技術レポート FFTアナライザーについて 1

The Fourier transform discovered by the French mathematician Fourier is theoretically based on Fourier series. The Fourier series is a theory that states that any complex waveform, if it has periodicity with a repeating shape, can be expressed as a series of multiple simple sine (Sin) and cosine (Cos) waves. The mathematical expression of this theory is called the Fourier series. Furthermore, the Fourier transform is an extension and development of this series over the range from -∞ to +∞. It is unclear how far one must observe a signal to determine its periodicity. Moreover, measuring over infinite time becomes an overwhelming task. Therefore, it is common to take a suitable segment of the observed waveform and assume that this segment is a signal that repeats infinitely, and then perform the Fourier transform on this waveform. Initially, the calculation of the Fourier transform required an enormous number of multiplications, but a method to reduce the number of calculations by taking the number of data points as a power of 2 was proposed by J.W. Cooley and J.W. Tukey. If the number of data points is 1024, the number of multiplications, which would be 1024 × 1024 = 1,048,576, is reduced to 10,240. This method is called the Fast Fourier Transform (FFT), and it has come to be commonly referred to by its initials.

The calculation of FFT specifically refers to obtaining the coefficients of the Fourier series (Fourier coefficients). An FFT analyzer can be described as a measuring instrument that digitally (discretely) samples the input signal waveform, stores it as data, and uses FFT to quickly compute the Fourier coefficients and display the results. Additionally, since the meaning of FFT involves decomposing signals into simple frequencies, it is also referred to as a frequency analyzer or a spectrum analyzer, which represents the magnitude of frequency components (spectrum). For example, when analyzing the sound of "A" with an FFT analyzer, a spectral waveform is displayed with frequency f on the X-axis and amplitude r on the Y-axis, as shown in Figure 1-1. This spectral waveform indicates that the sound of "A" consists of waves at frequencies f1, f2, f3, etc., with corresponding amplitudes r1, r2, r3, etc. Conversely, it also represents that synthesizing waves at frequencies f1, f2, f3, etc., with amplitudes r1, r2, r3, etc., will produce the sound of "A". Figure 1-2 shows the actual measured time waveform of "A" and its spectrum (bottom: time axis waveform, top: spectral waveform). The frequencies appearing as peaks on the left side of the spectral waveform correspond to f1, f2, f3, etc. Next, let us look at a more specific example.

Let's take a look at the vibration waveforms generated by actual machinery. An acceleration detector is installed on the bearing as shown in Figure 1-3, and the vibration waveforms obtained there are observed. Here too, similar complex time waveforms can be observed as in the previous section.

So, what are the benefits of performing frequency analysis (viewing in the frequency domain) using an FFT analyzer? The complex waveform seen in Figure 1-3 can be viewed as a waveform resulting from the combined vibrations of each component that makes up the machine (as shown in Figure 1-4).

Figure 1-5 shows a conceptual diagram illustrating the relationship between the complex vibration waveforms of this rotating machine, analyzed with an FFT analyzer, and the various parts that are the sources of the vibrations.

The frequency position of vibrations generated from various parts is determined by the structure of the machine. Traditionally, when performing maintenance management or abnormal diagnosis of equipment, the overall vibration level, or overall value, was measured using a vibration meter. However, since the overall value only indicates the magnitude of the vibration, it does not allow for identifying where the abnormality is located. Additionally, oscilloscopes, which are commonly used for waveform observation (time domain), can show the temporal changes of waveforms (time-axis waveforms), but it is difficult to determine what causes these temporal changes. Only through frequency analysis data using FFT can we examine how much change has occurred at which frequency level and consider which part of the machine is likely generating that frequency, thus enabling the estimation of the cause of the abnormality and its location. Particularly in the early stages of failure or in cases of minor abnormalities, there is often little change in the overall value or time-axis waveform, making detection difficult; however, by conducting frequency analysis (viewing in the frequency domain), it becomes possible to detect minor abnormalities. Recently, in addition to equipment management and abnormal diagnosis through vibration analysis, frequency analysis is being utilized in various fields, such as evaluating quietness and investigating noise causes and countermeasures in office machines and household appliances.

To understand the meaning of the data displayed by the FFT analyzer, it is necessary to grasp the concept of Fourier series, which is the fundamental idea behind FFT, along with its mathematical background. Here, we will discuss the representation of waveforms necessary for understanding Fourier series, as well as the characteristics of sine and cosine waves. Please read on while recalling what you learned in the past.

A waveform can be represented by three parameters: amplitude, frequency (or period), and phase (time difference).

Amplitude represents the size of a waveform. When thinking about sound, a loud sound has a large amplitude. Physical phenomena, such as vibrations detected by a vibration meter, sound measured by a sound level meter, force by a load cell, and pressure by a pressure gauge, are detected by various sensors. The signal is output as a voltage amplitude value proportional to the magnitude of the physical quantity. Measurements using FFT analyzers for time-domain waveform display, oscilloscopes, and pen recorders observe the passage of time of this voltage output.

Frequency refers to the number of waves that repeat in one second, with the unit being Hz. If we denote frequency as f and period as T, the following relationship exists.

(Equation 2-1)

When considering sound, waveforms with high frequencies correspond to high tones, while waveforms with low frequencies correspond to low tones.

The period of a wave is expressed in degrees (360 degrees) or in radians (2π radians). While degrees are used in explanations, radians are generally used in mathematical expressions and are denoted as 'rad'. Even for waveforms with the same frequency, the peak positions can be shifted at a given moment. This shift in peak position, expressed relative to a reference waveform, is called phase. When the peak is delayed relative to the reference waveform, it is expressed as negative, and when it is advanced, it is expressed as positive.