Random lines in a plane

RESOLUTION OF THE PARADOX

If we throw impulse functions randomly onto a plane, the power spectrum of the plane is the power spectrum of impulse functions, namely white.

Think of a 2-D Gaussian whose contour of half-amplitude describes an ellipse of great eccentricity. In the limit of large eccentricity, this Gaussian could be one of the lines that we sprinkle on the plane with random amplitudes and orientations. The spatial spectrum of such an eccentric Gaussian must be lower than that of a symmetrical point Gaussian because the spectrum along the long axis of the ellipsoid is concentrated at very low frequency.

Consider a single delta function along a line with an arbitrary slope and location in a plane. The autocorrelation of this dipping line is another dipping line with the same slope, but passing through the origin at zero lag. The polarity of the impulse function is lost in the autocorrelation; in the autocorrelation space, the amplitude of the dipping line is positive.

Now consider a superposition of many dipping lines on the plane. Its autocorrelation is the sum of the autocorrelations of individual lines. The autocorrelation of any individual line is a line of the same slope that is translated to pass through the origin. [The 2-D autocorrelation is not shown in the graphics here. You'll need to understand it from the words here. Sorry.] The autocorrelation is a superposition of lines of various slopes all passing through the origin, all having positive amplitude. This function would resemble a positive impulse function at the origin (and hence suggest a white spectrum). The function is actually not an impulse function, but, as we'll see, it is the pole $1/r$.

Consider an integral on a circular path around the origin. The circle crosses each line exactly twice. Thus the integral on this circular path is independent of the radius of the circle. Hence the average amplitude on the circumference is inverse with the circumference to keep the integral constant. Thus the autocorrelation function is the pole $1/r$.

Random lines in a plane