Orthogonal Frequency Division Multiplexing
When a signal with high bandwidth traverses through a medium, it tends to disperse more compared to a signal with lower bandwidth.A high-bandwidth signal comprises a wide range of frequency components. Each frequency component may interact differently with the transmission medium due to factors such as attenuation, dispersion, and distortion. OFDM combats the high-bandwidth frequency selective channel by dividing the original signal into multiple orthogonal multiplexed narrowband signals. In this way it, overcomes the inter-symbol interferences (ISI) issue.
Block Diagram
‘k’ indicates kth position in a input symbol
N is the number of subcarriers
Example: (OFDM using QPSK)
1.
Input Parameters:
N Number of Input bits: 128
Number of subcarriers (FFT length): 64
Cyclic prefix length (CP): 8
Step-by-Step Process:
QPSK Mapping:
Each QPSK symbol represents 2 bits.
‘With 128 bits, the number of QPSK symbols generated will be 64 symbols.
2. OFDM Symbol Construction:
The 64 QPSK symbols exactly fit into 64 subcarriers, meaning we form one OFDM
symbol
3. IFFT Operation:
Each OFDM symbol (composed of 64 QPSK symbols) is transformed from the
frequency domain to the time domain using a 64-point IFFT.
The output of the IFFT is 64 complex time-domain samples per OFDM symbol.
4. Adding Cyclic Prefix:
A cyclic prefix of length 8 is appended to each 64-sample time-domain OFDM
symbol.
Therefore, each OFDM symbol with the cyclic prefix becomes 64 + 8 = 72 samples
long.
5. Total Length of OFDM Modulated Signal:
Since we have only one OFDM symbol in this example, the total length of the OFDM
modulated signal is 72 samples.
MATLAB Code for a simple OFDM system
% The code is written by SalimWireless.Com
clc;
clear all;
close all;
% Generate random bits
numBits = 100;
bits = randi([0, 1], 1, numBits);
% Define parameters
numSubcarriers = 4; % Number of subcarriers
numPilotSymbols = 3; % Number of pilot symbols
cpLength = ceil(numBits / 4); % Length of cyclic prefix (one-fourth of the data length)
% Add cyclic prefix
dataWithCP = [bits(end - cpLength + 1:end), bits];
% Insert pilot symbols
pilotSymbols = ones(1, numPilotSymbols); % Example pilot symbols (could be any pattern)
dataWithPilots = [pilotSymbols, dataWithCP];
% Perform OFDM modulation (IFFT)
dataMatrix = reshape(dataWithPilots, numSubcarriers, []);
ofdmSignal = ifft(dataMatrix, numSubcarriers);
ofdmSignal1 = reshape(ofdmSignal, 1, []);
% Display the generated data
disp("Original Bits:");
disp(bits);
disp("Data with Cyclic Prefix and Pilots:");
disp(dataWithPilots);
disp("OFDM Signal:");
disp(ofdmSignal1);
%%%%%%%%%%%%%%%%%%%%%%%%%%% Demodulation %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Perform FFT on the received signal
%ofdmSignal = awgn(ofdmSignal, 1000);
ofdmSignal = reshape(ofdmSignal1, numSubcarriers, []);
rxSignal = fft(ofdmSignal, numSubcarriers);
%rxSignal = [rxSignal(1,:) rxSignal(2,:) rxSignal(3,:) rxSignal(4,:)];
% Remove cyclic prefix
rxSignalNoCP = rxSignal(cpLength + 1:end);
% Extract data symbols and discard pilot symbols
dataSymbols = rxSignalNoCP(numPilotSymbols + 1:end);
% Demodulate the symbols using thresholding
threshold = 0;
demodulatedBits = (real(dataSymbols) > threshold);
figure(1)
stem(bits);
legend("Original Information Bits")
figure(2)
% Plot real part
hReal = stem(real(ofdmSignal1), 'r', 'DisplayName', 'Real Part'); % 'r' for red color
hold on;
% Plot imaginary part
hImag = stem(imag(ofdmSignal1), 'b', 'DisplayName', 'Imaginary Part'); % 'b' for blue color
% Customizing the plots
set(hReal, 'Marker', 'o', 'LineWidth', 1.5); % Real part marker and line style
set(hImag, 'Marker', 'x', 'LineWidth', 1.5); % Imaginary part marker and line style
% Add grid and other plot customizations
grid on;
title('Real and Imaginary Parts of OFDM Signal');
xlabel('Index');
ylabel('Amplitude');
legend;
hold off;
figure(3)
stem(demodulatedBits);
legend("Received Bits")
Output
OFDM is a scheme of multicarrier modulation. It's utilized to make greater use of the spectrum. Multiple carriers are used to modulate the message signal in this case. According to Nyquist's law, if the highest operational frequency is fmax, we must sample the signal at a rate of at least 2*fmax in order for the signal to be retrieved at the receiver properly. The signal's bandwidth B, on the other hand, will be 2*fmax.
Multi-path components, or MPCs, are seen while transmitting a signal in a wireless environment. MPCs are numerous copies of the same transmitted signal that arrive at the receiver with time delay or dispersion. If we send the second symbol immediately after the first, the second symbol will interact with the first symbol's time delayed MPCs. Excess delay spread refers to the time gap between the first and last MPCs. However, for measuring the time dispersion of multi path components, or MPCs, RMS delay spread is the most appropriate word. However, the RMS delay spread and the excess delay spread are not the same. The RMS delay spread is the power delay spread's second central momentum. In a wireless context, you've probably noticed signal power delay spread owing to multi-path. The relevant power weightage associated with MPCs is also taken into consideration by RMS delay spread.
Assume that the total bandwidth available is B. The duration of the symbol will then be 1/B. Signals at higher frequencies are subjected to additional reflection and refraction. As a result, more multipath is created, and the signal reaches the receiver via several reflections and refractions. RMS delay spread (say, Td) is substantially more than symbol time length (say, Ts) or Td>>Ts in such circumstances (for very high frequency). When the RMS delay spread is greater than the symbol time length, the symbol interacts with the MPCs of other symbols. This is what we term it technically. Inter-symbol interferences, or ISI, is a better word for this.
We divide the entire available bandwidth B into N number of sub-bands to eliminate inter-symbol interference. The bandwidth of each sub-band will be B/N. The symbol duration, Ts, will be 1/(B/N) if we do this. The symbol duration, Ts, 1/(B/N), will be significantly larger than 1/B, according to the calculations. N = 256, 512, and so on are common values. In the OFDM approach, we employ N point FFT for multi carrier modulation, or MCM.
Let me explain using a mathematical example: the RMS delay spread for an outdoor communication channel is typically 2 to 3 microseconds. If we use single carrier transmission with a transmission bandwidth of 10 MHz, the symbol time duration is Ts = 1/B or 0.1 microsecond. Td (=2 to 3 microsecond) is greater than Ts (=0.1 microsecond). Inter-symbol interference, or ISI, is the result of this.
If we divide the broadband bandwidth, B, into N sub-bands, the bandwidth of each sub-band becomes B/N, increasing the symbol time duration, Ts. We normally keep symbol duration periods 10 times longer than RMS delay spread for seamless communication. This rule is also known as the sigma rule of communication.
Diagram:
In the diagram above, a traditional single carrier communication system is depicted. B is the total bandwidth. If B = 10 MHz, Ts = 1/(10 MHz) = 0.1 microsecond symbol duration or symbol time. RMS delay spread, Td = 2 - 3 microsecond. As a result, the RMS delay spread is greater than the symbol time. As a result, the desired signal is not recoverable. So, in the next diagram, we're attempting to demonstrate that the entire bandwidth B is divided into N (say, 1000) portions.
Each sub band's bandwidth is now B/N. Multicarrier modulation is used to modulate the sub band message signal. If N = 1000, then each sub band has a bandwidth of (100 MHz)/1000 = 10 KHz. Each sub band's symbol time, Ts, is now equal to 1/(10KHz) = 100 microsecond. The symbol time is significantly higher than the critical RMS delay spread in this case. Theoretically, That is enough to remove ISI.
Filter Bank Multi-Carrier (FBMC)
'Filter Bank Multi-Carrier' is the abbreviation for 'Filter Bank Multi-Carrier.' To obtain the desired data in an OFDM system, we use inverse fast Fourier transform (IFFT) at the transmitter side, or we use the opposite method or fast Fourier transform at the receiver side. For OFDM, we use the term Tsym, which stands for symbol duration time. As we all know, it's a multicarrier modulation system in which we send a single high data rate signal instead of multiple low data rate signals in parallel. To cancel inter-symbol interference (ISI) in a communication system caused by fading, we divide the entire bandwidth B into N sub bands.
The subcarrier filters of the IFFT/FFT filter banks in OFDM have poor containment, which is one of the main drawbacks of the system. As a result, there is a lot of noise from other users' transmissions.
On the other hand, when transmitting a symbol, we must not only use the Tsym time period, but also add a cyclic prefix. As a result, this phenomenon has an impact on bandwidth efficiency.
Another explanation is that when we send a signal over a multicarrier system, the carrier signal behaves like a sinc wave. As a result, every subcarrier can interfere with the subcarriers before and after it.
In this case, FBMC resolves the concerns with the OFDM system. To differentiate the sub carriers, we utilise a digital filter. We also don't require the cyclic prefix in this case. Digital filters are sharp in nature, reducing interference between other subcarriers significantly.
# OFDM delay spread channel to parallel fading channel conversion
Further Reading
