Unit Impulse Function

The unit impulse function, also known as the Dirac delta function, is a mathematical function that is used to model an idealized impulse or an infinitesimally short pulse. It is often denoted as δ(t) or δ[n], where t is a continuous variable and n is a discrete variable.

Limiting Process

The unit impulse function can be obtained by using a limiting process on the rectangular function. The rectangular function, also known as the boxcar function, is a function that is equal to 1 within a certain interval and 0 outside that interval.

Explanation

To understand how the unit impulse function is obtained from the rectangular function, let's consider a rectangular function with width 2a centered at the origin:

f(t) = 1 for -a ≤ t ≤ a
f(t) = 0 otherwise

Now, let's define a sequence of rectangular functions with decreasing widths:

f1(t) = 1 for -a/2 ≤ t ≤ a/2
f1(t) = 0 otherwise

f2(t) = 1 for -a/4 ≤ t ≤ a/4
f2(t) = 0 otherwise

f3(t) = 1 for -a/8 ≤ t ≤ a/8
f3(t) = 0 otherwise

...

We can observe that as the width of the rectangular function approaches zero, its height approaches infinity in such a way that the area under the curve remains constant. This is the key idea behind the limiting process.

Now, let's define the unit impulse function as the limit of this sequence of rectangular functions as the width approaches zero:

δ(t) = lim [f1(t), f2(t), f3(t), ...]

The unit impulse function is defined such that it is zero for all values of t except at t = 0, where it is infinitely high. However, the area under the curve of the unit impulse function is equal to 1.

Conclusion

In conclusion, the unit impulse function is obtained by using the limiting process on the rectangular function. As the width of the rectangular function approaches zero, its height approaches infinity in such a way that the area under the curve remains constant. This mathematical concept allows us to model an idealized impulse or an infinitesimally short pulse, which is useful in various fields of science and engineering, including signal processing and control systems.