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// engineering › Applications of Fourier

OFDM — 5G Sub-Carriers

See how orthogonal sub-carriers pack data side by side, and how breaking their spacing causes crosstalk.

s(t)=sum_{k} b_kcos(2pi k,Delta f,t)

A mind behind this: Joseph Fourier 1768–1830

Formula

\[ s(t)=\sum_{k} b_k\cos(2\pi k\,\Delta f\,t) \]

OFDM — How Your Phone Talks to a 5G Tower

Orthogonal Frequency-Division Multiplexing packs many data streams onto closely-spaced carriers that are mathematically orthogonal — so a receiver can pull them apart without crosstalk. WiFi, 4G, 5G, DVB-T television, ADSL — all of it is OFDM under the bonnet, all of it powered by the IFFT/FFT pair.

OFDM symbol synthesis

\[ s(t) = \sum_{k=1}^{K} b_{k}\;\cos\!\big(2\pi (k\,\Delta f)\,t\big),\qquad b_{k}\in\{-1,+1\} \]

Reference: Wikipedia — OFDM; Proakis, J. G., & Salehi, M. (2008). Digital Communications, 5th ed., McGraw-Hill, Ch. 12. Cimini, L. J. (1985). "Analysis and simulation of a digital mobile channel using OFDM." IEEE Trans. Comm. 33(7).

🧪 Try these experiments in order

  1. Click each bit toggle in turn. Watch the composite OFDM symbol reshape — every combination of 4 bits gives a unique waveform.
  2. Set all bits to +1. The carriers add constructively at t=0, producing the famous OFDM peak-to-average ratio problem (high amplitude spikes).
  3. Drop sub-carrier spacing Δf below the orthogonal value. Watch the decoded bits start to flip incorrectly — this is what happens when a Doppler shift on a fast-moving train breaks 5G orthogonality.
Data bits:

The four sub-carriers (each modulated by one data bit)

The transmitted OFDM symbol (what goes over the air)

Decoded bits at receiver: +1, +1, −1, +1 ✓

⚠ Watch out for

  • When you change Δf away from the integer-multiple condition, the sub-carriers are no longer orthogonal — the decoder readout will start showing wrong bits.
  • This is a BPSK-OFDM toy. Real 4G/5G uses QAM (constellations of 16, 64, 256 symbols) per sub-carrier, and 600–3300 carriers per symbol — same maths, far bigger arrays.
✅ Do

Use OFDM whenever the channel has frequency-selective fading (urban multipath, indoor reflections, fibre dispersion).

❌ Don't

Use OFDM where carriers move fast relative to symbol time — Doppler destroys orthogonality. Satellite-to-vehicle links need extra tricks.

Where this matters in industry

WiFi (802.11a/g/n/ac/ax), 4G LTE downlink, 5G NR uplink & downlink, DVB-T digital television, DAB digital radio, ADSL/VDSL broadband, powerline networking. Effectively every modern radio standard.

🎯 Learning checkpoint

Why does OFDM convert one wide-band frequency-selective channel into many narrow-band flat-fading channels? What does that simplification buy the receiver?

Frequently asked questions

What does OFDM have to do with my phone?

Everything — 4G, 5G, Wi-Fi and digital TV all use OFDM. Instead of sending data on one fast carrier, they split it across many slower sub-carriers packed tightly together. Fourier maths is what lets those carriers overlap without interfering, squeezing far more data into the same airwaves.

What does 'orthogonal' actually mean here?

It means the sub-carriers are spaced so precisely that, when you correlate the received signal with each one, the others contribute exactly zero. They overlap in frequency yet do not interfere — a Fourier property. Nudge the spacing slider off the ideal value and watch the decoded bits start to corrupt.

Why pack the carriers so close instead of spacing them out?

Spectrum is scarce and expensive — phone companies pay billions for it. Orthogonality lets carriers sit shoulder to shoulder with no guard gaps, so you fit the maximum data into the minimum bandwidth. It is one of the key tricks behind modern wireless speed.

What is crosstalk and why does the tool warn about it?

Crosstalk is when one sub-carrier leaks into another's decoding, scrambling the data. It happens when orthogonality breaks — wrong spacing, timing errors, or Doppler shift from a moving phone. The readout here flips to a warning so you can see the exact moment the channel becomes unreliable.

How does the receiver pull the bits back out?

It correlates the incoming sum with each carrier shape — essentially a Fourier analysis. Because the carriers are orthogonal, each correlation recovers just that carrier's bit and ignores the rest. Real chips do this with a Fast Fourier Transform thousands of times a second.