MATLAB Code
clc;
clear;
close all;
% Parameters
Rb = 1e6; % Bit rate (1 Mbps)
SNR_dB = 0:2:20; % SNR range in dB
beta_values = [0, 0.2, 0.5, 0.8, 1.0]; % Different roll-off factors
numBits = 1e5; % Number of bits
disp("For different roll-off (β) factors and a symbol rate of 1 MHz:");
% Simulation
BER = zeros(length(beta_values), length(SNR_dB));
for b = 1:length(beta_values)
beta = beta_values(b);
bandwidth = (1 + beta) * (Rb / 2); % Bandwidth calculation
timeBandwidthProduct = (1 + beta) / 2; % Time-bandwidth product calculation
fprintf('Beta = %.1f, Bandwidth = %.2f MHz, Time-Bandwidth Product = %.2f\n', beta, bandwidth / 1e6, timeBandwidthProduct);
for s = 1:length(SNR_dB)
snr = 10^(SNR_dB(s) / 10); % Convert dB to linear
EbN0 = snr * Rb / bandwidth; % Adjust for bandwidth
noiseVar = 1 / (2 * EbN0);
% Transmit random BPSK symbols
bits = randi([0, 1], numBits, 1);
symbols = 2 * bits - 1;
noise = sqrt(noiseVar) * randn(numBits, 1);
received = symbols + noise;
detectedBits = received > 0;
% Calculate BER
BER(b, s) = sum(detectedBits ~= bits) / numBits;
end
end
% Plot results
figure;
semilogy(SNR_dB, BER(1, :), 'o-', 'DisplayName', '\beta = 0'); hold on;
semilogy(SNR_dB, BER(2, :), 'x-', 'DisplayName', '\beta = 0.2');
semilogy(SNR_dB, BER(3, :), 's-', 'DisplayName', '\beta = 0.5');
semilogy(SNR_dB, BER(4, :), 'd-', 'DisplayName', '\beta = 0.8');
semilogy(SNR_dB, BER(5, :), '^-', 'DisplayName', '\beta = 1.0');
grid on;
xlabel('SNR (dB)');
ylabel('BER');
title('BER vs SNR for Different Roll-off Factors (\beta)');
legend('Location', 'southwest');
Output
For different roll-off (β) factors and a symbol rate of 1 MHz:
Beta = 0.0, Bandwidth = 0.50 MHz, Time-Bandwidth Product = 0.50
Beta = 0.2, Bandwidth = 0.60 MHz, Time-Bandwidth Product = 0.60
Beta = 0.5, Bandwidth = 0.75 MHz, Time-Bandwidth Product = 0.75
Beta = 0.8, Bandwidth = 0.90 MHz, Time-Bandwidth Product = 0.90
Beta = 1.0, Bandwidth = 1.00 MHz, Time-Bandwidth Product = 1.00
Copy the aforementioned MATLAB code from here03
Bandwidth Efficiency in Raised Cosine Filter
Raised cosine filters are bandwidth efficient, especially for low roll-off factors. They achieve a balance between bandwidth use and ISI suppression, which makes them ideal for baseband and pulse shaping applications in communication systems.
Occupied Bandwidth of a Raised Cosine Filter
| Roll-off β | Bandwidth Efficiency | Implementation |
|---|---|---|
| Low (e.g., 0.1) | High (narrow BW) | Harder (sharp filter) |
| High (e.g., 1.0) | Low (wider BW) | Easier (smooth filter) |
Further Reading
Time-Bandwidth Product and Pulse Shaping
The Time-Bandwidth Product (TBP), defined as $B \times T$, serves as the fundamental "energy budget" for signal processing. Governed by the limit $TBP \ge 0.5$, it dictates that a signal cannot be simultaneously narrow in both time and frequency. Modern pulse shaping techniques like Raised Cosine and Gaussian Filtering are the primary tools used to navigate this trade-off.
In high-speed data communications, the Raised Cosine Filter is utilized to eliminate Intersymbol Interference (ISI) by optimizing spectral efficiency. Similarly, Gaussian Filters in GMSK (typically with a $BT$ product of 0.3) prioritize spectral smoothness and a constant envelope, which is essential for power-efficient mobile transmitters.
The interconnection between these technologies is defined by system-specific requirements:
- Low TBP (0.3 – 1.5): Employed in 5G, Wi-Fi, and GSM to pack maximum data into narrow frequency bands.
- High TBP (> 10): Essential for Radar and Satellite systems. By using high TBP through techniques like Pulse Compression, these systems can transmit long, high-energy pulses (for range) while maintaining wide bandwidths (for resolution).
Ultimately, TBP is the master variable: engineers minimize it for spectral density in commercial wireless, but maximize it for robustness and sensitivity in radar and deep-space exploration.