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,
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.