Sigmoidal function is any function with the following properties:

• Monotonically increasing
• Differentiable
• Bounded

Sigmoidal function is useful in fuzzy logic, where it gives a value between 0 and 1 to tell how is the membership of a member to a set, with 1 means the full membership and 0 means not a member.

Usually $$x$$ is a measure (e.g. length) between two quantity, $$\ell_1$$ and $$\ell_2$$, where they are the hard boundaries. In other words, $$x<\ell_1$$ implies membership quality $$\mu$$=0 and $$x\ge\ell_2$$ implies $$\mu$$=1. Between the boundaries, membership quality of measure $$x$$ is determined by a sigmoidal function.

Examples of sigmoidal functions are:

• $$\sin^2\alpha$$ where $$\alpha=\frac{\pi}{2}\frac{x-\ell_1}{\ell_2-\ell_1}$$
• $$1/(1+e^{-\lambda x})$$ with $$\lambda>0$$, here $$\ell_1=-\infty$$ and $$\ell_2=\infty$$
• this one is called the “unipolar sigmoidal function”
• $$\dfrac{1-e^{-\lambda x}}{1+e^{-\lambda x}}$$ with $$\lambda>0$$
• this one is called the “bipolar sigmoidal function”
• with $$\ell_1=0, \ell_2=\infty$$
• signum function: $$(x)=1$$ if $$x\ge 0$$ and $$f(x)=-1$$ if $$x<0$$
• saturation function: $$f(x)=2x$$ when $$-0.5<x<0.5$$, $$f(x)=-1$$ when $$x<-0.5$$ and $$f(x)=1$$ when $$x>0.5$$
• the op-amp

## Reference

MathWorks’ Documentation on Fuzzy logic toolbox: Membership functions, http://www.mathworks.com/access/helpdesk/help/toolbox/fuzzy/fp608.html