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## A Closer Look At The WinDaq Derivative AlgorithmDeriving the rate of change from a waveform yields benefits, and potential pitfalls. Download this application note. The derivative is a mathematical tool used to obtain the rate of change from any given function. When the function to be differentiated is expressed as an equation, we would apply the appropriate derivative formula to attain the rate of change in a similar equation format. When the function is a waveform, an electronic device called a differentiator amplifier can be used to calculate the derivative of the input signal. The result is a second waveform which may be recorded, digitized, or otherwise used to provide additional information about the original waveform. In a computer-based instrumentation environment, it is possible to differentiate a waveform previously recorded to disk by passing the waveform through a derivative function constructed in software. This latter approach has several advantages. Unlike its hardware counterpart, a software differentiator is driftless, yielding enhanced accuracy and repeatability. Calibration of the resulting differentiated waveform is automatic, in units of the input waveform divided by seconds. Finally, where a hardware solution generates one, take-it-or-leave-it derivative signal, the software differentiator allows enhanced flexibility through multiple derivative operations performed on the same, or a different channel. Each operation can have its own set of parameters providing a valuable "what if" approach to waveform analysis. This application note seeks to explore the intricacies of software-based waveform differentiation using the derivative utility of DATAQ Instruments' Advanced CODAS analysis package as its model. All elements of waveform differentiation discussed in this piece are incorporated in this product. ## Differentiation and Integration BasicsThe relationship between differentiation and integration is so close that discussing one naturally leads to a discussion of the other. One of the more fundamental applications of these powerful tools is units transformation. For example, suppose we're acquiring the velocity waveform from an automobile calibrated in feet per second (ft/sec). Using this signal, we can determine the speed of the car at any instantaneous point during the entire test. Useful information, to be sure. But suppose we're curious about the acceleration of the vehicle. In other words, we want the rate of change of the velocity waveform with respect to time. Such information is provided by the derivative function by transforming a waveform calibrated in ft/sec into one calibrated in ft/sec/sec, or ft/sec
## Generating a Differentiated WaveformTurning again to our example velocity waveform and our need to derive acceleration, we could determine the change in velocity over any given range and divide by elapsed time for a rough approximation. But this is a tedious approach to waveform differentiation that can easily lead to errors, and fails to provide an overall graphic picture of acceleration that could help us identify other areas of interest. These problems can be solved by the software differentiator. The most basic approach such products could use to derive the first derivative of a waveform is to calculate the difference between any given point on the velocity waveform, and the next adjacent sample. This difference would then be divided by the elapsed time increment separating the two points to yield the rate of change in units of ft/sec Least squares as applied to linear regression is a well known technique used for deriving a predictive equation of the form. y = mx + b from a randomly sampled group. From basic calculus, we know that the first derivative of this equation (
## Other Waveform Differentiation IssuesThere are a number of issues you should consider when choosing an analysis package to perform waveform differentiation. First, make certain that the generated waveform is scaled into meaningful engineering units automatically. The units to which the generated waveform should be scaled are the original waveform units divided by time (usually in seconds). Such capability dramatically simplifies differentiation operations. Second, waveform differentiators
The differentiator should perform calculations using fast, integer math where a co-processor does not offer an advantage. The only floating-point operation required is the single step of calculating a scaling constant which will be used to scale the derivative waveform into meaningful units when required. Finally, check the data space memory requirements of your analysis package. Some analysis software utilities require that the waveform to be operated on reside entirely in semiconductor memory. This is no problem if the waveform to be analyzed is only 10Kb in length, but could be a real problem when attempting to tackle a 2Mb or larger channel. To insulate yourself from such problems, look for an analysis package that is disk-based. Such systems stream the waveform to be analyzed off disk, through the differentiator, and back to disk as a calculated channel. The only limit to the size of waveform information that can be analyzed using this approach is the size of the hard drive. |