Data Recording and EDM Post Analyzer – the Thinking Man’s Alternative

CI offers EDM Post Analyzer software, a powerful adjunct to your Spider-based analysis tool kit, allowing you to analyze recordings made using your Spider front-end modules. The beauty of this approach is that it lets you analyze and reanalyze digitally recorded data after the recording event. “But wait,” you say. “Isn’t that the way we used to work when we had to lug 75 pound FM recorders around?” The answer is yes and no.

Yes, in that the recording is made before analysis methods and details must be fully decided upon. No because the recorded data may include things the FM recorder could never capture, is database-filed and SQL searchable, can be initiated from a remote site via internet, has dynamic range and frequency span unheard of when analog tape was king and is fully compatible with all of your EDM analysis software.

Recording first and analyzing second makes great sense to first-responding problem solvers. Simply recording does not require all of the tactical measurement decisions be made before data is taken. Often a new problem requires some “get acquainted” measurements to really define the difficulty and its root cause. We are often not smart enough to guess what causes our new challenge. We need to look at some representative measurements from different analytic viewpoints to begin to understand the problem and home in on its solution. The approach is eminently suitable for a team effort. A recording technician can acquire data using minimum equipment while the analyst can remain on post with his analytic workstation.

Consider a typical scenario. Your NVH team is part of a broader coalition of engineers evaluating a prototype SUV as its release time draws near.  Several drivers have reported annoying “boom periods” during their durability loops. You grab a recorder-programmed Spider-80x, some microphones and cables and head to the track. A small DC-DC converter lets you power the small 1.3 kg 8-channel /2-tachometer Spider module from vehicle power (a mere 10 Watt draw). You have enough channels to measure a microphone at every seating position plus a couple of accelerometers or conditioned chassis strain gages. Having two dedicated tachometer channels let you monitor both engine RPM and drive shaft RPM, a real boon in this age of automatic and continuously variable transmissions (CVT)!

Spider-80X is a very capable recorder. It has 4 GByte of on-board flash memory, enough to store more than ¼ hour at a blazing 102.4 kHz sample rate for all 8 inputs and both tachs. Reduce the sample rate on the input channels and recording time increases proportionately (while the pulse tachometers are still sampled at maximum resolution). Its inputs span ±20 Volts with an unrivaled 150 dB dynamic range provided by dual 24-bit ADCs and our patented stitching algorithm, so there is no need to fiddle with input attenuator settings. All data is recorded in 32-bit single-precision floating-point format (per IEEE 754-2008). Recording may be controlled using the front panel Start and Stop buttons, or you can use your iPad® tablet running our EDM App for iPad.

Your recordings can find their way to “analysis central” by any means ranging from sneaker-net to internet. Now EDM Post Analyzer can go to work for you, using any and all of the DSA analysis tools available for live data analysis. The data streams recorded can be analyzed and reanalyzed in any manner you choose. If you wish to examine the full recorded bandwidth – no problem!  If you want to analyze a smaller bandwidth, or zoom into a far narrower frequency span – no problem – the necessary anti-aliasing filtration and data down-sampling occur automatically when you select an analysis bandwidth. Use FFT and spectrum averaging, employ nth-Octave analysis. Order-normalize your measurements based on either recorded tachometer. Recursively conjure those “what if” scenarios and make the appropriate measurements to prove or disprove each hypothesis. Do all of the proper “detective work” you would do if you were on-station with an analyzer and had the prototype exclusively available to you.  But do all of these things in the comfort of your lab without having to acquire additional field measurements while other test occupy the prototype hardware.

EDM Post Analyzer offers a new way to focus your intellect upon the problems of your enterprise. Now you can exhaustively reanalyze recorded questions as you learn from one analysis and formulate the next. Try this thinking man’s alternative – you’re sure to like it. It was designed to support and augment your natural curiosity and intuition when confronting a new problem. Think of it as a structural detective’s newest best friend.

The Coherence Function - A Brief Review

An important application of Dynamic Signal Analysis is characterizing the input-output behavior of physical systems. In linear systems, the output time history, y(t), can be predicted from a known input time history, x(t) if the Frequency Response Function (FRF) of the system is known.

The Fourier Transform and its inverse relate the observable time histories and their spectra. Specifically:

A linear process is defined by its Impulse Response, hxy(τ), or by the  Fourier Transform of its Impulse Response which is called the Frequency Response Function, Hxy(f). In the time domain the system output, y(t), is predicted by the convolution of x(t) and Hxy(τ). Specifically:

This same relationship is far simpler to describe in the frequency domain. The Frequency Response Function, Hxy(f), relates the Fourier Transform of the input X(f) to the Fourier Transform of the output Y(f) by simple multiplication:

Multiplying both sides of this equation by the conjugate of the input spectrum and ensemble averaging explains the importance of the power and cross power spectra as they allow Hxy(f) to be directly measured. That is:

In simpler notation:

… where Gxx(f) is the average auto-spectrum of X and Gxy(f) is the averaged cross spectrum between input X and output Y. From which:

The fact that Y(f) is directly dependent on the input X(f) is what makes the system linear. However, when measuring the input-output behavior of a system, there is always noise present that obscures the input and output measurements. An important measure is how much of the measured output is actually caused by the measured input and a linear process best estimated by Hxy(f). This is indicated by another important spectrum called the (ordinary) Coherence Function. This Coherence Function is also defined in terms of the cross spectrum and the power spectra. Specifically:

γ2 (f) is a real-valued spectrum. Its amplitude is dimensionless and positive and limited between 0 and 1. Note that the coherence formulation can also be stated as the product of an FRF with its inverse function. That is, if Hxy(f) measures a process going from input, x, to output, y, Hyx(f) characterizes the same process, but treats y as the input and x as the output.

This product definition indicates the coherence characterizes an “energy round trip” or a reflection through the process. We apply Gxx to Hxy and get Gxy at the output. Then we conjugate Gxy (equivalent to flipping or reflecting x(t) in time) and pass it back through Hyx. In a perfect world, this would result in exactly Gxx as the output of Hyx.

If the system is linear and none of our measurements are contaminated by noise, the round-trip is perfect and we get back everything we put in. That is, the Coherence will be exactly 1.0. If the system is non-linear or if extraneous noise has been interjected at the input or output, the round-trip will be less efficient and the Coherence will be less than one (but never more).

Thus, the coherence magnitude at every frequency is always between 0 and 1. A coherence of 1.0 means the output is perfectly explained by the input (i.e. the system is linear and the Hxy(f) measurement is perfect). A coherence of 0 means the output and input are totally unrelated. Values in-between state the fraction of measured output power explained by the measured input power and a linear process. Departures from unity can indicate input noise, output noise, a non-linear process or any combination of these things. Experienced analysts always use the Coherence measurement to quantify the quality of an FRF measurement at every frequency.

Note that ensemble averaging must be used for the Coherence to have meaning. The coherence computation for a single set of input and output “snapshots” will always evaluate to 1.0 at every frequency. This can be understood by expanding the Coherence equation to show the averaging process.

Now if we specify a single frame measurement (N=1):

Clearly then, the Coherence becomes more definitive as the number of spectra averaged increases. We will revisit this point shortly.

Let’s examine the effect of experiment-unrelated input and output noise on the Coherence function. As shown above, unintentional noise may be added to the input and output signals measured by our analyzer. This may be caused by a bad cable, sensor frailties or a host of other experimental “gremlins”. Further, if the process being tested exhibits some non-linear behavior, this will generate harmonics and other distortion products at the output; these appear as components of the output noise signal, o(t).

While the system on test sees an input, x(t), and produces an output, y(t), the analyzer measures the noise-contaminated signals:

The measured power spectra and the cross spectrum evaluate to:

In the (very likely) event that the input noise, i(t), is uncorrelated (unrelated) to the input stimulus, x(t), and the output noise, o(t), is completely unrelated to the system response, y(t), and i(t) and o(t) are uncorrelated, then all of cross spectra except Gxy(f) have an expected value of zero. Thus we find: 

Computing the coherence from these results in:

That is, noise contamination of either signal (or equivalent output noise due to a non-linear process) causes the coherence to be reduced from that of a noise-free measurement. But, there is no way to distort or contaminate the signals that will cause γ2(f) to exceed unity.