A Note on Fourier Transform Conventions Used in Wave Analyses Jerald L. Bauck* July 12, 2019 Abstract Temporal and spatial Fourier transforms are natural tools in the study of propagating waves in many applications. For example, the inverse spatial Fourier transform specifies how any wave can be built by summing plane waves. However, sign conventions necessary to describe waves are at odds with sign conventions used in spatio-temporal Fourier transforms. This note describes the problem and shows several ways that authors deal with it. 1 Introduction While any function s as described will work as a wave, it is often convenient to use a sinusoidal expression called HEREIS a slightly uncomfortable relationship between a monochromatic wave. In that case, for forward traveling T how plane waves are expressed mathematically and waves and wavenumber vector k described below, we have how the Fourier transforms used to describe and analyze i(!t k x) s (t,x) p (!t k x) e§ ¡ ¢ (2) them are expressed. This note will discuss some of the vari- Æ ¡ ¢ Æ ous approaches that authors take. which is used with the usual caveat that the real part is to be Authors of “wave” books tend to describe a wave as some taken, or in the case of radar and communication systems, function s (t,x) p (kx !t) or s (t,x) p (kx !t) with that quadrature processing is used in the receiver to recover Æ ¡ Æ Å p ( ) a kind of base function and where the former form de- both real and imaginary parts. ! is identified as the tem- ¢ scribes a wave propagating in the forward x direction as poral radian frequency of the sinusoid in radians/s and in t increases and the latter form a wave propagating in the one dimensional cases k k is identified as the spatial ra- Æ j j x direction with increasing t, both assuming that k and dian frequency in radians/m; we allow their cyclic frequency ¡ ! are positive. Some other authors, perhaps of the signal counterparts f and º defined by ! 2¼f and k 2¼º as Æ Æ processing bent, insert a negative sign in the argument, as well, and we will use either form at will, for example, express- in s (t,x) p (!t kx) with again the form propagating ing the same monochromatic wave above as Æ § ¡ forward and the form propagating backward1. While all of i2¼(f t º x) Å s (t,x) e§ ¡ ¢ . (3) these are solutions to the one-dimensional wave equation Æ Spatial radian frequency is commonly called the wavenum- @2s 1 @2s ber. The three-dimensional counterpart to k is the wavenum- @x2 Æ c @t 2 ¡ ¢ ber vector k k ,k ,k 0 with each component indicating Æ x y z with the constraint that k !/c as can be verified by direct the wavenumber in each direction and k is the wavenum- Æ j j substitution, the “wave” forms reverse the base function p in ber in the direction of propagation which is indicated by the time and thus if the wave s is considered to be emitted from unit vector kˆ k/ k ; the cyclic frequency vector º is defined ¡ Æ ¢j j an antenna or a loudspeaker, it will come out backwards by k º ,º ,º 0 2¼º. If loci of constant function value— Æ x y z Æ which is probably not as intended. In three dimensions, the phase for the monochromatic case—can be constructed as generic wave can be described with a spatial wave vector straight lines or planes orthogonal to k then the wave is ¡ ¢ x x, y,z 0 as s (x,t) and the wave equation as called a plane wave. Æ One-dimensional Fourier transforms are traditionally de- @2s @2s @2s 1 @2s 2s . (1) fined with a “sign-i” convention as follows, first for the for- r Æ @x2 Å @y2 Å @z2 Æ c @t 2 ward transform from time t to frequency f and then for the inverse transform (sometimes the substitution ! 2¼f is *Originally published on July 27, 2018. Available at engrxiv.org. This pa- Æ per has not been peer reviewed. Jerry Bauck welcomes comments and can used but that does not concern us here). These temporal be reached by e-mail at this address after left-shifting each alpha character transforms are calculated at a fixed location in space, letting by one position, e.g., f becomes e: [email protected]. © 2018 Jerald L. Bauck t and f be variable. 1Some authors use forms such as p [ (ct x)] or p [ (t x/c)] but since § § § § these are mere scalings of the argument we shall not consider them to be ¡ ¢ i2¼f t S f ,x s (t,x)e¡ dt (4) essentially different than the cases presented. f Æ ˆ 1 2 APPROACHES 2 ¡ ¢ i2¼f t The approach used in [3] defines a plane wave as in (8) s (t,x) S f f ,x e d f . (5) Æ ˆ but an alternate wavenumber vector which is the same as de- Spatial Fourier transforms are also useful and we would like fined herein except with a negative sign. Thus, this alternate to maintain the same sign-i convention. Thus we hopefully vector points in the direction opposite to the propagation write direction. While this might seem odd, it is used in a chap- i2¼º x ter in which an antenna array is described and the reversed S (t,º) s (t,x)e¡ ¢ dx (6) º Æ ˆ vector points in the direction of arriving plane waves, op- posite to their direction of propagation if transmitted. It is i2¼º x difficult to find an actual Fourier transform definition in this s (t,x) Sº (t,º)e ¢ dk, (7) Æ ˆ multi-author book but one supposes that this negation of now fixing time and letting the spatial and wavenumber the wavenumber vector has the further advantage of using variables become the transform variables. The latter equa- the traditional i-signs of (4) through (7), i.e., for both time tion is of special interest because it shows that any wave s and space transforms. can be expressed as an infinite summation, an integral, of Reference [4] is an acoustics book describing radiating plane waves over all directions and wavenumbers. This is surfaces and defines plane waves as the important concept of the plane wave spectrum and is the i(k x !t) s (t,x) e ¢ ¡ , (9) spatial corollary of composing any time function from an Æ infinity of sinusoids as in (5) above. the back-ward-in time form. The i-sign problem is dealt Note the similarity between the exponential form describ- with by keeping the traditional form of the spatial Fourier ing the monochromatic wave (3) and the exponential form transform in (6) and (7) but defining a non-traditional, sign- inside the Fourier transforms. This is important because reversed form for the time-frequency transforms, wave fields often can be advantageously studied by using their Fourier transforms. The uncomfortable situation arises S ¡f ,x¢ s (t,x)ei2¼f t dt when one considers that it would be interesting to take the f Æ ˆ Fourier transform of a wave with respect to either time t or space x, but those variables have different signs in the wave ¡ ¢ i2¼f t s (t,x) S f ,x e¡ d f . function—how do we make the inverse Fourier transforms Æ ˆ f fit our notion of a plane wave? A perusal of a few books The same method is used in [5], another “wave” book. In [6] reveals that authors deal with this in different ways. waves are initially described by both time and spatial vari- ables as in (9) but quickly the time dependence is dropped, 2 Approaches referring only to the “amplitude distribution” of the field; time-frequency Fourier transforms are never used and the In [1] a four-dimensional transform pair is used, combining traditional sign convention of (6) and (7) prevails. time and three spatial dimensions. A plane wave is repre- Many authors structure their presentations of waves so sented by that the time dependence of the wave s (t,x) is carefully ig- i(!t k x) nored, or rather, put in the background. This can be done s (t,x) e ¡ ¢ (8) Æ through the use of a version of phasors that electrical en- and, separating the authors’ four-dimensional transform gineers typically learn in circuit and system analysis, or by into time and space transforms, the time-frequency inverse simply stating that the time dependence is “understood” transform is as in (5) and thus the forward transform is as as in [7], in any case, carrying along only the magnitude in (4). (Note that we are freely mixing radian and cyclical and phase of sinusoids. In the study of waves, removing frequency variables as promised, and focusing on the signs the time dependence leaves a phasor that is the complex in the complex exponentials.) However, to accommodate the amplitude as a function of the spatial coordinates x. An as- spatial variables’ reversed sign, the definition of the spatial pect of this approach involves taking the Fourier transform transforms is redefined as with respect to time of the wave equation (1) resulting in the time-independent Helmholtz equation [8] i2¼º x Sº (t,º) s (t,x)e ¢ dx Æ ˆ 2s k2s 0. r Å Æ and It should be remembered that the time dependence such i2¼º x s (t,x) S (t,º)e¡ ¢ dk, as exp(i!0t) can always be reinserted as needed, in either Æ ˆ º circuit or wave studies, to recover the actual signal or wave.