OFDM Symbols and Subcarriers Explained

This article explains how OFDM (Orthogonal Frequency Division Multiplexing) symbols and subcarriers work. It covers modulation, mapping symbols to subcarriers, subcarrier frequency spacing, IFFT synthesis, cyclic prefix, and transmission.

Step 1: Modulation

First, modulate the input bitstream. For example, with 16-QAM, each group of 4 bits maps to one QAM symbol. Suppose we generate a sequence of QAM symbols:

s0, s1, s2, s3, s4, s5, …, s63

Step 2: Mapping Symbols to Subcarriers

Assume N_sub = 8 subcarriers. Each OFDM symbol in the frequency domain contains 8 QAM symbols (one per subcarrier):

Mapping (example)

1. OFDM symbol 1 → s0, s1, s2, s3, s4, s5, s6, s7
2. OFDM symbol 2 → s8, s9, s10, s11, s12, s13, s14, s15
3. …
4. OFDM symbol 8 → s56, s57, …, s63

Note

Each QAM symbol occupies one FFT bin (frequency bin). You assign (map) symbols to these bins before computing the IFFT.

Step 3: Subcarrier Frequencies (Clarified)

Each element of an OFDM symbol (the frequency-domain vector) corresponds to a subcarrier. The subcarriers are spaced by:

$$\Delta f = \dfrac{1}{T_u} = \dfrac{f_s}{N}$$

• \(f_s\) = sampling frequency
• \(N\) = number of FFT/IFFT points
• \(T_u = \dfrac{N}{f_s}\) = useful symbol duration

The frequency assigned to bin index \(k\) is:

$$f_k = k\dfrac{f_s}{N},\quad k = 0,1,\dots,N-1$$

(If you center bins with an FFT shift, indices run from \(-N/2\) to \(N/2-1\), and negative frequencies appear for bins \(k \ge N/2\).)

You map QAM symbols to these frequency bins before taking the IFFT. The IFFT then synthesizes the time-domain OFDM symbol, whose sinusoidal components oscillate at the \(f_k\) frequencies. This spacing \(\Delta f\) guarantees orthogonality between subcarriers over the symbol duration \(T_u\).

Step 4: Time-Domain Conversion

Take the IFFT of each frequency-domain OFDM symbol to convert it to the time domain. The discrete IFFT synthesis is:

\[ x[n] = \dfrac{1}{N} \sum_{k=0}^{N-1} X[k] e^{j 2 \pi \frac{k}{N} n}, \quad n=0,\dots,N-1 \]

1. Add a Cyclic Prefix (CP) to mitigate inter-symbol interference.
2. Convert the baseband signal to passband by modulation with a carrier.
3. Transmit the passband OFDM signal.

• The cyclic prefix length should exceed the channel delay spread to prevent inter-symbol interference.

At Transmitter Side

At Receiver Side

Further Reading

1. OFDM (Theory)
2. OFDM in MATLAB
3. OFDM Spectrum Analysis Using MATLAB
4. Single Carrier OFDM (SC‑OFDM): Benefits over OFDM in LTE/5G Uplink
5. OFDM vs SC-OFDM
6. DFTs-OFDM vs OFDM: Why DFT-Spread OFDM Reduces PAPR Effectively