FFT Analysis Basic Terminology Glossary: "Ta-row", "Na-row", "Ha-row", "Ma-row", "Ra-row"

The damping factor is the damping ratio for each frequency. This allows you to obtain the desired damping ratio for a given frequency, and along with the natural frequency, it influences the selection of materials.
The value is expressed as a percentage. A large damping factor means that the attenuation is fast at that frequency, and a small damping factor means that the attenuation is slow at that frequency. This damping factor (ζ) can be calculated for each resonant frequency using an FFT analyzer, from the 3 dB reduced frequency width Δf and the resonant frequency f0 using the following formula (half-width method).

When measuring the transfer function of acoustic or mechanical systems, if the signal propagation time in the system is long and a time lag occurs between the input and output signals, accurate measurement of the transfer function becomes impossible. (This causes a decrease in the coherence function.)

The delay function compensates for time differences between channels by delaying the start of sampling on the slave channel relative to the start of sampling on the master channel.

Voltage and sound pressure are often expressed in dB (decibels), but dB is a unit that uses the common logarithm (log10).
As a review, here are some properties of logarithms that you learned in high school mathematics:

Furthermore, logarithmic operations are:

It will be.

dB (decibels) is:

It is defined as follows: P0 is the reference level, which is 1 V for voltage and 20 μPa (Pascals) for sound pressure. dB represents the ratio of squares, i.e., the ratio of power. If P is the same as the reference level, then P2/P02 = 1, so log10 1 = 0, which is 0 dB. Now, assuming 0 dB = 1 (P0 = 1), let's calculate what dB 10 times that would be:

Therefore, it comes out to 20 dB. Since squaring it every time is troublesome, we can use the mathematical formula mentioned earlier and bring the square to the front, resulting in 20 log:

This eliminates the need to square the numbers, making it simpler. The formula may not be familiar, but essentially, if the actual values (a and b before converting to dB) are multiplied by 10, it becomes 10¹, so it changes by 20 dB. If they are reduced to 1/10, it becomes 10⁻¹, so it changes by -20 dB.

The trigger function is a feature that starts sampling at a specific point in the input signal or when an external signal is detected.

There are two types of triggers: an internal trigger that uses the input signal itself as the sampling start signal, i.e., the trigger signal, and starts sampling when it reaches a set voltage; and an external trigger that takes an externally input pulse signal and starts sampling based on that point in time.

This feature allows you to efficiently capture and analyze the desired portion of the waveform. Furthermore, when averaging time waveforms, the trigger function synchronizes the waveforms.

A Nyquist diagram is a plot of a frequency response function (transfer function) with the real part on the horizontal axis and the imaginary part on the vertical axis, plotted with respect to frequency. It is primarily used to determine the stability of control systems.

The frequency response function (transfer function) is displayed with gain on the vertical axis and phase on the horizontal axis.

It is defined as the ratio of the peak value to the RMS value of a waveform (peak value / RMS value). The crest factor of a DC waveform is "1", and the crest factor of a sine wave is √2 = 1.414.

For example, while peak and RMS vibration values change relatively depending on the size of the bearing (larger bearings have larger RMS vibration values, and even larger peak values in abnormal conditions), the crest factor value calculates the ratio of the peak value to the RMS value. Therefore, the vibration value is not affected by the size of the bearing, making it possible to more accurately judge the degree of abnormality such as damage. A larger measured crest factor value indicates a greater degree of abnormality.

A power spectrum is a representation of a signal's power divided into fixed frequency bands, where the power for each band is expressed as a function of frequency. The unit is the square of the amplitude (V² rms).

In an FFT analyzer, the frequency-domain waveform is obtained from the time-domain waveform using the Fourier transform. The Fourier transform pair of the time function x(t) is expressed by the following equation:

(Fourier transform)

(Inverse Fourier Transform)

Enter a number in half-width characters in the box below and press the "Original Unit" button. The converted value in each unit will appear in the row below.

V rms: RMS value when the power spectrum is displayed linearly.
V0-p: Zero-peak value when the power spectrum is displayed linearly.
dBVr: RMS value when displaying the power spectrum in log format.
dBV0-p: Zero peak value when displaying the power spectrum in log format.

The calculation is:

V0-p = √2×V rms

dBVr = 10 LOG{Vrms / (reference)}2 = 20 LOG (Vrms) [reference is the RMS value of 1 Vrms]

We are doing it this way.

Please note that the calculated value may not perfectly match the true value due to the limitations of significant figures.

In digital FFT analyzers, the bandwidth (Δf) varies depending on the analysis frequency range. For example, with a resolution of 1/800, the bandwidth for a 20 kHz range is 20 kHz/800 = 25 Hz.

When a broadband (distributed) signal, such as white noise, is frequency-analyzed, its power is obtained as an integral value for each bandwidth. Therefore, changing the analysis range alters this value, making comparison impossible. To address this, the power spectrum per unit frequency (1 Hz) is determined using the following method, which is called the power spectral density (PSD). However, this method is meaningless for line-spectrum signals.

The power spectral density is calculated as follows:

Here

In other words, the power obtained with the bandwidth corresponding to each window is normalized. Furthermore, when determining this power spectral density, please measure using the Hanning window or rectangular window whenever possible.

The Hilbert transform g(t) of a real function f(t) is defined by equation (1), and the inverse Hilbert transform is defined as shown in equation (2).

Here, * represents convolution.

Here, we define the analytic signal (complex number) Z(t) in equation (3) using the Hilbert transform from the real function f(t).

Since Z(t) is a complex number, we can express it as a vector.

Here

r(t) is called the amplitude (envelope) of f(t), and θ(t) is called the instantaneous phase. That is, any real function f(t) is

It can be expressed as follows.

In this way, by using the Hilbert transform, we can find the envelope of f(t) and from that, calculate the logarithmic decay rate (and even the decay ratio).

The envelope represents the instantaneous time evolution of a system's energy. Furthermore, f(t) can be observed not only in terms of amplitude, but also in terms of another parameter: instantaneous phase.

Differentiating θ(t)

It is also possible to view this as an instantaneous frequency.

The axis represents frequency, and the Y-axis is displayed on a logarithmic scale where 1 (V^2 rms) corresponds to 0 (dBV rms).

When a time function x(t) defined in the domain -T/2≦t≦T/2 can be expressed as in equation (2) using the complex Fourier coefficients of equation (1), this x(t) is a periodic function that repeats the same form outside the domain as it does inside the domain, due to the periodicity of the complex exponential function. Therefore, a periodic function x(t) that repeats the same waveform with period T can be expressed as a complex Fourier series of equation (1).

Equation (4) is called the Fourier transform of x(t), and equation (5) is called the inverse Fourier transform of X(f). This pair of equations is called the Fourier transform pair, or Fourier integral pair. Note that X(f), which is a function of frequency f, is also called the complex amplitude (or Fourier spectrum). As is clear from equations (4) and (5), the Fourier transform finds the corresponding frequency function from the time function, and the inverse Fourier transform finds the time function from the frequency function.

Furthermore, the following transformation pair is sometimes used, where the starting point of the time function x(t) is the origin of the time axis, x(t) is considered to be 0 in the interval t < 0, and the integration interval of the Fourier transform is a semi-infinite interval.

Now, if we express the complex frequency function X(f) defined by equation (4) as shown in equation (8), then the frequency f components that form the time function x(t) are given by equation (9) from the definition in equation (5).

In other words, the amplitude of the component with frequency f is 2|X(f)|, and its phase is the argument ∠X(f) of X(f). Thus, since the amplitude and phase of the component with frequency f of the time function x(t) can be obtained from the complex number X(f), X(f) is called the frequency spectrum of x(t).

Furthermore, the Fourier coefficients obtained from equation (1) have values only at frequency n/T and are always 0 at other frequencies, hence they are called line spectra. In contrast, X(f) in equation (4) is a continuous function of frequency, hence it is called a continuous spectrum.

Types of averaging processes

N-times of average addition

In the case of exponential averaging, you set a weighting value for the latest data, rather than setting the number of counts.

This number corresponds to the time constant of an analog RC filter.

When N=4

This function performs a peak hold on the power spectrum. It holds (memorizes) the maximum value for each frequency line from the peak hold start to the pause.

Furthermore, there is a Max Overall function that works in conjunction with this Peak Hold function. This can be thought of as an Overall Peak Hold function, and it is a function that memorizes the instantaneous power spectrum when the overall value was at its maximum.

This peak hold mode does not have a setting for the number of averaging operations. Therefore, it requires start and pause (stop) operations in averaging mode. Even if the number of averaging operations has already been set, it does not affect peak hold. Also, the number of CRT operations increases while peak hold is running. This indicates the number of FFT calculations.

Note
In peak hold mode, cross-channel calculations and other operations cannot be performed, therefore the average results of the following functions cannot be displayed.

Cross-spectrum, frequency response function, coherence function, coherent output power, impulse response

Subtractive averaging is a function that subtracts the power spectrum from the power spectrum after additive averaging.

[Calculation formula]

Example: Subtractive Average N=20

Si: Subtraction average result of the i-th iteration (displayed)

S: Subtracted power spectrum (power spectrum previously obtained from averaging)

Pi: i-th power spectrum

Starting with subtractive averaging
↓
Average 1st time S1 = S -(1/20) x P1
Average 2nd time S2 = S1 -(1/20) x P2
Average 3rd time S3 = S2 -(1/20) x P3
Average 20th time S20 = S19 -(1/20) x P20

We use a sine signal and sweep it from low to high frequencies, then perform an FFT operation according to the signal.

In sweep averaging, the master channel detects the maximum spectrum (1 line) for each acquisition, and calculations are performed only for that single line. Only that line is updated.

Please note that if the sweep speed of the external sweep signal is faster than the calculation processing time, undesirable spectral lines will appear (gap lines).

Areas and types that can be averaged

Averaging average, exponential average

Note
Time-domain averaging is performed using synchronous addition with a trigger function.

Synchronous summation has the advantage of separating the analysis signal, which is synchronized with the trigger included in the input, from random noise.

Time-domain averaging includes phase information, so it's necessary to specify the timing of the data acquisition. While time-domain averaging can be performed without using the trigger function (free-running), the phase will be random, rendering the averaging meaningless. Always use the trigger function.

Additive average, exponential average, peak hold, subtractive average, sweep average

Average

A Bode plot is a frequency response representation of a frequency response function H(f) that includes both the gain characteristic and the phase characteristic as a pair. The vertical axis of the gain is expressed in decibels (dB) of 20log10 (H(f)), and the phase is expressed in degrees or radians.

Every structure (machinery, buildings, automobiles, bicycles, home appliances, etc.) has its own natural frequency. Therefore, it is necessary to understand how structures vibrate at their natural frequencies and other frequencies. Modal analysis is software that simulates the state of various structures when subjected to vibrations of various frequencies. Currently, modal analysis can be easily performed on a PC using a combination of an FFT analyzer, exciter, and vibration pickup to obtain the transmission characteristics on each structure. This allows for the identification of structural weaknesses and the effective implementation of countermeasures such as vibration isolation and soundproofing.

Real-time analysis refers to an analysis state where FFT calculations are performed continuously on sampled data without any gaps between windows.

In typical FFT analysis, after sampling the signal's analysis data length (1024 or 2048 points), the FFT calculation is performed on that data. However, the next data is acquired during this process, and the next calculation is performed immediately after the previous one finishes. If the FFT calculation time (including display time) is shorter than the sampling time, real-time analysis can be performed. Conversely, if the calculation time is longer than the sampling time, more than one frame's worth of new signal will be sampled during the calculation, resulting in signal loss. On the other hand, if the sampling time is longer than the calculation time, a portion of the window can be overlapped with the previous window (overlap processing).

The envelope of the logarithmic power spectrum can be obtained by performing an inverse Fourier transform on the short-kefrency portion of the cepstrum. (Lifted envelope) This envelope is system-specific and does not depend on the spectrum of the input signal.

Applications include extracting fundamental frequencies and spectral envelopes from audio waves, bio-waves, and other sources.