Fourier Transforms

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 -t £ t £ t, we will treat it as a periodic function x_P(t) with period T (where T > t).

If we consider one frequency within the spectrum, w = kw₀ (= 2pk / T), as T increases, w₀ decreases, but we will say that k will increase so that the frequency we are examining stays the same.

From the expression of c_k earlier, we have:

Remember that this integral is zero outside of -t £ t £ t, 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 > t. Therefore, if we increase T to infinity, k goes to infinity so that w still equals kw₀, so this becomes

where we have replaced c_k T by X(w). This definition of X(w) 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 x^P(t) in terms of c _k:

where we have replaced c _k by X(w), where w = kw₀ once more. Now, if we say the difference between successive values of w, dw, is w₀ (= 2p / T), we can substitute for T:

 

Now, here comes the trick. If we take T to infinity, k goes to infinity so that w still equals kw₀ again. The difference between successive values of k will go to zero. Therefore, it is reasonable to assume that dw goes to dw, and the summation becomes an integral:

 

Since we now have an exact expression, not a periodic function, x_p(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 p is no exception. More commonly known as 3.1415 (approximately), or in its capital form ( P ) as the product operator (like S), there is also a function p( t/t ). This is a function where p( t/t ) = 1 for |t| < t/2, and zero everywhere else. (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 [-t/2, t/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 xp] / xp.

Next, we'll look at the properties of these transforms.

Copyright © David McCabe, 1998 - 2001. All rights reserved.

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