App-13
App
IM 701830-01E
Appendix
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 x Gx
*
=(Ry+jIy)(Rx-jIx)=Ryx+jIyx
where Ryx=RyRx+IyIx
Iyx=RxIy-RyIx
The following spectrums can be determined with this instrument.
Item
Expression
Computation
Real
CS-REAL
Ryx/2
Imaginary
CS-IMAG
Iyx/2
Magnitude
CS-MAG
(Ryx
2
+Iyx
2
)
/2
Log magnitude
CS-LOGMAG
10xlog
(Ryx
2
+Iyx
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 input spectrum (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 x Gx
*
)/(Gx x Gx
*
)=Gyx/Gxx
=(Ryx+jIyx)/(Rx
2
+Ix
2
)
The following spectrums can be determined with this instrument.
Item
Expression
Computation
Real
TF-REAL
Ryx/(Rx
2
+Ix
2
)
Imaginary
TF-IMAG
Iyx/(Rx
2
+Ix
2
)
Magnitude
TF-MAG
(Ryx
2
+Iyx
2
)
/(Rx
2
+Ix
2
)
Log magnitude
TF-LOGMAG
10xlog
(Ryx
2
+Iyx
2
)
/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.
Coherence function
This expresses the ratio of the output power generated with the input signal to the transfer
system and the total output power.
Coherence function=Gyx x Gyx
*
/(Gxx x Gyy)
Item
Expression
Computation
Magnitude
CH-MAG (Ryx
2
+Iyx
2
)/(GxxxGyy)
If the output signal is due entirely to the input signal, the coherence function becomes 1. As the
ratio decreases, it falls below 1. Thus, the coherence function always takes on a value between
0 and 1.
Note
On 1 data acquisition, the coherence function becomes 1 across all frequencies. Also, make sure to take the
frequency average of the computation.
Appendix 6 About User Defined Computations
