The Fourier transform makes it possible to obtain the components that make up a signal in the transformed domain. In the most common case, the Fourier transform of temporal signals provides the frequencies (ω) that make up the signal. Equivalently, the spatial Fourier transform, applied on a signal defined in space, provides its components typically known as wavenumbers (k). These are also sometimes referred to as propagation constants (β).
This article shows the equivalence of the spatial Fourier transform with the temporal Fourier transform. It also explains the application of these transforms to propagating signals defined simultaneously in space and time.
The content is distributed according to the following table of contents:
1. From Time / Space to Frequency / WaveNumber
The Fourier Transform decomposes functions into sinusoidal periodic components (namely phasors). On the other hand, a generic signal propagating on an x-axis along a time t is defined by the function f(x,t). Such a signal can be decomposed into doubly periodic waves in space and time, as explained in this link. Therefore the transformed domains of both variables can be obtained separately.
1.1 Temporal Fourier Transform
In communications, time signals defined at a fixed point in space are typically used, e.g. the input of a receiver xo. Indeed, the signal f(xo,t) depends only on the variable t, time.
The Fourier Transform \footnotesize \mathcal{F} of a temporal signal obtains the representation of the signal in the frequency domain (ω) such that:
\begin{equation} f(t) \xtofrom[\mathcal{F}^{-1}]{\mathcal{F}} F(\omega) \end{equation}
1.2 Spatial Fourier Transform
Similarly, it is evident that Fourier can also be applied to signals defined in space, for a fixed time instant to. In this case f(x,to) represents a snapshot of a signal propagating through a medium.
The spatial Fourier Transform obtains the representation of the signal in the wavenumber (k) domain such that:
\begin{equation} f(x) \xtofrom[\mathcal{F}^{-1}]{\mathcal{F}} F(k) \end{equation}
1.3 Equivalence
1.3.1 Mathematical
The analogy of the two cases is evident in the following table showing the mathematics of the direct and inverse Fourier transform for spatial and temporal signals:
| \footnotesize Notation | \footnotesize Direct (\mathcal{F}) | \footnotesize Inverse (\mathcal{F}^{-1}) |
|---|---|---|
| \footnotesize Time (t) \xtofrom[\mathcal{F}^{-1}]{\mathcal{F}} Frequency (\omega) \footnotesize f(t) \xtofrom[\mathcal{F}^{-1}]{\mathcal{F}} F(\omega) | \footnotesize F(\omega) = \int_{-\infty}^\infty f(t)e^{-j\omega t}\,dt | \footnotesize f(t) = \cfrac{1}{2\pi}\int_{-\infty}^\infty F(\omega)e^{j\omega t}\,d\omega |
| \footnotesize Space (x) \xtofrom[\mathcal{F}^{-1}]{\mathcal{F}} Wavenumber (k) \footnotesize f(x) \xtofrom[\mathcal{F}^{-1}]{\mathcal{F}} F(k) | \footnotesize F(k) = \int_{-\infty}^\infty f(x)e^{-jkx}\,dx | \footnotesize f(x) = \cfrac{1}{2\pi}\int_{-\infty}^\infty F(k)e^{jkx}\,dk |
Note that, according to the inverse transform, signals in time or space are formed by a sum of phasors or periodic components. More precisely, these components are frequencies for temporal signals and wavenumbers for spatial signals. In a sense, it can be said that the wavenumbers are the frequencies that make up the signal in the spatial domain.
1.3.2 Graphical
Moreover, it is also clear that the pair of variables (t,ω) is interchangeable by the pair (x,k). In other words, any function produces the same periodic components regardless of whether it is defined in time or space. This effect is shown in the following graph:
2. Application to Signal Propagation
Signal propagation theory uses the spatial and temporal Fourier transform in a very illustrative way. Therefore, this section extends the concepts explained above by applying them to propagating signals. Such signals propagate at time t along an x-axis, and are therefore defined by a function f(x,t).
The frequency analysis of f(x,t) can be performed using two strategies, which are explained below. In both cases the following concepts must be taken into account:
- Signal Frequencies: f(x,t) is composed of waves of different amplitudes, frequencies and phases. Leaving amplitude and phase shift aside, each frequency component is governed by the expression \footnotesize \cos(\omega t - k x) = \cos(kx -\omega t).
- Propagation Function: the transmission medium implicitly determines a function ω(k) or k(ω) for the frequencies that make up the signal.
2.1 Temporal Evolution of the Spatial Fourier Transform
2.1.1 Mathematical Approach
In this case the Spatial Fourier Transform is used to define the signal on the x-axis at the time origin f(x,0). Using the direct and inverse transforms:
\begin{equation} F(k) = \int_{-\infty}^\infty f(x,0)e^{-jkx}\,dx \end{equation}
\begin{equation} f(x,0) = \cfrac{1}{2\pi}\int_{-\infty}^\infty F(k)e^{jkx}\,dk \end{equation}
Given the double periodicity of the waves in space and time, the evolution of the signal in time is obtained by adding the frequency shift in the inverse Fourier transform. From equation (4) it is obtained that:
\begin{equation} f(x,t) = \cfrac{1}{2\pi}\int_{-\infty}^\infty F(k)e^{j(kx-\omega(k)t)}\,dk \end{equation}
This methodology was used in this link for the generic calculation of the group velocity of a signal.
2.1.2 Graphical Representation
The following graph illustrates this analysis applied to the propagation of a pulse. It can be seen that this method starts from the initial representation of the signal in space and then the effects of propagation through the medium are applied.
This technique allows to visualize very clearly the effect of the transmission on the x-axis. However, for this strategy to be very accurate, the effect of the medium must also be taken into account during the excitation/signal generation at the time origin f(x,0).
2.2 Spatial Evolution of the Temporal Fourier Transform
2.2.1 Mathematical Approach
In this case the temporal Fourier Transform is used to define the signal at a point in space, typically the origin of the transmission, over time, f(0,t). Using the direct and inverse transforms:
\begin{equation} F(\omega) = \int_{-\infty}^\infty f(0,t)e^{-j\omega t}\,dt \end{equation}
\begin{equation} f(0,t) =\cfrac{1}{2\pi}\int_{-\infty}^\infty F(\omega)e^{j\omega t}\,d\omega \end{equation}
Due to the double periodicity of waves in space and time, the evolution of the signal in space is obtained by adding the displacement due to the wavenumber in the inverse Fourier transform. From equation (7) it is obtained that:
\begin{equation} f(x,t) =\cfrac{1}{2\pi}\int_{-\infty}^\infty F(\omega)e^{j(\omega t-k(\omega)x)}\,d\omega \end{equation}
This methodology is typically used to calculate the dispersion in an optical fiber [1].
2.2.2 Graphical Representation
This analysis applied to the propagation of a pulse is illustrated below. The method starts from the initial representation of the time signal at a point in space and then applies the effects of propagation through the medium.
This method can be considered more accurate than the previous one, since the effect of the medium is applied to the signal from the very moment of transmission at the source.
3. Conclusions
The conclusions of this article are as follows:
- The spatial Fourier Transform and the temporal Fourier Transform are equivalent. Both decompose a signal or function into periodic sinusoidal components.
- The components of the time transform are the frequencies (ω), while the components of the spatial transform are the wavenumbers (k).
- These transforms are commonly applied in signal propagation theory.
- There are two strategies for performing propagating signal transform analysis: time evolution of the inverse spatial transform, or spatial evolution of the inverse temporal transform.
Bibliography
[1] Fiber-Optic Communication Systems, Govind P. Agrawal
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