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, …, s63Step 2: Mapping Symbols to Subcarriers
Assume Nsub = 8 subcarriers. Each OFDM symbol in the frequency domain contains 8 QAM symbols (one per subcarrier):
- OFDM symbol 1 → s0, s1, s2, s3, s4, s5, s6, s7
- OFDM symbol 2 → s8, s9, s10, s11, s12, s13, s14, s15
- …
- OFDM symbol 8 → s56, s57, …, s63
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:
\[ x[n] = \dfrac{1}{N} \sum_{k=0}^{N-1} X[k] e^{j 2 \pi \frac{k}{N} n}, \quad n=0,\dots,N-1 \]
- Add a Cyclic Prefix (CP) to mitigate inter-symbol interference.
- Convert the baseband signal to passband by modulation with a carrier.
- 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
In the above diagram, the graphical representation of the subcarrier signals is shown only for illustration purposes. In a real OFDM system, an IFFT is applied to the modulated baseband symbols (such as BPSK, QPSK, or QAM). This operation converts the signal into the time domain, while the subcarriers are automatically arranged with orthogonal frequency spacing in the frequency domain.
The resulting time-domain signal is then transmitted after being modulated onto a high-frequency carrier to convert it into a passband signal using:
Q(t)cos(ωct) + I(t)sin(ωct)
Run the interactive OFDM online simulator for a hands-on experience (click here)