Wide Band Integrators – What Are They?

For some time now it has been conventional ‘wisdom’ that using time based digital integration may cause amplitude errors in the result and that these get worse as the frequency increases. As a result of this, integration using Omega arithmetic has been prevalent by using Fourier Transforms of the signal. This, of course, remains a valid approach and is particularly useful if the data is already in the frequency domain, which was its prime purpose.

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Calculating Velocity Or Displacement From Acceleration Time Histories

It is quite straightforward to apply “classical” integration techniques to calculate either a velocity time history from an acceleration time history or the corresponding displacement time history from a velocity time history. The standard method is to calculate the area under the curve of the appropriate trace. If the curve follows a known deterministic function then a numerically exact solution can be found; if it follows a non-deterministic function then an approximate solution can be found by using numerical integration techniques such as rectangular or trapezoidal integration. Measured or digitized data falls in to the latter category. However, if the data contains even a small amount of low frequency or DC offset components then these can often lead to misleading (although numerically correct) results. The problem is not caused by loss of information inherent in the digitisation process; neither is it due to the effects of amplitude or time quantisation; it is in fact a characteristic of integrated trigonometric functions that their amplitudes increase with decreasing frequency.

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