Rapid construction of solid-state magnetic resonance powder spectra from frequencies and amplitudes as applied to ESEEM

In many Fourier-transform spectroscopies, such as pulse magnetic resonance (NMR, EPR), time-domain signals are acquired. Parameters are extracted from these signals by fitting numerical simulations to the experimental data. At present, simulations are often performed in frequency domain (FD). These computations generate a list of frequencies and amplitudes associated with the complex exponential components evolving during one or several variable time intervals. In order to compare simulations with experiments, this peak list is converted to a finite-length time-domain (TD) signal. This can be achieved either by directly evoluting the exponentials in time (direct method) or by rounding their frequencies and binning their amplitudes into a frequency-domain array (histogram method). The first approach is equivalent to a brute-force TD simulation and is slow for a large number of peaks. The second approach is a fast, but very crude approximation and is usually applied without considering in detail the errors involved. A third method introduced and illustrated here is based on the convolution and deconvolution of a short finite impulse response filter kernel. This convolution approach is much faster than the direct method and by orders of magnitude more accurate than the histogram method. For both TD and FD signals a detailed analysis of the errors and of the associated computational costs is presented. The convolution approach is applicable to any simulation problem where TD signals consist of a large number of complex exponentials. In particular, it is the method of choice for simulating 1D and 2D electron spin echo envelope modulation (ESEEM) spectra of disordered systems.