# Capacity Analysis Of Wireless Communication Systems Computer Science Essay

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An approach to demonstrate Capacity Analysis in Wireless Communication Systems, using a series of MATLAB simulations. Following a brief introduction to fading in general, models for flat fading are developed and simulated using MATLAB in which the CSI (channel state information) is known by the receiver only . And the comparison between the AWGN(Additive White Gaussian Noise) and Rayleigh Flat Fading Channel for Capacity, SNR (Signal to Noise Ratio) and BER(Bit error rate)subject to an average transmit power constraint. And our numerical comparison conducted shows that the Rayleigh Channel capacity has a better performance.

In communications, the additive white Gaussian noise (AWGN) channel model is one in which the only impairment is a linear addition of wideband or white noise with a constant spectral density (expressed as watts per hertz of bandwidth) and a Gaussian distribution of amplitude. The model does not account for the phenomena of fading, frequency selectivity, interference, nonlinearity or dispersion. However, it produces simple and tractable mathematical models which are useful for gaining insight into the underlying behavior of a system before these other phenomena are considered. The AWGN channel is a good model for many satellite and deep space communication links.

Radio waves propagate from a transmitting antenna, and travel through free space undergoing absorption, reflection, refraction, diffraction, and scattering. They are greatly affected by the ground terrain, the atmosphere, and the objects in their path, like buildings, bridges, hills, trees, etc. These multiple physical phenomena are responsible for most of the characteristic features of the received signal. A radio channel exploits an extremely random characteristic, which does not allow us to use the simple AWGN channel model mentioned above to analyze the channel capacity. Radio signals propagate by means of reflection, diffraction, and scattering, which result in three effects a radio signal experiences: attenuation, large-scale shadowing, and small-scale fading. It was proved that these three effects are independent of each other. Signal attenuation is mainly introduced by the location of a receiver (distance between the transmitter and the receiver), which can be predicted by a deterministic model. Large-scale shadowing of a signal is mainly caused by multiple reflections and/or diffractions of the signal while propagating, whose characteristics can be captured with a log-normal distribution model. Smallscale fading of a signal is caused by multiple versions of a transmitted signal with different delay times such that it has both time and location varying property.

One type of channels with the fading effect caused by the multi-path time delay spread is flat fading channels in which the period of the transmitted signal is larger than the multi-path delay spread. Since the received signal power varies significantly (in 20 or 30 dB) in a flat fading channel, it is critical to precisely capture the distribution of the channel gain for designing a radio communication system . The most common used signal amplitude distribution in flat fading channels is the Rayleigh distribution.

Rayleigh fading is a statistical model for the effect of a propagation environment on a radio signal, such as that used by wireless devices. Rayleigh fading models assume that the magnitude of a signal that has passed through such a transmission medium (also called a communications channel) will vary randomly, or fade, according to a Rayleigh distribution - the radial component of the sum of two uncorrelated Gaussian random variables. It is viewed as a reasonable model for tropospheric and ionospheric signal propagation as well as the effect of heavily built-up urban environments on radio signals. Rayleigh fading is most applicable when there is no dominant propagation along a line of sight between the transmitter and receiver. If there is a dominant line of sight, Rician fading may be more applicable.

For a rayleigh flat fading channel capacity we consider the following model shown below.

where Vt, Ut, and Nt are the same as defined above âˆšAtexp(jÏ•) is a complex channel gain with amplitude At and phase Ï• at time t. The phase Ï• is uniformly distributed in [0, 2Ï€), and the signal amplitude âˆšAt is a random variable with a Rayleigh probability density function (pdf). At is referred to as the channel power gain with the time-varying property, which is independent of the channel input Ut, and could be either independent or correlated over time.

The instantaneous SNR at time t is given by

Where,

S be the average transmit power, At is the channel power gain, the channel bandwidth and the power spectral density of the noise are B and N0, respectively.

By using Shannon theorem we find out the Shannon capacity of Rayleigh fading channel can be expressed by

Where,

C is the channel capacity

B is the bandwidth

## Zero Forcing Equalizer

The Zero-Forcing Equalizer applies the inverse of the channel to the received signal, to restore the signal before the channel. In a zero forcing equalizer the equalizer coefficients are chosen to force the samples of the combined channel to become zero, hence by letting the number of coefficients increase without bound, and infinite length equalizer with zero ISI at the output can be obtained.

Thus an infinite length, zero, the ISI equalizer Simply Becomes and inverse filter which inverts the folded frequency response of the channel.

For example-:

For an AWGN r= s + n and for an Rayleigh fading r= h*s + n,

Where, r = receive data, h = frequency response, n= noise, s = signal

Therefore when we use zero forcing Technique to transmit data we get,

## DESIGN

clear all; clc; close all; %clears the workspace, screen and aloses all the other windowws

N_loop = 1000; %No. of times the loop is to be executed

N_sym = 10000; %intitializes the value

Eb_dB=0:1:20; %taking a range of values from 0 to 20 with a step of 1

Eb = 10.^(Eb_dB/10); %coverting the values from dB to nats per second

N_bit=2*N_sym; %calculating nbits

awgn_sigma = sqrt(0.5); %initializing the value

sqrt_Es = sqrt(2*Eb); %value for Energy per symbol

BER1 = zeros(1,length(Eb)); %definig an array for bit error rate for awgn

BER2 = zeros(1,length(Eb)); %defining an array for rayleigh fading

normalized = 1/sqrt(2); %calculating the value for normalized

for e=1:length(Eb) %using a loop to find thw value of H for all the values ranging from 0 to 20dB

H = sqrt(0.5)*(randn(1,N_sym)+j*randn(1,N_sym)); %equation to find the value of H in rayleigh fading

for n=1:N_loop; %using a loop to run the program for N_loop times

ray_c(n)=log2(1+Eb(e)*(abs(H(n)).^2)); %using the capacity equation to get the value of capacity for a rayleigh fading

awgn_c(n)=log2(1+Eb(e)); %using the capacity equation to get the value of capacity for a AWGN Channel

end

C(e) = mean(ray_c); %calculating a mean for all the values of capacity for rayleigh fading

B(e) = mean(awgn_c); %calculating a mean for all the values of capacity for AWGN Channel

end

for n=1:N_loop; %using a loop to run the program for N_loop times

D = floor(2*rand(1,N_sym))+j*floor(2*rand(1,N_sym));%here we use D to calculate data and we use floor to round off all the values towards minus infinity

Tx_data = normalized * (2*D-(1+j)); %this equations shows the transmitted data

Noise=awgn_sigma*(randn(N_sym,length(Eb_dB))+j*randn(N_sym,length(Eb_dB))); %this equation shows calculates the noise

H = sqrt(0.5)*(randn(1,N_sym)+j*randn(1,N_sym)); %equation to find the value of H in rayleigh fading

Rx_data_awgn = transpose(Tx_data)*sqrt_Es + Noise; %recieveing data for awgn channel

Rx_data2 = transpose(Tx_data)*sqrt_Es .* (transpose(H) * ones(1,length(Eb_dB))) + Noise;%recieveing data for an rayleigh fading and equilizing the data using Zero Forcing technique

Rx_data_ray = Rx_data2.* (transpose(1./H) * ones(1,length(Eb_dB))); %after using zero forcing technique to equalize finding recieveing data for rayleigh fading

Recov_data_awgn = 0.5*(1+j+sign(real(Rx_data_awgn))+j * sign(imag(Rx_data_awgn))); %here we recover the data for an awgn channel to to 0 and 1 by using sign function

Recov_data_ray = 0.5*(1+j+sign(real(Rx_data_ray))+j * sign(imag(Rx_data_ray))); %here we recover the data for rayleigh fading to 0 and 1 by using sign function

X_awgn = abs(Recov_data_awgn-transpose(D)*ones(1,length(Eb_dB))).^2; %here multiplying and squareing to get the matrix in the same dimension and using abs to get the absolute values for awgn channel

X_ray = abs(Recov_data_ray-transpose(D)*ones(1,length(Eb_dB))).^2; %here multiplying and squareing to get the matrix in the same dimension and using abs to get the absolute values for rayleigh fading

BER1 = BER1+sum(X_awgn)/N_bit; %calculating bit error for awgn channel

BER2 = BER2+sum(X_ray)/N_bit; %calculating bit error for rayleigh fading channel

end;

BER_AWGN = BER1/N_loop; %calculating the bit error rate for awgn channel

BER_RAY = BER2/N_loop; %calculating the bit error rate for rayleigh fading

figure

plot(Eb_dB,C,'b*-',Eb_dB,B,'k*-'); %plotting a graph between the capacities and the SNR for both the channels

grid on;

legend ('Rayleigh Channel','AWGN Channel')

title('Capacity VS SNR');

xlabel('Eb/N0 (dB)(SNR)');

ylabel('CAPACITY');

figure

semilogy(Eb_dB,BER_RAY,'b*-',Eb_dB,BER_AWGN,'k*-'); %plotting a semilog y graph between BER and SNR for both the channels

grid on;

legend ('Rayleigh Channel','AWGN Channel')

title('BER VS SNR');

xlabel('Eb/N0 (dB) (SNR)');

ylabel('BER');

figure

semilogx(B,BER_AWGN,'k*-',C,BER_RAY,'b*-'); %plotting a semilog x graph between BER and Capacities for both the channels

grid on;

legend ('AWGN Channel','Rayleigh Channel')

title('Capacity VS BER');

xlabel('BER');

ylabel('CAPACITY');

## RESULTS

## Capacity VS SNR (Signal to Noise Ratio)

## BER (Bit Error Rate) Vs SNR (Signal to Noise Ratio)

## BER (Bit Error Rate) Vs Capacity

## SUSTAINABLE DEVELOPMENT ASPECTS OF THE DESIGN

## REGULATORY CONSIDERATIONS

WEEE have certain set of laws which set sights on the amount of electrical and electronic equipment produced, and encouraging the consumers to recycle and recuperate the product. It also sets its goal to increase the product's environmental performance by supporting the manufacturers, suppliers and consumers by making them recycle and recover the electrical and electronic equipment and reducing the amount of EEE going to junkyards.

RoHS or Restrictions of use of Hazardous Substances basically prohibits those Electrical and Electronic Equipments that have more than fixed amount of toxic substances like Lead, polybrominated biphenyl and other fire retardants.

The project CAPACITY ANALYSIS OF WIRELESS COMMUNICATION SYSTEMS is actually a matlab based program which does not comply itself with any hardware. But the software is a section of a system and according to Data Protection Act all the data is supposed to be destroyed or the hardware containing it must be destroyed as per WEEE and RoHS regulations as it later on can become vulnerable to the organisation.

## CONSIDERATION OF POTENTIAL USE OF PRODUCT

The program is a sustainable product which allows the user to find out the capacity of a high speed wireless network which can be a useful in heavily built-up centres or major cities where there is no line of sight between the transmitter and receiver and many buildings and other objects attenuate and diffract the signal. The program utilizes Shannon's capacity in a Rayleigh flat fading channel and compares the numerically computed capacity results with the proposed approximation capacity result; this helps us in setting up an ideal network which will help in a decrease of energy costs and consumption. The program makes it possible to transmit information nearly without error at any rate which will allow the probability of error at the receiver to be made arbitrarily small.

This program has high market opportunities as its algorithm can be used by many networking and telecommunication companies to revise and accumulate their networks so that modifications can be made in order to increase their bit rate transfer capacity which would make it more cost reliable and approximate fading system.

There is very less chance of receiving bit error rate in this program because no matter how contaminated the communication channel may be with noise interference, it would still be possible to communicate digital data through the channel.

## OPPORTUNITIES FOR SPECIFIC DESIGN IMPROVEMENTS

The program for "CAPACITY ANALYSIS OF HIGH SPEED WIRELESS COMMUNICATION SYSTEMS" is a successful sustainable product which is highly efficient and has no adverse effects on the society as well as the environment.

The efficiency of the program has more influence on the amount of resources required for its functioning and demanding of more high level programming skills. Hence the most suited design improvement would be to increase the algorithm of the program so that it would help in decreasing the time taken to complete the program and reduce the storage used by the program during its functioning hence saving time and space.

One more volatile solution for high fidelity of the signal would be to transmit the signal over multiple channels so that it would experience independent fading and then coherently combine them at the receiver, hence the probability of experiencing a fade in this composite channel would be a probability that all the component channels simultaneously experience a fade in a more unlikely event.

## CONCLUSION

Given an average transmit power; we have studied the capacity of a Rayleigh flat fading channel, in which the Channel State Information (CSI) is known by the receiver only. An approximation result of a Rayleigh fading channel capacity was proposed. The comparisons of the Shannon capacity to its approximation result were made, which verified our approximation result. And it also shows that Rayleigh fading has a better performance than the AWGN channel.

The zero forcing equalizer used in the above process has a disadvantage that the inverse filter may excessively amplify noise at frequencies where the folded channel spectrum, has high attenuation which thus neglect the effect of noise altogether and would not be preferred for wireless links however it performs well for static channels with high signal to noise ratio's such local wire telephone lines.