DiscreteDelta

DiscreteDelta[n₁,n₂,…]

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

Details

DiscreteDelta[0] gives 1; DiscreteDelta[n] gives 0 for other numeric n.
DiscreteDelta has attribute Orderless.

Examples

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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:

Multi‐argument 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:

TraditionalForm typesetting:

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 piecewise‐defined 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:

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DiscreteDelta

Details

Examples

Basic Examples (3)