Direction of arrival and distance to source

Suppose we have $3$ receivers $a,b,c$ and one wave source which are all located on the same $XY$ plane and we do not know the position of only $c$ on the plane.

Assume a plane wave is transmitted $s(t)$ from an aribitrary point source in far field.

• Using the received signal $s_a(t)$ and $s_b(t)$ by pattern matching would it be possible to infer the direction of arrival at $a$ and $b$ and time delay of signal to reach $a$ and $b$?

• Suppose the received signal at $c$ is $s_c(t)$ would it be possible to guess the distance and direction of source to $c$ and also $(x,y)$ coordinates of $c$ with $s_a(t),s_b(t)$ and $s_c(t)$? If not what other minimal information do we need to find these?

• Is there a reference for these kind of problems for multiple sources and $XYZ$ coordinates?

Your statements are a little confusing. First you say that the sensors and the source all are on the same XY plane and then you mention that the source signal $s(t)$ is transmitted from a point source in far field. I will go with the latter.

In array signal processing, the source's or the sensor's position is usually estimated using the concept of relative time delay. For a given source at location $\boldsymbol{\beta} = \left[\phi, \theta \right]$ where $\phi$ is the azimuth angle and $\theta$ is the elevation angle, the signal received by the $k^{\text{th}}$ sensor at position $\boldsymbol{p}^{\text{T}} = \left[x_k,y_k,z_k=0 \right]$ will experience a time delay of

$$\tau_k(\boldsymbol{\beta}) = \frac{1}{c}\boldsymbol{p}^{\text{T}} \begin{bmatrix}{\cos\phi \sin\theta \\ \sin\phi \sin\theta \\ \cos\theta} \end{bmatrix}$$

To simplify this setting, let us only work with the elevation angles. Let the sensor $a$ is at location $\left[x_a, y_a \right]$ and the sensor $b$ is at location $\left[x_b, y_b \right]$. So the distance between them will be $d = \sqrt{\left(x_a-x_b\right)^2 + \left(y_a-y_b\right)^2}$. Let the plane wave reaches sensor $a$ first. Solving a basic geometry will tell you that the extra distance the plane wave has to travel to reach $b$ is $\delta = d\tan\theta$. So the time delay will be

$$\tau = \frac{\delta}{c} = \frac{d\tan\theta}{c}$$

• So to answer your first question, Yes! It would be possible to estimate the direction of arrival at sensor $a$ and/or $b$. You do not need to get it for both the sensors as they are receiving the signal from the same source. You would need to know the time delay which you can get by cross-correlating the signals received by the two sensors. Once you know the time delay, you can use it to get the direction of source i.e., $\theta$.

• To answer your next question, Yes! Once you find out the position of the source. Find out the time delay between the sensors $a$ and $c$ and the sensors $b$ and $c$ and then apply the same equation. You should pay attention to the sign of the delay i.e., between sensors $a$ and $c$ whether is reaches $a$ first or $c$ first and same with $b$ and $c$. Since the formulation only has the distance $d$ and not the specific coordinates, the absolute value of the delay will be the same in the circle of radius $d$ with origin being the sensor $c$.

• I have included the formulation of the delay in terms of the 3-D coordinates. There are a lot of good resources available online to help you understand this in detail. A basic google search will go a long way.

There are a lot of other ways to solve the direction of arrival estimation problem. You should find out more and select the one based on your needs.

Hope this helps!

• when points $a$ and $b$ are two ears attached to the same head, the time difference (which you can get with cross-correlation) is called the interaural time difference (ITD) and is a function of $\phi$ and $\theta$ and wavespeed $c$. it is a much stronger location que than the pinna cues. but all points with the same difference of distance to the two are perceived identically, regarding ITD. that would be all points on a cone that has an axis with both ears lying on it. you need the pinna cues to tell you where the sound might be coming from. but with the third receiver... – robert bristow-johnson Dec 8 '16 at 23:59
• what the third receiver (we shan't call it "$c$" because that symbol has already has a definition) receives relative to either $a$ or $b$, the same ITD formula is used, but with different reference points, that can define another two cones and you want to look for points where all three cones coincide (or, due to estimation errors, look for little tiny "compartments" with walls that are bordered with the loci of points in those 3 cones). – robert bristow-johnson Dec 9 '16 at 0:03