Fourier Filtering Cardiac Time Activity Curves: Sharp Cutoff Filters

Abstract

Several problems may occur when analyzing cardiac left ventricular (LV) time activity curves (TACs) using the techniques of Fourier analysis. These difficulties are of course not unique to cardiac TACs, but rather are representative of the general problems encountered when applying Fourier analysis to discrete, sampled data. In the case of LV TAC’s created from equilibrium gated blood pool studies, one usually applies Fourier analysis in an attempt to filter counting statistical noise from the data. One of the common methods employed to accomplish this goal (albeit probably the least justified) is to Fourier transform the TAC into the frequency domain, filter the data with a sharp cutoff filter (i.e. multiplication in the frequency domain with a rectangular pulse) and invert the filtered data. The reason for the choice of a rectangular cutoff filter is usually simplicity. Whatever the reason, using a sharp cutoff in the frequency domain is identical to describing the data by a truncated Fourier series. Thus one is able to skip the Fourier transform step and instead directly calculate only those first few Fourier coefficients one is interested in. Several concerns need to be addressed. First, are the two constraints of the sampling theorem satisfied? That is, is the underlying TAC which has been sampled by the gated equilibrium procedure band limited, and if so is the sampling rate adequate (at least equal to the Nyquist rate).