App-8
IM DLM3054-01EN
Cross Spectrum
The cross spectrum is determined from two 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 spectra 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
This instrument can determine the following spectra.
Item
Formula
Computation
Real part
CS-REAL
DC component
Ryx
AC component
Ryx/2
Imaginary part
CS-IMAG
DC component
Iyx
AC component
Iyx/2
Bandwidth
CS-MAG
DC component
(Ryx2 + Iyx2)
AC component
(Ryx2 + Iyx2)
/2
Log magnitude
CS-LOGMAG
DC component
10×log
(Ryx2 + Iyx2)
AC component
10×log(
(Ryx2 + Iyx2)
/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).
Transfer function = Gy/Gx = (Gy × Gx*)/(Gx × Gx*) = Gyx/Gxx
= (Ryx + jIyx)/(Rx
2
+ Ix
2
)
The instrument can determine the following items.
Item
Formula
Computation
Real part
TF-REAL
Ryx/(Rx
2
+ Ix
2
)
Imaginary part
TF-IMAG
Iyx/(Rx
2
+ Ix
2
)
Bandwidth
TF-MAG
(Ryx2 + Iyx2)
/(Rx
2
+ Ix
2
)
Log magnitude
TF-LOGMAG
20×log
(Ryx2 + Iyx2)
/(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
Formula
Computation
Bandwidth
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
