Bluetooth Channel Sounding (CS) Ranging Calculator : Phase based Vs. RTT

As per Bluetooth Core Specification (v6.3), Bluetooth Low Energy (LE) now includes Channel Sounding (CS) for High Accuracy Distance Measurement. The specification defines two distinct but complementary methods for calculating distance between an Initiator and a Reflector device: Phase-Based Ranging (PBR) and Round-Trip Time (RTT).

Bluetooth Distance Calculation as per phase based ranging

Phase-Based Ranging measures the phase difference of radio signals transmitted across multiple frequencies to estimate distance. Because phase shifts as frequency changes, the rate of phase change over the frequency change yields the distance.

Inputs Clear Input

Measured Phase Difference between two frequencies, degrees (input1) :

Frequency Step or difference between two tones, MHz (input2) :

Outputs Calculate

Bluetooth Device Distance, in meters (output1) :

Scenario : Let’s assume a Bluetooth device steps its frequency by 1 MHz between two tones, and the receiver measures a phase difference of $\frac{\pi}{2}$ radians (90 degrees).

INPUTS:

• $\Delta f$ = $1 \text{ MHz} = 1 \times 10^6 \text{ Hz}$
• $\Delta\phi_{CH}$ = $\frac{\pi}{2} \approx 1.5708 \text{ radians}$
• $c$ = $3 \times 10^8 \text{ m/s}$

OUTPUT:

• Distance = 37.5 meters

Bluetooth Phase Based Ranging formula

Where,

• $x$ = Distance in meters (m)
• $\Delta\phi_{CH}$ = Measured phase difference between two frequencies (in radians)
• $\Delta f$ = Frequency step / difference between the two tones (in Hz)
• $c$ = Speed of light ($\approx 3 \times 10^8$ m/s)
• $4\pi$ = Constant accounting for the round-trip nature of the signal ($2x$ distance) and conversion of full cycle ($2\pi$).

Bluetooth Distance Calculation as per RTT based ranging

RTT simply calculates the time it takes for a packet to leave the Initiator, arrive at the Reflector, be processed, and return to the Initiator.

Inputs Clear Input

Total turnaround time measured by the initiator, ns (input1) :

Processing/Turnaround time used by the reflector, ns (input2) :

Outputs Calculate

Bluetooth Distance, in meters (output1) :

Scenario: Let’s assume the Initiator measures the total time from sending to receiving as 1,000 nanoseconds and the Reflector reports back that it took 800 nanoseconds to process and send the reply.

INPUTS:

• $T_{initiator}$ = 1000 ns
• $T_{reflector}$ = 800 ns

OUTPUT:

• Distance (x) = 30 meters

Bluetooth RTT Based Ranging formula

Where,

• $T_{initiator}$ = Total turnaround time measured by the initiator ($Time\ of\ Arrival - Time\ of\ Departure$)
• $T_{reflector}$ = Processing/Turnaround time used by the reflector ($Time\ of\ Departure - Time\ of\ Arrival$)
• $c$ = Speed of light ($\approx 3 \times 10^8$ m/s)
• Division by 2 accounts for the two-way (round-trip) flight.

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