4.6 Java1490-2d fourier transforms using java, part 1  (Page 7/15)

Response of the leftmost array

Now consider the leftmost pair of images where the two array elements are relatively close together. The wavenumber response of this array has a pair ofnull points about midway between the center and either end, as indicated by the blue and black colors. This is the wavenumber for which the element spacing isan exact multiple of one-half of the wavelength, causing the elements to move in equal but opposite directions in response to the wave motion. Thus, the outputfrom one element cancels the output from the other element producing zero voltage in the sum.

This same pair of images shows high responses at either end as indicated by the red areas. This is the wavenumber for which the element spacing is an exactmultiple of one wavelength.

The Nyquist folding wavenumber

If this were a frequency spectrum analysis, we would say that the end points are at the Nyquist folding frequency, which is one-half of the samplingfrequency. Thus, we can say that in the wavenumber domain, the end points on the two leftmostplots are at the Nyquist folding wavenumber, which is one-half the sampling wavenumber.

As in the frequency domain, the wavenumber spectrum is periodic. The section of the wavenumber spectrum that we are viewing represents one complete period of a periodic wavenumber spectrum ranging from minus the folding wavenumber on the left, through zero wavenumberat the center, to plus the folding wavenumber on the right.

(If you are already familiar with this sort of thing, you may have figured out that the separation between our elements in this example istwice the sampling distance that determines the location of the folding wavenumber.)

Estimating the array response from the colors

The leftmost array response shows a white area at the center.

(The white area is split by the vertical component of red axes drawn on the plot. The color of the axes has nothing to do with elevations on thesurface. I simply decided to draw them in red to make them stand out.)

This array response also shows the two black areas mentioned earlier. You can use the orange, yellow, green, and cyan locations on the calibration scale toestimate the response of the array to different wavenumber values between maximum and minimum.

Additional separation between array elements

Now consider the two images in the center of Figure 2 where the wavenumber range is the same as the wavenumber plot on the left. The elements for thisarray are separated more than the elements in the leftmost pair of images. Again, the wavenumber response for this array has a maximum value at the origin,which is at the center of the wavenumber response in the lower image. In addition, this array response has a high (red) response at four other wavenumber zones (for a total of five) , whereas the array on the left had only three red zones. Similarly, this array has a low response (black and blue) at four differentwavenumber zones whereas the array on the left has a low response at only two wavenumber zones.