Time domain uncertainty bound of signal reconstruction in the case of jittery and noisy measurements

Time domain measurements are distorted by the measurement system if the bandwidth of the system is not sufficiently high compared to that of the signal to be measured. If the distortion is known the measured signal can be compensated (inverse filtering or deconvolution). Since the measurement is always corrupted by noise the reconstruction is an estimation task. Our aim is to investigate the errors related to the signal reconstruction, and provide an error bound around the reconstructed waveform. Based on their nature we can distinguish between two types of errors, bias and variance. In this paper we investigate the stochastic type errors and suggest a method to calculate the uncertainty (variance) of the reconstruction. We developed the method for the calibration of high speed sampling systems. Beside the stationary additive noise of the measurement system (quantization, electromagnetic interferences etc.) the waveforms are distorted due to the uncertainty of the time base (jitter). The effect of the jitter can be described as a nonstationary additive noise. This noise causes both variance, and bias, since the mean value of the noise is not zero. We will take both stationary and jitter related nonstationary noise into account and provide an uncertainty bound around the reconstructed signal. Because of the nonstationary nature of the jitter the uncertainty bound is a function of time. The complete error analysis should also involve the investigation of the bias. This will be carried out in the future.