4.6 Java1490-2d fourier transforms using java, part 1  (Page 6/15)

How can we compute the array response?

This is where Fourier transforms come into play. We could compute the response of our one-dimensional array for waves propagating along the length ofthe wire by treating the elements of the array as samples in space and performing a one-dimensional Fourier transform on the elements of the array.

A sampled space series

In this case, we would consider the array elements to constitute samples taken in space in the same way that we consider a sampled time series toconstitute samples taken in time. In other words, the array elements constitute a sampled space series. The Fourier transform of a sampled time series is thefrequency spectrum of the time series. The Fourier transform of a sampled space series is the wavenumber spectrum of the space series.

Three example wavenumber spectra

To set the stage for what we will be seeing later, Figure 2 depicts three different two-element arrays with different spacing in the three (black and white) images across the top of the figure. The wavenumber response for each of the three arrays is shown inthe three images in the bottom row of images in Figure 2 .

(In Figure 2 , the array elements are represented by the white dots on the black background.)

Figure 2. Three example wavenumber spectra.

Plots of 3D surfaces

The images shown in Figure 2 were produced using the class named ImgMod29 , which I explained earlier in the module titled Plotting 3D Surfaces using Java . You can refer back to that module for a detailed explanation of the display format. Briefly, each of the sixindividual images in Figure 2 is a plot of a 3D surface, with the elevation of the surface at any particular point being indicated by the color at that pointbased on the colors on the calibration scale below each plot.

(The calibration scale is the strip that changes color in a smooth gradient from black through blue, cyan, green, yellow, red, and white with black at the leftend and white at the right end.

The three images in the top row of Figure 2 with the white dots on the black background represent the arrayelements for each of the three arrays. The three images in the bottom row of Figure 2 represent the wavenumber response of the corresponding arrays in the top row.)

Black, white, and the colors in between

The lowest (algebraic) elevation in the plot is colored black. Hence the backgrounds are black in the top three plots. The highest elevation is colored white. Thearray elements are white in the top three plots.

Between black and white, the elevation is given by the color scale below the plot with the lowest elevation on the left of the scale and the highestelevation on the right. Thus, a green elevation is about half way between the lowest and highest elevations. Blue elevations are near the low end. Redelevations are near the high end. Cyan and yellow elevations fall in between as shown by the calibration scale.

The maximum array output

All three wavenumber plots have a maximum response at the center, which is the zero wavenumber origin. In effect, this corresponds to infinite wavelength.If the wavelength is infinite, it doesn't matter what the separation between the elements is, they will all move up and down in unison and their electricaloutputs will add constructively to produce a maximum output.