Inverse Filters for Decomposition of Multi-Exponential and Related Signals

Decomposition of multi-exponential and related signals is generalized as a filtering problem on a logarithmic time or frequency scale and FIR filters operating with logarithmically sampled data are proposed to use for its implementation. The filter algorithms and types are found for various time-domain and frequencydomain mono-components. It is demonstrated that the ill-posedness in the multi-component decomposition manifests as high sampling-rate dependent noise amplification coefficients. The noise transformation control of a filter is provided by algorithm design, which integrates together the signal acquisition, the discrete-time filter design and the regularization based on choosing an optimum sampling rate. As an example, an algorithm is designed for the decomposition in the frequency-domain. Key-Words: Decomposition, Distribution of Time Constants, FIR Filters, Logarithmic Sampling, Illposedness, Regularization