Technical Report: About FFT Analyzers 1

The Fourier transform, discovered by the French mathematician Fourier, is theoretically based on the Fourier series. The Fourier series theory states that any complex waveform, as long as it is a periodic wave repeating the same shape, can be represented by a series of simple sine waves and cosine waves. The mathematical expression of this theory is called a Fourier series. The Fourier transform is an extension and development of this series from -∞ to +∞. In reality, it is unclear how far we need to observe a signal to determine its periodicity. Measuring to infinite time is an incredibly daunting task. Therefore, generally, a suitable time segment is taken from the observed waveform, and this segmented waveform is assumed to be an infinitely repeating signal. The Fourier transform is then performed on this waveform. Initially, calculating the Fourier transform required a vast number of multiplication operations. However, J.W. Cooley and J.W. Tukey proposed a method to reduce the number of calculations by using 2 to the power of n data points. If the number of data points is 1024, the number of multiplication operations is reduced from 1024 × 1024 = 1,048,576 to 10,240. This method is called the Fast Fourier Transform, and it is commonly referred to as FFT, taking its initials.

Specifically, FFT calculation refers to finding the coefficients of a Fourier series (Fourier coefficients). An FFT analyzer is a measuring instrument that stores data by digitally (discretely) sampling the input signal waveform, uses FFT to quickly find the Fourier coefficients from this data, and displays the results. It is also called a frequency analyzer because FFT decomposes a signal into simple frequencies, or a spectrum analyzer because it represents the magnitude (spectrum) of the frequency components. For example, if you analyze the sound "A" with an FFT analyzer, a spectral waveform will be displayed as shown in Figure 1-1 below, with frequency f on the X axis and amplitude r on the Y axis. This spectral waveform means that the sound "A" is made up of waves with frequencies f1, f2, f3... and amplitudes r1, r2, r3,... Conversely, this means that the sound "a" is produced when waves with frequencies f1, f2, f3, etc., and amplitudes r1, r2, r3, etc., are combined. Figure 1-2 below shows the time waveform and spectrum of the "a" sound as measured (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. Now let's look at a more specific example.

Let's look at the vibration waveform generated by an actual machine. An acceleration detector is installed on the bearing as shown in Figure 1-3, and the vibration waveform obtained there is observed. Here too, a complex time waveform similar to that of "A" above can be observed.

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

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

The frequency range at which vibrations originate from each part of a machine is determined by its structure. Traditionally, when maintaining equipment or diagnosing abnormalities, vibration meters were used to measure the overall vibration level, i.e., the overall value. However, the overall value only indicates the magnitude of the vibration, making it impossible to pinpoint the location of the abnormality. Similarly, while oscilloscopes, commonly used for waveform observation (time domain), can show the temporal changes in the waveform (time-domain waveform), it is difficult to determine what is causing these temporal changes. Only by using frequency analysis data obtained through FFT can we examine how much the level of each frequency has changed and which part of the machine is thought to be generating that frequency, thereby estimating the cause of the abnormality and its location. In particular, in the early stages of a failure or in the case of minute abnormalities, there is little change in the overall value or time-domain waveform, making detection difficult. However, by performing frequency analysis (viewing in the frequency domain), even minute abnormalities can be detected. Recently, in addition to equipment management and anomaly diagnosis using vibration analysis, frequency analysis is being used in various fields, such as evaluating noise levels and analyzing noise sources and countermeasures for office equipment and home appliances.

To understand the meaning of the data displayed by an FFT analyzer, it is necessary to understand the concept of Fourier series, which is the fundamental idea behind FFT, and its mathematical background. Here, we will discuss the ways in which waveforms are represented, which are necessary for understanding Fourier series, as well as the characteristics of sine and cosine waves. Please read along 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. In the case of sound, a loud sound has a large amplitude. Physical phenomena are detected by various sensors, such as vibrations with a vibration meter, sound with a sound level meter, force with a load cell, and pressure with a pressure gauge. The signal is output as a voltage amplitude value proportional to the magnitude of the physical quantity. Time-domain waveform displays on FFT analyzers, and measurements using oscilloscopes and pen recorders are observations of the time course of this voltage output.

Frequency represents the number of times a wave repeats per second, and its unit is Hz. If the frequency is f and the period is T, the following relationship holds:

(Formula 2-1)

In terms of sound, high-frequency waveforms correspond to high-pitched sounds, and low-frequency waveforms correspond to low-pitched sounds.

One period of a wave is expressed as 360 degrees or 2π radians. While "degrees" are used in explanations, "radians" are generally used in mathematical expressions and are written as "rad". Even waveforms of the same frequency will have different peak positions at a given point in time (instantaneous). This shift in peak position relative to a reference waveform is called phase. When the peak lags behind the reference waveform, it is expressed as a negative value, and when it leads, it is expressed as a positive value.