1.3.5. Orthogonal Frequency Division Multiplexing (OFDM)
Orthogonal Frequency Division Multiplexing (OFDM)
Comprehensive Technical Guide with Mathematical Foundations and Practical Implementations
Wireless Communications
Digital Modulation
4G/5G Technologies
Table of Contents
1. OFDM Fundamentals
1.1 What is OFDM?
Orthogonal Frequency Division Multiplexing (OFDM) is a digital multi-carrier modulation scheme that has become the foundation for most modern wireless communication systems. Its key characteristics include:
- Parallel Transmission: Divides a high-rate data stream into N parallel low-rate streams transmitted simultaneously on orthogonal subcarriers
- Spectral Efficiency: Subcarriers are packed closely together (Δf = 1/Tu) with overlapping spectra but zero interference due to orthogonality
- Robustness: Converts frequency-selective fading channels into multiple flat-fading subchannels
- Implementation: Efficient realization using Fast Fourier Transform (FFT) algorithms
Historical Note: OFDM concepts date back to the 1960s (Chang, 1966), but practical implementations only became feasible with the advent of digital signal processors capable of efficient FFT computation in the 1990s.
1.2 Mathematical Foundation
The OFDM signal can be represented as:
\[
s(t) = \sum_{k=0}^{N-1} X_k e^{j2\pi f_k t} \cdot \text{rect}\left(\frac{t-T_u/2}{T_u}\right)
\]
Where:
\(X_k\) = Complex symbol for subcarrier k
\(f_k = f_c + k/T_u\) = Frequency of subcarrier k
\(T_u\) = Useful symbol duration
\(N\) = Number of subcarriers
\(X_k\) = Complex symbol for subcarrier k
\(f_k = f_c + k/T_u\) = Frequency of subcarrier k
\(T_u\) = Useful symbol duration
\(N\) = Number of subcarriers
The orthogonality condition ensures:
\[
\frac{1}{T_u} \int_0^{T_u} e^{j2\pi(f_k-f_l)t} dt = \delta[k-l]
\]
Where \(\delta[k-l]\) is the Kronecker delta function (1 when k=l, 0 otherwise)
1.3 Why Use OFDM? Comparative Advantages
+=================================================================+
| Comparative Analysis of Modulation Schemes |
+=================================================================+
| |
| Feature | OFDM | SC-FDE | CDMA |
|------------------------+-----------+------------+---------------|
| Spectral Efficiency | High | Moderate | Low |
| Multipath Robustness | Excellent | Good | Poor |
| Implementation | FFT-based | Equalizer | Correlators |
| PAPR | High | Moderate | Low |
| Synchronization | Sensitive | Moderate | Robust |
| MIMO Compatibility | Excellent | Good | Limited |
+=================================================================+
Key advantages that make OFDM the preferred choice for modern systems:
- Multipath Immunity: Long symbol duration with cyclic prefix makes OFDM robust against delay spread
- Adaptive Modulation: Different modulation schemes (QPSK, 16QAM, 64QAM) can be used per subcarrier based on channel conditions
- Scalability: Easily extends to MIMO configurations (e.g., 802.11n/ac/ax, LTE, 5G)
- Computational Efficiency: FFT implementations reduce complexity from O(N²) to O(NlogN)
2. OFDM System Architecture
2.1 Detailed Transmitter Architecture
+====================================================================+
| OFDM Transmitter (Detailed) |
+====================================================================+
| |
| +---------+ +---------+ +-------+ +-------+ +------+ |
| | Input | -> | Channel | -> | QAM | -> | IFFT | -> | Cyclic| |
| | Bits | | Coding | | Mapper| | | | Prefix| |
| +---------+ +---------+ +-------+ +-------+ +------+ |
| | | | | | |
| v v v v v |
| Source Data Reed-Solomon Symbol Time Domain Guard |
| or LDPC Coding Mapping Signal Interval|
| |
| +---------+ +---------+ +-------+ +-------+ |
| | Pilot | -> | Channel | -> | Guard | -> | DAC | -> RF Chain |
| | Insert | | Est. | | Band | | | |
| +---------+ | Symbols | +-------+ +-------+ |
| +---------+ |
| |
| Key Parameters: |
| - FFT Size: 64/128/256/512/1024 |
| - CP Length: 1/4, 1/8, 1/16, 1/32 of symbol duration |
| - Pilot Pattern: Block/Comb type, density 5-20% |
+====================================================================+
2.1.1 Channel Coding
Modern OFDM systems employ sophisticated channel coding:
- Convolutional Codes: Used in early systems (802.11a/g) with constraint length 7
- Turbo Codes: Used in 3G/4G with iterative decoding
- LDPC: Used in 802.11n/ac/ax and 5G with near-Shannon limit performance
- Polar Codes: Used in 5G control channels
2.1.2 Constellation Mapping
Common modulation schemes in OFDM:
| Modulation | Bits/Symbol | SNR Requirement | Usage |
|---|---|---|---|
| BPSK | 1 | 6 dB | Control channels |
| QPSK | 2 | 9 dB | Robust data transmission |
| 16QAM | 4 | 16 dB | Medium-rate data |
| 64QAM | 6 | 22 dB | High-rate data (good SNR) |
| 256QAM | 8 | 28 dB | 802.11ac/ax, 5G |
2.2 Detailed Receiver Architecture
+====================================================================+
| OFDM Receiver (Detailed) |
+====================================================================+
| |
| +-------+ +-------+ +---------+ +-------+ +---------+ |
| | ADC | -> | Sync | -> | Remove | -> | FFT | -> | Channel | |
| | | | | | CP | | | | Est. | |
| +-------+ +-------+ +---------+ +-------+ +---------+ |
| | | | | | |
| v v v v v |
| Sampling Symbol Guard Interval Freq. Domain Pilot |
| & Quant. Timing Sync Removal Conversion Proc. |
| |
| +---------+ +---------+ +---------+ +---------+ |
| | Equalize| -> | Demod | -> | Channel | -> | Parallel| -> Bits |
| | | | | | Decoder | | to Serial| |
| +---------+ +---------+ +---------+ +---------+ |
| |
| Key Algorithms: |
| - Synchronization: Schmidl & Cox, Minn algorithms |
| - Channel Estimation: LS, MMSE, DFT-based |
| - Equalization: Zero-forcing, MMSE, Decision-directed |
+====================================================================+
2.2.1 Synchronization Challenges
OFDM receivers must address:
- Symbol Timing: Finding the correct FFT window position
- Carrier Frequency Offset (CFO): Caused by oscillator mismatches (±0.1 ppm typical)
- Sampling Clock Offset: Causing phase rotation across subcarriers
\[
\text{Normalized CFO} = \frac{\Delta f}{1/T_u} = \epsilon_I + \epsilon_f
\]
Where \(\epsilon_I\) is integer CFO and \(\epsilon_f\) is fractional CFO
2.3 Resource Allocation in OFDMA
Orthogonal Frequency Division Multiple Access (OFDMA) extends OFDM to support multiple users:
+====================================================================+
| OFDMA Resource Allocation Framework |
+====================================================================+
| |
| Allocation Type | Description | Example |
|--------------------+---------------------------------+-------------|
| Static | Fixed subcarrier assignment | Early WiMAX |
| Dynamic | Channel-aware allocation | LTE, 5G |
| Cluster-based | Groups of subcarriers | LTE PUSCH |
| Interleaved | Distributed subcarriers | LTE PUCCH |
| Contiguous | Adjacent subcarriers | 5G eMBB |
| |
| Scheduling Metrics: |
| - Channel quality (CQI reports) |
| - QoS requirements (latency, throughput) |
| - Fairness considerations (proportional fairness) |
+====================================================================+
3. Core OFDM Concepts
3.1 Orthogonality Principle in Depth
The orthogonality condition ensures that the integral of any two different subcarriers over the symbol period is zero:
\[
\int_0^{T_u} \exp\left(j2\pi\frac{k}{T_u}t\right) \exp\left(-j2\pi\frac{l}{T_u}t\right) dt =
\begin{cases}
T_u & \text{if } k = l \\
0 & \text{if } k \neq l
\end{cases}
\]
This property enables:
- Spectrum Overlap: Subcarriers can overlap without interference
- Efficient Demodulation: Simple correlation receiver can separate subcarriers
- Flexible Design: Subcarrier spacing can be adjusted based on channel conditions
3.2 Cyclic Prefix: Theory and Practice
The cyclic prefix (CP) is a guard interval that:
- Eliminates inter-symbol interference (ISI) from multipath
- Maintains circular convolution needed for FFT processing
- Simplifies channel equalization to per-subcarrier scaling
\[
\text{CP Overhead} = \frac{T_{CP}}{T_u + T_{CP}} \times 100\%
\]
Typical values range from 6.25% (1/16) to 25% (1/4)
Trade-off Warning: While longer CP provides better multipath immunity, it reduces spectral efficiency. System designers must balance these factors based on expected channel conditions.
3.3 FFT Implementation Considerations
Practical FFT implementations require attention to:
| Parameter | Considerations |
|---|---|
| FFT Size | Power of 2 for radix-2 algorithms; tradeoff between granularity and latency |
| Window Function | Raised cosine window reduces out-of-band emissions |
| Numerical Precision | 16-bit fixed point common in hardware implementations |
| Memory Requirements | Butterfly computation needs intermediate storage |
3.3.1 FFT Complexity Analysis
The computational complexity of N-point FFT is:
\[
\text{Complexity} = \frac{N}{2}\log_2 N \text{ complex multiplications}
\]
For N=2048 (5G NR), this requires 11,264 complex multiplications per OFDM symbol
4. Channel Handling in OFDM
4.1 Multipath Channel Modeling
Multipath channels are characterized by:
\[
h(\tau,t) = \sum_{i=0}^{L-1} \alpha_i(t)\delta(\tau - \tau_i(t))
\]
Where \(\alpha_i(t)\) are complex gains and \(\tau_i(t)\) are path delays
Key parameters:
- Coherence Bandwidth: \(B_c \approx 1/(5\tau_{rms})\)
- Delay Spread: \(\tau_{max}\) determines required CP length
- Doppler Spread: Causes ICI if significant compared to subcarrier spacing
4.2 Advanced Channel Estimation Techniques
Beyond simple pilot-based estimation, modern systems use:
+====================================================================+
| Channel Estimation Methods |
+====================================================================+
| |
| Method | Algorithm | Complexity | Accuracy |
|------------------+------------------------+------------+-----------|
| LS Estimation | Ĥ = Y/X | Low | Low |
| MMSE Estimation | RHH(RHH+σ²I)-1ĤLS | High | High |
| DFT-based | IDFT thresholding | Medium | Medium |
| Decision-Directed| Use decoded symbols | Variable | High (if |
| | | | low BER) |
+====================================================================+
4.2.1 Pilot Patterns
Common pilot arrangements:
- Comb Type: Pilots on certain subcarriers in all symbols
- Block Type: All subcarriers in certain symbols are pilots
- Staggered: Combination of comb and block (LTE CRS)
4.3 Equalization Techniques
Frequency-domain equalization options:
| Method | Operation | Advantages | Disadvantages |
|---|---|---|---|
| Zero Forcing | X̂ = Y/Ĥ | Simple | Noise enhancement |
| MMSE | X̂ = YĤ*/(|Ĥ|²+σ²) | Optimal SNR | Requires noise estimate |
| Decision Feedback | Iterative correction | Improved performance | Error propagation |
5. Peak-to-Average Power Ratio (PAPR)
5.1 PAPR Fundamentals
The PAPR of an OFDM signal with N subcarriers is:
\[
\text{PAPR} = \frac{\max_{0\leq t \leq T}|s(t)|^2}{E[|s(t)|^2]}
\]
For N subcarriers, theoretical maximum PAPR is 10log10(N) dB
PAPR distribution characteristics:
- For N=64, PAPR > 10 dB occurs with probability ~10-4
- PAPR increases with number of subcarriers
- Higher-order QAM shows slightly higher PAPR than QPSK
5.2 PAPR Reduction Techniques Comparison
+====================================================================+
| PAPR Reduction Techniques |
+====================================================================+
| |
| Technique | PAPR Reduction | Side Info | Complexity | BER |
|------------------+----------------+-----------+------------+-------|
| Clipping | 3-5 dB | No | Low | High |
| SLM | 4-6 dB | Yes | High | None |
| PTS | 5-7 dB | Yes | Very High | None |
| Tone Injection | 3-4 dB | No | Medium | None |
| DFT-spreading | 4-5 dB | No | Medium | None |
+====================================================================+
5.2.1 Selected Mapping (SLM) Algorithm
SLM implementation steps:
- Generate U different phase sequences Φ(u) = [ejφ0,...,ejφN-1]
- Create U candidate signals: s(u)(t) = IFFT{X·Φ(u)}
- Select signal with lowest PAPR: uopt = argminu(PAPR(s(u)))
- Transmit s(uopt)(t) along with side information about uopt
5.3 Inter-Carrier Interference (ICI) Analysis
Main ICI sources and their impact:
| Source | Effect | Mitigation |
|---|---|---|
| Frequency Offset | SINR degradation ~10log10(1+0.594(πε)2) | AFC loops, pilot tracking |
| Phase Noise | Common phase error + ICI | Phase tracking, robust designs |
| Doppler Spread | Frequency dispersion | Adaptive subcarrier spacing |
6. Real-World OFDM Implementations
6.1 Wi-Fi Evolution (802.11a to 802.11ax)
| Standard | FFT Size | Active Subcarriers | Bandwidth | Modulation | MIMO |
|---|---|---|---|---|---|
| 802.11a/g | 64 | 52 | 20 MHz | Up to 64QAM | 1×1 |
| 802.11n | 64 | 56 | 20/40 MHz | Up to 64QAM | 4×4 |
| 802.11ac | 256 | 234 | 80/160 MHz | Up to 256QAM | 8×8 |
| 802.11ax | 256/512 | 234/468 | 80/160 MHz | Up to 1024QAM | 8×8 |
6.2 Cellular Standards: LTE vs 5G NR
+====================================================================+
| LTE vs 5G NR OFDM Parameters |
+====================================================================+
| |
| Parameter | LTE | 5G NR |
|-----------------------+---------------------+---------------------|
| Subcarrier Spacing | 15 kHz fixed | 15/30/60/120/240 kHz|
| Frame Structure | 10 ms radio frame | Flexible (2n×0.5 ms) |
| Bandwidth | Up to 20 MHz | Up to 400 MHz |
| Waveform (UL) | SC-FDMA | DFT-s-OFDM/CP-OFDM |
| Numerology | Single (Δf=15 kHz) | Multiple (μ=0...4) |
| Max FFT Size | 2048 | 4096 |
+====================================================================+
6.2.1 5G Numerology
5G introduces scalable numerology with parameter μ:
\[
\Delta f = 2^\mu \times 15 \text{ kHz}, \quad \mu \in \{0,1,2,3,4\}
\]
μ=0: eMBB, μ=3: URLLC, μ=4: mmWave
6.3 Digital TV Standards
OFDM in broadcast systems:
- DVB-T/T2: 2k/8k modes, 4-256QAM, hierarchical modulation
- ATSC 3.0: 2048/4096 FFT, LDM (layered division multiplexing)
- ISDB-T: Band-segmented transmission, 4k mode
7. OFDM Simulation in MATLAB/Python
7.1 Basic OFDM Transmitter
# Python OFDM Transmitter Example
import numpy as np
def ofdm_tx(bit_stream, N, cp_len, mod_order=4):
# Parameters
M = mod_order # QAM order
bits_per_symbol = int(np.log2(M))
# Modulation
symbols = qam_modulate(bit_stream, M)
# Reshape to OFDM symbols
num_symbols = len(symbols) // N
ofdm_symbols = symbols[:num_symbols*N].reshape(num_symbols, N)
# IFFT
time_domain = np.fft.ifft(ofdm_symbols, axis=1)
# Add cyclic prefix
tx_signal = np.hstack([time_domain[:, -cp_len:], time_domain])
return tx_signal.flatten()
def qam_modulate(bits, M):
# Implement QAM modulation
# Returns complex symbols
pass
7.2 Channel Modeling
% MATLAB Multipath Channel Simulation
function [rx_signal, channel] = multipath_channel(tx_signal, snr_db)
% Typical urban channel model (3GPP TU)
path_delays = [0 50 120 200 230 500 1600 2300 5000]*1e-9; % in seconds
avg_path_gains = [-1.0 -1.0 -1.0 0.0 0.0 0.0 -3.0 -5.0 -7.0]; % in dB
% Convert to linear scale
path_gains = 10.^(avg_path_gains/10);
% Create channel impulse response
max_delay = max(path_delays);
channel = zeros(1, ceil(max_delay*fs)+1);
for i = 1:length(path_delays)
pos = round(path_delays(i)*fs)+1;
channel(pos) = sqrt(path_gains(i))*(randn + 1i*randn)/sqrt(2);
end
% Apply channel
rx_signal = conv(tx_signal, channel);
% Add AWGN
rx_signal = awgn(rx_signal, snr_db, 'measured');
end
7.3 PAPR Analysis
# Python PAPR Calculation
import numpy as np
import matplotlib.pyplot as plt
def calculate_papr(signal):
peak = np.max(np.abs(signal)**2)
avg = np.mean(np.abs(signal)**2)
return 10*np.log10(peak/avg)
def plot_ccdf(papr_values):
# Calculate CCDF
x = np.sort(papr_values)
y = 1 - np.arange(len(x))/float(len(x))
# Plot
plt.figure()
plt.semilogy(x, y)
plt.xlabel('PAPR [dB]')
plt.ylabel('Probability (PAPR > x)')
plt.title('PAPR CCDF')
plt.grid()
plt.show()
# Example usage
ofdm_signal = generate_ofdm_signal()
papr = calculate_papr(ofdm_signal)
print(f"PAPR: {papr:.2f} dB")
8. Advanced OFDM Topics
8.1 MIMO-OFDM
Combining OFDM with multiple antennas:
\[
\mathbf{Y}[k] = \mathbf{H}[k]\mathbf{X}[k] + \mathbf{N}[k]
\]
Where H[k] is the frequency-domain MIMO channel matrix at subcarrier k
Key techniques:
- Spatial Multiplexing: Transmit independent streams (V-BLAST)
- Space-Time Coding: Alamouti coding across OFDM symbols
- Beamforming: Precoding based on channel state information
8.2 Filter Bank Multicarrier (FBMC)
Alternative to CP-OFDM with better spectral containment:
\[
s(t) = \sum_{m=0}^{M-1}\sum_{n=-\infty}^{\infty} a_{m,n}g(t-nT)e^{j2\pi mF(t-nT)}e^{j\phi_{m,n}}
\]
Where g(t) is the prototype filter with good time-frequency localization
8.3 Non-Orthogonal Multiple Access (NOMA)
Power-domain multiplexing in OFDM systems:
- Superposition Coding: Users share same subcarriers with different power levels
- SIC Receiver: Successive interference cancellation
- Benefits: Improved spectral efficiency, massive connectivity
8.4 OTFS (Orthogonal Time Frequency Space)
2D modulation in delay-Doppler domain:
\[
x[\ell,k] = \frac{1}{NM}\sum_{n=0}^{N-1}\sum_{m=0}^{M-1}X[n,m]e^{j2\pi(\frac{nk}{N}-\frac{m\ell}{M})}
\]
Provides robustness in high-mobility scenarios
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