What is the relation between Bit Error Rate (BER), distance d and SNR(dB)?

We are considering Single-input-single-output system (SISO) in which Transmitted (Alice), Receiver (Bob) and Eavesdropping (Eve).

And Rayleigh channel and BPSK modulation technique is used.

System model: y = hx + n

where, y is the received signal, x is the transmitted signal and h,n are Rayleigh fading channel response and noise, both of which are circularly symmetric Gaussian noise with zero mean respectively.

For finding the BER vs SNR:

y = hx + n*(10^( -(SNRdB) / 20 ) )

from this above equation I can find BER vs SNR.

But I didn't get the relation between BER, distance and SNR.

Suppose SNR is fixed at 20dB and distance is vary d = 1, 2, 3, ..., 10 cm. And BER of EVE is small at d=1. BER of EVE is increases with increasing distance d. BER of Eve is large at d=10 cm.

My question is how BER of Eve is increses with increasing distance d? Relation between BER and distance d? Relation between BER, SNR and distance 'd'?

[Dyuman Joshi]

How is your question related to MATLAB?

Answer 1

[Hassaan]

1. Relationship between SNR and BER:The BER for BPSK in a Rayleigh fading channel is a function of the SNR, where higher SNR typically leads to lower BER. The Q-function is used to calculate BER from SNR in such scenarios.

2. Relationship between Distance and SNR:Received signal strength and thus SNR decrease as the distance between transmitter and receiver increases due to path loss, typically proportional to the square of the distance in free space.

3. Relationship between BER, SNR, and Distance:BER worsens as distance increases because SNR decreases due to path loss. For a fixed transmit power, a longer distance results in a higher BER due to a lower received SNR.

While it's true that increasing distance leads to lower SNR and higher BER, there are additional factors to consider:

• Modulation Scheme: Different modulation schemes have varying sensitivities to SNR changes. For example, QAM is generally more sensitive than BPSK, meaning its BER will increase more rapidly with decreasing SNR.
• Channel Characteristics: Factors like fading severity, multipath propagation, and interference in the channel can also impact the relationship between distance, SNR, and BER. A noisy or highly fading channel can amplify the BER increase despite maintaining the same distance andSNR compared to a clearer channel.

The relationship between BER, SNR, and distance for a SISO system using BPSK modulation in a Rayleigh fading channel

% Define parameters
Fs = 1000; % Sampling frequency
fc = 250; % Carrier frequency in Hz
Kf = 50; % Frequency sensitivity
SNRdB_fixed = 20; % Fixed SNR value in dB
distances = 1:10; % Distances in cm (converted to meters)
lambda = 3e8 / (fc * 1e6); % Wavelength in meters (assuming fc is in MHz)

% Preallocate arrays for BER
BER = zeros(size(distances));

% Loop over each distance
for i = 1:length(distances)
    d = distances(i) / 100; % Convert distance from cm to meters
    Pr = (lambda / (4 * pi * d))^2; % Friis transmission equation for received power
    SNR = 10^(SNRdB_fixed / 10) * Pr; % Calculate SNR for the given distance
    
    % Generate random BPSK signal
    data = randi([0, 1], 1, Fs); % Random data
    x = 2 * data - 1; % BPSK modulation (0 -> -1, 1 -> +1)
    
    % Generate Rayleigh fading channel response
    h = (randn + 1i * randn) / sqrt(2); % Rayleigh channel
    
    % Generate AWGN noise
    N0 = 1 / SNR; % Noise power spectral density
    n = sqrt(N0 / 2) * (randn(size(x)) + 1i * randn(size(x))); % Complex Gaussian noise
    
    % Received signal
    y = h * x + n;
    
    % Demodulation (assuming perfect channel state information)
    y_demod = real(y) / h;
    received_data = y_demod > 0; % Decision rule for BPSK
    
    % Calculate BER
    errors = sum(data ~= received_data);
    BER(i) = errors / length(data);
end

% Plot BER vs. Distance
figure;
plot(distances, BER, 'b-o');
xlabel('Distance (cm)');
ylabel('Bit Error Rate (BER)');
title('BER vs. Distance for fixed SNRdB = 20 dB in Rayleigh Channel');
grid on;

This script simulates the transmission of a BPSK signal over a Rayleigh fading channel with a fixed SNR of 20 dB. It calculates the received power using the Friis transmission equation and assumes an ideal free-space environment. It then computes the BER for different distances and plots the BER against the distance.

Please note that the simulation assumes perfect channel state information for demodulation, which is an idealization. In practice, channel estimation would be necessary. The Rayleigh fading is also simulated as a single coefficient, which is an oversimplification for a real multipath fading channel.

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[Himanshi]

Thank you so much sir