Spider DSA User’s Manual
185
Cross Spectrum
The Cross Spectrum characterizes the relationship between two spectra. For two
signals and , with frequency components X(f)and Y(f)it is defined as:
The Cross Spectrum reflect the correlation between the two signals. While the
Power Spectrum is real-valued, the Cross Spectrum is complex. This means that it
also describes the phase relationship between the two signals.
Frequency Response Function
An important application of Dynamic Signal Analysis is characterizing the input-
output behavior of physical systems. In linear systems, the output can be
predicted from a known input if the Frequency Response Function (FRF) of the
system is known. The Frequency Response Function, H(f), relates the Fourier
Transform of the input X(f) to the Fourier Transform of the output Y(f) by the
simple equation:
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 H(f) to be measured and calculated.
That is:
The fact that Y(f) is dependent on the input X(f) is what makes the system linear.
When measuring the input-output behavior of a system, there is always noise
present that obscures the output. An important measure is how much of the
output is actually caused by the input and a linear process. This is indicated by
another important real-valued spectrum called the (ordinary) Coherence
Function. This coherence function is also defined in terms of the cross spectrum
and the power spectra. Specifically:
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