Orthogonality

The Spacing Problem

We have multiple subcarriers transmitting in parallel. But how close can we pack them in frequency?

Traditional approach: Leave gaps between channels.

OFDM approach: Overlap the channels, but make them orthogonal.

The subcarriers overlap, but they don’t interfere. How?

The Magic of Orthogonality

Two signals are orthogonal if they don’t interfere when you measure one while the other is present.

Key insight: At the exact center of each subcarrier, all other subcarriers have zero amplitude.

Look at a single subcarrier in the frequency domain. It’s not a perfect rectangle. It has a shape called a sinc function:

• Peak at the center frequency
• Crosses zero at regular intervals
• Small ripples that fade away

The trick: Space subcarriers exactly 1/T Hz apart, where T is the symbol duration.

When you do this:

• Subcarrier 1’s peak is at subcarrier 2’s zero crossing
• Subcarrier 2’s peak is at subcarrier 1’s zero crossing
• And so on for all subcarriers

Every subcarrier’s peak aligns perfectly with every other subcarrier’s null.

Why 1/T Hz?

A symbol lasting T seconds has a frequency spectrum that crosses zero every 1/T Hz.

Subcarrier spacing = 1 / Symbol duration

This is not a coincidence. It’s a fundamental property of the Fourier transform.

The Result

OFDM packs subcarriers as tightly as mathematically possible.

Summary

Orthogonality = subcarriers overlap but don’t interfere because peaks align with zero crossings.

• The spacing of 1/T Hz is the key
• When you sample one subcarrier, all others contribute exactly zero
• This lets OFDM use spectrum 50% more efficiently than traditional systems