The degree of causality in a frequency response

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Unformatted text preview: er. It (5-12) +UC-SDRL-RJA CN-20-263-663/664 Revision: June 12, 2001 + should be emphasized, however, that low coherence does not necessarily imply poor estimates of the frequency response function, but simply means that more averaging is needed for a reliable result. The ordinary coherence function is computed as follows: COH pq = 2 γ pq | GXF pq |2 GXF pq GFX qp = = GFF qq GXX pp GFF qq GXX pp (5.20) When the coherence is zero, the output is caused totally by sources other than the measured input. In general, then, the coherence can be a measure of the degree of noise contamination in a measurement. Thus, with more averaging, the estimate of coherence may contain less variance, therefore giving a better estimate of the noise energy in a measured signal. This is not the case, though, if the low coherence is due to bias errors such as nonlinearities, multiple inputs or leakage. A typical ordinary coherence function is shown in Fig. 5.4 together with the corresponding frequency response function magnitude. In Fig. 5.4, the frequencies where the coherence is lowest is often the same frequencies where the frequency response function is at a maxima in magnitude or at a minima in magnitude. This is often an indication of leakage since the frequency response function is most sensitive to the leakage error at the lightly damped peaks corresponding to the maxima. At the minima, where there is little response from the system, the leakage error, even though it is small, may still be significant. (5-13) +UC-SDRL-RJA CN-20-263-663/664 Revision: June 12, 2001 + Ordinary 1 0.9 0.8 0.7 Coherence Magnitude 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10 20 Frequency 30 (Hz) 40 50 Frequency Response - Automotive Exhaust 200 150 100 50 0 -50 -100 -150 -200 0 Phase (deg) 10 20 30 Frequency (Hz) 40 50 10 1 Frequency Response - Automotive Exhaust 10 0 Magnitude 10 -1 10 -2 10 -3 0 10 20 30 Frequency (Hz) 40 50 Figure 5-4. FRF and Corresponding Ordinary Coherence Function In all of these cases, the estimated coherence function will approach, in the limit, the expected value of coherence at each frequency, dependent upon the type of noise present in the structure and measurement system. Note that, with more averaging, the estimated value of coherence will (5-14) +UC-SDRL-RJA CN-20-263-663/664 Revision: June 12, 2001 + not increase; the estimated value of coherence always approaches the expected value from the upper side. This is described in Figure 5-4. Note that a high value of coherence (0.9) after 16 averages has approximately the same possible variance of the frequency response function as a low value of coherence (0.1) after 256 averages. Figure 5-5. Ordinary Coherence Relationship - Averaging Two special cases of low coherence are worth particular mention. The first situation occurs when a leakage error occurs in one or both of the input and output measurements. This causes the coherence in the area of the peaks of the frequency response to be less than unity. This error can (5-...
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