Glossary of Basic Terms for FFT Analysis: 'Ta', 'Na', 'Ha', 'Ma', 'Ra'

The damping factor refers to the attenuation ratio for each frequency. This allows us to obtain the attenuation ratio for the desired frequency, which influences the selection of materials along with the natural frequency.
The value is displayed in percentage, and a larger damping factor indicates faster attenuation at that frequency, while a smaller damping factor indicates slower attenuation at that frequency. This damping factor (ζ) can be determined for each resonant frequency using the FFT analyzer, based on the frequency width Δf at which the power is down by 3 dB and the resonant frequency f ₀, using the following formula (half-width method).

When measuring transfer functions in audio and mechanical systems, if the signal propagation time within the system is long, a temporal shift occurs between the input and output signals, making accurate measurement of the transfer function impossible. (This leads to a decrease in the coherence function.)

The delay function is a feature that compensates for temporal shifts between channels by delaying the start of sampling for the slave channel in relation to the start of sampling for the master channel.

Voltage and sound pressure are often expressed in dB (decibels), and dB is an expression that uses the common logarithm (log ₁₀).
As a review, one of the properties of log that you learned in high school mathematics is;

Also, the operation of logarithms is;

will become.

dB (decibel) is;

It is defined as. P ₀ is the reference level, with 1 V for voltage and 20μPa (pascals) for sound pressure. dB represents the square ratio, that is, the ratio of power. If P is the same as the reference level, then P ² /P ₀² = 1, so log ₁₀ 1 = 0, which means 0 dB. Now, if we consider 0 dB = 1 (P ₀ = 1), let's calculate what 10 times would be in dB;

Therefore, it will be 20 dB. Since it is cumbersome to square each time during calculations, we can use the previous mathematical formula to bring the square out front and use '20 log'.

Thus, there is no need to square, making it simpler. It may not be familiar in mathematical terms, but essentially, if the argument (the value before converting to dB = a, b) increases by 10 times, it becomes 10 ¹, which means a change of 20 dB; if it decreases to 1/10, it becomes 10^-1, resulting in a change of -20 dB.

The trigger function is the ability to start sampling based on an input signal or an external signal.

There are two types of triggers for starting the sampling of the input signal itself: an internal trigger, which begins sampling based on the point at which the signal reaches a set voltage, and an external trigger, which starts sampling based on an external pulse signal.

This feature allows for the efficient capture and analysis of the desired parts of a waveform. Additionally, when averaging time waveforms, the trigger function synchronizes the waveforms.

The Nyquist plot is a graph where the real part of the frequency response function (transfer function) is plotted on the horizontal axis and the imaginary part on the vertical axis, and it is mainly used to determine the stability of control systems.

The vertical axis represents gain, and the horizontal axis represents phase, displaying the frequency response function (transfer function).

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

For example, the vibration values change relatively depending on the size of the bearings in terms of peak values and effective values (larger bearings have higher effective vibration values, and the peak values in abnormal conditions are even larger). However, since the crest factor value seeks the ratio of peak value to effective value, the vibration values are not influenced by the size of the bearings, allowing for a more accurate judgment of the degree of abnormalities such as damage. A larger measured crest factor value indicates a greater degree of abnormality.

The power of a signal is divided into specific frequency bands, and the power for each band is expressed as a function of frequency, which is called the power spectrum. The unit is the square of the amplitude (V ² rms).

The FFT analyzer obtains frequency-domain waveforms from time-domain waveforms through Fourier transform. The Fourier transform pair of the time function x(t) is expressed by the following formula.

(Fourier Transform)

Inverse Fourier Transform

Please enter a half-width numeric value in the box below and press the original unit button. The converted values for each unit will be displayed below.

V _rms: Effective value when the power spectrum is displayed linearly
V _0-p: Zero peak value when the power spectrum is displayed linearly
dBV _r: Effective value when the power spectrum is displayed logarithmically
dBV _0-p: Zero peak value when the power spectrum is displayed logarithmically

Calculation is;

　V_0-p = √2×V _rms

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

is being carried out.

Note that calculated values may not perfectly match the true values due to the relationship with significant figures.

In a digital FFT analyzer, the bandwidth (Δf) varies depending on the analysis frequency range. For example, at a resolution of 1/800, in the 20 kHz range, the bandwidth corresponds to 20 kHz/800 = 25 Hz.

Here, when performing frequency analysis on wideband signals such as white noise, the power is obtained as an integral value for each bandwidth. Therefore, changing the analysis range alters this value, making comparisons impossible. Thus, we calculate the power spectrum per unit frequency (1 Hz) using the following method, which is referred to as power spectral density (abbreviated as PSD). However, this has no meaning 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. Additionally, when calculating this power spectral density, please measure using a Hanning window or a rectangular window as much as possible.

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

Here, * represents convolution.

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

Since Z(t) is a complex number, when represented 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...

can be expressed as.

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

The envelope represents the instantaneous time variation of energy in the system. Furthermore, f(t) can be observed not only in terms of amplitude but also from another parameter known as instantaneous phase.

Differentiate θ (t)

It is also possible to view it as instantaneous frequency.

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

A time function x(t) defined in the interval -T/2 ≦ t ≦ T/2 can be expressed using the complex Fourier coefficients as shown in equation (2). Due to the periodicity of the complex exponential functions, this x(t) becomes a periodic function that repeats the same shape outside this interval. Therefore, the periodic function x(t), which repeats the same waveform with a period T, can be represented by the complex Fourier series in equation (1).

Equations (4) and (5) are called the Fourier transform of x(t) and the inverse Fourier transform of X(f), respectively. This pair of equations is referred to as the Fourier transform pair or Fourier integral pair. Furthermore, the function X(f), which is a function of frequency f, is also called complex amplitude (or Fourier spectrum). As is evident from equations (4) and (5), the Fourier transform seeks the corresponding frequency function from a time function, while the inverse Fourier transform seeks the time function from a frequency function.

Additionally, using the starting point of the time function x(t) as the origin of the time axis, and considering that x(t) is zero for the interval t < 0, a pair of transformations is sometimes used where the integration interval for the Fourier transform is taken as a semi-infinite interval.

Now, when the complex function of frequency X(f) defined by equation (4) is expressed as in equation (8), the components of frequency f that form the time function x(t) result in equation (9) from the definition in equation (5).

That is, the amplitude of the component at frequency f is 2|X(f)|, and the phase is the argument ∠X(f) of X(f). Thus, since the amplitude and phase of the frequency component f of the time function x(t) can be derived from the complex number X(f), X(f) is referred to as the frequency spectrum of x(t).

Furthermore, the Fourier coefficients obtained from equation (1) have values only at the frequency n/T and are always zero at other frequencies, which is why they are called a line spectrum. In contrast, X(f) from equation (4) is a continuous function of frequency, and therefore it is referred to as a continuous spectrum.

Types of averaging processes

Average of N additions

In the case of exponential averages, instead of setting the number of occurrences, we set the weighting values for the latest data.

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

When N=4

We will perform peak hold of the power spectrum. We hold (remember) the maximum value for each frequency line from the peak hold start to the pause.

Additionally, there is a max overall function linked to this peak hold. This can be considered a peak hold function of the overall, which is the ability to store the instantaneous power spectrum when the overall is at its maximum value.

In this peak hold mode, there is no setting for the number of averaging cycles. Therefore, operations for starting and pausing (stopping) are required in averaging mode. The peak hold is not affected even if the number of averaging cycles has already been set. Additionally, during the execution of this peak hold, the number of CRT executions will also increase. This indicates the number of FFT operations.

Note
In peak hold mode, operations such as inter-channel calculations cannot be performed, so the average results of the following functions cannot be displayed.

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

Subtraction average is a function that subtracts the power spectrum from the power spectrum after the addition average.

Calculation formula

Example) Subtraction Average N = 20

S _i: i-th subtraction average result (display)

S: The power spectrum to be subtracted (previously obtained by averaging through addition)

P _i: i-th power spectrum

Subtraction average start
↓
Average 1st time S ₁ = S -(1/20) x P ₁
Average 2nd time S ₂ = S ₁-(1/20) x P ₂
Average 3rd time S ₃ = S ₂-(1/20) x P ₃
Average 20th time S ₂₀ = S ₁₉-(1/20) x P ₂₀

Using a sine signal, sweep from low frequency to high frequency and perform FFT calculations based on that signal.

In the sweep average, the maximum spectrum (one line) is detected on the master channel side for each capture, and calculations are performed only for that one line. Only that line is updated.

Please note that if the sweep speed of the external sweep signal is faster than the processing time, missing spectral lines (gaps) will occur.

Areas and types that can undergo averaging processes

Additive average, index average

Note
Time domain averaging performs synchronous addition using the trigger function.

Synchronous addition has the advantage of separating analysis signals synchronized with triggers included in the input from random noise.

Averaging in the time domain requires a timing trigger because it includes phase information. Although time domain averaging can be performed without using the trigger function (free-running), the phase will become random, making the averaging meaningless. Always use the trigger function.

Addition average, index average, peak hold, subtraction average, sweep average

cumulative average

The frequency response function H(f) is represented in a Bode plot, which consists of two characteristics: gain and phase. The vertical axis of gain is expressed in decibels (dB) as 20log ₁₀ (H(f)), while phase is represented in degrees or radians.

Every structure (machines, buildings, cars, 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 at different frequencies. Currently, by combining FFT analyzers with exciters and vibration pickups, modal analysis can be easily performed on a computer based on the transfer characteristics obtained from each structure. This allows for the identification of structural weaknesses and the effective implementation of measures such as vibration and noise reduction.

Real-time analysis refers to the state of analysis in which FFT operations are performed continuously on sampled data without any gaps between windows.

In a typical FFT analysis, data is sampled for a length corresponding to the analysis data (1024 points or 2048 points), and then FFT computation is performed on that data. During this time, the next data is collected, allowing for the next computation to begin immediately after the previous one finishes. Real-time analysis can be executed if the FFT computation time (including display time) is shorter than the sampling time. Conversely, if the computation time is longer than the sampling time, more than one frame of new signals may be sampled during the computation, leading to signal loss. On the other hand, if the sampling time is longer than the computation time, overlapping of windows (overlap processing) can be performed.

By performing an inverse Fourier transform on the short cepstral frequency part of the cepstrum, the envelope of the logarithmic power spectrum (lifted envelope) can be obtained. This envelope is specific to the system and does not depend on the spectrum of the input signal.

As an application, there is the extraction of fundamental frequencies and spectral envelopes from sound waves and biological waves.