Glossary of Basic Terms Related to FFT Analysis

Isolation refers to the process of insulating electrical signals, which is necessary when the common of the signal source differs from the ground potential. If the difference between the signal common and the ground potential (common-mode voltage) is small, measurements can be made using differential inputs; however, when the potential difference exceeds 100 V, it results in excessive input, making measurement impossible. In such cases, isolation amplifiers may be used, and for FFT analyzers, the analog input section up to the amplifier, anti-aliasing filter, and AD converter is isolated from the chassis ground using phot couplers for each channel. This isolation is advantageous for eliminating ground loops or avoiding connections to the common of the signal source.

The response of the system when a unit impulse δ (t ) is added to a linear system is called the impulse response h (t). The impulse response represents the characteristics of the system in the time domain, while the representation in the frequency domain is referred to as the frequency response function.

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

Our company's FFT analyzer calculates the impulse response by performing an inverse Fourier transform on the frequency response function.

(1) formula'sIn this context, it is specifically referred to as the (complex) Fourier spectrum.Since it is a complex function, it can be expressed in terms of amplitude and phase from its real and imaginary parts.

The Wigner distribution was proposed by E. Wigner in the field of quantum mechanics. It has properties similar to an extended power spectrum for nonstationary signals. With conventional FFT, time resolution and frequency resolution are complementary (increasing frequency resolution requires a longer sampling time), making it difficult to obtain instantaneous spectra of nonstationary signals with good resolution. In contrast, the Wigner distribution is not directly subject to this complementary constraint between time and frequency resolution, allowing good time-frequency resolution of the power spectrum on the frequency-time plane. However, it was not practical because it requires far more computation points than FFT. Using the OS-0263 Time-Frequency Analysis Software option for the OS-2000 series time-series data analysis tool (software), you can analyze the Wigner distribution.​

FFT processing is performed on a sampled numerical data series over a certain interval (for example, 1024 points or 2048 points), and this process of cutting out a part of the waveform is referred to as windowing (time windowing). The Fourier transform is defined by processing infinite-length data. This remains true for the Discrete Fourier Transform (DFT), where the FFT analyzer assumes that when the waveform is cut using a window, the waveform in that interval is repeated infinitely for signal analysis. At this time, if the analysis data length (the length of the window) is an integer multiple of the period of each frequency, the waveform assumed by the FFT analyzer will match the actual input waveform, resulting in a single line spectrum. However, if the analysis data length does not match an integer multiple of the period (which does not apply to frequency resolution, where the start and end do not connect), the processing will result in a distorted waveform, and the spectrum will spread out without concentrating power, leading to the occurrence of side lobes. This power leakage is referred to as leakage error. To prevent this leakage error, window processing is employed. By applying a bell-shaped function that makes both ends of the frame zero, the start and end can connect, reducing errors. Such a function is called a window function, and the process of synchronizing the analysis signal using the window function is referred to as window processing. As a result, the shape of the spectrum approaches that of a line spectrum.

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

According to the sampling theorem, it is necessary to sample at a rate greater than twice the highest frequency component of the signal. The frequency that is half of the sampling frequency is called the Nyquist frequency, and if components above the Nyquist frequency are included in the signal, aliasing occurs.

For impulse-like finite energy due to methods such as impact methods, this is normalized and displayed using energy. This is determined by further incorporating the time duration (window length, T = 1/Δf) into the power spectral density.

Real-time analysis can perform FFT analysis by overlapping the window when below the frequency.

For example, we perform FFT processing on data points of 1024, and at this time, we execute FFT analysis by overlapping the newly sampled data with the previous data. A larger overlap means that we can measure changes in signal time more precisely.

The figure formed by synthesizing two signals on orthogonal x and y axes is called an orbit or Lissajous figure, and it exhibits visual characteristics based on the combination of the amplitudes, frequency ratios, and phase differences of the two signals. Additionally, when the frequency ratio is an integer, the trajectory of the drawn figure returns to its original position at a constant period.

While the power spectrum divides the analysis frequency into fixed-width bands (fixed-width type) to represent the power for each band, frequency analyzers in the acoustic field often take the frequency axis on a logarithmic scale and perform frequency analysis by passing through band filters of constant relative width that are equally divided on the logarithmic scale. Bandwidths of 1 octave and 1/3 octave are common, and this type of analysis is referred to as octave analysis.

In general, on the frequency axis, when the lower cutoff frequency f ₁ is taken, and the upper cutoff frequency f ₂ is taken at twice the frequency, that is,Then, the spacing between f ₁ and f ₂ is an octave.

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