App-24 IM 701450-01E
Cross Spectrum
Cross spectrum is determined from 2 signals. It is found by taking the product of the
linear spectrum of one signal(Gy) and the complex conjugate (Gx*) of the linear
spectrum of the other signal (Gx). If the linear spectrums of the 2 signals are
represented by
Gx = Rx+jIx
Gy = Ry+jIy
then the cross spectrum Gyx is
Gyx= Gy×Gx*
= (Ry+jIy)(Rx–jIx) = Ryx+jIyx
where Ryx = RyRx+IyIx
Iyx = RxIy–RyIx
The following spectrums can be determined with this instrument.
Item Equation Computation
Real part CS-REAL DC component Ryx/
AC component Ryx/2
Imaginary part CS-IMAG DC component Iyx
AC component Iyx/2
Amplitude CS-MAG DC component
(Ryx
2
+ lyx
2
)
AC component
(Ryx
2
+ lyx
2
)/2
Log magnitude CS-LOGMAG DC component 10×log
(Ryx
2
+ lyx
2
)
AC component 10×log(
(Ryx
2
+ lyx
2
)/2
)
Phase CS-PHASE tan
–1
(Iyx/Ryx)
Transfer Function
The transfer function expresses the frequency characteristics between the input to the
transfer system and the output. The transfer function is determined by the ratio of the
output linear spectrum (Gy) and the inpvIS‚-ectrum (Gx) at each frequency. Also, as can
be seen from the next equation, the transfer function can be defined as the ratio of the
cross spectrum of the input and output (Gyx) and the input power spectrum (Gxx).
Transfer Function = Gy/Gx = (Gy×Gx*)/(Gx×Gx*) = Gyx/Gxx
= (Ryx+jIyx)/(Rx
2
+Ix
2
)
The following items can be determined with this instrument.
Item Equation Computation
Real part TF-REAL Ryx/(Rx
2
+Ix
2
)
Imaginary part TF-IMAG Iyx/(Rx
2
+Ix
2
)
Amplitude TF-MAG
(Ryx
2
+ lyx
2
)
/(Rx
2
+Ix
2
)
Log magnitude TF-LOGMAG 20×log
(Ryx
2
+ lyx
2
)
/(Rx
2
+Ix
2
)
Phase TF-PHASE tan
–1
(Iyx/Ryx)
The magnitude of the transfer function shows the ratio of the magnitudes of the output
linear spectrum and the input linear spectrum while the phase shows the phase
difference of the two.
Appendix 4 User-Defined Computation
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