1- and 2-Sided Decaying Exponential

The decaying exponential function arises often in engineering and physical analysis of systems, in which the magnitude of a function dies off over time. Examples include the voltage across a capacitor as a function of time when a resistor is present, and from information theory, the expected amount of money you have left after optimal gambling in a casino with non-favorable odds. Right-Sided Decaying Exponentials The one-sided (or right-sided) decaying exponential is given in Equation [1], and plotted in Figure 1.         [1] Figure 1. Plot of Equation [1], the right-sided decaying exponential, for |a|=1. Note that in Equation [1], and throughout this page, we use |a|, because this parameter must be positive, or the Fourier Transform will diverge (and therefore not exist). The Fourier Transform for the right-sided exponential can be found from the Fourier Transform definition:       [2] Left-Sided Decaying Exponentials The left-sided decaying exponential is described by equation [3] and is plotted in Figure 2:         [3] Figure 2. Plot of Equation [3], the left-sided decaying exponential, for |a|=1. The Fourier Transform for the left-sided decaying exponential can be found exactly as done for the right-sided in equation [2], giving:       [4] Two-Sided Decaying Exponentials The two-sided decaying exponential is given in equation [5], and plotted in Figure 3:       [5] Figure 3. Plot of Equation [5], the two-sided decaying exponential, for |a|=1. Note that k(t) = f(t) + g(t). Hence, the Fourier Transform can be easily found with the linearity property:       [6] That wraps up the study of the decaying exponential. Give yourselves a round of applause.