Heaviside Function
The Heaviside function, H is a non-continuous function whose value is zero for negative argument and one for positive argument.
The function is used in the mathematics of control theory to represent a signal that switches on at a specified time and stays switched on indefinitely. It was named after the Englishman Oliver Heaviside.
The Heaviside function is the integral of the Dirac delta function: H′ = δ. This is sometimes written as
- [math]\displaystyle{ H(x) = \int_{-\infty}^x { \delta(t)} \mathrm{d}t }[/math]
Discrete form
We can also define an alternative form of the Heaviside step function as a function of a discrete variable n:
- [math]\displaystyle{ H[n]=\begin{cases} 0, & n \lt 0 \\ 1, & n \ge 0 \end{cases} }[/math]
where n is an integer.
Or
- [math]\displaystyle{ H(x) = \lim_{z \rightarrow x^-} ((|z| / z + 1) / 2) }[/math]
The discrete-time unit impulse is the first difference of the discrete-time step
- [math]\displaystyle{ \delta\left[ n \right] = H[n] - H[n-1]. }[/math]
This function is the cumulative summation of the Kronecker delta:
- [math]\displaystyle{ H[n] = \sum_{k=-\infty}^{n} \delta[k] \, }[/math]
where
- [math]\displaystyle{ \delta[k] = \delta_{k,0} \, }[/math]
is the discrete unit impulse function.
Representations
Often an integral representation of the Heaviside step function is useful:
- [math]\displaystyle{ H(x)=\lim_{ \epsilon \to 0^+} -{1\over 2\pi \mathrm{i}}\int_{-\infty}^\infty {1 \over \tau+\mathrm{i}\epsilon} \mathrm{e}^{-\mathrm{i} x \tau} \mathrm{d}\tau =\lim_{ \epsilon \to 0^+} {1\over 2\pi \mathrm{i}}\int_{-\infty}^\infty {1 \over \tau-\mathrm{i}\epsilon} \mathrm{e}^{\mathrm{i} x \tau} \mathrm{d}\tau. }[/math]
H(0)
The value of the function at 0 can be defined as H(0) = 0, H(0) = ½ or H(0) = 1.
- [math]\displaystyle{ H(x) = \frac{1+\sgn(x)}{2} = \begin{cases} 0, & x \lt 0 \\ \frac{1}{2}, & x = 0 \\ 1, & x \gt 0. \end{cases} }[/math]