fourier transform

An Example: A Pie By Any Other Name or The Kitchen Functions

True to form, mathematics has half a dozen uses for each Greek letter, and π is no exception. More commonly known as 3.1415 (approximately), or in its capital form (Π) as the product operator (like Σ), there is also a function Π( t/τ ).

(See HLT p12.)

So, let's derive the frequency spectrum for this nice and simple function. Remember that:

In this case, x(t) is only defined in the interval [-τ/2, τ/2], and so these become our limits of integration, and the integral becomes

which we can now integrate to

This has therefore given us a function of the form (sin x) / x, which is translated to the sinc function in HLT.

Note that sinc x = [sin xπ] / xπ.