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.
Example: OFDM using QPSK
1. Input Parameters:
- Number of Input bits: 128
- Number of subcarriers (FFT length): 64
- Cyclic prefix length (CP): 8
2. 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.
OFDM Symbol Construction: The 64 QPSK symbols exactly fit into 64 subcarriers, forming one OFDM symbol.
IFFT Operation: The symbol is transformed from the frequency domain to the time domain using a 64-point IFFT, resulting in 64 complex time-domain samples.
Adding Cyclic Prefix: A cyclic prefix of length 8 is appended. The final symbol length is 64 + 8 = 72 samples.
MATLAB Code for OFDM System
% Updated MATLAB Code by SalimWireless.Com
clc; clear all; close all;
numBits = 128;
bits = randi([0, 1], 1, numBits);
% Parameters
numSubcarriers = 64;
cpLength = 16;
% 1. QPSK Modulation (Mapping bits to symbols)
symbols = (2*bits(1:2:end)-1) + j*(2*bits(2:2:end)-1);
% 2. Reshape into OFDM symbols
dataMatrix = reshape(symbols, numSubcarriers, []);
% 3. IFFT (Frequency to Time Domain)
ofdmTime = ifft(dataMatrix, numSubcarriers);
% 4. Add Cyclic Prefix (Correct way: copy end of time signal to front)
ofdmWithCP = [ofdmTime(end-cpLength+1:end, :); ofdmTime];
ofdmSignal = ofdmWithCP(:).';
% Display Signal
figure(1); stem(real(ofdmSignal)); title('OFDM Time Domain Signal (Real Part)');
OFDM is a scheme of multicarrier modulation used to make better use of the spectrum. According to Nyquist's sampling theorem, to perfectly reconstruct a baseband signal with a maximum frequency component of fmax, we must sample at 2*fmax.
In wireless environments, Multi-path components (MPCs) cause time dispersion. RMS delay spread (Td) is used to measure this. If Td > Ts (Symbol duration), symbols overlap, causing Inter-Symbol Interference (ISI). OFDM mitigates this by dividing the total bandwidth B into N sub-bands, which increases the symbol duration significantly (Ts = N/B), making it much larger than the delay spread.
Q & A and Summary
1. What is the main advantage of using OFDM?
The main advantage is its ability to overcome ISI caused by frequency-selective fading. It transforms a wideband frequency-selective channel into multiple narrowband frequency-flat channels.
3. What role does the IFFT play in OFDM?
IFFT converts frequency-domain data symbols into time-domain signals. It ensures that subcarriers remain orthogonal. Mathematical representation:
$$ y(t) = \sum_{k=0}^{N-1} X_k e^{j2\pi k t / T} $$
4. What is the function of the cyclic prefix?
The cyclic prefix provides a guard interval that prevents overlap between symbols due to multipath propagation, thus maintaining orthogonality. It must be longer than the maximum delay spread.
5. How does high Doppler shift affect OFDM?
High Doppler shift causes frequency shifts in subcarriers, leading to Inter-Carrier Interference (ICI) and loss of orthogonality. Frequency shift equation:
$$ \Delta f = f_0 \cdot \frac{v}{c} $$
6. What is "orthogonality" in OFDM?
It ensures subcarriers do not interfere with each other even though they overlap in frequency. The integral of any two subcarriers over a symbol period is zero.
7. What are the transmitter steps?
1. Data Encoding. 2. Serial-to-Parallel. 3. IFFT. 4. Cyclic Prefix Addition. 5. D/A Conversion. 6. Upconversion.
9. Explain frequency-selective fading.
It occurs when different frequencies experience different levels of attenuation. OFDM solves this by using narrowband subcarriers that experience "flat" fading instead.