Subsample interpolation bias error in time of flight estimation by direct correlation in digital domain

This work presents an investigation of the bias error introduced in time of flight estimation realized by subsample interpolation in digital domain. The time of flight estimation is accomplished based on the evaluation of the peak position of the cross correlation function. In order to cope with the discrete nature of the cross-correlation function, subsample estimation exploits three time domain interpolation techniques: parabolic, cosine, Gaussian and frequency domain interpolation using phase angle. An empirical equation relating the maximum value of the bias error to sampling frequency and signal parameters (center frequency and envelope bandwidth) has been derived. It is found that the maximum value of the bias error is in inverse cubic relation to sampling frequency and in quadratic relation envelope bandwidth for cosine interpolation. The maximum value of the bias error is in inverse cubic relation to sampling frequency and in quadratic relation to center frequency and envelope bandwidth for parabolic interpolation. The coefficients related to the approximation technique are given. Results can be applied for bias errors estimation or correction when fast subsample interpolation is used and application of phase domain interpolation is unacceptable due to processing speed limitations. The equations for minimum required sampling frequency are derived by balancing the interpolation error against Cramer–Rao lower bound.