Determination of phase angles in complex modulated ultrasonic signals using probability density distributions

An ultrasonic wave crossing a fluid perpendicularly will be complex modulated by the turbulent structures in the fluid. Because the modulation frequency is low in comparison to the carrier frequency of ultrasound the sidebands are very close to the carrier frequency. The modulation frequency can be separated by demodulation which is achieved by undersampling. The imaginary part of the signal is determined by Hilbert Transformation . The phase angles can reach high values exceeding the range of the arctan-function. It is necessary to compensate the zero-offset due to tolerances at installing the transducers. A new differential procedure using the derivative of phase function is presented for high phase angles. The receiving response of the ultrasonic transducers causes an additional nonlinear amplitude modulation.