A new 2d feature spectrum of an EKG signal,
showing several distinct components.

As in the previous analysis, the following is not really "new", since it also uses some of the same math as that behind the discrete Fourier transform. However, in this case the analysis provides not only an indication of the matching strength of a filter function with respect to a signal, but also an indication of the location within the signal where the filter matching occurs. In this way it is also somewhat like a wavelet analysis.

Here is an image of a component software analysis system which generates a single sine pulse of arbitrary frequency and delay (controlled by the sliders), then forms the inner product of that pulse with an input signal of the same length, integrates over the length to produce a single real number, and then repeatedly varies the delay of the sine pulse throughout the length of the input and performs additional products and integrals in order to accumulate a 1d set of values. The frequency of the sine pulse is then varied and the accumulation of product integrals is repeated to create additional 1d curves. All of these frequency-delay 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 pulse at a single frequency, but at different delays, into a 1d curve. During the accumulation, the pulse is rolled through the length of the input signal, so that the value of the integral at any point indicates the strength of the pulse feature at that location in time.

As with the previous analysis, I have coded the entire process shown above into the convolution array component, which performs all operations as a single step, and which can compute the resulting 2d array to any resolution.

Here is the result of applying the above frequency-delay process to the average V(3,3) EKG shown previously. The 2d result is shown here as a filled surface. The frequency, from 0-32 Hz, is along the right axis, while the delay from 0-1 sec (the duration of the input signal), is along the left axis. The height of the surface shows the presence of features in the input signal which match the shape of the filter function. The distribution of these features, in both time and frequency, is shown by the shape of the surface. Note that these features appear to be well localized in time, but may consist of a wide range of frequencies.

Here is the same result shown as an image. Also shown is a 2d vector plot of the gradient of the frequency-delay surface, which better shows the geometry of the individual features, each of which appears as a distinct basin of attraction. The gradient also shows the process of feature bifurcation, in which several new features at higher frequencies appear to be created by branching off from a single feature present at lower frequencies.

In these plots there appear to be two major features (in red at left), and several other smaller ones. These two primary features correspond to the QRS "complex", and the T "wave" in the EKG signal. Note that the QRS feature is very broad-band in frequency, but narrow in time, while the T feature is broad in time, but narrower in frequency. This duality in frequency and time is to be expected from Fourier analysis as well.

Also note that the maximum "ridge" of the QRS feature is surrounded by two very sharp minimum "troughs" on either side. This suggests that the QRS feature of the EKG (the "power stroke") is very stable, as it must be in order to operate properly under a wide variety of conditions.

(Click for larger image)

Here is the frequency-delay surface shown at a larger Z axis scale. Expansion along this axis brings out the details of the smaller features in the signal. It is possible that the time distribution of features at a given frequency forms a fractal set, which is suggested by the shape of the left edge of the surface. If this is indeed the case, I not-so-humbly suggest the name "Coyote's Canyon" for this property of the EKG and other systems.

Here are image and gradient plots at larger scale settings. As the scale is expanded, more features become visible in the image.

(Click for larger image)

Here are details from all previous plots, for frequencies from 0-8 Hz, and a duration of just the first 1/2 second of the signal.

(Click for larger image)

©Copyright Sky Coyote, 2001-2002.