Computation of signal-threshold crossing times directly from higher order moments

This paper introduces a simple method for calculating the times at which any signal crosses a prespecified threshold voltage (e.g., 10%, 20%, 50%, etc.) directly from the moments. The method can use higher order moments to asymptotically improve the accuracy of the estimated crossing times. This technique bypasses the steps involved in calculating poles and residues to obtain time-domain information. Once q moments are calculated, only 2q, multiplications and (q-1) additions are required to determine any threshold-crossing time at a vermin node. Moreover, this technique avoids other problem such as pole instability. The final outcome of this paper is a set of empirical expressions relating the moments to different threshold-crossing times in analogy to the t_d=-0.693m₁ formula. The presented methodology can also be used with other user defined forms of empirical expressions relating the moments to different threshold-crossing times. Several orders of approximations an presented for different threshold-crossing times, depending on the number of moments involved. For example, the worst-case error of a first- to seventh-order (single to seven moments) approximation of 50% RC delay is 1650%, 192.26%, 11.31%, 3.37%, 2.57%, 2.56%, and 1.43%, respectively. This technique is very useful to obtain information about certain signal metrics, such as delay and rise time directly without having to compute the whole time domain waveform. In addition, if the whole waveform is required it can be easily determined by interpolation between different threshold-crossing points. The presented technique works for both step and nonstep inputs, including piecewise-linear waveforms.