Geoid Determination by FFT Techniques

Abstract

This chapter introduces Fourier-based methods, and in particular the fast Fourier transform (FFT), as a tool for the efficient evaluation of the convolution integrals involved in geoid determination. An attempt was made to make this document as self-contained as possible for the benefit of readers inexperienced in spectral methods. Therefore, the Fourier transform and its properties are presented in the appendix following the chapter (Appendix A), and reference is made to the particular formulas and properties employed in geoid determination.

Appendix A: Definition, Properties and Application of the Fourier Transform

1.1 A.1 Basic Definitions

1.1.1 A.1.1 Sinusoids

A real sinusoid of amplitude A, cyclic frequency ω _o and phase angle θ _o is a function of the form:

$$s(t) = A\cos ({\omega }_{o}t + {\theta }_{o}),$$

(10.110)

where t is time or, usually in geodetic applications, distance. The cyclic frequency is related to the period T and the (linear) frequency f _o by the expression

$${\omega }_{o} = 2\pi /T = 2\pi {f}_{o}.$$

(10.111)

Expanding the cosine term in (10.110) yields

$$s(t) = a\cos {\omega }_{o}t + b\sin {\omega }_{o}t,\quad a = A\cos {\theta }_{o},\quad b = -A\sin {\theta }_{o}.$$

(10.112)

which allows for the computation of A and θ _o from the coefficients a and b:

$$\begin{array}{rcl} A& =& {({a}^{2} + {b}^{2})}^{1/2},\end{array}$$

(10.113)

$$\begin{array}{rcl}{ \theta }_{o}& =& \arctan (-b/a).\end{array}$$

(10.114)

With i being the imaginary unit, a complex sinusoid has the form

$${s}_{c}(t) = a\cos {\omega }_{o}t \pm ia\sin {\omega }_{o}t = a{e}^{\pm i{\omega }_{o}t},$$

(10.115)

which can be used to express a real sinusoid as a function of complex sinusoids:

$$s(t) = A\cos ({\omega }_{o}t+{\theta }_{o}) = A\frac{{e}^{i({\omega }_{o}t+{\theta }_{o})} + {e}^{-i({\omega }_{o}t+{\theta }_{o})}} {2} = \frac{A} {2} {e}^{i{\omega }_{o}t}{e}^{i{\theta }_{o} }+ \frac{A} {2} {e}^{-i{\omega }_{o}t}{e}^{-i{\theta }_{o} }.$$

(10.116)

1.1.2 A.1.2 Fourier Series

If a function g(t) is periodic with period T, i.e., if

$$g(t) = g(t + T);\quad \int\limits_{0}^{T}g(t)dt =\int\limits_{{t}_{o}}^{{t}_{o}+T}g(t)dt,$$

(10.117)

then, making use of the orthogonality properties of sine and cosine, g(t) can be expanded into the following series with coefficients a _n and b _n :

$$\begin{array}{rcl} g(t)& =& {\sum }_{n=0}^{\infty }\left [{a}_{ n}\cos \left (\frac{2\pi n} {T} t\right ) + {b}_{n}\sin \left (\frac{2\pi n} {T} t\right )\right ],\end{array}$$

(10.118)

$$\begin{array}{rcl}{ a}_{n}& =& \frac{2} {T}\int\limits_{-T/2}^{T/2}g(t)\cos \left (\frac{2\pi n} {T} t\right )dt, \\ {b}_{n}& =& \frac{2} {T}\int\limits_{-T/2}^{T/2}g(t)\sin \left (\frac{2\pi n} {T} t\right )dt,n = 0,1,2,\ldots.,\qquad \qquad \end{array}$$

(10.119)

provided that g(t) has a finite number of maxima and minima in a period, a finite number of finite discontinuities in a period, and is absolutely integrable over a period (Dirichlet’s conditions; see also Sect. A.2.1, Chap. 10).

Making use of (10.115) and (10.116), the above Fourier series expansion can be written in the following complex form:

$$\begin{array}{rcl} g(t)& =& \frac{1} {T}{\sum }_{n=-\infty }^{\infty }{G}_{ n}{e}^{i{\omega }_{n}t},{\omega }_{ n} = \frac{2\pi n} {T} = {\omega }_{o}n,\end{array}$$

(10.120)

$$\begin{array}{rcl}{ G}_{n}& =& \int\limits_{-T/2}^{T/2}g(t){e}^{-i{\omega }_{n}t}dt=\frac{1} {2}({a}_{n} - i{b}_{n}),n = 0,\pm 1,\pm 2,\ldots.,\end{array}$$

(10.121)

which shows that a Fourier expansion decomposes a periodic function into a sum of sinusoids with cyclic frequencies 2πn ∕ T.

Denoting by $\Delta \omega$ the frequency ‘spacing’ or ‘step’ 2π ∕ T, we get

$$\Delta \omega = \frac{2\pi } {T} ,\quad {\omega }_{n} = n\Delta \omega ,\quad \frac{1} {T} = \frac{\Delta \omega } {2\pi } ,$$

(10.122)

and thus (10.120) is finally written as a series, i.e., linear combination, of complex sinusoids:

$$g(t) = \frac{1} {2\pi }{\sum }_{n=-\infty }^{\infty }{G}_{ n}{e}^{i{\omega }_{n}t}\Delta \omega.$$

(10.123)

1.2 A.2 The Continuous Fourier Transform and Its Properties

1.2.1 A.2.1 Definition of the Continuous Fourier Transform

We give here a heuristic definition of the continuous Fourier transform (CFT), or continuous spectrum, based on the Fourier series. By letting T → ∞, the periodic function g(t) becomes non-periodic. Also, n → ∞, ω _o becomes vanishingly small, say ω _o  = Δω → 0 and ω _n  = nΔω → ω. Then at the limit, T → ∞, Δω → dω the summation becomes integration, i.e, G _n  = G(ω),  ∑Δω =  ∫dω, and (10.121) and (10.123), respectively, become

$$\begin{array}{rcl} G(\omega )& =& \int\limits_{-\infty }^{\infty }g(t){e}^{-i\omega t}dt,\end{array}$$

(10.124)

$$\begin{array}{rcl} g(t)& =& \frac{1} {2\pi }\int\limits_{-\infty }^{\infty }G(\omega ){e}^{i\omega t}d\omega ,\end{array}$$

(10.125)

which define the direct and inverse CFT. Since ω = 2πf, the factor 1 ∕ 2π can be avoided by expressing the spectrum as a function of f instead of ω as follows:

$$\begin{array}{rcl} G(f)& =& \int\limits_{-\infty }^{\infty }g(t){e}^{-2\pi ift}dt = \mathbf{F}\{g(t)\},\end{array}$$

(10.126)

$$\begin{array}{rcl} g(t)& =& \int\limits_{-\infty }^{\infty }G(f){e}^{2\pi ift}df ={ \mathbf{F}}^{-\mathbf{1}}\{G(f)\},\end{array}$$

(10.127)

where F and F ^− 1 denote the direct and inverse Fourier transform, respectively. The direct and inverse CFT are called a Fourier transform pair and are usually abbreviated as

$$g(t) \leftrightarrow G(f).$$

(10.128)

G(f) is, in general, a complex function with real part G _R (f) and imaginary part G _I (f). It thus contains information both about the amplitude | G(f) | and the phase angle θ(f). Similarly to (10.112)–(10.116), these quantities are

$$\begin{array}{rcl} G(f)& =& {G}_{R}(f) + i{G}_{I}(f) = \left \vert G(f)\right \vert {e}^{i\theta (f)},\end{array}$$

(10.129)

$$\begin{array}{rcl} \left \vert G(f)\right \vert & =& {[{G}_{R}^{2}(f) + {G}_{ I}^{2}(f)]}^{1/2},\end{array}$$

(10.130)

$$\begin{array}{rcl} \theta (f)& =& Arg\{G(f)\} =\arctan \frac{{G}_{I}(f)} {{G}_{R}(f)}.\end{array}$$

(10.131)

G(f) exists when g(t) is absolutely integrable, i.e., the integral of | g(t) | from − ∞ to ∞ exists (is < ∞), and g(t) has only finite discontinuities. If g(t) is periodic or impulse, G(f) does not exist unless the theory of distributions is introduced. This leads to the definition of the impulse function that is given below.

1.2.2 A.2.2 The Impulse Function

The unit impulse or Dirac delta function δ(t) is usually defined by the relationships

$$\delta (t - {t}_{o}) = 0,t\neq {t}_{o};\int\limits_{-\infty }^{\infty }\delta (t - {t}_{ o})dt = 1.$$

(10.132)

Other definitions are based on treating the impulse function as a distribution or a generalized limit of a sequence of functions. An alternative definition is

$$\delta (t) ={ \lim }_{a\rightarrow 0}f(t,a),$$

(10.133)

where f(t, a) is a function in a series of functions that progressively increase in amplitude, decrease in duration, and have a constant area of unit; see Fig. 10.7. Using f(t, a) = sin(at) ∕ πt, the following expression for δ(t) is obtained (Papoulis 1977, 1984), which is of importance in evaluating the otherwise non-existent CFT of periodic and other particular functions:

$$\int\limits_{-\infty }^{\infty }\cos (2\pi ft)df =\int\limits_{-\infty }^{\infty }{e}^{i2\pi ft}dt = \delta (t).$$

(10.134)

Fig. 10.7

The impulse function as the limit of a function sequence

As an example, the CFT of the sinusoid function of (10.110) will be derived for θ _o  = 0; see Fig. 10.8. Equation 10.126, using (10.111), (10.117) and (10.134) gives

$$\begin{array}{rcl} S(f)& =& \mathbf{F}\{s(t)\} =\int\limits_{-\infty }^{\infty }A\cos (2\pi {f}_{ o}t){e}^{-i2\pi ft}dt \\ & =& \frac{A} {2} \int\limits_{-\infty }^{\infty }({e}^{i2\pi {f}_{o}t} + {e}^{-i2\pi {f}_{o}t}){e}^{-i2\pi ft}dt \\ & =& \frac{A} {2} \int\limits_{-\infty }^{\infty }({e}^{-i2\pi (f-{f}_{o})t} + {e}^{-i2\pi (f+{f}_{o})t})dt \\ & =& \frac{A} {2} \delta (f - {f}_{o}) + \frac{A} {2} \delta (f + {f}_{o}), \\ A\cos (2\pi {f}_{o}t)& \leftrightarrow & \frac{A} {2} \delta (f - {f}_{o}) + \frac{A} {2} \delta (f + {f}_{o}). \end{array}$$

(10.135)

Fig. 10.8

The Fourier transform pairs of the cosine and the sine function

Similarly, for the sine function (see Fig. 10.8), it can be proven that

$$A\sin (2\pi {f}_{o}t) \leftrightarrow i\frac{A} {2} \delta (f + {f}_{o}) - i\frac{A} {2} \delta (f - {f}_{o}).$$

(10.136)

Important properties of the impulse function are listed below:

$$\begin{array}{rcl} & & \delta ({t}_{o})h(t) = h({t}_{o})\delta ({t}_{o})\end{array}$$

(10.137)

$$\begin{array}{rcl} & & \int\limits_{-\infty }^{\infty }\delta (t - {t}_{ o})h(t)dt = h({t}_{o}),\end{array}$$

(10.138)

$$\begin{array}{rcl} & & \delta (at) ={ \left \vert a\right \vert }^{-1}\delta (t),\end{array}$$

(10.139)

$$\begin{array}{rcl} & & \mathbf{F}\{a\delta (t)\} = a,\end{array}$$

(10.140)

$$\begin{array}{rcl} & & \Delta (t) ={ \sum }_{n=-\infty }^{\infty }\delta (t - nT) \leftrightarrow \Delta (f) = \frac{1} {T}{\sum }_{n=-\infty }^{\infty }\delta \left (f - \frac{n} {T}\right ).\end{array}$$

(10.141)

The last expression describes a sequence of impulse functions, sometimes called ‘comb’ function, which repeat at intervals T in the time (space) domain and 1/T in the frequency domain. The multiplication of any continuous function with $\Delta (t)$ produces digitization. Thus, $\Delta (t)$ is very important for sampling and for deriving formulas for the discrete Fourier transform from those for the continuous Fourier transform.

1.2.3 A.2.3 The Rectangle and the Sinc Functions

Also important for deriving formulas for the discrete Fourier transform from those for the continuous Fourier transform are the rectangle and the sinc functions, which actually form a Fourier transform pair. The rectangle function of base T _o and amplitude A is defined as follows:

$$\Pi (t) = \left \{\begin{array}{ll} A, &\left \vert t\right \vert = {T}_{0}/2 \\ A/2,&t = \pm {T}_{0}/2 \\ 0, &\left \vert t\right \vert > {T}_{0}/2\\ \end{array} \right..$$

(10.142)

The sinc function, which is very important in interpolation problems, is defined as

$$\mathrm{sinc}(t) = \frac{\sin (\pi t)} {\pi t} ,$$

(10.143)

and the Fourier transform pair (see Fig. 10.9) is

$$\Pi (t) \leftrightarrow 2A{T}_{\mathrm{o}}\mathrm{sinc}(2{T}_{o}f).$$

(10.144)

Fig. 10.9

The rectangle function and its Fourier transform, the sinc function

1.2.4 A.2.4 Interpretation of the Fourier Transform and the Fourier Series

Equation 10.120 indicates that a periodic function can be represented as a sum of harmonics of amplitudes G _n and cyclic frequencies ω _n , with fundamental frequency ω _o . Comparing (10.125) to (10.120), G(ω)dω ∕ 2π can be viewed as the infinitesimal magnitude of a ‘harmonic’ with cyclic frequency ω. These ‘harmonics’ have zero fundamental frequency (ω _o  → dω) and are ‘spaced’ infinitesimally far apart. In other words, a non-periodic function can be represented as a sum of exponentials (harmonics) with fundamental frequency tending to zero!

From the above interpretation, and also from (10.129) to (10.131), it is clear that the Fourier transform contains information regarding the amplitude and the phase of the ‘harmonics’ that constitute the function. This becomes easily apparent in the examples of Fig. 10.8, where the spectra show both the amplitude and frequency of the sine and cosine functions and the fact that they have a phase difference of π ∕ 2 [recall that cos( − x) = cosx while sin( − x) =  − sin(x)]. Basically, we assume that any given function has two equivalent representations: one in the time (or space) domain and another one in the frequency domain. Equation 10.126 analyzes the time (space) function into a frequency spectrum (in terms of magnitude and phase or, equivalently, in terms of real and imaginary part; see 10.129), while (10.127) synthesizes the frequency spectrum to regain the time (space) function. Equation 10.130 gives the magnitude spectrum while (10.131) gives the phase spectrum of the function.

1.2.5 A.2.5 Properties of the CFT

The following properties are listed here without proof. The proofs, based directly on the definition equations of the CFT, can be found in Brigham (1988).

$$\begin{array}{rcl} & & ah(t) + bg(t) \leftrightarrow aH(f) + bG(f)\qquad \qquad \,\,\,\,\,\mathrm{Linearity}\end{array}$$

(10.145)

$$\begin{array}{rcl} & & H(t) \leftrightarrow h(-f)\qquad \qquad \qquad \qquad \qquad \qquad \,\,\,\,\,\mathrm{Symmetry}\end{array}$$

(10.146)

$$\begin{array}{rcl} & & h(at) \leftrightarrow \frac{1} {\left \vert a\right \vert }H\left (\frac{f} {a}\right )\qquad \qquad \qquad \qquad \qquad \,\text{ Time scaling}\end{array}$$

(10.147)

$$\begin{array}{rcl} & & h(t - {t}_{o}) \leftrightarrow H(f){e}^{-i2\pi f{t}_{o} }\qquad \qquad \qquad \qquad \,\,\text{ Time shifting}\end{array}$$

(10.148)

$$\begin{array}{rcl} & & \frac{{\partial }^{n}h(t)} {\partial {t}^{n}} \leftrightarrow {(i2\pi f)}^{n}H(f)\qquad \qquad \qquad \qquad \quad \mathrm{Differentiation}\end{array}$$

(10.149)

$$\begin{array}{rcl} & & \int\limits_{-\infty }^{t}h(x)dx \leftrightarrow \frac{1} {i2\pi f}H(f) + \frac{1} {2}H(0)\delta (f)\qquad \mathrm{Integration}\end{array}$$

(10.150)

$$\begin{array}{rcl} & & \int\limits_{-\infty }^{\infty }h(t)dt = H(0)\qquad \qquad \qquad \qquad \qquad \quad \,\,\,\mathrm{DC - value}\end{array}$$

(10.151)

$$\begin{array}{rcl} & & {h}_{E}(t) \leftrightarrow {H}_{E}(f) = {R}_{E}(f)\qquad \qquad \qquad \quad \,\,\,\,\,\text{ Even function}\end{array}$$

(10.152)

$$\begin{array}{rcl} & & {h}_{O}(t) \leftrightarrow {H}_{O}(f) ={ \mathit{iI}}_{O}(f)\qquad \qquad \qquad \quad \,\,\,\,\,\text{ Odd function}\end{array}$$

(10.153)

$$\begin{array}{rcl} & & h(t) = {h}_{R}(t) \leftrightarrow H(f) = {R}_{E}(f) + i{I}_{O}(f)\quad \text{ Real function}\end{array}$$

(10.154)

$$\begin{array}{rcl} & & h(t) = i{h}_{I}(t) \leftrightarrow H(f)={R}_{O}(f)+i{I}_{E}(f)\quad \,\,\,\,\text{ Imaginary function}\quad \quad \quad \end{array}$$

(10.155)

In the above formulas, R and I stand for the real and imaginary part of H, respectively, and the subscripts E, O, R, I stand for even, odd, real and imaginary function, respectively.

1.2.6 A.2.6 Convolution and Correlation

The convolution and correlation of two functions g(t) and h(t), denoted by ∗ and  ⊗ , respectively, are defined as follows:

$$\begin{array}{rcl} x(t)& =& \int\limits_{-\infty }^{\infty }g(\tau )h(t - \tau )d\tau = g(t) {_\ast} h(t) = h(t) {_\ast} g(t) \\ & =& \int\limits_{-\infty }^{\infty }h(\tau )g(t - \tau )d\tau , \end{array}$$

(10.156)

$$\begin{array}{rcl} y(t)& =& \int\limits_{-\infty }^{\infty }g(\tau )h(t + \tau )d\tau = g(t) \otimes h(t)\neq h(t) \otimes g(t) \\ & =& \int\limits_{-\infty }^{\infty }h(\tau )g(t + \tau )d\tau.\qquad \quad \end{array}$$

(10.157)

The most important property of the convolution is that its spectrum is the product of the spectra of the two functions. Similarly, correlation transforms to multiplication of the complex conjugate of the second spectrum, denoted by superscript ^∗, with the spectrum of the first function. These constitute the convolution theorem and the correlation theorem, respectively, which in abbreviated form are

$$\begin{array}{rcl} x(t)& =& g(t) {_\ast} h(t) \leftrightarrow X(f) = G(f)H(f),\end{array}$$

(10.158)

$$\begin{array}{rcl} y(t)& =& g(t) \otimes h(t) \leftrightarrow Y (f) = G(f){H}^{{_\ast}}(f).\end{array}$$

(10.159)

The process of convolution in the time (space) domain comprises four steps: (1) folding, i.e., taking the mirror image of h(τ) about the ordinate axis;(2) displacement, i.e., shifting h( − τ) by the amount t; (3) multiplication of h(t − τ) by g(τ); and (4) integration, i.e., computation of the area under the product of h(t − τ) and g(τ). In correlation, the procedure is the same without the folding step; see Fig. 10.10.

Fig. 10.10

Graphical illustration of time-domain convolution and correlation (After Brigham 1988)

Although this four-step process shows what needs to be done to evaluate a convolution (or a correlation) integral numerically, it really gives no clear ‘physical’ interpretation of what a convolution is. This, however, becomes rather obvious from the frequency domain representation of convolution. The multiplication of the two spectra indicates that the whole process is nothing else but filtering of one of the functions by the other. In other words, regions of the spectrum of one of the functions are either attenuated, or amplified, or otherwise altered according to the shape of the spectrum of the other function. This interpretation is important in the frequency-domain evaluation of gravity field convolution integrals like, e.g., Stokes’s integral.

The simple spectral representations of (10.158) and (10.159) are of great practical importance. It is now obvious that instead of computing the tedious convolution and correlation integrals by numerical integration one could evaluate them by multiplication of the spectra and use of the inverse Fourier transform. Two direct and one inverse Fourier transforms are needed in each case, and the process is made clear by the following equations:

$$\begin{array}{rcl} x(t)& =& g(t) {_\ast} h(t) ={ \mathbf{F}}^{-\mathbf{1}}\{X(f)\} ={ \mathbf{F}}^{-\mathbf{1}}\{G(f)H(f)\} \\ & =&{ \mathbf{F}}^{-\mathbf{1}}\{\mathbf{F}\{g(t)\}\mathbf{F}\{h(t)\}\}, \end{array}$$

(10.160)

$$\begin{array}{rcl} y(t)& =& g(t) {_\ast} h(t) ={ \mathbf{F}}^{-\mathbf{1}}\{Y (f)\} ={ \mathbf{F}}^{-\mathbf{1}}\{G(f){H}^{{_\ast}}(f)\} \\ & =&{ \mathbf{F}}^{-\mathbf{1}}\{\mathrm{\mathbf{F}}\{g(t)\}{[\mathbf{F}\{h(t)\}]}^{{_\ast}}\}.\qquad \qquad \end{array}$$

(10.161)

Important properties of convolution are listed below:

$$\begin{array}{rcl} & & g(t) {_\ast} h(t) = g(t) \otimes h(t),\text{ if either }g(t)\,\mathrm{or}\,h(t)\,\text{ is even};\end{array}$$

(10.162)

$$\begin{array}{rcl} & & \delta (t + {t}_{o}) {_\ast} h(t) = h(t + {t}_{o}),\,\,\delta (t) {_\ast} h(t) = h(t);\end{array}$$

(10.163)

$$\begin{array}{rcl} & & \frac{\partial x(t)} {\partial t} = \frac{\partial [g(t) {_\ast} h(t)]} {\partial t} = \frac{\partial g(t)} {\partial t} {_\ast} h(t) = g(t) {_\ast}\frac{\partial h(t)} {\partial t} ;\end{array}$$

(10.164)

$$\begin{array}{rcl} & & g(t)h(t) \leftrightarrow G(f) {_\ast} H(f).\end{array}$$

(10.165)

1.3 A.3 The Discrete Fourier Transform

1.3.1 A.3.1 From the Continuous to the Discrete Fourier Transform: Aliasing and Leakage

In the practical implementation of the Fourier transform formulas, two approximations are employed: (a) the continuous integrations are replaced by discrete summations and (b) the infinite limits of summation are replaced by finite ones. Obviously, such approximations will introduced errors due to the digitization and the truncation of the series that may or may not be significant depending of the properties of the transformed function. Figure 10.11 illustrates graphically the process of going from the continuous to the discrete Fourier transform (DFT).

Fig. 10.11

From the continuous to the discrete Fourier transform (After Brigham 1988)

First, the function h(t) is sampled or digitized with a sampling interval $\Delta t\,=\,T$ by multiplying it with a comb function ${\Delta }_{o}(t)$. According to (10.141) and (10.165), this leads to the convolution of the spectrum of h(t) with the spectrum of ${\Delta }_{o}(t)$ which is another comb function consisting of impulses at intervals 1 ∕ T. $H(f){_\ast}{\Delta }_{o}(f)$ is thus a repeating, i.e., periodic, version of the true spectrum. Depending on the value of T, this repetition can cause overlap, which alters the spectrum producing an error caused aliasing. The next step is to limit the extent of the function to a finite length, say T _o , containing N sampled points. This is accomplished by multiplying the discretized function by a rectangular function of base T _o and unit height, denoted x(t) in Fig. 10.11, which leads to the multiplication of $H(f){_\ast}{\Delta }_{o}(f)$ by a sinc function (see 10.142, 10.143 and 10.144). Consequently, another distortion is introduce to the resulting spectrum $H(f) {_\ast} {\Delta }_{o}(f) {_\ast} X(f)$ called leakage. The last step is to discretize the resulting spectrum by multiplying it by a frequency-domain comb function ${\Delta }_{1}(f)$ with frequency spacing $\Delta f = 1/{T}_{o}$, which of course leads to the repetition of the discretized time (or space) domain function. The DFT is thus periodic in both domains:

$$\begin{array}{rcl} & & H(m\Delta f) ={ \sum }_{j=-\infty }^{\infty }H(m\Delta f + jF),\end{array}$$

(10.166)

$$\begin{array}{rcl} & & h(k\Delta t) ={ \sum }_{\mathcal{l}=-\infty }^{\infty }h(k\Delta t + \mathcal{l}T),\end{array}$$

(10.167)

and can be defined as follows:

$$\begin{array}{rcl} & & H(m\Delta f)={ \sum }_{k=0}^{N-1}h(k\Delta t){e}^{-i2\pi k\Delta tm\Delta f}\Delta t ={ \sum }_{k=0}^{N-1}h(k\Delta t){e}^{-i2\pi km/N}\Delta t,\end{array}$$

(10.168)

$$\begin{array}{rcl} & & h(k\Delta t) ={ \sum }_{m=0}^{N-1}H(m\Delta f){e}^{i2\pi k\Delta tm\Delta f}\Delta f ={ \sum }_{m=0}^{N-1}H(m\Delta f){e}^{i2\pi km/N}\Delta f.\end{array}$$

(10.169)

In discrete form, the functions have arguments either their wavelengths ${t}_{k} = k\Delta t$ or simply their wavenumbers k in the time (space) domain, and ${f}_{m} = m\Delta f$ or simply m in the frequency domain. We will use these representations interchangeably, i.e., we will defined the DFT pair in any one of the following three forms:

$$h(k\Delta t) \leftrightarrow H(m\Delta f)\quad \mathrm{or}\quad h({t}_{k}) \leftrightarrow H({f}_{m})\quad \mathrm{or}\quad h(k) \leftrightarrow H(m).$$

(10.170)

The time period T _o , the frequency period F _o , the time spacing $\Delta$ t, the frequency spacing $\Delta$ f and the number of discrete points N are related as follows:

$${T}_{o} = \frac{1} {\Delta f} = N\Delta t,\quad {F}_{o} = \frac{1} {\Delta t} = N\Delta f.$$

(10.171)

The above equations show that there is a certain maximum frequency (shortest wavelength) and a certain minimum frequency (longest wavelength) that can be recovered from the DFT. Frequencies beyond these limits cannot be recovered due to the aliasing and leakage effects. The maximum frequency that can be recovered is F _o  ∕ 2, depends on $\Delta$t, and is called the Nyquist frequencyf _N . From (10.172)

$$\left \vert {f}_{N}\right \vert = \frac{{F}_{o}} {2} = \frac{1} {2\Delta t}.$$

(10.172)

The aliasing error, say H _e , can be shown mathematically by rewriting (10.166) as

$$\begin{array}{rcl}{ H}_{P}(m\Delta f)& =& {\sum }_{j=-\infty }^{\infty }H(m\Delta f + jF) = H(m\Delta f) +{ \sum }_{{ j=-\infty \atop j\neq 0} }^{\infty }H(m\Delta f + jF) \\ & =& H(m\Delta f) + {H}_{e}(m\Delta f), \end{array}$$

(10.173)

where a subscript P has been added to the left-hand side to indicate the periodic nature of the DFT. Thus to minimize aliasing, the function must be sampled as densely as possible and to eliminate it $\Delta t$ should be selected such that $1/2\Delta t$ is larger that the highest frequency present in the data. However, the user cannot always select $\Delta t$ and minimize aliasing, as is the case when gravity or terrain data are only available on regular grids. In such cases, aliasing can be minimized by removing the high-frequency information from the data by, e.g., applying terrain reductions to gravity anomalies.

The minimum frequency that can be recovered depends on T _o , i.e., on both N and $\Delta t$, and is $\Delta f = 1/{T}_{o} = 1/N\Delta t$. Given $\Delta t$, N should be chosen so that it provides the required frequency resolution $\Delta f$. In practice, N is much easier to control than $\Delta t$ but it will always be a finite number and thus leakage will always be present. From Fig. 10.11e, the altered spectrum due to leakage only will be

$$H \prime (m\Delta f) = {T}^{-1}H(m\Delta f) {_\ast} {T}_{ o}\mathrm{sinc}({T}_{o}m\Delta f).$$

(10.174)

This error does nor occur only when T _o is infinite, i.e., when the Π-function becomes a unit constant from − ∞ to ∞. In this case, from (10.140) and (10.146) we obtain F{1} = δ(f), and (10.174), using (10.163), becomes ${H}^{{\prime}}(m\Delta f) = H(m\Delta f) {_\ast} \delta (m\Delta f) = H(m\Delta f)$. In practice of course this is not possible and, in order to minimize leakage, the truncation of the infinite function is done by functions other than the Π-function, called window functions (Harris 1978). These functions have spectra with smaller side lobes than the sinc function, i.e., their spectra are better approximations to an impulse function than the sinc function is. In gravity field applications, leakage can be minimized by removing the low-frequency information from the data by, e.g., removing the contribution of a global geopotential model from gravity anomalies.

1.3.2 A.3.2 Discrete Convolution and Correlation: Circular Convolution and Correlation

Discretization of (10.146) and (10.157) for both functions given at N points results in the following expressions for discrete convolution and correlation:

$$\begin{array}{rcl} & & x(k) ={ \sum }_{l=0}^{N-1}g(l)h(k - l)\Delta t = g(k) {_\ast} h(k),\end{array}$$

(10.175)

$$\begin{array}{rcl} & & y(k) ={ \sum }_{l=0}^{N-1}g(l)h(k + l)\Delta t = g(k) \otimes h(k).\end{array}$$

(10.176)

When these equations are evaluated by numerical summation the results are correct and correspond to linear convolution and linear correlation. If, however, the discrete form of (10.160) and (10.161) are used instead, i.e.,

$$\begin{array}{rcl}{ x}_{P}(k)& =&{ \mathbf{F}}^{-\mathbf{1}}\{{X}_{ P}(m)\} ={ \mathbf{F}}^{-\mathbf{1}}\{{G}_{ P}(m){H}_{P}(m)\} \\ & =&{ \mathbf{F}}^{-\mathbf{1}}\{\mathbf{F}\{{g}_{ P}(k)\}\mathbf{F}\{{h}_{P}(k)\}\}, \end{array}$$

(10.177)

$$\begin{array}{rcl}{ y}_{P}(k)& =&{ \mathbf{F}}^{-\mathbf{1}}\{{Y }_{ P}(m)\} ={ \mathbf{F}}^{-\mathbf{1}}\{{G}_{ P}(m){H}_{P}^{{_\ast}}(m)\} \\ & =&{ \mathbf{F}}^{-\mathbf{1}}\{\mathbf{F}\{{g}_{ P}(k)\}{[\mathbf{F}\{{h}_{P}(k)\}]}^{{_\ast}}\},\qquad \qquad \end{array}$$

(10.178)

both functions are treated as periodic (hence the subscript P), the results are incorrect and correspond to circular convolution and circular correlation. Equations 10.175 and 10.176 indicate that if g(k) and h(k) have N values (or support N) each, then x(k) and y(k) will each have 2N − 1 values. On the other hand, when (10.177) and (10.178) are evaluated by the (periodic) DFT, it is clear that the resulting x(k) and y(k) will each have support N and will be periodic, as well. Mathematically, circular convolution can be viewed as linear convolution contaminated by aliasing (see Fig. 10.12), i.e.,

$${x}_{P}(k) ={ \sum }_{r=-\infty }^{\infty }x(k + rN),\quad 0 \leq k \leq N - 1.$$

(10.179)

Fig. 10.12

Illustration of circular convolution as linear convolution plus aliasing (After Oppenheim and Schafer 1989)

Circular convolution can be avoided by a procedure called zero-padding by which zeros are appended to g(k) and h(k) as follows:

$$g \prime (k) = \left \{\begin{array}{ll} g(k),&0 \leq k \leq N\\ 0, &N \leq k \leq 2N\\ \end{array} \right.;\qquad h \prime (k) = \left \{\begin{array}{ll} h(k),&0 \leq k \leq N\\ 0, &N \leq k \leq 2N\\ \end{array} \right..$$

(10.180)

The required steps are: (1) Form g ^′(k) and h ^′(k); (2) compute G ^′(m) and H ^′(m) via the DFT; (3) compute ${X}^{{\prime}}(m) = {G}^{{\prime}}(m){H}^{{\prime}}(m)$; and (4) compute x ^′(k) by applying the inverse DFT to X ^′(m). Now x ^′(k) is a 2N − 1 sequence and is exactly the same as x(k) because no aliasing due to overlapping occurs; see again Fig. 10.12. This procedure is the same for computing correlation. For more details on circular convolution and correlation, Oppenheim and Schafer (1989) should be consulted.

1.3.3 A.3.3 Correlation, Covariance, and Power Spectral Density Functions

The discrete correlation function R _gh (t _k ) of two functions h(t _k ) and g(t _k ) is defined as

$${ R}_{gh}({t}_{k}) = \mathbf{E}\{g({t}_{l})h({t}_{k}-{t}_{l})\}={\lim }_{N\rightarrow \infty }{ 1 \over N} {\sum }_{l=0}^{N-1}g({t}_{ l})h({t}_{k}-{t}_{l})={\lim }_{{T}_{o}\rightarrow \infty } \frac{1} {{T}_{o}}g({t}_{k})\otimes h({t}_{k}),$$

(10.181)

where we have made use of (10.171) and (10.176) which defines the discrete correlation. When the mean values $\overline{g}$, $\overline{h}$ are subtracted, the formula for the discrete covariance function C _gh (t _k ) is obtained:

$$\begin{array}{rcl}{ C}_{gh}({t}_{k})& =& \mathbf{E}\{\{g({t}_{l}) -\overline{g}][h({t}_{k} - {t}_{l}) -\overline{h}]\} ={ \lim }_{N\rightarrow \infty } \frac{1} {N}{\sum }_{l=0}^{N-1}[g({t}_{ l}) -\overline{g}][h({t}_{k} - {t}_{l})-\overline{h}] \\ & =&{ \lim }_{{T}_{o}\rightarrow \infty } \frac{1} {{T}_{o}}g({t}_{k}) \otimes h({t}_{k}) -\overline{g}\overline{h} = {R}_{gh}({t}_{k}) -\overline{g}\overline{h}, \end{array}$$

(10.182)

where, using the discrete version of (10.151), $\overline{g}$ (and similarly $\overline{h})$ can be expressed as

$$\overline{g} ={ \lim }_{N\rightarrow \infty }{ 1 \over N} { \sum }_{k=0}^{N-1}g({t}_{k}) ={ \lim }_{{T}_{o}\rightarrow \infty } \frac{1} {{T}_{o}}G(0).$$

(10.183)

When h(t _k ) and g(t _k ) are the same function, we talk about the auto-covariance and the auto-correlation function. When they are different, we talk about the cross-covariance and the cross-correlation function. The spectrum of the correlation function is called the power spectral density (PSD) function P _gh (f _m ) and, by (10.178), it has the form

$${P}_{gh}({f}_{m}) = \mathbf{F}\{{R}_{gh}({t}_{k})\} ={ \lim }_{{T}_{o}\rightarrow \infty } \frac{1} {{T}_{o}}G({f}_{m}){H}^{{_\ast}}({f}_{ m}).$$

(10.184)

In practice, of course, we only have a finite number of data and P _gh (f _m ) is approximated by the biased estimate F ^− 1{G(f _m )H ^∗(f _m )}. If ν records are available each containing N data values, an unbiased estimate for the PSD function is obtained by averaging over all records (Bendat and Piersol 1986) as follows:

$$\hat{{P}}_{gh}({f}_{m}) = \frac{1} {\nu {T}_{o}}{ \sum }_{\lambda =1}^{\nu }{G}_{ \lambda }({f}_{m}){H}_{\lambda }^{{_\ast}}({f}_{ m}).$$

(10.185)

The normalized standard error $\epsilon$ of $\hat{{P}}_{gh}$ computed from ν sample records or, more generally, using ν number of averages is

$$\epsilon = \frac{\sigma (\hat{{P}}_{gh})} {\hat{{P}}_{gh}} = \frac{1} {\sqrt{\nu }},$$

(10.186)

where σ denotes the standard error. Thus, 100 averages are required for a 10% error. When only one sample record is available, the estimated PSD is called the periodogram. Although very noisy, it might be the only estimate that can be obtained from a single record.

By applying the inverse Fourier transform to the PSD function, an efficient way of estimating correlation and covariance functions of gridded data is obtained:

$$\begin{array}{rcl} & & \hat{{R}}_{gh}({t}_{k}) = \mathbf{F}\{\hat{{P}}_{gh}({f}_{m})\},\end{array}$$

(10.187)

$$\begin{array}{rcl} & & \hat{{C}}_{gh}({t}_{k}) = \mathbf{F}\{\hat{{P}}_{gh}({f}_{m}) -\overline{g}\overline{h}\delta ({f}_{m})\}.\end{array}$$

(10.188)

We end this section by some useful properties of the correlation, covariance, and PSD functions:

$$\begin{array}{rcl} & & {R}_{gh}(-{t}_{k}) = {R}_{hg}({t}_{k}),\quad {C}_{gh}(-{t}_{k}) = {C}_{hg}({t}_{k}),\end{array}$$

(10.189)

$$\begin{array}{rcl} & & {R}_{gh}(0) = {\psi }_{gh} = \mathbf{E}[g({t}_{k})h({t}_{k})],\quad {C}_{gh}(0) = {\sigma }_{gh} = \mathbf{E}[(g({t}_{k}) -\overline{g})\qquad \\ & & \qquad \qquad \qquad \cdot (h({t}_{k}) -\overline{h})] = {\psi }_{gh} -\overline{g}\overline{h}, \end{array}$$

(10.190)

$$\begin{array}{rcl} & & {R}_{gh}(\infty ) = \overline{g}\overline{h},\quad {C}_{gh}(\infty ) = 0,\end{array}$$

(10.191)

$$\begin{array}{rcl} & & {P}_{gh}\left (-{f}_{m}\right ) = {P}_{gh}^{{_\ast}}\left ({f}_{ m}\right ) = {P}_{hf}\left ({f}_{m}\right ),\end{array}$$

(10.192)

$$\begin{array}{rcl} & & {P}_{gh}(0) = T\overline{g}\overline{h}.\end{array}$$

(10.193)

​Note that σ _gh is nothing else but the usual covariance while, when g = h, σ _gg is the variance and ψ _gg is the mean square value.

1.3.4 A.3.4 The DFT in Computers

In most computer software for DFTs, such as the FFT subroutines in the IMSL library, the DFT is simply defined by using the wavenumber k instead of the ‘wavelength’ x _k and also by omitting the period (record length) T _o . This means that the time (space) interval $\Delta t$ is taken as unit and all other parameters dependent on it are omitted. Thus, in a computer, hence the subscript c, the DFT pair is defined as

$$\begin{array}{rcl} & & {H}_{c}(m) = \frac{1} {N}{\sum }_{k=0}^{N-1}h(k){e}^{-i2\pi km/N} ={ \mathbf{F}}_{\mathbf{ c}}\{h(k)\},\end{array}$$

(10.194)

$$\begin{array}{rcl} & & {h}_{c}(k) ={ \sum }_{m=0}^{N-1}{H}_{ c}(m){e}^{i2\pi km/N} ={ \mathbf{F}}_{\mathrm{\mathbf{ c}}}^{-\mathbf{1}}\{{H}_{ c}(m)\}.\end{array}$$

(10.195)

A comparison of the above equations to (10.168) and (10.169) shows that their exist the following relationships:

$$\begin{array}{rcl} & & {H}_{c}(m) = \frac{1} {N\Delta x}H({f}_{m}) = \frac{1} {T}H({f}_{m}),\end{array}$$

(10.196)

$$\begin{array}{rcl} & & {h}_{c}(k) = h({x}_{k})\end{array}$$

(10.197)

Consequently, when the DFT of h(x _k ) is computed by (10.192) the results must be rescaled by T _o to get the correct values. Of practical importance is that

$${H}_{c}(0) = \overline{h}$$

(10.198)

and that, for the computation of discrete convolution and correlation, the following relations hold:

$$\begin{array}{rcl} & & x({t}_{k}) = g({t}_{k}) {_\ast} h({t}_{k}) = {T}_{o}{x}_{c}({t}_{k}) = {T}_{o}{\mathbf{F}}_{c}^{-1}\{{G}_{ c}(m){H}_{c}(m)\},\end{array}$$

(10.199)

$$\begin{array}{rcl} & & y({t}_{k}) = g({t}_{k}) \otimes h({t}_{k}) = {T}_{o}{y}_{c}({t}_{k}) = {T}_{o}{\mathbf{F}}_{c}^{-1}\{{G}_{ c}(m){H}_{c}^{{_\ast}}(m)\}.\end{array}$$

(10.200)

Another point that requires attention is the location of the coordinate origin. Usually, computer subroutines consider as origin the first point from the left in both domains. When the points of the sample record are referred to an origin being at the centre of the record, the discrete version of (10.148) must be used to correct the computed spectrum. In such a case, ${t}_{o} = N\Delta t/2 = {T}_{o}/2$ and thus ${e}^{-i2\pi m\Delta f{T}_{o}/2} = {e}^{-i\pi m} =\cos (m\pi ) = {(-1)}^{m}$, which results in

$$h({t}_{k} - {T}_{o}/2) \leftrightarrow {(-1)}^{m}H({f}_{ m}).$$

(10.201)

Consequently, when we are after H(f _m ), we should multiply the result of the DFT subroutine by ( − 1)^m. Notice that in the product of two spectra obtained by,e.g., (10.199), ( − 1)^m cancels out. Hence, to avoid the origin shift when we compute convolutions, the product of the two spectra should first be multiplied by ( − 1)^m and then entered into the inverse DFT subroutine. In the same fashion, special care must be taken for the computation of covariance, correlation and PSD functions.

Finally, in order to avoid extra aliasing errors, no sample should be taken at the end point of the record length. Since the DFT is periodic, the missing end point of a period is considered to be the starting point of the next period. This fact is graphically illustrated in Fig. 10.13, where the function h(x) is sampled at M = 10 points per period T _x and is thus represented by the discrete values h(x _k ) or simply h(k). It is important to note that the even symmetry of the function is not upset.

Fig. 10.13

DFT sampling: the end point of a period (After Sideris 1984)

1.3.5 A.3.5 The Fast Fourier Transform

The fast Fourier transform (FFT) is an algorithm for computing the DFT much faster (number of required complex multiplications proportional to Nlog ₂ N) than by the conventional Fourier transform (number of required complex multiplications proportional to N ²). To illustrate the FFT algorithm, the intuitive development presented in Brigham (1988) for the 1D FFT is explained in the following.

Suppose that the DFT of a function f(k) with N = 4 is required. Omitting, for simplicity, the constants in front of the summation symbol, we have

$$H(m) ={ \sum }_{k=0}^{N-1}h(k){e}^{-i2\pi km/N} ={ \sum }_{k=0}^{N-1}h(k){W}^{km},\quad m = 0,1,2,3,$$

(10.202)

or, equivalently,

$$\left (\begin{array}{l} H(0) \\ H(1) \\ H(2) \\ H(3)\\ \end{array} \right ) = \left (\begin{array}{llll} {W}^{0} & {W}^{0} & {W}^{0} & {W}^{0} \\ {W}^{0} & {W}^{1} & {W}^{2} & {W}^{3} \\ {W}^{0} & {W}^{2} & {W}^{4} & {W}^{6} \\ {W}^{0} & {W}^{3} & {W}^{6} & {W}^{9}\\ \end{array} \right )\left (\begin{array}{l} h(0) \\ h(1) \\ h(2) \\ h(3)\\ \end{array} \right ).$$

(10.203)

Since

$${W}^{km} = {e}^{-i2\pi km/N} = {W}^{km\,\,mod(N)},$$

(10.204)

where km mod(N) is the remainder of the division of nk by N, (10.203) become

$$\left (\begin{array}{l} H(0) \\ H(1) \\ H(2) \\ H(3)\\ \end{array} \right ) = \left (\begin{array}{llll} 1&1 &1 &1 \\ 1&{W}^{1} & {W}^{2} & {W}^{3} \\ 1&{W}^{2} & {W}^{0} & {W}^{2} \\ 1&{W}^{3} & {W}^{2} & {W}^{1}\\ \end{array} \right )\left (\begin{array}{l} h(0) \\ h(1) \\ h(2) \\ h(3)\\ \end{array} \right ).$$

(10.205)

By bit-reversing the indices of H(m) and by factorizing the matrix of W coefficients into log ₂ N = 2 matrices, the above system becomes

$$\left (\begin{array}{l} H(0) \\ H(2) \\ H(1) \\ H(3)\\ \end{array} \right ) = \left (\begin{array}{l@{\quad }l@{\quad }l@{\quad }l} 1\quad &{W}^{0}\quad &0\quad &0 \\ 1\quad &{W}^{2}\quad &0\quad &0 \\ 0\quad &0 \quad &1\quad &{W}^{1} \\ 0\quad &0 \quad &1\quad &{W}^{3}\\ \quad \end{array} \right )\left (\begin{array}{l@{\quad }l@{\quad }l@{\quad }l} 1\quad &0\quad &{W}^{0}\quad &0 \\ 0\quad &1\quad &0 \quad &{W}^{0} \\ 1\quad &0\quad &{W}^{2}\quad &0 \\ 0\quad &1\quad &0 \quad &{W}^{2}\\ \quad \end{array} \right )\left (\begin{array}{l} h(0) \\ h(1) \\ h(2) \\ h(3)\\ \end{array} \right ).$$

(10.206)

Finally, because W ² =  − W ⁰ and W ³ =  − W ¹, it follows that

$$\begin{array}{rcl} \left (\begin{array}{l} H(0) \\ H(2) \\ H(1) \\ H(3)\\ \end{array} \right )& =& \left (\begin{array}{l@{\quad }l@{\quad }l@{\quad }l} 1\quad &{W}^{0} \quad &0\quad &0 \\ 1\quad &-{W}^{0}\quad &0\quad &0 \\ 0\quad &0 \quad &1\quad &{W}^{1} \\ 0\quad &0 \quad &1\quad &-{W}^{1}\\ \quad \end{array} \right )\left (\begin{array}{l} {h}_{1}(0) \\ {h}_{1}(1) \\ {h}_{1}(2) \\ {h}_{1}(3)\\ \end{array} \right ), \\ \left (\begin{array}{l} {h}_{1}(0) \\ {h}_{1}(1) \\ {h}_{1}(2) \\ {h}_{1}(3)\\ \end{array} \right )& =& \left (\begin{array}{l@{\quad }l@{\quad }l@{\quad }l} 1\quad &0\quad &{W}^{0} \quad &0 \\ 0\quad &1\quad &0 \quad &{W}^{0} \\ 1\quad &0\quad &-{W}^{0}\quad &0 \\ 0\quad &1\quad &0 \quad &-{W}^{0}\\ \quad \end{array} \right )\left (\begin{array}{l} h(0) \\ h(1) \\ h(2) \\ h(3)\\ \end{array} \right ).\end{array}$$

(10.207)

From the system of (10.207), a flow graph of operations is constructed and shown in Fig. 10.14.

Fig. 10.14

Flow graph of FFT operations for N = 4 (After Sideris 1984)

The matrix factorization introduces zeroes into the sub-matrices and results in an appreciable reduction of multiplications. Figure 10.14 indicates that not only the number of multiplications is reduced but the number of additions is reduced as well, since each h ₁(k) is computed only once and then used for the computations of all H(m) in which it takes part. These are the main reasons that the FFT is much faster that the conventional Fourier transform. An extensive discussion of the computational aspects of the FFT algorithm can be found in IEEE (1967) and in Brigham (1988).

1.4 A.4 The Two-Dimensional Discrete Fourier Transform

The multi-dimensional continuous Fourier transform pair is defined as follows:

$$\begin{array}{rcl} & & G(\underline{f}) =\int\limits_{-\infty }^{\infty }g(\underline{t}){e}^{-2\pi i{\underline{f}}^{T}\underline{t} }d\underline{t} = \mathbf{F}\{g(\underline{t})\},\end{array}$$

(10.208)

$$\begin{array}{rcl} & & g(\underline{t}) =\int\limits_{-\infty }^{\infty }G(\underline{f}){e}^{-2\pi i{\underline{f}}^{T}\underline{t} }d\underline{f} ={ \mathbf{F}}^{-1}\{G(\underline{f})\},\end{array}$$

(10.209)

The vectors {t} and {f} comprise the time (space) and frequency coordinates, respectively. For example, in the case of the three-dimensional (3D) CFT, {t_} = (x, y, z)^T, {f} = (u, v, w)^T, {f}^T{t_} = ux + vy + wz, d{t} = dxdydz and d{f} = dudvdw, where u, v, w are the frequencies corresponding to x, y, z, respectively, and the integrals in (10.208) and (10.209) are triple integrals. Notice that the above definition indicates that the multi-dimensional CFT is separable, i.e. it consists of consecutive applications of the 1D CFT, one for each dimension (or direction), which is of great importance in practical applications. In a similar manner, the properties of the 1D CFT and the convolution and correlation theorems can be extended to many dimensions and will not be repeated here; formulas and more details can be found in Dudgeon and Mersereau (1984), Sideris (1984), Bracewell (1986a), Brigham (1988), Oppenheim and Schafer (1989), and Schwarz et al., (1990). Instead, we will concentrate here on the 2D DFT, which is used in most of the physical geodesy problems.

For a function h(x _k , y _l ) given at M ×N gridded points in an area T _x ×T _y with grid spacing $\Delta x$ and $\Delta y$, the two-dimensional discrete Fourier transform pair is

$$\begin{array}{rcl} & & H({u}_{m},{v}_{n}) ={ \sum }_{k=0}^{M-1}\,{\sum }_{1=0}^{N-1}h({x}_{ k},{y}_{l}){e}^{-i2\pi (mk/M+nl/N)}\Delta x\Delta y,\end{array}$$

(10.210)

$$\begin{array}{rcl} & & h({x}_{k},{y}_{l}) ={ \sum }_{k=0}^{M-1}\,{\sum }_{1=0}^{N-1}H({u}_{ m},{v}_{n}){e}^{i2\pi (mk/M+nl/N)}\Delta u\Delta v,\end{array}$$

(10.211)

with (see Fig. 10.15)

$$\begin{array}{rcl} & & \Delta u = \frac{1} {{T}_{y}} = \frac{1} {M\Delta x},\qquad \qquad \quad \ \Delta v = \frac{1} {{T}_{y}} = \frac{1} {N\Delta y},\end{array}$$

(10.212)

$$\begin{array}{rcl} & & \Delta x = \frac{1} {{F}_{u}} = \frac{1} {M\Delta u} = \frac{1} {2{u}_{N}},\qquad \Delta y = \frac{1} {{F}_{v}} = \frac{1} {N\Delta v} = \frac{1} {2{v}_{N}},\end{array}$$

(10.213)

where u _N and v _N are the Nyquist frequencies corresponding to x and y, respectively. Note that in Fig. 10.15 the end points (here, end row and end column) of a period have not been plotted (see Sect. A.3.5, Chap. 10). Due to the seperability of the 2D Fourier transform, the 2D DFT can be evaluated in computers with limited memory be applying the 1D DFT twice, first along rows and then along columns or vise versa.

Fig. 10.15

Two-dimensional grids in the space and frequency domain

1.5 A.5 Efficient DFT for Real Functions

In most applications, the data being processed are real but the FFT algorithm is designed for complex functions. Thus, if we only consider a real function, the imaginary part of the algorithm is wasted. In this section, we will give a method to compute the Fourier transform of two real functions via a single DFT and the convolutions of two real functions with the same function simultaneously.

1.5.1 A.5.1 DFT of Two Real Functions Via a Single FFT

If g(k, l) and h(k, l) are two real functions, we can compute their Fourier transform via a single FFT. Let us construct a complex function y(k, l) as the sum of g(k, l) and h(k, l), where one of these is taken to be imaginary, i.e.,

$$y\left (k,l\right ) = g\left (k,l\right ) + ih\left (k,l\right ),$$

(10.214)

Applying the DFT to (10.214) yields

$$Y (m,n) = \mathbf{F}\{y(k,l)\} ={ \sum }_{k=0}^{M-1}{\sum }_{l=0}^{N-1}[g(k,l)+i\,h(k,l)]{e}^{-i2\pi (mk/M+n1/N)}.$$

(10.215)

Expanding the right-hand side of (10.215) and denoting R(m, n) and I(m, n) as the real and the imaginary parts of Y (m, n) respectively, we get

$$\begin{array}{rcl} R(m,n)& =& {\sum }_{k=0}^{M-1}{\sum }_{l=0}^{N-1}g(k,l)\cos 2\pi \left (\frac{mk} {M} + \frac{nl} {N}\right ) \\ & & -\,{\sum }_{k=0}^{M-1}{\sum }_{l=0}^{N-1}h(k,l)\sin 2\pi \left (\frac{mk} {M} + \frac{nl} {N}\right ),\qquad \qquad \end{array}$$

(10.216)

$$\begin{array}{rcl} I(m,n)& =& {\sum }_{k=0}^{M-1}{\sum }_{l=0}^{N-1}g(k,l)\sin 2\pi \left (\frac{mk} {M} + \frac{nl} {N}\right ) \\ & & +{\sum }_{k=0}^{M-1}{\sum }_{l=0}^{N-1}h(k,l)\cos 2\pi \left (\frac{mk} {M} + \frac{nl} {N}\right ),\qquad \qquad \end{array}$$

(10.217)

and we can easily verify that the DFT of g(k, l) and h(k, l) can be evaluated as

$$\begin{array}{rcl} G(m,n)& =& [R(m,n) + R(M - m,N - n)]/2 \\ & & +i[I(m,n) - I(M - m,N - n)]/2,\end{array}$$

(10.218)

$$\begin{array}{rcl} H(m,n)& =& [I(m,n) + I(M - m,N - n)]/2 \\ & & -i[R(m,n) - R(M - m,N - n)]/2.\end{array}$$

(10.219)

If we divide R(m, n) and I(m, n) into even and odd part as

$$\begin{array}{rcl} & & R(m,n) = {R}_{e}(m,n) + {R}_{o}(m,n),\end{array}$$

(10.220)

$$\begin{array}{rcl} & & I(m,n) = {I}_{e}(m,n) + {I}_{o}(m,n),\end{array}$$

(10.221)

from (10.218) and (10.219) we can see that

$$\begin{array}{rcl} & & G(m,n) = {R}_{e}(m,n) + i{I}_{o}(m,n),\end{array}$$

(10.222)

$$\begin{array}{rcl} & & H(m,n) = {I}_{e}(m,n) - i{R}_{o}(m,n).\end{array}$$

(10.223)

The DFT of the convolution of the two real functions can be directly evaluated from R(m, n) and I(m, n) as

$$\begin{array}{rcl} \mathbf{F}\{g(k,l) {_\ast} h(k,l)\}& =& [R(m,n)I(m,n) \\ & & +R(M - m,N - n)I(M - m,N - n)]/2 \\ & & -i[{R}^{2}(m,n) - {R}^{2}(M - m,N - n) \\ & & -{I}^{2}(m,n) + {I}^{2}(M - m,N - n)]/2\end{array}$$

(10.224)

By using this technique, geoid undulations, i.e., (10.23), can be efficiently computed via one direct and one inverse DFT.

1.5.2 A.5.2 Simultaneous Convolution of Two Real Functions with the Same Function

In the computation of terrain corrections, see, e.g., (10.53a), we have to evaluate the convolutions of three real functions with the same kernel function. To save computer time, two of the three convolutions can be done simultaneously via one convolution as

$$p(k,l) = x(k,l) {_\ast} y(k,l),$$

(10.225)

where y(k, l) is the sum of two real functions as defined by (10.214), x(k, l) represents the kernel function of terrain correction and, as we know, its Fourier transform is an even real function, i.e.

$$X\left (m,n\right ) = Xe\left (m,n\right ).$$

(10.226)

The spectrum of p(k, l) is

$$P(m,n) = X(m,n)Y (m,n) = X(m,n)[G(m,n) + iH(m,n)].$$

(10.227)

Considering (10.222), (10.223) and (10.226), we get

$$\begin{array}{rcl} & & X(m,n)G(m,n) = {X}_{e}(m,n){R}_{e}(m,n) + i{X}_{e}(m,n){I}_{o}(m,n),\end{array}$$

(10.228)

$$\begin{array}{rcl} & & X(m,n)H(m,n) = {X}_{e}(m,n){I}_{e}(m,n)\mathrm{} - i{X}_{e}(m,n){R}_{o}(m,n).\end{array}$$

(10.229)

By using the properties of the Fourier transform of even and odd functions, we can verify that F ^− 1{X(m, n)G(m, n)} is a real function and equal to the real part of F ^− 1{P(m, n)}, i F ^− 1{X(m, n)H(m, n)} is an imaginary function and equal to the imaginary part of F ^− 1{P(m, n)}, i.e.,

$$\begin{array}{rcl} & & x(k,l) {_\ast} g(k,l) ={ \mathbf{F}}^{-1}\{X(m,n)G(m,n)\} = real\{{\mathbf{F}}^{-1}\{P(m,n)\}\},\qquad \qquad \end{array}$$

(10.230)

$$\begin{array}{rcl} & & x(k,l) {_\ast} h(k,l) ={ \mathbf{F}}^{-1}\{X(m,n)H(m,n)\} = imag\{{\mathbf{F}}^{-1}\{P(m,n)\}\}.\qquad \qquad \end{array}$$

(10.231)

1.6 A.6 Use of the Fast Hartley Transform

FFT-based spectral techniques, as standard and indispensable tools for the evaluation of gravity field convolutions, make it possible to perform the computations, such as geoid undulations and terrain reductions, etc., in a large area simultaneously. However, the fact that all signals are real and the FFT is a complex operation makes half of the computer core memory required by an FFT-based program useless, and the complex mathematical operations, such as addition and multiplication, take twice as much time as real operations. To avoid such shortcomings related to the FFT method, this chapter will introduce the use of the fast Hartley transform, and discuss other methods of efficient FFT convolutions.

1.6.1 A.6.1 The Discrete Hartley Transform

Hartley (1942) proposed the use of a new kind of transform that is expressed in a more symmetrical form between the function of the original real variable and its transform, which forms the basis for the present Hartley transform and the fast Hartley transform (FHT). The FHT is as fast as or faster than the FFT, and serves for all the uses, such as the convolution operations and spectral analysis, to which the FFT is at present applied.

This section will discuss the properties of the discrete Hartley transform, and show how the gravity convolutions can be performed by FHT. For more details about the basic principles of the Hartley transform, Hartley (1942) and Bracewell (1986a) can be consulted. For the applications of the fast Hartley transform in physical geodesy, Li and Sideris (1992) is recommended.

1.6.2 A.6.2 Definition of the 1D Discrete Hartley Transform

Hartley (1942) defined a more symmetrical one-dimensional Fourier transform as follows:

$$\begin{array}{rcl} & & H(\omega ) =\int\limits_{-\infty }^{\infty }h(t)\,\,\mathrm{cas}2\pi \omega t\,\,dt,\end{array}$$

(10.232)

$$\begin{array}{rcl} & & h(t) =\int\limits_{-\infty }^{\infty }H(\omega )\,\,\mathrm{cas}2\pi \omega t\,\,d\omega ,\end{array}$$

(10.233)

where

$$\mathrm{cas}\,x =\cos \, x +\sin \, x.$$

(10.234)

Because the transform pair (10.232) and (10.233) was first defined by Hartley, (10.232) is called the direct Hartley transform and (10.232) is called the inverse Hartley transform (Bracewell 1984, 1986b).

For a real function h(k) given at M gridded points with grid spacing $\Delta x$, the one-dimensional discrete Hartley transform pair is defined as

$$\begin{array}{rcl} & & H(m) = \Delta x{\sum }_{k=0}^{M-1}\,h(k\Delta x)\,\,\mathrm{cas}\frac{2\pi mk} {M} ,\end{array}$$

(10.235)

$$\begin{array}{rcl} & & h(k) = \frac{1} {M\Delta x}{\sum }_{m=0}^{M-1}\,H(m\Delta u)\,\,\mathrm{cas}\frac{2\pi mk} {M} ,\end{array}$$

(10.236)

where $\Delta u = 1/(M\Delta x$).

1.6.3 A.6.3 Definition of the 2D Discrete Hartley Transform

For a real function h(k, l) given at M ×N gridded points with grid spacing $\Delta x$ and $\Delta y$, the two-dimensional discrete Hartley transform pair is defined as

$$\begin{array}{rcl} & & H(m\Delta u, n\Delta v) = \Delta x\Delta y{\sum }_{k=0}^{M-1}\,{\sum }_{1=0}^{N-1}h(k\Delta x,1\Delta y)\,\,\mathrm{cas}\frac{2\pi mk} {M} \,\,\mathrm{cas}\frac{2\pi nl} {N} ,\end{array}$$

(10.237)

$$\begin{array}{rcl} & & h(k\Delta x,1\Delta y) = \frac{1} {M\Delta xN\Delta y}{\sum }_{m=0}^{M-1}\,{\sum }_{n=0}^{N-1}H(m\Delta u,n\Delta v)\mathrm{cas}\frac{2\pi mk} {M} \,\,\mathrm{cas}\,\,\frac{2\pi nl} {N} ,\end{array}$$

(10.238)

where

$$\Delta u = \frac{1} {M\Delta x},\quad \Delta v = \frac{1} {N\Delta y}$$

(10.239)

For convenience, (10.237) and (10.238) can be simply expressed as

$$\begin{array}{rcl} & & H(m,m) = \Delta x\Delta x{\sum }_{k=0}^{M-1}{\sum }_{l=0}^{N-1}h(k,l)\,\,\mathrm{cas}mk\,\,\mathrm{cas}nl,\end{array}$$

(10.240)

$$\begin{array}{rcl} & & h(k,l) = \frac{1} {{T}_{x}{T}_{y}}{ \sum }_{m=0}^{M-1}{\sum }_{n=0}^{N-1}H(m,n)\,\,\mathrm{cas}mk\,\,\mathrm{cas}nl.\end{array}$$

(10.241)

The Hartley transform pair is denoted as

$$h\left (k,1\right ) \Leftrightarrow H\left (m,n\right ).$$

(10.242)

Similar to the discrete Fourier transform, very efficient operations can be developed for the evaluation of the Hartley transform, which results in the fast Hartley transform.

1.6.4 A.6.4 Properties of the Discrete Hartley Transform

The following properties of the two-dimensional discrete Hartley transform can be derived directly from the definition and, therefore, most of them are listed below without proof.

• (a) Linearity

  $$a\,\,h(k,1) + b\,\,g(k,1) \Leftrightarrow a\,\,H(m,n) + b\,\,\mathrm{}G(m,n).$$

  (10.243)

• (b) Spacing shifting

$$\begin{array}{rcl} & & h(k - \lambda ,l - \mu ) \Leftrightarrow H(m,n)\cos m\lambda \cos n\mu - H(m,-n)\cos m\lambda \sin n\mu \\ & & \quad - H(-m,n)\sin m\lambda \cos n\mu + H(-m,-n)\sin m\lambda \sin n\mu. \end{array}$$

(10.244)

If λ = M ∕ 2 and μ = N ∕ 2, then

$$h(k - M/2,l - N/2) \Leftrightarrow {(-1)}^{m+n}H(m,n).$$

(10.245)

• (c) Even function

If h(k, l) is even with respect to the two variables, i.e.,

$${h}_{e}(k,l) = {h}_{e}(-k,l) = {h}_{e}(k,-l),$$

(10.246)

then,

$${h}_{e}(k,l) \Leftrightarrow {H}_{e}(m,n) = \Delta x\,\,\Delta y{\sum }_{k=0}^{M-1}{\sum }_{1=0}^{N-1}{h}_{ e}(k,l)\cos mk\,\cos nl,$$

(10.247)

therefore,

$${H}_{e}(m,n) = {H}_{e}(-m,n) = {H}_{e}(m,-n)\mathrm{} = {H}_{e}(-m,-n).$$

(10.248)

• (d) Odd function

If h(k, l) is odd with respect to the two variables, i.e.,

$$\begin{array}{l} {h}_{o}(k,l) = -{h}_{o}(-k,l) = -{h}_{o}(k,-l), \\ {h}_{o}(0,l) = {h}_{o}(M/2,l) = {h}_{o}(k,0)\mathrm{} = {h}_{o}(k,N/2) = 0,\\ \end{array}$$

(10.249)

then,

$${h}_{o}(k,l) \Leftrightarrow {H}_{o}(m,n) = \Delta x\Delta y{\sum }_{k=0}^{M-1}{\sum }_{l=0}^{N-1}{h}_{ o}(k,l)\sin mk\,\sin nl,$$

(10.250)

therefore,

$${H}_{o}(m,n) = -{H}_{o}(-m,n) = -{H}_{o}(m,-n) = {H}_{o}(-m,-n).$$

(10.251)

• (e) Odd-even function

If h(k, l) is odd with respect to one variable and even with respect to the other, i.e.,

$${h}_{oe}(k,l) = -{h}_{oe}(-k,l) = {h}_{oe}(k,-l)\ \mathrm{and}\,\,h(0,l) = h(M/2,l) = 0,$$

(10.252)

then,

$${h}_{oe}(k,l) \Leftrightarrow {H}_{oe}(m,n) = \Delta x\,\,\Delta y{\sum }_{k=0}^{M-1}{\sum }_{l=0}^{N-1}{h}_{ oe}(k,l)\sin \,mk\,\cos nl,$$

(10.253)

therefore,

$${H}_{oe}(m,n) = -{H}_{oe}(-m,n) = {H}_{oe}(m,-n) = -{H}_{oe}(-m,-n).$$

(10.254)

• (f) Two-dimensional convolution theorem

$$\begin{array}{l} h(k,1) {_\ast} g(k,1) \Leftrightarrow \mathrm{}G(m,n){H}_{1}(m,n) + G(-m,-n){H}_{2}(m,n) \\ \quad + G(-m,n){H}_{3}(m,n) + \mathrm{}G(m,-n){H}_{4}(m,n),\\ \end{array}$$

(10.255)

where

$$\begin{array}{rcl}{ H}_{1}(m,n)& =& \frac{1} {4}[H(m,n) + H(-m,-n) + H(m,-n) + H(-m,n)], \\ {H}_{2}(m,n)& =& \frac{1} {4}[H(m,n) + H(-m,-n) - H(m,-n) - H(-m,n)], \\ {H}_{3}(m,n)& =& \frac{1} {4}[H(m,n) - H(-m,-n) + H(m,-n) - H(-m,n)], \\ {H}_{4}(m,n)& =& \frac{1} {4}[H(m,n) - H(-m,-n) - H(m,-n) + H(-m,n)]. \end{array}$$

(10.256)

If h(k, l) is even, the convolution theorem simplifies to

$${h}_{e}(k,l) {_\ast} g(k,l) \Leftrightarrow H(m,n)\,G(m,n).$$

(10.257)

If h(k, l) is odd, the convolution theorem simplifies to

$${h}_{o}(k,l) {_\ast} g(k,l) \Leftrightarrow H(m,n)\,G(-m,-n).$$

(10.258)

If h(k, l) is odd in k and even in l, the convolution theorem simplifies to

$${h}_{oe}(k,l) {_\ast} g(k,l) \Leftrightarrow H(m,n)\,G(-m,n).$$

(10.259)

Proof.The convolution of h(k, l) with g(k, l) is defined by

$$f(k,l) = h(k,l) {_\ast} g(k,l) ={ \sum }_{i=0}^{M-1}{\sum }_{j=0}^{N-1}h(i,j)g(k - i,l - j)$$

(10.260)

and the Hartley transform of f(k, l) is

$$\begin{array}{rcl} F(m,n)& =& {\sum }_{k=0}^{M-1}{\sum }_{l=0}^{N-1}\left ({\sum }_{i=0}^{M-1}{\sum }_{j=o}^{N-1}h(i,j)g(k - i,l - j)\right )cas\ \ mk\ \ cas\ \ nl \\ & =& {\sum }_{i=0}^{M-1}{\sum }_{j=0}^{N-1}h(i,j){\sum }_{k=0}^{M-1}{\sum }_{l=0}^{N-1}g(k,l)\cos m(k+i)\cos n(l+j).\qquad \end{array}$$

(10.261)

With the following identity,

$$\cos (x + y) =\cos x\cos y +\sin (-x)\sin y,$$

(10.262)

F(m, n) can be expressed as

$$\begin{array}{rcl} F(m,n)& =& {\sum }_{i=0}^{M-1}{\sum }_{j=0}^{N-1}h(i,j)[G(m,n)\cos mi\sin nj + G(-m,-n)\sin mi\sin nj \\ & & +\,G(-m,n)\sin mi\cos nj + G(m,-n)\cos mi\sin nj]. \end{array}$$

(10.263)

With the following notations,

$$\begin{array}{rcl} & & {H}_{1}(m,n) ={ \sum }_{i=0}^{M-1}{\sum }_{j=0}^{N-1}h(i,j)\cos mi\,\cos nj \\ & & {H}_{2}(m,n) ={ \sum }_{i=0}^{M-1}{\sum }_{j=0}^{N-1}h(i,j)\sin mi\,\sin nj \\ & & {H}_{3}(m,n) ={ \sum }_{i=0}^{M-1}{\sum }_{j=0}^{N-1}h(i,j)\sin mi\,\cos nj \\ & & {H}_{4}(m,n) ={ \sum }_{i=0}^{M-1}{\sum }_{j=0}^{N-1}h(i,j)\cos mi\,\sin nj\end{array}$$

(10.264)

(10.263) becomes

$$\begin{array}{rcl} F(m,n)& =& G(m,n){H}_{1}(m,n) + G(-m,-n){H}_{2}(m,n) \\ & & +\,G(-m,n){H}_{3}(m,n) + G(m,-n){H}_{4}(m,n),\end{array}$$

(10.265)

which results in the convolution theorem as expressed in (10.255).

With the equation

$$H(m,n) = {H}_{1}(m,n) + {H}_{2}(m,n) + {H}_{3}(m,n) + {H}_{4}(m,n),$$

(10.266)

the convolution theorem can be simplified as follows.

If h(k, l) is an even function, then

$$H(m,n) = {H}_{1}\left (m,n\right )\mathrm{and}\,{H}_{2}\left (m,n\right ) = {H}_{3}\left (m,n\right ) = {H}_{4}\left (m,n\right ) = 0.$$

(10.267)

Inserting (10.267) into (10.265) gives the Hartley transform pair of (10.257).

If h(k, l) is an odd function, then

$$H(m,n) = {H}_{2}\left (m,n\right )\mathrm{and}\,{H}_{1}\left (m,n\right ) = {H}_{3}\left (m,n\right ) = {H}_{4}\left (m,n\right ) = 0.$$

(10.268)

Combining (10.268) with (10.265) yields the Hartley transform pair (10.258).

If h(k, l) is an odd in k and even in l, then

$$H(m,n) = {H}_{3}\left (m,n\right )\mathrm{and}\,{H}_{1}\left (m,n\right ) = {H}_{2}\left (m,n\right ) = {H}_{4}\left (m,n\right ) = 0,$$

(10.269)

and, consequently, (10.265) results in the Hartley transform pair of (10.259). □ 

• (g) Cross correlation

$$\begin{array}{l} h(k,l) \otimes g(k,1) \Leftrightarrow G(m,n){H}_{1}(m,n) + G(-m,-n){H}_{2}(m,n) \\ - G(-m,n){H}_{3}(m,n) - G(m,-n){H}_{4}(m,n).\end{array}$$

(10.270)

If g(k, l) is an even function, then

$$h(k,l) \otimes {g}_{e}(k,1) \Leftrightarrow G(m,n)\mathrm{}H(-m,-n).$$

(10.271)

If h(k, l) is an even function, then

$${h}_{e}(k,l) \otimes g(k,1) \Leftrightarrow G(m,n)H(m,n).$$

(10.272)

• (h) DC value

$$\begin{array}{rcl} H(0,0) = \frac{{T}_{x}{T}_{y}} {MN} {\sum }_{k=0}^{M-1}{\sum }_{l=0}^{N-1}h(k,1) = {T}_{ x}{T}_{y}{\mu }_{h},& &\end{array}$$

(10.273)

$$\begin{array}{rcl} h(0,0) = \frac{1} {{T}_{x}{T}_{y}}{ \sum }_{k=0}^{M-1}{\sum }_{l=0}^{N-1}H(m,n),& & \end{array}$$

(10.274)

where μ _h is the mean value of h(k, l).

• (i) The quadratic content theorem

$$\frac{{T}_{x}{T}_{y}} {MN} {\sum }_{k=0}^{M-1}{\sum }_{1=0}^{N-1}{[h(k,1)]}^{2} = \frac{1} {{T}_{x}{T}_{y}}{ \sum }_{k=0}^{M-1}{\sum }_{1=0}^{N-1}{[H(m,n)]}^{2}.$$

(10.275)

1.7 A.7 Relationship Between the DHT and the DFT

1.7.1 A.7.1 Computation of the 1D DFT Via the 1D DHT

Comparing the definition of the one-dimensional discrete Fourier transform

$${H}^{F}(m) = \frac{{T}_{x}} {M}{\sum }_{k=0}^{M-1}h(k)\left [\cos \frac{2\pi mk} {M} - j\sin \frac{2\pi mk} {M} \right ],$$

(10.276)

and that of the one-dimensional discrete Hartley transform

$$H(m) = \frac{{T}_{x}} {M}{\sum }_{k=0}^{M-1}h(k)\left [\cos \frac{2\pi mk} {M} +\sin \frac{2\pi mk} {M} \right ],$$

(10.277)

we can see that

$$\begin{array}{rcl} & & \mathit{real}\,({H}^{F}(m)) = [H(m) + H(-m)]/2,\end{array}$$

(10.278a)

$$\begin{array}{rcl} & & \mathit{imag}\,({H}^{F}(m)) = [H(m) - H(-m)]/2.\end{array}$$

(10.278b)

Equation 10.278 indicates that the real and the imaginary parts of the one-dimensional discrete Fourier transform are equal to the even and the odd parts of the discrete Hartley transform, respectively. If h(k) is an even function, considering that H(m) = H( − m), (10.278) can be simplified as

$$\begin{array}{rcl} & & \mathit{real}\,({H}^{F}(m)) = H(m),\end{array}$$

(10.279a)

$$\begin{array}{rcl} & & \mathit{imag}\,({H}^{F}(m)) = 0.\end{array}$$

(10.279b)

On the other hand, if h(k) is an odd function, with the relation H(m) =  − H( − m), (10.278) becomes

$$\begin{array}{rcl} & & real({H}^{F}(m)) = 0,\end{array}$$

(10.280a)

$$\begin{array}{rcl} & & imag({H}^{F}(m)) = H(m).\end{array}$$

(10.280b)

When the power spectrum is the desired product, it may be obtained directly from the DHT without first calculating the real and the imaginary part of the DFT as in the usual way of calculating power spectrum, i.e.

$${[{H}^{F}(m)]}^{2} = {[H(m)]}^{2}.$$

(10.281)

1.7.2 A.7.2 Computation of the 2D DFT Via the 2D DHT

The real and the imaginary part R(m, n) and I(m, n) of the two-dimensional discrete Fourier transform are

$$\begin{array}{rcl} & & R(m,n) = \frac{{T}_{x}{T}_{y}} {M\,N} {\sum }_{k=0}^{M-1}{\sum }_{l=0}^{N-1}h(k,l)\cos \left (\frac{2\pi mk} {M} + \frac{2\pi nl} {N} \right )\end{array}$$

(10.282a)

$$\begin{array}{rcl} & & I(m,n) = \frac{{T}_{x}{T}_{y}} {M\,N} {\sum }_{k=0}^{M-1}{\sum }_{l=0}^{N-1}h(k,l)\sin \left (\frac{2\pi mk} {M} + \frac{2\pi nl} {N} \right )\end{array}$$

(10.282b)

Compared with (10.264) and (10.26), and (10.282) becomes

$$\begin{array}{rcl} & & R(m,n) = {H}_{1}(m,n) - {H}_{2}(m,n) = [H(m,-n) + H(-m,n)]/2,\qquad \end{array}$$

(10.283a)

$$\begin{array}{rcl} & & I(m,n) = {H}_{3}(m,n) + {H}_{4}(m,n) = [H(m,n) - H(-m,-n)]/2.\qquad \end{array}$$

(10.283b)

If h(k, l) is an even function, with (10.267) and (10.282) becomes

$$\begin{array}{rcl} & & R(m,n) = H(m,n),\end{array}$$

(10.284a)

$$\begin{array}{rcl} & & I\left (m,n\right ) = 0.\end{array}$$

(10.284b)

If h(k, l) is an odd function, with (10.268) and (10.282) becomes

$$\begin{array}{rcl} & & R\left (m,n\right ) = -H\left (m,n\right ),\end{array}$$

(10.285a)

$$\begin{array}{rcl} & & I\left (m,n\right ) = 0.\end{array}$$

(10.285b)

If h(k, l) is an odd in k and even in l, with (10.269), and (10.282) becomes

$$\begin{array}{rcl} R\left (m,n\right )& =& 0,\end{array}$$

(10.286a)

$$\begin{array}{rcl} I\left (m,n\right )& =& H\left (m,n\right ).\end{array}$$

(10.286b)

So, the real and imaginary parts of the two-dimensional discrete Fourier transform can be easily computed via the discrete Hartley transform.

1.7.3 A.7.3 Advantages Unique to the FHT

The Hartley transform is superior to the Fourier transform with respect to the requirements in both computer time and computer memory. The Hartley transform is symmetric according to the transformation formula and its inverse. The transformation kernel (cas-function) is real; i.e., the Hartley spectrum of a real signal is also real. So, using the Hartley transform instead of the Fourier transform, we can save half of the computer core memory, or, for the same computer system, the Hartley transform can handle an amount of data twice as large as that handled by the Fourier transform. This is very important if we want to compute a large area of geoid undulations or terrain corrections simultaneously.

These properties have led to the use of the Hartley transform for time-efficient Fourier analysis of real signals. For a data length N being an integer power of 2, i.e., N = 2ⁿ, the FHT algorithm can be developed in just the same way as the FFT algorithm. As the FHT uses only real operations, it is about twice as fast as the FFT. In practice, typically only 20–30% of the total execution time is consumed in butterfly execution, and the remainder is spent in interpretation, indexing, etc. (Bold 1985). The computer time saved by the FHT may be less than 50% and about one-third. It should be mentioned that very time-efficient real FFT algorithms are available now, but the programming effort increases with increasing speed of computation.