Heaviside function

The Heaviside step function, using the half-maximum convention

The Heaviside step function, H is a discontinuous 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 English Oliver Heaviside.

The Heaviside function is the integral of the Dirac delta function: H′ = δ. This is sometimes written as

$H(x)=\int _{-\infty }^{x}{\delta (t)}\mathrm {d} t$

Discrete form

We can also define an alternative form of the Heaviside step function as a function of a discrete variable n:

$H[n]={\begin{cases}0,&n<0\\1,&n\geq 0\end{cases}}$

where n is an integer.

Or

$H(x)=\lim _{z\rightarrow x^{-}}((|z|/z+1)/2)$

The discrete-time unit impulse is the first difference of the discrete-time step

$\delta \left[n\right]=H[n]-H[n-1].$

This function is the cumulative summation of the Kronecker delta:

$H[n]=\sum _{k=-\infty }^{n}\delta [k]\,$

where

$\delta [k]=\delta _{k,0}\,$

is the discrete unit impulse function.

Analytic approximations

For a smooth approximation to the step function, one can use the logistic function

$H(x)\approx {\frac {1}{2}}+{\frac {1}{2}}\tanh(kx)={\frac {1}{1+\mathrm {e} ^{-2kx}}}$,

where a larger k corresponds to a sharper transition at x = 0. If we take H(0) = ½, equality holds in the limit:

$H(x)=\lim _{k\rightarrow \infty }{\frac {1}{2}}(1+\tanh kx)=\lim _{k\rightarrow \infty }{\frac {1}{1+\mathrm {e} ^{-2kx}}}.$

There are many other smooth, analytic approximations to the step function. They include:

$H(x)=\lim _{k\rightarrow \infty }\left({\frac {1}{2}}+{\frac {1}{\pi }}\arctan(kx)\right)$

$H(x)=\lim _{k\rightarrow \infty }\left({\frac {1}{2}}+{\frac {1}{2}}\operatorname {erf} (kx)\right).$

These limits hold pointwise and in the sense of distributions. In general, however, pointwise convergence need not imply distributional convergence, and vice-versa distributional convergence need not imply pointwise convergence.

Representations

Often an integral representation of the Heaviside step function is useful:

$H(x)=\lim _{\epsilon \to 0^{+}}-{1 \over 2\pi \mathrm {i} }\int _{-\infty }^{\infty }{1 \over \tau +\mathrm {i} \epsilon }\mathrm {e} ^{-\mathrm {i} x\tau }\mathrm {d} \tau =\lim _{\epsilon \to 0^{+}}{1 \over 2\pi \mathrm {i} }\int _{-\infty }^{\infty }{1 \over \tau -\mathrm {i} \epsilon }\mathrm {e} ^{\mathrm {i} x\tau }\mathrm {d} \tau .$

H(0)

The value of the function at 0 can be defined as H(0) = 0, H(0) = ½ or H(0) = 1.

$H(x)={\frac {1+\operatorname {sgn}(x)}{2}}={\begin{cases}0,&x<0\\{\frac {1}{2}},&x=0\\1,&x>0.\end{cases}}$

See also

• Rectangular function
• Step response
• Sign function
• Negative and non-negative numbers
• Laplace transform
• Iverson bracket