New ideas for TDoA multi-lateration

(1) Finding the common propagation mode

Let's consider the case of three receivers receiving the same signal.

• Each receiver receives the signal at times t₁, t₂, t₃.
• The cross-correlations produce time differences Δt_1,2, Δt_1,3, Δt_2,3.
• Trivially, Δt_1,2 = Δt_1,3 - Δt_2,3 (and cyclically) (*)

This situation is illustrated in the plot below.

Common propagation mode

As a second step let's consider the situation when one of the three receivers receives the common signal via two propagation paths, and the cross-correlations to the signals received at the two other receivers have approximately the same magnitude.

Currently the TDoA software performs for each cross-correlation a clustering algorithm, and in the situation shown in the plot below it may well be that the following time differences are obtained: Δt_1,2 = -50, Δt_1,3 = -100, Δt_2,3 = -140. Note that these do not fulfill the identity (*).

Two propagation modes for one receiver

As a consequence, a trivial stage of pre-processing is given by selecting those cross-correlation peaks which are consistent according to (*).

Update 9/22/2018

(2) Using constraints (number of degrees of freedom)

Now let us count the number of degrees of freedom for the case of N receivers receiving the same signal: there are

• 2 unknowns, (lat,lon), for the most likely position
• N unknown virtual heights of which only (N-1) are observable in the time differences (adding a common time offset does not change Δt_i,j)
• N(N-1)/2 hyperbolæ (=constraints)

Note that instead of virtual heights we might as well add time offsets δt_j  to the time differences, Δt_i,j+δt_j, j≠1, and in this way become agnostic w.r.t. the specifics of ionospheric propagation.

As a summary there are N+1 unknowns and N(N-1)/2 constraints:

#receivers	#unknowns	#constraints	#constraints-#unknowns
3	2(pos)+2(δtj, j≠1)	3 hyperbolæ	-1
4	2(pos)+3(δtj, j≠1)	6 hyperbolæ	1
N	2(pos)+(N-1)(δtj, j≠1)	N(N-1)/2 hyperbolæ	N(N-1)/2-N-1

So for 3 receivers there is a 1-dimensional family of solutions while for N≥4 receivers the system of equations is overdetermined.  This sounds good enough to make me suspicious if it is indeed true.

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It will be interesting to try these ideas on real data obtained from KiwiSDRs. If anyone spots an error in the logic above please do let me know.

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When using N stations there are N-1 independent measurements contained in the time differences δt_ij, j>i. Two of these are needed to determine the likely position, so N-3 out of the N-1 δt_i, i≠1, can be found:

#receivers	#unknowns	#measurements	#fixed/total δtj
3	2(pos)+2(δtj, j≠1)	2 time differences	0/2
4	2(pos)+3(δtj, j≠1)	3 time differences	1/3
5	2(pos)+4(δtj, j≠1)	4 time differences	2/4
6	2(pos)+5(δtj, j≠1)	5 time differences	3/5
N	2(pos)+(N-1)(δtj, j≠1)	(N-1) time differences	(N-3)/(N-1)