Rayleigh vs. CRLB — why “resolution” and “precision” are not the same, and what that means for real-world indoor ranging.

Introduction

Bluetooth® Channel Sounding enables precise distance estimation by measuring the radio channel across frequency and reconstructing a channel impulse response (CIR). In practice, achievable performance is shaped by two fundamental (and very different) limits:

This page explains both limits in a practical, application-focused way and puts realistic indoor accuracy expectations into context.

Rayleigh Bound

Can we separate paths?

Rayleigh bound intuition (CIR from 1024-point IFFT, first 80 frequency bins filled, zero-padded, no window): paths closer than the Rayleigh limit blend together.

The Rayleigh bound describes the minimum separation required to distinguish two signal paths (e.g., direct path and reflection). It depends primarily on the effective bandwidth BBB:

  • Time resolution ≈ 1 / B
  • Distance resolution (2-way ranging) ≈ c / (2 × B)
  • Defines multipath separability (resolution), not “accuracy”
  • With ~80 MHz bandwidth, reflections within roughly ~1.8 m can blend together
  • More bandwidth → finer time resolution → better path separation

Cramér–Rao Lower Bound (CRLB)

How precisely can we measure in noise?

CRLB intuition: the CIR shape is the same, but uncertainty shrinks as SNR increases—higher SNR enables higher precision.

CRLB describes the best achievable statistical precision for an (unbiased) estimator in the presence of noise. A simplified takeaway:

Distance precision ∝ 1 / (B × √SNR)

  • Defines noise-limited precision (accuracy in noise)
  • Higher SNR → significantly better precision (improves with √SNR)
  • Explains why sub-Rayleigh accuracy can be physically possible

What the Rayleigh bound means at typical bandwidths

Typical 2-way Rayleigh distance resolution scales approximately with c/(2B). Example values:

Rayleigh does not set the achievable accuracy directly — it sets how well close multipath components can be separated in the CIR.

What the CRLB means at ~80 MHz bandwidths

CRLB depends on both bandwidth and SNR. For ~80 MHz effective span, rule-of-thumb noise-limited distance precision is:

This explains why a system can achieve sub-Rayleigh accuracy in clean LOS conditions: the Rayleigh bound limits path separability, while CRLB limits noise-driven precision.

Why SNR is critically important

Bandwidth influences both resolution and precision. However, SNR sets a fundamental information limit: if useful signal content is buried in noise, no signal processing algorithm can fully recover it. While we can try to improve resolution (e.g., with wider frequency span or multipath-mitigation techniques), low SNR directly limits achievable accuracy.

Why sub-meter error can still be a strong result indoors

Office spaces are highly dynamic RF environments: reflections from walls, glass, metal frames, furniture, and people moving around create time-varying multipath, often alongside interference from Wi-Fi and other 2.4 GHz systems. In these conditions, performance is frequently limited more by environmental physics (multipath + interference) than by estimator quality alone. Achieving accuracy within sub-meter in a typical office environment can therefore represent strong performance, even when strict requirements are challenging.

Big picture

Talk to us about your environment and requirements

If you’d like to discuss expected performance for your application (accuracy, robustness, power), please contact us.