Match analytical and simulation values of SER vs SNR curve.

% Parameters
M = 4; % MQAM order (e.g., 16-QAM)
N = 1e5; % Number of transmit symbols

% Define SNR range
SNR_dB_range = 0:10;
num_SNR_points = length(SNR_dB_range);

% Initialize SER array
SER_values = zeros(num_SNR_points, 1);

% Create MQAM constellation
constellation = createMQAM(M);
% Loop over each SNR value
for idx = 1:num_SNR_points
SNR_dB = SNR_dB_range(idx);

% Generate transmit symbols
transmitSymbols = generateSymbols(constellation, N);

% Transmit through AWGN channel
receivedSymbols = transmitSymbolsThroughAWGN(transmitSymbols, SNR_dB);

% Demodulate symbols
decodedSymbols = demodulateSymbols(receivedSymbols, constellation);

% Compute Symbol Error Rate
SER = computeSER(transmitSymbols, decodedSymbols);

% Store SER value
SER_values(idx) = SER;

% Compute Theoretical SER
SER_theoretical = computeTheoreticalSER(M, SNR_dB_range);
SER_theoretical_values = SER_theoretical;
end

% Plot SER vs SNR
figure;
semilogy(SNR_dB_range, SER_values, ‘o-‘, ‘DisplayName’, ‘Simulated SER’);
hold on;
semilogy(SNR_dB_range, SER_theoretical_values, ‘r–‘, ‘DisplayName’, ‘Theoretical SER’);
xlabel(‘SNR (dB)’);
ylabel(‘Symbol Error Rate (SER)’);
title(‘SER vs. SNR for MQAM’);
legend;
grid on;

function constellation = createMQAM(M)
% Create an MQAM constellation manually
% M: Number of symbols (e.g., 16, 64, etc.)

% Check that M is a perfect square
k = sqrt(M);
if mod(k,1) ~= 0
error(‘M must be a perfect square’);
end

% Generate constellation points
k = round(k); % Number of points per side
[X, Y] = meshgrid(linspace(-k+1, k-1, k), linspace(-k+1, k-1, k));
constellation = X(:) + 1i*Y(:);
end

function symbols = generateSymbols(constellation, N)
% Generate N random transmit symbols
% constellation: The MQAM constellation points
% N: Number of symbols to generate

numPoints = length(constellation);
indices = randi([1 numPoints], [N, 1]); % Randomly choose indices
symbols = constellation(indices); % Select symbols from the constellation
end

function received = transmitSymbolsThroughAWGN(symbols, SNR_dB)
% Transmit symbols through a SISO AWGN channel
% symbols: Transmit symbols
% SNR_dB: Signal-to-noise ratio in dB

% Convert SNR from dB to linear scale
SNR_linear = 10^(SNR_dB / 10);

% Calculate noise variance
noise_variance = 1 / SNR_linear;

% Generate noise
noise = sqrt(noise_variance / 2) * (randn(size(symbols)) + 1i*randn(size(symbols)));

% Received symbols
received = symbols + noise;
end

function decodedSymbols = demodulateSymbols(received, constellation)
% Demodulate symbols using minimum distance decoding
% received: Received symbols
% constellation: MQAM constellation points

numSymbols = length(received);
numPoints = length(constellation);

% Initialize decoded symbols
decodedSymbols = zeros(size(received));

for i = 1:numSymbols
% Compute distances to all constellation points
distances = abs(received(i) – constellation).^2;
% Find the closest constellation point
[~, idx] = min(distances);
decodedSymbols(i) = constellation(idx);
end
end

function SER = computeSER(originalSymbols, decodedSymbols)
% Compute symbol error rate
% originalSymbols: Transmit symbols
% decodedSymbols: Received and decoded symbols

numErrors = sum(originalSymbols ~= decodedSymbols);
totalSymbols = length(originalSymbols);

SER = numErrors / totalSymbols;
end

function SER_theoretical = computeTheoreticalSER(M, SNR_dB)
% Compute the theoretical SER for MQAM in AWGN
% M: Number of symbols (e.g., 16, 64, etc.)
% SNR_dB: SNR in dB

% Convert SNR from dB to linear scale
SNR_linear = 10.^(SNR_dB / 10);

% Compute theoretical SER
k = sqrt(M); % Number of points per side
if mod(k,1) ~= 0
error(‘M must be a perfect square’);
end
% Calculate SER using the Q-function
SER_theoretical = 4*(1-1./sqrt(M))*qfunc(sqrt(3./(M-1)*SNR_linear));
end

In the above code when I plot the SER v/s SNR and compare the simulation and analytical values don’t match. Can someone tell me what is the error in my code and how can I match the curves ?% Parameters
M = 4; % MQAM order (e.g., 16-QAM)
N = 1e5; % Number of transmit symbols