# DiscreteDelta

DiscreteDelta[n1,n2,]

gives the discrete delta function , equal to 1 if all the ni are zero, and 0 otherwise.

# Details # Examples

open allclose all

## Basic Examples(3)

Evaluate numerically:

Use in sums:

Plot over a subset of the integers:

## Scope(24)

### Numerical Evaluation(4)

Evaluate numerically:

Complex number inputs:

DiscreteDelta always returns an exact result irrespective of the precision of the input:

Evaluate efficiently at high precision:

### Specific Values(4)

Value at zero:

Multiargument form gives 1 when all inputs are zero:

Value at infinity:

Evaluate symbolically:

### Visualization(3)

Plot the single-argument DiscreteDelta using integer-width bins:

Visualize DiscreteDelta over the reals. Except for a jump at , it is indistinguishable from a zero function:

Plot DiscreteDelta in three dimensions:

### Function Properties(9)

DiscreteDelta is defined for all real and complex inputs:

Function range of DiscreteDelta:

The function range for complex values is the same:

DiscreteDelta is not an analytic function:

Has both singularities and discontinuities:

DiscreteDelta is neither nondecreasing nor nonincreasing:

DiscreteDelta is not injective:

DiscreteDelta is not surjective:

DiscreteDelta is non-negative:

DiscreteDelta is neither convex nor concave:

### Differentiation and Integration(4)

First derivative with respect to x:

Series expansion at a generic point:

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

More integrals:

## Applications(4)

Use in sums to pick out terms:

Pick out elements:

Patch pointwise values of piecewisedefined functions:

Define some finite duration signals:

Plot the signals in the time domain:

To find the convolution of these signals, first calculate the product of the transforms:

Then, perform inversion back to the time domain:

Plot the convolution in the time domain:

Alternatively, find the convolution using DiscreteConvolve:

## Properties & Relations(3)

Reduce an equation containing DiscreteDelta:

The support of DiscreteDelta has measure zero:

DiscreteDelta can be represented as a DifferenceRoot:

## Possible Issues(2)

DiscreteDelta can stay unevaluated with numeric arguments:

A larger setting for \$MaxExtraPrecision can be needed: Equality testing of the arguments takes numerical precision into account: