# A Unit Impulse in Continuous-Time

This post treats the signals in continuous time which is different than the approach I adopted in my book. The book deals exclusively in discrete time.

A unit impulse is defined as

The result is an impulse with zero width and infinite height, but a consequence of defining it in this way is that the area under the curve is unity.

This is shown in Figure below.

Stated in another way,

A consequence of this property is that theoretically a particular value of a signal can be extracted in the following manner.

(1)

The Fourier Transform of a unit impulse can be derived through the help of Eq (1).

which is a constant for all frequencies from to . This is shown in Figure below.

Here, there is a chance of not emphasizing the underlying concept of a unit impulse — a rectangle with width approaching zero. An alternative and beautiful approach in this context is transforming a rectangular signal and taking the limit . Since the Fourier Transform of a rectangle is a sinc function (),

where we used the approximation for small . With becoming small and approaching zero, the mainlobe of the sinc becomes wider and approaches . Imagine an eagle spreading its wings. This illustrated in Figure below and perfectly demonstrates the concept that a signal narrow in time domain has a wide frequency domain representation and vice versa.