l=1:3 delj(l)=outj(l)*(1-outj(l))*delk*wjk(l); end for l=1:3 delwjk(l)=(-eta)*delk*outj(l); wjk(l)=wjk(l)+delwjk(l); end for l=1:3 delwij(l)=(-eta)*delj(l)*inp10(i);wij(l)=wij(l)+delwij(l); end for l=1:3 delbias(l)=(-eta)*delj(l);bias(l)=bias(l)+delbias(l); end coun=coun+1; ek=abs(ek); if coun==5000 wij=[1.5 2.5 3.5]; wjk=[1.55 2.55 3.55];bias=[1.1 1.2 1.3];

end if coun==50000; wij=[10.5 11.5 12.5]; wjk=[10.55 11.55 12.55];bias=[5.1 5.2 5.3]; end if ek>=0 && ek<=0.004 || coun>=1000000 c=0;finout5(i)=outk; end else c=0; finout5(i)=targ10(i); end end coun=1; c=1;endendif f==193.4for i=1:len4 while c if inp11(i)~=targ11(i) inpij=(inp11(i)*wij)+bias; for l=1:3 outj(l)=(1/(1+exp(-inpij(l)))); end inpjk=outj*wjk'; outk=(1/(1+exp(-inpjk)));ek=targ11(i)-outk;delk=outk*(1-outk)*(outk-targ11(i)); for l=1:3 delj(l)=outj(l)*(1-outj(l))*delk*wjk(l); end for l=1:3 delwjk(l)=(-eta)*delk*outj(l);wjk(l)=wjk(l)+delwjk(l); end for l=1:3 delwij(l)=(-eta)*delj(l)*inp11(i); wij(l)=wij(l)+delwij(l); end for l=1:3 delbias(l)=(-eta)*delj(l); bias(l)=bias(l)+delbias(l); end coun=coun+1; ek=abs(ek); if coun==5000 wij=[1.5 2.5 3.5]; wjk=[1.55 2.55 3.55]; bias=[1.1 1.2 1.3];end if coun==50000; wij=[10.5 11.5 12.5]; wjk=[15.55 20.55 25.55];bias=[5.1 5.2 5.3]; end; if ek>=0 && ek<=0.0040 || coun>=1000000 c=0;finout6(i)=outk; end else c=0; finout6(i)=targ11(i); end end coun=1;c=1;endwij=[0 0 0];wjk=[0 0 0];bias=[0.051 0.052 0.053];c=1;coun=1; for i=1:len4 while c if inp12(i)~=targ12(i) inpij=inp12(i)*wij; for l=1:3 outj(l)=(1/(1+exp(-inpij(l))));

end inpjk=outj*wjk';outk=(1/(1+exp(-inpjk)));ek=targ12(i)-outk; delk=outk*(1-outk)*(outk-targ12(i)); for l=1:3 delj(l)=outj(l)*(1-outj(l))*delk*wjk(l); end for l=1:3 delwjk(l)=(-eta)*delk*outj(l);wjk(l)=wjk(l)+delwjk(l); end for l=1:3 delwij(l)=(-eta)*delj(l)*inp12(i);wij(l)=wij(l)+delwij(l); end for l=1:3 delbias(l)=(-eta)*delj(l); bias(l)=bias(l)+delbias(l); end coun=coun+1; ek=abs(ek); if coun==5000 wij=[1.5 2.5 3.5]; wjk=[1.55 2.55 3.55];bias=[1.1 1.2 1.3]; end if coun==50000; wij=[10.5 11.5 12.5]; wjk=[10.55 11.55 12.55]; bias=[5.1 5.2 5.3]; end if ek>=0 && ek<=0.004 || coun>=1000000 c=0; finout7(i)=outk; else else c=0;finout7(i)=targ12(i); end end coun=1; c=1;endendfinout=finout.*sign1;finout1=finout1.*sign2;finout2=finout.*sign3;finout3=finout1.*sign4;finout4=finout.*sign5;finout5=finout1.*sign6;finout6=finout.*sign7;finout7=finout1.*sign8;finout1=(0+1i)*finout1;finout3=(0+1i)*finout3;finout5=(0+1i)*finout5;finout7=(0+1i)*finout7;finout=finout+finout1;finout2=finout2+finout3;finout4=finout4+finout5;finout6=finout6+finout7;finout=0.1*finout;finout2=0.1*finout2;finout4=0.1*finout4;finout6=0.1*finout6;OutputPort1.Sampled.Signal=finout;OutputPort2.Sampled.Signal=finout2;OutputPort3.Sampled.Signal=finout4;OutputPort4.Sampled.Signal=finout6;

CASE 5 :case 4 MIMO but optical fiber used after multiplexer.

For the 5 t h case since all four frequency components are available simultaneously MATLAB COMPONENT is unable to differentiate between all 4 frequencies components. In my code I have assigned four different block of codes to corresponding four frequencies .

CONCLUSION

First of all I have studied MIMO channel modelling and simulated statist ical

results of various MIMO channel models using MATLAB.I studied test

procedures ,such as the autocorrelation function ,average duration of

fades ,probabili ty density function and the level-crossing rate , in order to test

and to confirm correctness of the implemented channel simulators, simulated

time ACF function and seen reference model result and simulation results are

coming almost same.

All the parameters used during simulation of one ring geometrical model

statist ical results are taken from LPNM parameter computation method.

In 2nd part I have implemented MIMO concept on optical fibre channel using

optisystem software. For the improvement of signal strength at the receiver side

equalizer concept is used which is trained using back propagation technique.

My result for single channel for Q value at 180km distance of optical fibre is

as follows-without equalizer=2.18,with equalizer=275.8

For the case of MIMO channel , wavelength division multiplexing is used where

four input optical signals at frequencies 193.1Thz,193.2Thz,193.3Thz and

193.4Thz are mult iplexed and transmitted over 180km length of fibre. I have

performed it in four different schemes 1.single channel with & without

equalizer for 180km length,2.MIMO without equalizer and without amplifier for

180km length3.MIMO without equalizer with amplifier(pre amplification and

inline amplification) 4.MIMO with equalizer and amplifier out of which fourth

scheme is better because here in place 4 EDFA amplifiers only one amplifier is

used makes it economic and Q factor for four outputs is 84 , 55.58 , 48.51 , 63.1

makes output bit error rate low.

REFERENCES

1. Modelling and simulation of MIMO channels by Matthias P¨atzold BEATS/CUBAN workshop 2004.2. MIMO-OFDM wireless communications with MATLAB by Yang Soo Cho | Jaekwon Jim, Won Young Yang, Chung G. Kang WILEY Publications. Print ISBN: 978-0-470-82561-7, Edition 1st.Chapter 33. G. J. Foschini and M. J. Gans. On limits of wireless communications in a fading environment. Wireless Personal Communications, 6: Page 311-335, 1998.

4. Matthias P¨atzold and Bjorn Olav Hogstad ,” A Wideband Space-Time MIMO Channel Simulator Based on the Geometrical One-Ring Model” Vehicular technology conference, 2006.VTC-2006, FALL2006.IEEE64th 5. Lee, W. (1973) “Effect on correlation between two mobile radio base-station antennas”. IEEE Trans. Commun., volume 21, issue11, pages 1214–1224, NOVEMBER 1973.6. Matthias Pätzold, Cheng-Xiang Wang, and Bjørn Olav Hogstad Two New Sum-of-Sinusoids-Based Methods for the Efficient Generation of Multiple Uncorrelated Rayleigh Fading Waveforms. IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 6, JUNE 20097. M. Pätzold: “Mobile Radio Channels”, 2nd Edition, and Chichester: John Wiley & Sons, 2011. Print ISBN: 9780470517475.8. Sebastian de la Kethulle, 27 September 2004. An Overview of MIMO systems in Wireless . Communications. Available: http://www.iet.ntnu.no/projects/beats/Documents/mimo.pdf