Glossary of Basic Terms Related to FFT Analysis

Isolation refers to the process of isolating electrical signals, and it is necessary for measurements when the common potential of the signal source differs from the ground potential. When the difference between the signal common potential and the ground potential (common-mode voltage) is small, measurement is possible even with differential input, but when the potential difference is 100 V or more, it results in an overload and measurement is not possible. In this case, isolation amplifiers may be used, but in the case of FFT analyzers, the analog input section from the amplifier to the anti-aliasing filter and AD converter is isolated from the chassis ground for each channel using a photocoupler. As a result, the digital circuit and the analog circuit are isolated, which is advantageous when it is necessary to eliminate ground loops or connections to the common potential of the signal source.

The response h(t) of a linear system when a unit impulse δ(t) is applied is called the impulseless response. The impulse response expresses the characteristics of a system in the time domain, while the frequency response function expresses it in the frequency domain.

If the impulse response of a system is known, the output y(t) when x(t) is input to that system can be obtained by a convolution operation.

Our FFT analyzer obtains the impulse response by performing an inverse Fourier transform on the frequency response function.

(1)In this instrument, this is specifically called the (complex) Fourier spectrum.Since it is a complex function, it can be expressed as amplitude and phase, given its real and imaginary parts.

The Wigner distribution, proposed by E. Wigner in the field of quantum mechanics, possesses properties such as an extended power spectrum for non-stationary signals. In conventional FFT, time resolution and frequency resolution are complementary (increasing frequency resolution increases the sample time), making it difficult to obtain the instantaneous spectrum of a non-stationary signal with good resolution. In contrast, the Wigner distribution is not directly subject to the complementary constraints of time resolution and frequency resolution, thus enabling the acquisition of good time-frequency resolution for the power spectrum on the frequency-time plane. However, it was not practical due to the significantly larger number of calculation points compared to FFT.Oscope Time Series Data Analysis Tool (Software) Option OS-0263 Time-Frequency Analysis SoftwareBy using this, you can analyze the Wigner distribution.

FFT processing is performed on a certain interval (for example, 1024 or 2048 points) of a sampled numerical data sequence. This act of cutting out a portion of the waveform is called windowing (cutting the waveform with a time window), or applying a window. The Fourier transform is defined as processing infinitely long data. This remains true for the Discrete Fourier Transform (DFT), and in an FFT analyzer, signal analysis is performed under the assumption that when the waveform is cut with a window, the waveform in that interval is repeated infinitely. In this case, if the analysis data length (window length) is an integer multiple of the period of each frequency, the waveform assumed by the FFT analyzer matches the actual input waveform, and a single line spectrum is obtained. However, if the analysis data length does not match an integer multiple of the period (i.e., it does not meet the frequency resolution, and the start and end are not connected), a distorted waveform is processed, and the spectrum does not concentrate power, resulting in spread to the left and right (side ropes). This power leakage is called leakage error. Windowing processing prevents this leakage error. By multiplying the frame by a bell-shaped function such that both ends of the frame are zero, the start and end points connect, reducing errors. Such a function is called a window function, and the process of synchronizing the analysis signal using a window function is called windowing. As a result, the shape of the spectrum approaches that of a line spectrum.

The Hanning window is a typical example of a window used for analysis, but other windows are used depending on the signal being analyzed.

The sampling theorem states that it is necessary to sample a signal at a rate at least twice the rate of the highest frequency component. The frequency that is half the sampling frequency is called the Nyquist frequency, and if the signal contains components above the Nyquist frequency, aliasing (aliasing distortion) occurs.

For impulsive, finite energy signals such as those generated by striking techniques, this is normalized and displayed using energy. This is calculated by multiplying the power spectral density by the acquisition time (window length, T = 1/Δf).

If the real-time analysis frequency is below the specified frequency, you can overlap the windows and perform the FFT analysis.

For example, when performing an FFT analysis on data at intervals of 1024 points, the newly sampled data is overlaid with the previously sampled data during the FFT analysis. A larger overlap allows for more detailed measurement of changes in signal time.

The figure formed by combining two signals on orthogonal x and y axes is called an orbit or Lissajous curve, and its visual characteristics are determined by the combination of the amplitude, frequency ratio, and phase difference of the two signals. Furthermore, when the frequency ratio is an integer, the trajectory of the drawn figure returns to its original position with a constant period.

While power spectra divide the analysis frequency into constant-width segments (constant-width type) to represent the power of each band, frequency analyzers in the field of acoustics often use a logarithmic scale for the frequency axis and perform frequency analysis by passing the signal through a bandpass filter with a constant-ratio width that divides the logarithmic scale equally. Bandwidths of one octave and one-third of an octave are common, and this type of analysis is called octave analysis.

Generally, the lower limit of the cutoff frequency is on the frequency axis. For ​_f1, the upper limit cutoff frequency ​_f2 When we take twice the frequency, that isTherefore, the interval between ​_f1 and ​_f2 is an octave.

Acoustic intensity is the energy of sound passing through a unit cross-sectional area containing a point in the sound field within a unit time, and is expressed as the sound pressure p(t) at that point and the particle velocity in the r direction.It is a vector quantity defined as the time average of the product of .