A Fourier Transform is a mathematical operation that indicates “how much” of the input function occurs at a particular “frequency.” The resulting function is sometimes called the frequency spectrum.
To make this more concrete, let’s consider an AC time-varying voltage V(t) = A_0 \cos(\omega_0 t). This voltage has only one pure frequency component at \omega = \omega_0. The Fourier Transform of V(t), let’s call the resulting function g(\omega), would be zero for all frequencies except for \omega = \omega_0.
What is the value of g(\omega) at \omega= \omega_0? This is a delta function. A delta function resembles a “peak” (think Gaussian or Lorentzian) with an infinite height, zero width, but a finite area (i.e. height times width is a nonzero, non-infinite constant).
In this example, the area of the delta function is A_0 which implies that g(\omega) \propto A_0 \delta(\omega-\omega_0). Our time-varying voltage could have been more complicated, for example V(t) = A_0 \cos(\omega_0 t) + A_1 \cos(\omega_1 t). In this case, Fourier Transform involves two frequency components which can be represented with two delta functions: g(\omega) \propto A_0 \delta(\omega-\omega_0) + A_1 \delta(\omega-\omega_1). (I use \propto and not = because of various factors of 2\pi that float around depending on the exact convention that is being followed for the Fourier Transform, see discussion below.)
In these simple examples, we could just “read off” the frequency components from the explicit form of voltage V(t). In general, this is not the case. The voltage could be a more complicated function of time and in general is not written explicitly written in terms of cosine functions. In this more general case, the Fourier Transform of the voltage will likely contain many more frequency components. The amount that a frequency component at \omega contributes to V(t) is given simply by g(\omega).
In order to determine the exact functional form of g(\omega), we perform a Fourier Transform and, in Boas notation, this is g(\omega) = \frac{1}{2\pi} \int_{-\infty}^{+\infty} \exp(-i \omega t) V(t)\ dt. This is an interesting integral and I’ll point out some of its features:
- The Fourier Transform g(\omega) is only a function of \omega and had no dependence on t. The more general name for \omega is the conjugate variable or reciprocal variable of t. When t is time, then \omega is just the angular frequency. If we replaced t with x, where x is a spatial position, then the conjugate or reciprocal variable is often labeled as k. In this case, k is called the spatial frequency and it closely related to the wave number. When t is a time variable, we say that V(t) represents the voltage in the time domain, while g(\omega) represents the voltage in the frequency domain. In MRI, we refer to the spatial domain and the corresponding K-space. In quantum mechanics, because of the concept of De Broglie wavelength, we often talk about position space and the corresponding momentum space, where the particle’s momentum is proportional to its spatial frequency.
- The result of the Fourier Transform is in general a complex number. This is deeply connected to the idea that our input function could be represented by both cosine and sine frequency components. The real part of the Fourier transform provides information about just the cosine components and the imaginary part provides information about the sine components. In other words, we need to specify both the Amplitude and Phase of each frequency component: A_0 \cos(\omega_0 t + \phi_0) = [A_0 \cos(\phi_0)]\cos(\omega_0t) + [-A_0\sin(\phi_0)]\sin(\omega_0 t).
- This definition of the Fourier Transform is not unique and the various conventions differ by how to treat the factor of 2\pi, see for example wikipedia for details.
- The original function can be recovered by an inverse Fourier Transform, which in Boas notation, is V(t) = \int_{-\infty}^{+\infty} \exp(+i\omega t) g(\omega)\ d\omega. Note the lack of the 2\pi and the presence of the positive i in the inverse transform.