A Lower Bound on the Estimation Variance of Direction-of-Arrival and Skew Angle of a Biaxial Velocity Sensor Suffering from Stochastic Loss of Perpendicularity

The biaxial velocity sensor comprises two nominally perpendicular particle velocity sensors and a collocated pressure sensor. Due to real-world imperfections in manufacturing or setup errors, the two axes may suffer from perpendicularity losses. To analytically study how skewness affects its direction-finding performance, the hybrid Cramér-Rao bound (HCRB) of the directions-of-arrival for the polar angle, azimuth angle and the skew angle of a biaxial velocity sensor that suffers from stochastic loss of perpendicularity were derived in closed form. The skew angle was modeled as a zero-mean Gaussian random variable of a known variance, which was assumed to be very small, to capture the uncertainty in the orthogonality of the biaxial velocity sensor. The analysis shows that for the polar and azimuth angle, the loss of perpendicularity introduces the variation of the HCRB along the azimuth angle axis, which is independent of the skew angle, but on its variance. The dynamic range of this variation increases as the variance of the skew angle increases. For the estimation of the skew angle, the HCRB of the skew angle is bounded upwards by the variance of the skew angle and varies with the azimuth angle. The hybrid maximum likelihood- maximum a posterior (hybrid ML/MAP) estimator was used to verify the derived bounds.