# Separation of a delayed mixtures of signals

I have three emitters in three different positions that emite three signals $$S_1(t), S_2(t), S_3(t)$$ with $$S_1(t)=S_2(t)=S_3(t)$$ Two receivers recive $$R_1(t), R_2(t)$$ with \begin{align} R_1(t)&=r_1(t-D_1)+r_2(t-D_2)+r_3(t-D_3)\\ R_2(t)&=r_1(t-D_1)+r_2(t-D_2)+r_3(t-D_3)\\ r(t)&=\operatorname{conv}\big(S(t),\rm{Imp_r}\big) \end{align} Where $$\rm Imp_r$$ is a realization of the channel impulse response.

My goal is to separate $$R(t)$$ and get $$r_1(t-D_1), r_2(t-D_2), r_3(t-D_3)$$ with the exacte delays. ICA and SCA methods works only on instatinous mixtures, in this case we have a delayed mixtures. is there any other methods to solve this problem ?

• My first reaction is there are two many unknowns given what you wrote: transmitting $S1(t)=S2(t)=S3(t)$ from three different locations over a dispersive channel $h[n]$ is identical to transmitting from one location with a new channel that has the two additional multipath elements. However $h[n]$ will also be different for both receivers wouldn't it? And it would be different for each individual tx-rx path as well I would guess. I'd be interested in seeing your data and the list of assumptions that can be made.Do your transmit signals have good spectral occupancy for doing channel estimation? – Dan Boschen Jun 25 at 12:50
• Your model is both having delayed signals and non LPF channel? I think it will be too much for anything practical. – Royi Jun 25 at 17:24
• To explain more, i'm using UWB IEEE 802.15.4a channel model and the impulse response of the channel is different for each tx-rx because of the different positions of the three emitters. – Nouali Ibrahim Yassine Jun 26 at 10:20