1- and 2-Sided Decaying Exponential

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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.

one sided exponential decay         [1]

plot of decaying exponential

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:

derivation for fourier transform of decaying exponential       [2]

Left-Sided Decaying Exponentials

The left-sided decaying exponential is described by equation [3] and is plotted in Figure 2:

left sided exponential decay         [3]

plot of decaying exponential

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:

derivation for fourier transform of left-sided decaying exponential       [4]

Two-Sided Decaying Exponentials

The two-sided decaying exponential is given in equation [5], and plotted in Figure 3:

two sided exponential       [5]

plot of two-sided decaying exponential

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:

two sided exponential fourier transform       [6]

That wraps up the study of the decaying exponential. Give yourselves a round of applause.


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