Does a heaviside step function exist?

I’m not sure if it’s there out-of-the-box, but you can always write one:

def heaviside(x):
    if x == 0:
        return 0.5

    return 0 if x < 0 else 1

It’s part of sympy, which you can install with pip install sympy

From the docs:

class sympy.functions.special.delta_functions.Heaviside


Heaviside Piecewise function. Heaviside function has the following properties: 

1) diff(Heaviside(x),x) = DiracDelta(x)    ( 0, if x<0 )
2) Heaviside(x) = < [*] 1/2 if x==0        ( 1, if x>0 )

You would use it like this:

In [1]: from sympy.functions.special.delta_functions import Heaviside

In [2]: Heaviside(1)
Out[2]: 1

In [3]: Heaviside(0)
Out[3]: 1/2

In [4]: Heaviside(-1)
Out[4]: 0

You could also write your own:

heaviside = lambda x: 0.5 if x == 0 else 0 if x < 0 else 1

Although that may not meet your needs if you require a symbolic variable.

Not sure if the best way getting things done… but here is function that I hacked up.

def u(t):
    unit_step = numpy.arange(t.shape[0])
    lcv = numpy.arange(t.shape[0])
        for place in lcv:
            if t[place] == 0:
               unit_step[place] = .5
            elif t[place] > 0:
               unit_step[place] = 1
            elif t[place] < 0:
    unit_step[place] = 0
    return unit_step

Probably the simplest method is just

def step(x):
    return 1 * (x > 0)

This works for both single numbers and numpy arrays, returns integers, and is zero for x = 0. The last criteria may be preferable over step(0) => 0.5 in certain circumstances.

def heaviside(xx):
    return numpy.where(xx <= 0, 0.0, 1.0) + numpy.where(xx == 0.0, 0.5, 0.0)

Or, if numpy.where is too slow:

def heaviside(xx):
    yy = numpy.ones_like(xx)
    yy[xx < 0.0] = 0.0
    yy[xx == 0.0] = 0.5
    return yy

The following timings are with numpy 1.8.2; some optimisations were made in numpy 1.9.0, so try this yourself:

>>> import timeit
>>> import numpy
>>> array = numpy.arange(10) - 5
>>> def one():
...  return numpy.where(array <= 0, 0.0, 1.0) + numpy.where(array == 0.0, 0.5, 0.0)
... 
>>> def two():
...  yy = numpy.ones_like(array)
...  yy[array < 0] = 0.0
...  yy[array == 0] = 0.5
...  return yy
... 
>>> timeit.timeit(one, number=100000)
3.026144027709961
>>> timeit.timeit(two, number=100000)
1.5265140533447266
>>> numpy.__version__
'1.8.2'

On a different machine, with a different numpy:

>>> timeit.timeit(one, number=100000)
0.5119631290435791
>>> timeit.timeit(two, number=100000)
0.5458788871765137
>>> numpy.__version__
'1.11.1'
>>> def three():
...  return 0.5*(numpy.sign(array) + 1)
... 
>>> timeit.timeit(three, number=100000)
0.313539981842041

As of numpy 1.13, it is numpy.heaviside.

Easy Solution:

import numpy as np
amplitudes = np.array([1*(x >= 0) for x in range(-5,6)])