Covariance analysis of time delay estimates for strained signals

Estimates of time-varying delays from the ultrasonic echo signals of compressed biological media are the basis of a new type of medical imaging known as elastography. This paper is focused on predicting the covariance between time delays estimated from sequential Gaussian-weighted echo segments that overlap. The accuracy of the analysis was tested and improved by comparisons with measurements involving ultrasonic waveforms simulated from independent band-limited Gaussian signal and noise spectra. Data were generated to explore the dependence of time delay covariance on ultrasonic signal-to-noise ratio, time-bandwidth product W, fractional bandwidth, window separation, and the amount of strain (a is the time rate of change in delay). The relationship between a and other experimental parameters was crucial for understanding how signal decorrelation affects time delay error and, ultimately, elastographic noise. For echo waveforms without strain (a=0), delay variance was found to decrease with W. However, when waveform segments were strained (a≠0) a minimum was found in the plot of time delay variance versus W, where delay errors from additive noise equaled those from signal decorrelation caused by strain. Delay covariance decreased monotonically with increasing window separation (less overlap) when a=0. When a≠0, however, the covariance became negative for large separations. Properties of strain image noise were predicted from knowledge of the experimental parameters and time delay errors