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## Re: [Discuss-gnuradio] Determining distance from OFDM signals

 From: Qasim Chaudhari Subject: Re: [Discuss-gnuradio] Determining distance from OFDM signals Date: Sun, 9 Jun 2019 23:23:13 +1000

Hi Marcus,
Thanks for your nice words. GNU Radio is an amazing project and the email archive is our first resource for finding help. Benefited me numerous times.

Regarding the necessity of back and forth transmission for ranging, you already know this stuff but I write the below note for later searchers.

The fundamental question to ask is the following. What are the conditions under which finding the distance is possible? It turns out that unsynchronized nodes have a phase offset Theta and a delay Tau between them. So we have two unknowns.

Now if we implement one way transmission, we get only one equation and hence it becomes impossible to separate the phase offset from delay.

Time_2 = Time_1 + Theta + Delay

However, there are several ways in which an extra equation can be provided to this system for range determination (since phase is just another manifestation of time, so the underlying principles stay the same).

Method 1: Synchronize the nodes and hence Theta becomes zero. One equation and one unknown, i.e., delay.

Method 2: Provide an extra independent equation through back and forth transmission where phase offset Theta appears with a negative sign (because measurement is at the initiator now).

Time_4 = Time_3 - Theta + Delay

Here, we have two equations that can solve the two unknowns.

Method 3: Move one node around that doesn't affect the phase offset Theta but provides an extra equation for delay Tau. For example, if we take it twice as far,

Time_4 = Time_3 + Theta + 2*Delay

Again, two equations and two unknowns. The OFDM radar you mentioned is a generalization of this concept where we do graphs in 2 dimensions and find the range/velocity through the respective slopes. So basically we provide the second equation through movement that changes the delay.

As a side remark, I just mention that the large number of subcarriers doesn't provide independent equations (multiply both sides of the above equations with 2\piF_k instead of 2\piF_1 to check). However, they help in solving the range ambiguity problem, e.g., with 2.4 GHz, the maximum range determination is only 12.5cm. This comes from one wavelength (c/F).

Method 4: Place extra anchor nodes around the Tx. Now we have several extra equations but exactly the same number of extra phase offsets! One node (even an anchor) then can respond with a reply message where phase offsets appear with negative signs making the system solvable. This is usually known as ranging through one-way transmission (because target node has to transmit only once) or Blink Mode by some.

Cheers,
Qasim

On Sat, Jun 8, 2019 at 11:33 PM Müller, Marcus (CEL) <address@hidden> wrote:
Hi Qasim,

a) it's so nice to see you drop in here from time to time :)
b) that's true! But reality is even better; the back and forth exchange
isn't strictly necessary.
c) I finally find the time to write down what I wanted to write.

## First, agreeing with you:

One can basically emulate the principle of a correlation-based radar by
making the other end "reflect" info with a known delay.

All one needs to do is ensure that a reply packet is sent a *fixed*
amount of time after a packet is received. That interval doesn't have
to be necessarily short, just known.

>From the reply, and its knowledge of the original transmission's time,
the original transmitter can deduce the double of the one-way
transmission latency; that can, given enough bandwidth, SNR and
processing gain, be enough to resolve integer wavelength ambiguities.
In practice, passband bandwidth will often be the fundamentally
limiting factor.

With the phase estimate done on the reply, and info on the phase
estimate done by the replying party, both sides would have relative
phase CSI, and the original transmitter a distance estimate.

## Then, disagreeing with you (just a little bit):

Now, while the above works with any transmission that allows phase
recovery, OFDM frames have the advantage of allowing a two-dimensional
DFT to be used to estimate range through the time-shift property of the
DFT. However, that requires knowledge of the transmitted symbols at the
receiver – which luckily is recoverable in case of successful OFDM
communication¹.

In fact, that's how OFDM radar works very nicely; it's explained in
[1].

Imagine a receiver with knowledge of the transmitted signal. While
lacking a common time base, a receiver can infer distance from the
development of the phases of entries of a sufficiently large (in both
number of OFDM symbols and number of subcarriers) OFDM frame.

The idea is simple: assume you know the symbols at the transmitter. The
speed-of-light induced delay is constant across all subcarriers.
The resulting phase shift, thus, is proportional to the subcarrier
frequency, and hence the subcarrier number.
Therefor, when you observe linear channel phase change over subcarrier,
you can get a distance estimate. Phase being a linear function of index
implies we're dealing with a sinusoid – and a DFT in subcarrier
direction will give us a range plot.

Same idea for Doppler, but with phase on the same subcarrier, but for
consecutive and hence constant-interval OFDM symbols; do another DFT
for each subcarrier across OFDM symbols, and get a doppler plot.

Overall: Write down your received OFDM symbols as column vectors of a
matrix, point-wise divide by the transmitted symbols (normalize
amplitude if helpful); the result is a matrix full of complex numbers
with the channel phase for each subcarrier at each symbol time.
Do an appropriate 2D-DFT, get a range/doppler plane "image". Find the
peak; use clever interpolation / post-processing to increase resolution
and/or reduce estimate variance.

Cheers,
Marcus

¹  (unless someone inexplicably decided to use differential PSK on the
subcarriers – looking at DAB, there.)

[1] https://publikationen.bibliothek.kit.edu/1000038892

On Sat, 2019-06-08 at 14:21 +1000, Qasim Chaudhari wrote:
> Phase information is preserved whether the Rx architecture is zero-IF
> or not. The OP I guess is already using a back and forth exchange
> between two USRPs, from which the distance information can be
> extracted in case of OFDM signals.
>
> Cheers,
> Qasim