Basic Relations

The time Fourier transform is defined as

Figure 1. EQUATION_DISPLAY

{F T}_{t} [h (x; t)] = \int_{- \infty}^{\infty} h (x : t) e^{- i ω t} ⅆ t = g (x; ω) \forall ω \in [- \infty, \infty]

(488)

where $h (x; t)$ is a real function which depends on the position $x$ and time $t \cdot ω$ is angular frequency.

The integration limits are over the whole time domain $t \in [\infty, - \infty]$ . However, the integration is over a given period $t \in [- T / 2, T / 2]$ in practice. The function $h (x; t)$ is periodic in time (or is altered to be periodic).

More commonly, a “finite” Fourier transform for a specific time block $T$ is used:

Figure 2. EQUATION_DISPLAY

{F T}_{t} [h (x; t)] = \int_{- T / 2}^{T / 2} h (x : t) e^{- i ω t} ⅆ t = g (x; ω, T) \forall ω \in [- \frac{m π}{T}, \frac{m π}{T}]

(489)

where $m$ is the number of sampling points.

If the finite transform is conducted on a $k^{t h}$ record of block $T$ , then the transform is:

Figure 3. EQUATION_DISPLAY

{F T}_{t} [h (x; t)] = \int_{- (t_{k} + (T / 2))}^{(t_{k} + (T / 2))} h (x : t) e^{- i ω t} ⅆ t = g (x; ω, T) \forall ω \in [- \frac{m π}{T}, \frac{m π}{T}]

(490)

The inverse time Fourier transform is

Figure 4. EQUATION_DISPLAY

{F T}_{t}^{- 1} [g (x; ω)] = \frac{1}{2 π} \int_{- \infty}^{\infty} g (x; ω) e^{- i ω t} ⅆ ω = h (x; t) \forall t \in [ - \infty, \infty]

(491)

Similarly, although the integration domain is infinite, it is understood that this integration is over a finite frequency space.

The space Fourier transform is defined as

Figure 5. EQUATION_DISPLAY

{F T}_{s} [g (x; ω)] = \int_{x} g (x; ω) e^{- i k \cdot x} | ⅆ x | = f (k; ω) \forall k \in K; ω \in [ - \infty, \infty]

(492)

The function is periodic is space (or is altered to be periodic in space).

Using a compact notation

Figure 6. EQUATION_DISPLAY

g (x; ω) e^{- i k \cdot x} | d x | \equiv g (x (ξ); ω) e^{- i k \cdot s (ξ)} | \frac{\partial x}{\partial ξ_{1}} \times \frac{\partial x}{\partial ξ_{2}} | \partial ξ_{1} \partial ξ_{2}

(493)

The space Fourier transform is actually in the parametric space instead of the three-dimensional space of the coordinates of a point on a surface. This parametric representation maps to a two-dimensional space which is both planar and regularly sampled.

The inverse space Fourier transform is

Figure 7. EQUATION_DISPLAY

{F T}_{s}^{- 1} [f (x; ω)] = \frac{1}{{(2 π)}^{n}} \int_{K} f (x; ω) e^{- i k \cdot x} | ⅆ k | = g (x; ω) \forall x \in S; ω \in [ - \infty, \infty]

(494)

Here, $n$ is the dimensionality of the wavenumber space $K$ . For transforms on a surface $k = {k_{1,} k_{2}} \in K, n = 2$ . For transforms on a curve, $n = 1$ .