The use of uniform circular array UCA microphones for beamforming has recently been proposed as an alternate to cascaded quadrilateral topologies. As presented in numerous literature, the direction of arrival (DOA) of the acoustic signal of interest is most times critical in any beamforming algorithm that is being used. We resolve the DOA problem for a UCA.
Consider a far field source impinging N UCA microphones as shown in Figure 1:
Figure 1: N UCA microphones
Then it can be shown that the DOA estimate between any two pairs of microphones 
will obey
![\tau_{ij} = -2d \cos{\left(\frac{(j-i)\psi}{2}\right)} \cos{\left(\arcsin\left(\frac{\sin{\left((j-i)\psi\right)}}{2 \cos{\left((j-i) \frac{\psi}{2} \right)}} \right) +\theta\right)}, j > i , \{i,j\} \in [1 N]](https://s0.wp.com/latex.php?latex=%5Ctau_%7Bij%7D+%3D+-2d+%5Ccos%7B%5Cleft%28%5Cfrac%7B%28j-i%29%5Cpsi%7D%7B2%7D%5Cright%29%7D+%5Ccos%7B%5Cleft%28%5Carcsin%5Cleft%28%5Cfrac%7B%5Csin%7B%5Cleft%28%28j-i%29%5Cpsi%5Cright%29%7D%7D%7B2+%5Ccos%7B%5Cleft%28%28j-i%29+%5Cfrac%7B%5Cpsi%7D%7B2%7D+%5Cright%29%7D%7D+%5Cright%29+%2B%5Ctheta%5Cright%29%7D%2C+j+%3E+i+%2C+%5C%7Bi%2Cj%5C%7D+%5Cin+%5B1+N%5D&bg=ffffff&fg=000000&s=0 "\tau_{ij} = -2d \cos{\left(\frac{(j-i)\psi}{2}\right)} \cos{\left(\arcsin\left(\frac{\sin{\left((j-i)\psi\right)}}{2 \cos{\left((j-i) \frac{\psi}{2} \right)}} \right) +\theta\right)}, j > i , \{i,j\} \in [1 N]")
Any of the well known approaches such as least squares or SRP-PHAT can be used to combine all estimates between paired microphones to estimate 
.
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