Unit impulse vs Kronecker delta vs Dirac delta?

Question

Apparently these 3 terms are similar, but what is exactly difference between them and what is exact value of each of them at time zero,1 or infinity?

Answer 1

UPDATE: After seeing MattL's comment below, I am shortening the answer and tried to provide a more accurate description of Unit Impulse and Dirac Delta function.

Kronecker Delta is defined for the discrete time domain. It is defined as a function of 2 indices. $$ \delta_{ij} = \begin{cases} 0 \quad \text{ if } i \ne j \\ 1 \quad \text{ if } i = j \\ \end{cases} $$.

Unit Impulse function is defined for both discrete and continuous time values as independent variable. For discrete time, it is defined as $\delta[n]$, $$ \delta[n] = \begin{cases} 0 \quad \text{ if } n \ne 0 \\ 1 \quad \text{ if } n = 0 \\ \end{cases} $$.

Now, Unit impulse and Dirac Delta are defined for continuous time as independent variable and are used interchangeably sometimes. They are not technically a function but are defined as limiting values of function as the width around $t=0$ reduces.

Area under integral under $-\infty \lt t \lt +\infty$ is 1. ie. $$ \int_{-\infty}^{+\infty}\delta(t) = 1 $$

As an example, consider a rectangle function from $-T/2$ to $+T/2$ with a height of $1/T$ and $0$ everywhere else. The area of this rectangle $\int_{-\infty}^{+\infty}x_T(t) = 1$. As the $\lim_{T \rightarrow 0}$ height goes to $\infty$ as $T \rightarrow 0$. $$ \lim_{T\rightarrow 0}\int_{-\infty}^{+\infty}x_T(t) = 1 $$

Similarly, we can define Dirac Delta as limiting case of Gaussian, Sinc, Exponential functions etc. where the limit of the parameter which defines the width is taken such that the width goes to 0.

Answer 2

Kronecker Delta used mainly in context of discrete time signals is defined as $$\delta_{ij} $$ is 1 only when $i=j$ or $i-j=0$, $0$ otherwise. So alternatively it is also written as $\delta[i-j]$. This value is exactly 1 and finite (refer below for the distinction with Dirac Delta)

The unit impulse is just the kronecker Delta with $j=0$ hence we only refer to unit impulse with one parameter $\delta_i$. Since $j=0$, this is alternatively written as $\delta[i ]$.Hence is 1 at $i=0$, $0$ otherwise.

The continuous time unit impulse can be thought of derivative of continuous time unit step (modified to have a slightly less steep than the sudden step at $t=0$). Refer to Oppenheim's book on signals and systems Chapter 1.

The Dirac Delta is used in context of continuous time signals to define impulses. It is defined as having an infinitely small width and hence a large magnitude, however the area under the curve integrates to 1. Refer to Oppenheim's book on signal and systems for various shapes that a Dirac Delta function could take, in all cases the area being 1.