For a typical wireless communication system, we use modulation schemes and filters before transmitting the signal. The main purpose of using it is to transmit a proper waveform so that we can recover the signal at the receiving end more accurately.
If the roll-off factor is α, then
Bandwidth (B) = (1 + α) / (2 * T)
where T is the time interval. The filter response is zero outside that.
The roll-off factor is a parameter used to shape the spectrum of a digital signal in communication systems, and it is not just the product of time and bandwidth. It affects both the time and frequency domain characteristics of the signal.
Example
According to the Nyquist criterion, the sampling frequency of a signal must be at least twice the highest frequency present in the message signal. Conversely, during signal transmission, the bandwidth of the transmitted signal must be at least half the symbol rate to ensure inter-symbol interference (ISI)-free transmission. A raised cosine filter facilitates this requirement.
For example, if the symbol rate is 100 symbols per second, the minimum bandwidth required for ISI-free transmission is: 100 / 2 = 50 Hz
In simple terms, the symbol rate indicates that symbols are changing 100 times per second. To recover the transmitted signal at the receiver end without ISI, the minimum transmission bandwidth required is 50 Hz.
The bandwidth of a raised cosine filter is given by the formula:
where α is the roll-off factor of the filter. If the roll-off factor α is 0.25, the bandwidth is calculated as:
This bandwidth (62.5 Hz) exceeds the minimum requirement of 50 Hz for transmitting a signal at a symbol rate of 100 symbols per second.
MATLAB Code for the example above
clc;
clear;
close all;
% Parameters
fs = 1000; % Sampling frequency in Hz
symbolRate = 100; % Symbol rate (baud)
span = 6; % Filter span in symbols
alpha = 0.25; % Roll-off factor for raised cosine filter
% Generate random data symbols
numSymbols = 100; % Number of symbols
data = randi([0 1], numSymbols, 1) * 2 - 1; % Generate random binary data (BPSK symbols: -1, 1)
% Upsample the data to match sampling rate
samplesPerSymbol = fs / symbolRate; % Samples per symbol based on fs and symbol rate
dataUpsampled = upsample(data, samplesPerSymbol);
% Create a raised cosine filter
rcFilter = rcosdesign(alpha, span, samplesPerSymbol, 'sqrt'); % Square root raised cosine filter
% Apply the filter to the upsampled data
txSignal = conv(dataUpsampled, rcFilter, 'same');
figure;
subplot(4,1,1)
stem(data);
title('Original Message signal');
grid on;
subplot(4,1,2)
plot(dataUpsampled);
title('Upsampled Message signal');
grid on;
subplot(4,1,3)
plot(rcFilter);
title('Raise Cosine Filter Coefficient');
grid on;
subplot(4,1,4)
plot(txSignal);
title('Transmitted Signal after Raised Cosine Filtering');
grid on;
Output
MATLAB code for raise-cosine filter
Result
Application
A raised cosine filter is used for pulse shaping. You might have noticed in most of the diagrams of 'communication systems.' It is common to use this type of filter after the modulation module.
Further Reading
Time Bandwidth Product (TBP)
The Time-Bandwidth Product (TBP) is a fundamental parameter in signal processing and digital communications that quantifies the relationship between a signal's temporal duration and its occupied bandwidth. It is mathematically expressed as TBP = Δt × Δf, where Δt represents the effective signal duration and Δf denotes its frequency bandwidth. The TBP illustrates the inherent trade-off between time localization and frequency localization, as described by Heisenberg's uncertainty principle. A signal cannot be made arbitrarily short in time while simultaneously occupying an extremely narrow bandwidth.
This study examines the Time-Bandwidth Product using two commonly employed pulse-shaping techniques: the raised cosine filter and the Gaussian pulse. For a raised cosine filter, the bandwidth depends on the symbol rate and roll-off factor, while the signal duration is determined by the filter span. Increasing either the roll-off factor or the filter span improves pulse shaping and reduces intersymbol interference (ISI); however, it also increases the TBP, thereby lowering spectral efficiency. Conversely, smaller TBP values indicate more efficient utilization of the available spectrum but require more precise system design.
Gaussian pulses represent the theoretical optimum because they achieve the minimum achievable time-bandwidth product permitted by the uncertainty principle. Their energy is optimally concentrated in both the time and frequency domains, making them valuable in applications such as optical communications, radar, and ultra-wideband systems. MATLAB simulations are employed to compute bandwidth, signal duration, and TBP, while interactive visualizations demonstrate how variations in filter parameters influence signal characteristics. Overall, the analysis highlights the importance of the Time-Bandwidth Product in balancing bandwidth efficiency, pulse duration, interference suppression, and overall communication system performance.