The previous explanations focused on the relationship between the time waveform of channel 1 and its frequency analysis results. From this point on, we will explain the function that describes the relationship between two channels.
The main purpose of finding a function between two channels is to apply an input such as an excitation force to the object of interest (generally called a system), simultaneously measure its response, and determine the relationship between the input and output.
Let x(t) and y(t) be the input and output signals to a system, and let X(f) and Y(f) be their Fourier transforms, respectively. As explained previously, X(f) and Y(f) are generally complex numbers. Their power spectrum Gxx(f) is:
Gxx(f)＝｜X(f)｜2＝XR(f)2+XI(f)2
Furthermore, if we denote the complex conjugate of a complex number X(f) (the complex number with the sign of the imaginary part reversed) as X(f)*, then
Gxx(f)＝X(f)* ・ X(f)
It can also be expressed as follows. The power spectrum of the output signal y(t) is similarly expressed as
Gyy(f)＝Y(f)* ・Y(f)
This is how it is expressed. The power spectrum represents the power components for each frequency and is a real-valued function (not a complex number), so it can be graphed with frequency on the horizontal axis and power (magnitude of the square of the amplitude) on the vertical axis.
Next, we define the cross (power) spectrum Gxy(f) as a function that describes the relationship between the input signal x(t) and the output signal y(t), as follows:
Gxy(f) = X(f) * Y(f)
The cross spectrum represents the common power component in two 2-channel signals, x(t) and y(t), as a function of frequency, and is a complex function. If we denote its real part as CR(f) and its imaginary part as CI(f), it can also be expressed in terms of absolute value (amplitude) and phase.
Amplitude |Gxy(f)|= √{CR(f) 2 +CI(f) 2}
The contents of { } represent what is inside the square root.
Phase θ(f) = arctan (CI(f)/CR(f))
While it's difficult to definitively define the meaning of amplitude information, the phase represents the phase difference between channels at each frequency, making it extremely important information.
This document summarizes the meaning and applications of cross-spectrum analysis.
-
While cross-spectrums themselves don't have many practical applications, they can be used to calculate important functions (such as transfer functions, coherence functions, and cross-correlation functions) using quadratic computation.
-
This can be used to determine the phase difference between channels.
-
This is used when measuring acoustic intensity using the two-microphone method.
-
Although it's a specialized application, cross-spectrum analysis can be used to detect the period and power spectrum of periodic signals that are buried in noise.
(Excerpt from the email newsletter issued on June 20, 2003)
