Frequency Analysis from the Basics (16) - "Cross Spectrum"

• This series, "Frequency Analysis from the Basics," has so far covered 15 installments;
• Fundamentals of Fourier Transform
• Fundamentals of DFT (FFT)
• Power Spectrum

We have explained these points. In particular, the power spectrum is the most basic and important 1-channel function of an FFT analyzer as an electrical measuring instrument, but over the next few installments, we will talk about the 2-channel function. This time, we will discuss the cross spectrum (cross power spectrum), which is the frequency function between 2 channels, corresponding to the power spectrum, which is the frequency function of 1 channel.

If X(f) is the Fourier transform of the time signal x(t), then the power spectrum _Px (f) of x(t) is:

Here, * represents the complex conjugate. Also, X(f) here is considered as the complex Fourier coefficient, not the general Fourier spectrum.

Similarly, we define the cross spectrum of two time signals x(t) and y(t) as shown in equation (2). If the Fourier transforms of the two time signals x(t) and y(t) are X(f) and Y(f), respectively, then the cross spectrum Cxy(f) between the two channels is:

　　　　　　　　　c_xy(f)=X(f)*Y(f)

It will be.

Now let's consider the physical meaning of the cross spectrum. As explained earlier, the power spectrum is the power of a 1-channel time signal at each frequency, but the cross spectrum is the amplitude component at the common frequency component contained in the 2-channel x(t) and y(t) signals, expressed as a function of frequency. Since this is usually a complex number, equation (2) can be rewritten as follows.

　　　　　　　　　c_xy(f)=|cxy(f)|e^jθ(f)

Here, |fC _xy (f)| represents the amplitude component of the cross spectrum, and θ(f) represents the phase difference between the two channels (referenced to the complex conjugate channel, which is Channel 1 in this case). Thus, while it is difficult to give a clear meaning to the amplitude information, the phase represents the phase difference between channels at each frequency and is very important information.

Similar to power spectra, averaging is essential in actual cross-spectral measurements to improve the statistical accuracy of the spectrum. The averaging method used here does not involve adding power (amplitude squared) as in power spectra, but rather averaging the real and imaginary parts while preserving the phase information. Therefore, averaging reduces noise components unrelated to the signal, improving the signal-to-noise ratio.

Since the cross spectrum is a complex function, it can be represented as a vector on the complex plane, and Figures 1 and 2 show three additions.

Figure 1 shows the case where there is correlation between the two channels and no noise. In this example,

This is the result (the bar above indicates the average).

Figure 2 shows the case where the correlation between the two channels is less than 1 or there is noise; in this example,

The ratio of the left-hand side to the right-hand side in equation (4) or equation (5) is called the coherence function, and it represents the degree of frequency correlation between the two input/output channels.

In actual FFT analyzers, you don't often explicitly display and use cross-spectrum graphs, but they are important functions necessary for calculating the transfer function and this coherence function, which will be explained next time.

Here's a summary of the main applications of cross-spectral imaging.

1. Calculating AC Power: In the field of electricity, power (W) can be calculated by taking the product of AC voltage (V) and AC current (i). In this calculation, the real part represents the effective power, the imaginary part represents the reactive power, and the phase cosine (cos θ) represents the power factor. (See the measurement column below.)
   Fundamentals of Digital Measurement - Part 11: "Power Factor of AC Power"
2. Calculation of Acoustic Intensity In an acoustic system, the product of sound pressure and particle velocity is the acoustic power per unit area, or acoustic intensity. However, in the two-microphone method, acoustic intensity can be determined from the imaginary part of the cross spectrum.
3. It is used in FFT analyzers to calculate important functions between channels, such as the transfer function, coherence function, and cross-correlation function.
4. Phase information between channels is obtained.

Finally, here's a summary.

1. The power spectrum is the frequency function of one channel, while the cross spectrum is the frequency function between two channels.
2. The cross-spectrum is generally a complex function that represents the amplitude component of the common frequency component contained in a 2-channel signal, and the phase represents the phase difference between the channels.
3. Cross-spectrum averaging preserves phase information, and unlike power spectrum averaging, it has a noise reduction effect.
4. Applications of cross-spectrum technology include calculating AC power and acoustic intensity power (electricity).
5. In FFT analyzers, important applications of cross-spectrum analysis include the calculation of transfer functions and coherence functions.

【keyword】
Cross-spectrum, cross-power spectrum, coherence function, transfer function, effective power, reactive power, power factor, acoustic intensity, cross-correlation function

[Reference materials]

1. "Spectral Analysis," by Mikio Hino, Asakura Shoten (1977)
2. "Statistical Processing of Random Data," by Bendatt and Peersol, Baifukan (1985).

(Excerpt from the email newsletter issued on July 17, 2014)