So, we now have equations that define the amplitude of harmonics in a periodic wave. In fact, this is called the frequency spectrum of the wave, and tells us all about the frequency content.
Now it’s time to move onto aperiodic functions, and in fact to processing arbitrary functions. If we are sampling a signal, we don’t really want to be concerned about what the basic unit of the wave is – we just want to feed the function in and get a frequency spectrum out. And this is where the functions in HLT come in.
Towards a Less Periodic Signal
To derive the Fourier transform of an aperiodic function x(t), defined over -τ <= t <= τ, we will treat it as a periodic function xP(t) with period T (where T > t).
If we consider one frequency within the spectrum, ω = kω0 (= 2πk / T), as T increases, ω0 decreases, but we will say that k will increase so that the frequency we are examining stays the same.
From the expression of ck earlier, we have:
Remember that this integral is zero outside of -τ <= t <= τ, so we don’t have any problems here. This integral is going to be the same value whatever the value of T, so long as T > τ. Therefore, if we increase T to infinity, k goes to infinity so that ω still equals kω0, so this becomes
where we have replaced ck T by X(ω). This definition of X(ω) is familiar if you look in HLT p10.
The converse is obtained from our familiar Fourier series expression for x(t), writing an expression for xP(t) in terms of c k:
where we have replaced ck by X(ω), where ω = kω0 once more. The difference between values of ω in each part of the summation is ω0, so we will instead call it δω.
Substituting for T and kδω:
Now, here comes the trick. If we fix ω, as we take T towards infinity, ω0 will go towards zero. Therefore, it is reasonable to assume that δω goes to dω, and the summation becomes an integral:
Since we now have an exact expression, not a periodic function, xp(t) has become x(t), and this should be familiar from HLT as well.
The upshot of all this is that a frequency spectrum of any signal can be derived from these integrals. This is quite fantastic, since we can now dispense with all that theory and just use these integrals!
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.