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I have a reference signal $r(t)$ and the correlation between that reference signal and the received signal : $C_{XR}(\tau)$. The signal I receive contains reflections on walls. I have to build a synthetic signal $s(t)$ with the addition of attenuated and time-shifted versions of the reference signal. The goal is to build the synthetic signal in a way that it is as close to the received signal as possible. If I do that, I get informations about the walls on which the signal has been reflected.

My idea is to write the synthetic signal as follow : $s(t) = \sum_{i=0}^{N} A_i r(t-T_i)$ where $N$ is the number of reflections we consider.

Then, the correlation between $s(t)$ and $r(t)$ is :

$$C_{RS}(\tau) = \sum_i A_i \ IFFT(R(\omega) R^*(\omega) e^{j\omega T_i}) = \sum_i A_i C_{R}(\tau+T_i) \simeq C_{XR}(\tau)$$

After that, I don't know how to determine $A_i$ and $T_i$ in order to get the last equality.

Jon
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  • Would you consider using well established equalization / channel estimation approaches that should be well suited as an alternate approach to your problem? – Dan Boschen Mar 28 '20 at 20:43
  • See this post for an example: https://dsp.stackexchange.com/questions/31318/compensating-loudspeaker-frequency-response-in-an-audio-signal/31326#31326 – Dan Boschen Mar 28 '20 at 20:44
  • This will only work if you either have a very narrow band signal or the reflection coefficient of the wall is constant with frequency. That would certainly NOT be the case for an acoustic reflection. – Hilmar Mar 28 '20 at 21:16
  • Thank you all. I'm working with signals between 3.2 GHz and 4.8 GHz. The data I have has been sampled at 38.4 Gsamples/s. I don't think I could use channel estimation for this problem. – Jon Mar 28 '20 at 21:44
  • @Jon why do you say that? Certainly the received signal after having been reflected has effectively gone through a "channel" and the channel estimation approaches are ideal for determining the effects of reflections (similar to multipath). If you have a reference signal and the reflected signal, that is all you need and the channel estimation will give you all the information on the various reflections within your bandwidth of interest. With the referenced approach you can replicate the transmit signal from the received signal (which is equalization) using the least-squared algorithm. – Dan Boschen Mar 28 '20 at 21:46
  • Stated simpler it is the least squared result to the inverse convolution problem: given your received signal is the convolution of your transmitted signal with the unknown "channel" you can use the Wiener-Hopf equation referenced to efficiently solve for the channel through a least squared inverse convolution technique, or in your case the reverse: given the replication of the transmit signal is the convolution of your received signal with the correct equalizer filter (which will undo the channel), the solution for the equalizer can also be found using the same approach. – Dan Boschen Mar 28 '20 at 22:00
  • I have nothing to add other than sampling 40Gs/s is extraordinary engineering. – FourierFlux Aug 27 '20 at 00:22
  • qno 1 in a multiple channel,signal reflection s give rise to multiple copies of the signal arriving at the receiver at different times depending on time difference be three different arrivals the signal may or may not be recoverable consider the time limited signal x(t) defined by suppose that the signal is passed through a channel with frequency response resulting in channel output y(t) (a) sketch y(t) for t0=3t how would you recover x(t) from y(t) ? – Abubakarmemon Nov 12 '23 at 10:56

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This is a classical deconvolution problem in that your received signal consists of multiple delayed and scaled copies of your transmitted signal, no different than what could be modeled with an FIR filter (which is the sum of delayed and scaled copies of the input signal), and your goal is to reconstruct the original transmitted signal from the received signal.

The received signal is the convolution of transmitted signal with the effective channel given by the "filter" described above which in this case would be the multiple different reflections from the surface of the wall. So we have $r[n] = t[n]\ast h[n]$ (received signal $r[n]$ is the convolution of the transmitted signal $t[n]$ and the channel $h[n]$), with two knowns $r[n]$ and $t[n]$ and we wish to solve for $h[n]$. This is a classical deconvolution problem typically solved using the Wiener-Hopf equations resulting in a least squared solution when there are sufficient samples for an overdetermined equation. This is explained in detail along with MATLAB code at this post here which was specific to an audio signal but would equally apply to the OP's case of a broadband microwave signal from 3.2 GHz to 4.8 GHz as he explained in comments:

Compensating Loudspeaker frequency response in an audio signal

As described above this would provide for a solution to the actual channel modeled as an FIR filter, but the process can equally be used to solve for an equalizer $c[n]$ which would optimally in a least squared sense return the received signal to the closest copy of what was transmitted simply by swapping $t[n]$ and $r[n]$ since in the receiver, the copy of the transmitted signal $t[n]$ is $r[n]\ast c[n]$.

Some precautions when using "sounding" signals to determine a channel:

The transmitted signal would ideally be white over the frequency band of interest. Certainly if there is any frequency portion with no signal content, no solution can be determined for that portion of frequency (since there would be no input/output relationship). If the signal is not white, the least squared solution would be weighted according to the respective spectral density over the entire bandwidth of interest.

If the delay spread is long compared to the inverse of the bandwidth (so becomes a greater problem for wideband signals), deep nulls can occur in the received spectrum due to the cancellation effects of the various delay paths. This is known as "Frequency Selective Fading" and makes equalization and deconvolution challenging as noise enhancement would occur at the locations where deep spectral nulls exist. The received spectrum should be reviewed to confirm that such deep nulls do not exist. For communication systems this is typical motivation for use of non-linear equalizers such as decision-feedback equalizers which would perform better for equalization under such conditions. (see non linear equalizer vs linear equalizer)

In this related post I showed how the channel estimation formulas can be used to determine the group delay of a channel:

How determine the delay in my signal practically

Dan Boschen
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