MATHEMATICAL SERIES FOR SIGNAL MODELING USING EXPONENTIALLY MODIFIED FUNCTIONS

The simple recursive generating function of eq 8, with a time-dependent function, E(t), a convoluted function, U(t), corresponding to E(t), and the time interval between two measurements, DELTA-t, and a dimensionless parameter, A, linked to the time constant tau by A = tau/DELTA-t + 0.5, is demonstrated to produce U(t), which is the discrete representation of the exponentially modified E(t) function. If E(t) is a Gaussian function, then U(t) is the discrete representation of the corresponding exponentially modified Gaussian (EMG) function. The EMG function is widely used in chromatography and flow injection analysis (FIA) to model talling peaks. The proposed method can exponentially modify any Gaussian functions and/or series expansion approximations commonly used. The convolution series U(t) is defined. The link with the exponentially modified function is demonstrated for any E(t) function. The properties of the convoluted U(t) function are given. Three examples are developed: (i) the single square function whose exponentially modified form (EMS) is shown to be the equation of the one tank model in FIA; (ii) the triangle function; (iii) the Gaussian function producing the EMG function. The exponential modification of very different functions produces tailing peaks of similar general shape.