OFDM Basics

Orthogonal Frequency Division Multiplexing (OFDM) is a multicarrier modulation technique that divides a high-rate data stream into multiple lower-rate streams, each transmitted simultaneously on separate subcarriers. It is the backbone of modern communication standards including 4G LTE, 5G NR, Wi-Fi, and digital broadcasting. Understanding OFDM is essential for GATE aspirants and students studying advanced digital communications.

Core Concept of OFDM

In a conventional single-carrier system, if the channel bandwidth is large relative to the coherence bandwidth, different frequency components of the signal experience different fading, causing intersymbol interference (ISI). OFDM solves this by splitting the total bandwidth into N narrowband subcarriers, each occupying a bandwidth much smaller than the coherence bandwidth. This converts a frequency-selective channel into multiple flat-fading subchannels.

The defining property of OFDM is orthogonality between subcarriers. Two sinusoids are orthogonal if their inner product over a symbol period T is zero. Mathematically, if subcarrier frequencies are chosen as fk = f0 + k/T for k = 0, 1, ..., N-1, then any two subcarriers fk and fm satisfy the orthogonality condition, meaning their spectra overlap yet do not interfere with each other when sampled correctly.

The subcarrier spacing is delta_f = 1/T where T is the useful OFDM symbol duration. At the peak of any one subcarrier's spectrum, all other subcarriers have exactly zero amplitude. This is why subcarriers can overlap in frequency without causing inter-carrier interference (ICI), making OFDM spectrally far more efficient than traditional frequency division multiplexing (FDM) where guard bands are wasted between channels.

Mathematical Expression

An OFDM signal in the time domain is expressed as the sum of all modulated subcarriers. If Xk is the complex data symbol on subcarrier k, the baseband OFDM signal over one symbol period is:

x(t) = (1/N) * sum(k=0 to N-1) [ Xk * exp(j * 2*pi * k * t / T) ]

This equation is exactly the Inverse Discrete Fourier Transform (IDFT) of the sequence {X0, X1, ..., XN-1}. At the receiver, a DFT recovers the original data symbols. In practice, the IDFT and DFT are implemented efficiently using the IFFT and FFT algorithms, making OFDM computationally practical even for large N.

The cyclic prefix (CP) is a guard interval added at the start of each OFDM symbol by copying the last Ncp samples of the symbol to the front. If CP length exceeds the channel's maximum delay spread, ISI is completely eliminated. The CP also converts linear convolution with the channel into circular convolution, ensuring the FFT-based equalization works correctly.

Practical Understanding

In 4G LTE, the subcarrier spacing is 15 kHz and FFT sizes range from 128 to 2048 depending on bandwidth. The CP is typically 4.7 microseconds, which handles most urban multipath delays. In 5G NR, numerology is flexible, allowing subcarrier spacings of 15, 30, 60, 120, or 240 kHz to accommodate different use cases from IoT to mmWave.

OFDM has some practical drawbacks. The Peak to Average Power Ratio (PAPR) of an OFDM signal can be very high because many subcarriers occasionally add constructively. High PAPR forces the power amplifier to operate with large back-off, reducing efficiency. Another concern is sensitivity to frequency offset and phase noise, which destroys orthogonality and introduces ICI. These are active areas of design in modern wireless systems.

Given:
N = 64 subcarriers, Symbol duration T = 4 microseconds, CP length = 0.8 microseconds

Why this formula applies:
Subcarrier spacing = 1/T (orthogonality condition)
Total OFDM symbol duration = T + TCP

Formula:
delta_f = 1 / T
Total symbol time = T + TCP
Spectral efficiency = N * log2(M) / (T + TCP)  [for M-QAM on each subcarrier]

Substitution:
delta_f = 1 / (4 x 10^-6) = 250,000 Hz = 250 kHz
Total symbol time = 4 + 0.8 = 4.8 microseconds
For QPSK (M=4): bits per OFDM symbol = 64 * 2 = 128 bits

Calculation:
Bit rate = 128 bits / 4.8 microseconds = 26.67 Mbps
Total bandwidth = 64 * 250 kHz = 16 MHz

Final Answer:
Subcarrier spacing = 250 kHz
Achievable bit rate = 26.67 Mbps
Spectral efficiency = 26.67 / 16 = 1.67 bits/s/Hz

Exam Tip: GATE often asks about subcarrier spacing. Always remember delta_f = 1/T where T is the useful symbol duration only, not including the cyclic prefix. Also, the FFT size N equals the number of subcarriers, and OFDM output is the IDFT of the input symbols.

Quick Revision

• OFDM splits high-rate serial data into N parallel low-rate streams on N orthogonal subcarriers.
• Subcarrier spacing: delta_f = 1/T, where T is the useful OFDM symbol duration.
• OFDM modulation and demodulation use IFFT and FFT respectively.
• Cyclic prefix eliminates ISI if CP length is greater than or equal to the maximum channel delay spread.
• Key drawback: high PAPR, which causes power amplifier inefficiency.
• Trap: Do not include CP duration when computing subcarrier spacing. CP only affects total symbol time and overhead.
• OFDM converts a frequency-selective channel into multiple flat-fading subchannels, simplifying equalization.