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I have a simulation to make where I have an array of transmitters that transmit the same signal. At a random point, which I have to consider as a receiver I have to measure the phase shift of the signals and also measure the total signal strength. The requirement is that the transmitter emits the same signal but with different phase offsets.

  • As far I understood because the transmitters are separated apart, the received signal is already offset by some phase because they different travel distances. Is my assumption right?

  • I know the formula to find the phase offset given the distance travelled of two signals - how do I calculate it for multiple signals? Or am I understanding the question wrong?

  • How do I calculate the total signal strength for all signals combined? I know for one signal assuming a free space, line of sight connection I can use friis equation. But how do I use it for multiple signals?

Any help is very much appreciated. I am Software Engineer who have recently taken up Wireless communications course, and I'm not sure if I understand the motivation and meaning behind the simulation right. Thanks!

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Only thing which can help you to find absolute phases is that all of them were transmitted simultaneously (or in other words) have a clock reference for these signals otherwise there is no way to know their absolute phases.

Fir ex: let's take the example if only direct line of sight for all signals, even if you know that all transmitters are essentially transmitting the same thing, but one transmitter1 (further away from receiver) trasmits earlier and second trarsnmitter2 (closer to the receiver) transmits later, and both of them arrive at receiver at pretty much the same time, then there is no way to know their actual phases, not knowing about a reference (time or clock) of their transmission

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This is simplified assuming there is not multipath occurring in the transmission (where the receiver would typically receive multiple copies of the same transmit signal at different delays resulting in multipath distortion which are typically addressed using channel estimation and equalization).

The OP's phase shift would all be relative since there is no common reference (clock) with the transmitter, so the comparison would be of the phase of the received signals to each other. A cross-correlation of the received signals in complex notation with the reference signal of what you know was transmitted would provide this.

This is accomplished by doing a multiply and accumulate of the received signal with the complex conjugate of the known ideal transmit signal, repeating the computation below for each possible offset $m$ (thus is the cross-correlation function specifically, correlating at each possible offset in time).

$$XCorr[m] = \sum r[n-m]t^*[n]$$

Where (*) indicates a complex conjugate.

The magnitude of the correlation peak would be proportional to the signal strength, and the shift $m$ is proportional to the delay. With two signals at the same shift $m$ (in the same bin), the phase of the correlation is proportional to the carrier phase, giving a fine delay indication by comparing the phase result from two different receiver signals if they were in the same bin. In this case the signals would need to not be correlated themselves, otherwise the result would be the vector addition of the two (appearing as a single received signal). For this reason the transmit signals from each transmitter would be uncorrelated by separating them in frequency, time or code; and you can identify each one separately in receiver using correlation.

Note the above computation for the cross correlation function can be done directly with FFTs given the FFT correlation property (which results in a circular cross-correlation):

$$XCorr[m] = \text{ifft}\{\text{fft}(r[n])\}\text{fft}^*\{(t[n])\}$$

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