Compare Kronecker and Dirac Deltas

Section 16.6 Compare Kronecker and Dirac Deltas

Comparing the Kronecker and Dirac Deltas.

In Section 16.1, we defined the Kronecker delta as

\begin{equation*} \delta_{ij} = \begin{cases}0\quad \amp i\not= j\\ 1\quad\amp i=j \end{cases} \end{equation*}

In Section 16.4, we defined the Dirac delta as

\begin{equation*} \delta(x-a) = \begin{cases} 0\quad \amp x\not= 0\\ \infty\quad\amp x=a \end{cases} \end{equation*}

These definitions look very much the same. What is the difference?

One difference comes from the input variables (the domain). For the Kronecker delta, the input variables \(i\) and \(j\) are discrete variables; for the Dirac delta, the input variable \(x\) is a continuous variable. You should always choose to use the form of the delta that is appropriate for the input variables.

Definition 16.10. Discrete vs. Continuous Variables.

A variable is called a continuous variable if for any two values you can always find another in between them (think about the real numbers). The values of these variables are typically obtained by measuring; common examples are distance and time. If the values of the variable are spaced apart (think about the integers), so that there is always a gap between one value and its nearest neighbor(s), the variable is called a discrete variable. These values of these variables are typically obtained by counting; simple examples include the sets \(\{0, 1, 2, 3\}\) and the even integers. These variables can be either finite or countably infinite. Examples can also include more abstract sets such as \(\{x, y, z\}\) since the elements could be assigned discrete numbers according to their position in the list.

Another difference between the Kronecker delta and the Dirac delta comes from the output variables (the range). The Kronecker delta compares the discrete input variables \(i\) and \(j\text{.}\) It takes the value one whenever these two input variables are the same. The Dirac delta compares the continuous input variable \(x\) to the constant parameter \(a\text{.}\) It takes the value \(\infty\) whenever \(x\) is the same a \(a\text{.}\) This choice between one and \(\infty\) comes from the ways in which the deltas are used. The Kronecker delta is used to pick out a particular term in a sum (see Section 16.1). For example, if we want to pick out the term \(a_N\) in a sum, we use \(\delta_{iN}\)

\begin{equation*} \sum_{i=1}^{\infty} a_i\, \delta_{iN} =a_N \end{equation*}

On the other hand, the Dirac delta is used to pick out a particular term in an integral (see Section 16.4). For example, if we want to pick out the value of the function \(f(x)\) when \(x=a\) in an integral, we use \(\delta(x-a)\)

\begin{equation*} \int_{-\infty}^{\infty} f(x)\, \delta(x-a)\, dx =f(a) \end{equation*}

Visualization of the Delta Function.

To understand why the delta function under an integral sign picks out a particular value of the integrand, it may help to think of the delta function as the limit of a sequence of steps \(\delta_{\epsilon}(x)\text{,}\) Each step is narrower (width \(2\epsilon\)) and higher (height \(1/(2\epsilon)\)) than the previous step, such that the area under each step is always one; see Figure Figure 16.11.

Figure 16.11. The function \(\delta(x)\) can be approximated by a series of steps that get progressively thinner and higher in such a way that the area under the curve is always equal to one.

Then

\begin{equation} \int_{-\infty}^{\infty} f(x)\, \delta_{\epsilon}(x-a)\, dx\tag{16.6.1} \end{equation}

give the average value of \(f(x)\) on the interval determined by \(\epsilon\text{.}\) In the limit that \(\epsilon\) becomes infinitesimally small, then the peak becomes infinitely narrow and infinitely high in just the right way to pick out the value of the function at \(x=a\text{.}\)

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