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Orthogonal Frequency Division Multiplexing (OFDM)

1. System Overview & Block Diagram

OFDM Block Diagram

  • OFDM splits total bandwidth \(B\) into \(N\) narrow, orthogonal subcarriers.
  • Each subcarrier transmits a low-rate QAM/PSK symbol.
  • Subcarrier spacing: \(\Delta f = \frac{1}{T}\), where \(T\) = useful OFDM symbol duration (without cyclic prefix).

Time-domain OFDM symbol (without CP):

\[ s[n] = \sum_{k=0}^{N-1} S_k \exp\left(j2\pi \frac{k n}{N}\right), \quad n = 0, 1, ..., N-1 \]

2. Discrete Implementation & DFT/IDFT

  • IFFT: Converts frequency domain QAM symbols to time domain OFDM symbol.
  • FFT: At receiver, recovers subcarrier symbols.
  • Cyclic prefix (CP): Appends last \(L_{cp}\) samples of each OFDM symbol to its front.

DFT Properties:

  • DFT: \(X[i] = \frac{1}{\sqrt{N}} \sum_{n=0}^{N-1} x[n] e^{-j2\pi n i / N}\)
  • IDFT: \(x[n] = \frac{1}{\sqrt{N}} \sum_{i=0}^{N-1} X[i] e^{j2\pi n i / N}\)
  • Circular convolution: In frequency, becomes pointwise multiplication.

3. ISI and Cyclic Prefix

  • ISI (Inter-Symbol Interference) occurs in frequency-selective fading.

  • Cyclic Prefix:

  • If \(L_{cp} \geq L_{ch}\) (channel length), linear convolution becomes circular, eliminating ISI.

  • Received symbol after FFT:

$$ Y_k = H_k S_k + W_k $$

Where \(H_k\) = channel response at subcarrier \(k\), \(W_k\) = noise.

  • Spectral efficiency penalty: CP overhead reduces net data rate by \(N/(N + L_{cp})\).

4. Inter-Carrier Interference (ICI)

a. Ideal Synchronization

  • Subcarriers are orthogonal:

$$ \frac{1}{N} \sum_{n=0}^{N-1} e^{j2\pi (k-l)n/N} = \delta_{kl} $$ * No ICI when time/frequency perfectly aligned.

b. ICI Due to Frequency Offset

  • Frequency offset (\(\epsilon\), normalized to subcarrier spacing):

$$ r[n] = s[n] e^{j2\pi \epsilon n / N} $$ * After FFT:

$$ Y_k = \sum_{l=0}^{N-1} S_l \cdot \mathrm{ICI}(k, l, \epsilon) $$

with

$$ \mathrm{ICI}(k, l, \epsilon) = \frac{\sin(\pi (l - k + \epsilon))}{N \sin \left( \frac{\pi}{N}(l - k + \epsilon) \right)} \exp \left( j\pi \frac{N-1}{N}(l - k + \epsilon) \right) $$ * Effect: Frequency offset causes energy leakage (ICI), rotating/distorting constellation points.

c. ICI Due to Timing Offset

  • Timing offset (\(\tau\)):

$$ r[n] = s[n-\tau] $$

$$ Y_k = S_k e^{-j2\pi k \tau / N} + \text{ICI terms} $$ * If \(\tau < L_{cp}\): Only phase rotation; no ISI. * If \(\tau \geq L_{cp}\): Both ISI and ICI appear.

5. Constellation Rotation & SNR Degradation

  • Frequency offset (\(\epsilon\)): Rotates constellation by \(2\pi \epsilon\) per symbol.
  • Timing offset (\(\tau\)): Rotates constellation by \(2\pi k \tau / N\) per subcarrier.
  • Both effects increase error rate and reduce SNR.

6. Peak-to-Average Power Ratio (PAPR)

\[ \mathrm{PAPR} = \frac{\max |s[n]|^2}{\mathbb{E}[|s[n]|^2]} \]
  • OFDM signals have high PAPR (up to \(N\) for \(N\) subcarriers).
  • Requires linear power amplifiers, reducing power efficiency.
  • Mitigation: Clipping, peak cancellation, special coding, etc.

7. Adaptive Loading & Water-Filling

  • Water-filling algorithm: Allocates power/rate based on subcarrier SNR \(\gamma_i\):
\[ P_i = \left( \frac{1}{\gamma_0} - \frac{N_0 B_N}{|H_i|^2} \right)^+ \]
\[ C = \sum_{i=1}^N B_N \log_2 \left(1 + \frac{|H_i|^2 P_i}{N_0 B_N}\right) \]
  • Improves overall rate in frequency-selective channels.

8. Diversity, Coding & Equalization

  • Channel coding and interleaving spread errors over subcarriers (frequency diversity).
  • Precoding and equalization can compensate for fading, at the cost of complexity and required channel state information (CSI).

9. Summary Table

Feature Effect/Formula
OFDM Signal \(s[n] = \sum_k S_k e^{j2\pi k n / N}\)
ISI Elimination CP: \(L_{cp} \geq L_{ch}\)
ICI from Freq Offset Increases with offset; see formula above
ICI from Timing Offset Phase shift; ISI if offset \(> CP\)
PAPR High (\(\sim N\)); reduces PA efficiency
Water-filling ( P_i = \left( \frac{1}{\gamma_0} - \frac{N_0 B_N}{ H_i ^2} \right)^+ )
Constellation Effect Rotated/distorted; higher error rates

10. Key Drawbacks of OFDM

Drawback Explanation
High PAPR Requires highly linear, less efficient amplifiers.
Sensitivity to Freq/Phase Errors Frequency/phase offset and Doppler cause ICI, degrade orthogonality.
CP Overhead Reduces net spectral efficiency due to redundant samples.
Complexity Needs FFT/IFFT, accurate synchronization, and digital processing.
Nonlinearity Sensitivity High PAPR increases distortion from analog/RF components.
Spectral Leakage Significant sidelobes increase out-of-band emissions unless windowing/filtering is used.
Channel Estimation Required Accurate and fast channel tracking needed for mobility/fast fading.

11. OFDM vs. Vector Coding

  • Vector Coding (VC): SVD-based decomposition for ISI-free parallel channels.

  • Theoretically optimal for ISI elimination, but requires full channel knowledge and is computationally expensive.

  • Not practical for real-time wireless systems.
  • OFDM: Efficient, does not need full CSI at transmitter, uses CP for simple ISI removal, widely adopted.

12. Case Study: IEEE 802.11a (WiFi)

  • Bandwidth: 20 MHz (5 GHz band)
  • Subcarriers: 64 (48 data, 4 pilots, 12 null)
  • CP Length: 16 samples (\(\approx 0.8\,\mu\text{s}\))
  • OFDM Symbol Time: 4 \(\mu\)s (80 samples at 20 MHz)
  • Data Rates: 6–54 Mbps (varies with modulation and coding)
  • Modulations: BPSK, QPSK, 16QAM, 64QAM
  • Coding rates: 1/2, 2/3, 3/4 (convolutional)

13. References

  • D. Tse, P. Viswanath, Fundamentals of Wireless Communication
  • J. Proakis, Digital Communications
  • Wikipedia: OFDM

_Last updated: June 06, 2025