Measuring time domain waveforms can be difficult if the bandwidth of the signal to be measured is higher then that of the measuring system. In this case the signal will be distorted according to the transfer function of the measuring system. If the distortion is not acceptable, there are two possibilities to overcome on the problem. The fist one is to use a higher bandwidth measuring system. This is the appropriate solution, if this can be followed. However, there might be either financial or technical/technological difficulties (metrological or calibration application) which do not allow the first solution. In this case the numerical post-processing of the measured data can extend the bandwidth of the measurement. The process is called inverse filtering, since the frequency dependent distortion of the measurement system is compensated. The basic difficulty of inverse filtering is its ill-posedness, namely that the measurement noise is highly amplified during the reconstruction process. Suppression of the noise leads to bias of the useful signal. The solution is a tradeoff between the distorted and noisy estimates. The inverse filter is thus an adaptive filter, which adapts its shape to the signal to be measured and the measurement noise. To be able to compensate the measurement system one needs to know its transfer function. The transfer function is determined from measurement (this is called identification), which is also a inverse filtering process. The two inverse filtering processes (identification and signal reconstruction) are usually optimized separately, although they influence each other. This approach does not lead us to the global optimum. Our aim is to join the two optimization processes and to find the global optimum from the point of view of the signal reconstruction.