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DiscreteMath`KroneckerDelta`

This package introduces two functions defined on the integers, It is automatically loaded by the ZTransform package in the DiscreteMath subdirectory. KroneckerDelta and DiscreteStep.


DiscreteStep and KroneckerDelta.

  • This loads the package.
  • In[1]:= <<DiscreteMath`KroneckerDelta`

  • DiscreteStep[n] has a step of size unity at

    . The discrete step is zero for nonintegral arguments.
  • In[2]:= Table[DiscreteStep[n], {n, -2, 2, 1/2}]

    Out[2]=

  • KroneckerDelta[n] is zero for all nonzero . The value of KroneckerDelta[

    n] at

    is unity.
  • In[3]:= Table[KroneckerDelta[n], {n, -2, 2, 1/2}]

    Out[3]=

  • This demonstrates the sampling property of the Kronecker delta. It is assumed that f[n] is defined on the integers and

    is integer-valued.
  • In[4]:= Sum[KroneckerDelta[n - a] f[n],
    {n, -Infinity, Infinity}]

    Out[4]=

  • This demonstrates the sampling property of differences of the Kronecker delta.
  • In[5]:= Sum[( (KroneckerDelta[n] - KroneckerDelta[n-1]) -
    (KroneckerDelta[n-1] - KroneckerDelta[n-2]) ) f[n],
    {n, -Infinity, Infinity}]

    Out[5]=


    Multidimensional DiscreteStep and KroneckerDelta.

  • A multidimensional discrete unit step may be expressed as DiscreteStep[n,m] or as the product DiscreteStep[n]DiscreteStep[m].
  • In[6]:= Sum[f[n m] DiscreteStep[n, m],
    {n, -Infinity, n1}, {m, -Infinity, m1}]

    Out[6]=

  • Since the discrete unit step is separable, the value of DiscreteStep[n,m] at is DiscreteStep[

    m].
  • In[7]:= DiscreteStep[0, m]

    Out[7]=

  • This is a plot of the characteristic or indicator function for the set of integers



    n,m

    :n^2+m^2<=100,n+m>=10

    .
  • In[8]:= (r = Range[0, 10]; t = Transpose[{r + .5, r}];
    ListDensityPlot[
    Table[DiscreteStep[100 - n^2 - m^2, n + m - 10],
    {n, 0, 10}, {m, 0, 10}], FrameTicks -> {t, t}])



    Simplifying expressions involving DiscreteStep.

  • Complicated expressions involving DiscreteStep can sometimes be simplified using SimplifyDiscreteStep.
  • In[9]:= DiscreteStep[n - 2] (KroneckerDelta[n - 3] +
    DiscreteStep[n, m] DiscreteStep[m - 1]) //
    SimplifyDiscreteStep

    Out[9]=


    Simplifying expressions involving KroneckerDelta.

  • Expressions containing terms equivalent to zero can sometimes be simplified using SimplifyKroneckerDelta.
  • In[10]:= (f[n] + KroneckerDelta[n]) DiscreteStep[n-3] //
    SimplifyKroneckerDelta

    Out[10]=