Derivation of the Fourier Transform

Module by: Don Johnson

Summary: This module shows how to derive the scintillating and useful Fourier transform.

Fourier series clearly open the frequency domain as an interesting and useful way of determining how circuits and systems respond to periodic input signals. Can we use similar techniques for nonperiodic signals? What is the response of the filter to a single pulse? Addressing these issues requires us to find the Fourier spectrum of all signals, both periodic and nonperiodic ones. We need a definition for the Fourier spectrum of a signal, periodic or not. This spectrum is calculated by what is known as the Fourier transform.

Let s T t s T t be a periodic signal having period T T. We want to consider what happens to this signal's spectrum as we let the period become longer and longer. We denote the spectrum for any assumed value of the period by c k T c k T . We calculate the spectrum according to the familiar formula

where we have used a symmetric placement of the integration interval about the origin for subsequent derivational convenience. Let ff be a fixed frequency equaling kT k T ; we vary the frequency index k k proportionally as we increase the period. Define making the corresponding Fourier series As the period increases, the spectral lines become closer together, becoming a continuum. Therefore, with Sf S f is the Fourier transform of st s t (the Fourier transform is symbolically denoted by the uppercase version of the signal's symbol) and is defined for any signal for which the integral (Equation 5) converges.

Figure 1 shows how increasing the period does indeed lead to a continuum of coefficients, and that the Fourier transform does correspond to what the continuum becomes. The quantity sintt t t has a special name, the sinc (pronounced "sink") function, and is denoted by sinct sinc t . Thus, the magnitude of the pulse's Fourier transform equals |ΔsincπfΔ| Δ sinc f Δ .

The Fourier transform relates a signal's time and frequency domain representations to each other. The direct Fourier transform (or simply the Fourier transform) calculates a signal's frequency domain representation from its time-domain variant (Equation 6). The inverse Fourier transform (Equation 7) finds the time-domain representation from the frequency domain. Rather than explicitly writing the required integral, we often symbolically express these transform calculations as ℱs ℱ s and ℱ-1S ℱ S , respectively.

We must have st=ℱ-1ℱst s t ℱ ℱ s t and Sf=ℱℱ-1Sf S f ℱ ℱ S f , and these results are indeed valid with minor exceptions. Showing that you "get back to where you started" is difficult from an analytic viewpoint, and we won't try here. Note that the direct and inverse transforms differ only in the sign of the exponent.

Properties of the Fourier transform and some useful transform pairs are provided in the accompanying tables (Table 1 and Table 2). Especially important among these properties is Parseval's Theorem, which states that power computed in either domain equals the power in the other.

Of practical importance is the conjugate symmetry property: When st s t is real-valued, the spectrum at negative frequencies equals the complex conjugate of the spectrum at the corresponding positive frequencies. Consequently, we need only plot the positive frequency portion of the spectrum (we can easily determine the remainder of the spectrum).

Note that the mathematical relationships between the time domain and frequency domain versions of the same signal are termed transforms. We are transforming (in the nontechnical meaning of the word) a signal from one representation to another. We express Fourier transform pairs as st↔Sf ↔ s t S f . A signal's time and frequency domain representations are uniquely related to each other. A signal thus "exists" in both the time and frequency domains, with the Fourier transform bridging between the two. We can define an information carrying signal in either the time or frequency domains; it behooves the wise engineer to use the simpler of the two.

A common misunderstanding is that while a signal exists in both the time and frequency domains, a single formula expressing a signal must contain only time or frequency: Both cannot be present simultaneously. This situation mirrors what happens with complex amplitudes in circuits: As we reveal how communications systems work and are designed, we will define signals entirely in the frequency domain without explicitly finding their time domain variants. This idea is shown in another module where we define Fourier series coefficients according to letter to be transmitted. Thus, a signal, though most familiarly defined in the time-domain, really can be defined equally as well (and sometimes more easily) in the frequency domain. For example, impedances depend on frequency and the time variable cannot appear.

We will learn that finding a linear, time-invariant system's output in the time domain can be most easily calculated by determining the input signal's spectrum, performing a simple calculation in the frequency domain, and inverse transforming the result. Furthermore, understanding communications and information processing systems requires a thorough understanding of signal structure and of how systems work in both the time and frequency domains.

The only difficulty in calculating the Fourier transform of any signal occurs when we have periodic signals (in either domain). Realizing that the Fourier series is a special case of the Fourier transform, we simply calculate the Fourier series coefficients instead, and plot them along with the spectra of nonperiodic signals on the same frequency axis.