Signal Processing of Random Physiological Signals - Charles S. Lessard

FIGURE 18.4: Test setup. The possible results in the evaluation would be (a) normal transfer function response or (b) transfer function is contaminated by vibration from loose hardware during testing of specimen

was 0.83/(1 − 0.83), so the coherence function is an alternative method of measuring the signal-to-noise ratio or an input–output system.

Lessard [3] used the coherence function to determine the cause-and-effect relationship between the output of the vestibular system and pseudorandom angular velocity input. The subject was subjected to pseudorandom angular velocity the saccade of the horizontal eye movement was converted to a signal. A full analysis of the output frequencies was performed. The coherence function analysis was performed as the last step in the analysis. This project was done with the objective of determining the nature and causes of space sickness.

18.5PROBLEMS THAT LIMIT THE APPLICATION OF COHERENCE

There are numerous problems that limit the application. The major problems will be listed without discussion. Among these problems are

1.inaccurate source measurements,

2.source measurement interference,

212SIGNAL PROCESSING OF RANDOM PHYSIOLOGICAL SIGNALS

3.correlation among sources,

4.input–output time-delay errors,

5.input measurement noise,

6.input–output nonlinearities,

7.input–output feedback,

8.reverberation effects, and

9.common source periodicities.

18.6CONCLUSION

The coherence function, like many of the other mathematical operations used in signal processing, is susceptible to abuses by researchers who intentionally misapply its use or is not well informed with regard to its limitations. The coherence function should be used to establish a relationship between input and output of a system. None of the literature supports the idea of using the coherence function to compare two output signals such as the electromyograms (EMG) from two different muscles. Finally, as in all types of scientific tools, all assumptions must be met if the results are to be interpreted correctly.

18.7REFERENCES

1.Bendat, J. S. and Piersol, A. G. Random Data: Analysis and Measurement Procedures. Wiley Interscience, 1971.

2.Bendat, J. S. “Statistical Errors in Measurement of the Coherence Function and Input–Output Quantities,” Journal of Sound and Vibration, September 1978.

3.Lessard, C. S. “Analysis of Nystagmus Response to Pseudorandom Velocity Input,”

Compute Methods and Programs in Biomedicine, Vol. 23, 1986.

4.Neter, J., Wasserman, W., and Hunter M.H., Applied Linear Regression Models. Irwin, 1983.

COHERENCE FUNCTION FROM SPECTRAL ANALYSIS 213

5.Roth, P. R. “Measurements Using Digital Signal Analysis,” IEEE Spectrum, April 1971.

6.Roth, P. R. “How to Use the Spectrum and Coherence Function,” Sound and Vibration, January 1971.

7.Winer, B. J. Statistical Principles in Experimental Design. McGraw-Hill, 1971.

216 SIGNAL PROCESSING OF RANDOM PHYSIOLOGICAL SIGNALS

records x(t) and y (t) of unlimited length T, the one-sided cross-spectrum is given by (19.1).

where X( f, T ) is the conjugate of the input and Y ( f, T) are the finite Fourier transforms of x(t) and y (t), respectively.

It should be noted that the raw spectrum (no averages) of a finite length has the same equation as (19.1), without the limit as given by (19.2) and will have a resolution

The ordinary coherence function is defined by (19.4).

The corresponding variance errors for “smooth” estimates of all of the “raw” es-

timates in (19.4) will be reduced by a factor of nd when averages are taken over nd statistically independent “raw” quantities. To be specific, the variance expressions are given by (19.5), (19.6), and (19.7).

218 SIGNAL PROCESSING OF RANDOM PHYSIOLOGICAL SIGNALS

TABLE 19.2: Multiple-Input/Output Model Estimates

Equations (19.11) and (19.12) are used to calculate the variance and the normalized random error of the transfer function, respectively.

Then from (19.12), the normalized random error in the gain factor estimates at frequencies f1and f2 will be given by (19.13) and (19.14).

ERROR IN RANDOM DATA ESTIMATE ANALYSIS 219

Applying (19.15), the results also constitute approximations for the standard deviation (not normalized) in the phase factor estimate calculation as follows.

Hence, the estimates made with a coherence of γx2y ( f2) = 0.90 are almost ten times as accurate as those made with a coherence of γx2y ( f1) = 0.10.

19.2 SUMMARY

A summary is given in Table 19.2 on the main normalized random error formulas for single-input/output model estimates.