The following is not really a "new" analysis, since it uses the same math as that behind the discrete Fourier transform. However, it provides a continuous set of 2d results, whereas the DFT only calculates 2 discrete 1d results.
Here is an image of a component software analysis system which generates a sine wave of arbitrary frequency and phase (controlled by the sliders), then forms the inner product of that wave with an input signal of the same length, integrates over the length to produce a single real number, and then repeatedly varies the phase of the sine wave throughout a given range (usually 0-2 pi) and performs additional products and integrals in order to accumulate a 1d set of values. The frequency of the sine wave is then varied and the accumulation of product integrals is repeated to create additional 1d curves. All of these frequency-phase integral curves are then accumulated into a single 2d array which can be displayed as a surface or an image.
The following image shows the result of accumulating the integral of a sine wave of a single frequency, but at different phases, into a 1d curve.
As the frequency of the sine wave is varied, additional phase spectrum curves are accumulated into a 2d array. This array is shown below as a set of 33 phase cross-section lines.
In the next image, the 2d array will be rotated by 90 degrees and shown as a set of 100 frequency curves.
Here is the result of applying the above frequency-phase process to the average V(3,3) EKG shown previously. The 2d result is shown both as a surface (drawing only constant phase cross sections) and as an image. The frequency, from 0-32 Hz in 1 Hz increments, is along the horizontal axis, while the phase from 0-2 pi in 100 steps, is along the vertical axis from top to bottom. The real and imaginary parts of the DFT are 2 constant phase cross sections (horizontal frequency curves in plot) of this array which are 1/4 and 1/2 way down the phase axis (since they represent the cos() and -sin() integrals of this analysis).
Note in particular the banded structure of the image. In fact, there seem to be two distinct banded regions: one at frequencies less than 5 Hz, and another at frequencies above 5 Hz. This will be examined in more detail below.
Since the frequency-phase accumulation process does involve some manual work, as well as several different components, I have coded the entire process into a single component which performs all operations as a single step, and which can compute the resulting 2d array to any resolution. I call this new component a "convolution array", since the process of taking the inner product and integral is a convolution. This component can form the 2d convolution result of varying an input filter function in frequency and phase, and applying this filter to another input signal.
Here is the convolution array result of varying the sine filter frequency from 0-32 Hz in 0.25 Hz increments, and its phase from 0-2 pi in 128 steps. Shown below are results for different frequency ranges.
Once again note the two different banded regions. Each region appears to contain multiple frequency-phase bands having a constant slope. This suggests that there are two different dynamic systems in operation, and that these systems interact at the 5 Hz "gap" in the line spectra shown previously. Across the 5 Hz boundary, the dynamics of the overall system make a transition from one sub-system to the other.
The slopes of each banded region can be measured graphically, and are observed to have the following values:
Slope 1 = -3.590 radians/Hz
Slope 2 = -1.733 radians/Hz
S1 / S2 = 2.071
Thus, the phase velocity of band-1 is just over 2 times that of band-2. This suggests that the overall dynamics are governed by two major sub-systems, with a phase velocity ratio of almost exactly 2:1. There are many examples of entrained 2-body non-linear dynamic systems with harmonic ratios like this in the literature (for example, see Abraham & Shaw). It appears that the EKG may be another example of such a system.
Here is a close-up of the previous analysis for 3-7 Hz, showing the "null line" in the spectrum where the two systems interact and destructively interfere. The image plot has had its scale expanded to show the 0 line (green) just above 5 Hz. The overall geometry is that of a saddle region between two subsystems, where the dynamics change from one subsystem to the other.
(Click surface for larger filled-surface plot)
The following two plots show more detail of the saddle region. In the left plot, a wireframe of the surface from 3-7 Hz has been rotated by 90 degrees at 0 altitude to show the interior of the saddle. In this region, the frequency response of the combined system makes a transition from left to right (or vice versa), from the lower part of the upper left subsurface nearest the blue 0-line to the upper part of the lower right subsurface, across the face of the saddle (yellow path).
The right image shows a 2d vector plot of the gradient of the surface also from 3-7 Hz. Two unstable saddle points are visible in the plot: one at the upper middle showing convergence in phase and divergence in frequency, and the other at the lower middle showing convergence in frequency and divergence in phase (both outlined in yellow).
As a controlled comparison, the following image shows a randomly generated signal and its associated frequency-phase spectrum. This result is typical for random signals. The 2d result does not show the coherent banding, nor a transition from one dynamic to another at a specific frequency, that the EKG signal above exhibits. This suggests that the observed behavior of the EKG signal is "really there", is significantly different from random signals, and is not an artifact of the analysis.
©Copyright Sky Coyote, 2001-2002.