YOKOGAWA ScopeCorder DL850E - Page 243

App-9
IM DL850E-01EN
Cross Spectrum
The cross spectrum is determined from two signals. It is 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 two 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 DL850E/DL850EV can determine the following spectrums.
Item Expression Computation
Real part CS-REAL Ryx/2
Imaginary part CS-IMAG Iyx/2
Magnitude CS-MAG
(Ryx
2
+Iyx
2
)/2
Log magnitude
CS-LOGMAG 10 × log (
(Ryx
2
+Iyx
2
)/2
)
Phase CS-PHASE tan
−1
(Iyx/Ryx)
Transfer Function
The transfer function expresses the frequency responses of the input to and the output from the transfer system.
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 following 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).
T
ransfer function = Gy/Gx = (Gy × Gx
*
)/(Gx × Gx
*
) = Gyx/Gxx
= (Ryx + jIyx)/(Rx2 + Ix2)
The DL850E/DL850EV can determine the following items.
Item Expression Computation
Real part TF-REAL Ryx/(Rx
2
+ Ix
2
)
Imaginary part TF-IMAG Iyx/(Rx
2
+ Ix
2
)
Magnitude TF-MAG
(Ryx
2
+Iyx
2
)
/(Rx
2
+ Ix
2
)
Log magnitude
TF-LOGMAG 20 × log
(Ryx
2
+Iyx
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.
Coherence Function
The coherence function expresses the ratio of the output power generated by the input signal to the transfer
system and the total output power.
Coherence function = Gyx × Gyx
*
/(Gxx × Gyy)
Item Expression Computation
Phase CH-MAG (Ryx
2
+ Iyx
2
)/(Gxx × Gyy)
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.
On one data acquisition, the coherence function becomes 1 across all frequencies. Make sure to take the
frequency average of the computation.
Appendix