Frequency Analysis from the Basics (26) - "Transfer Function and Signal Output"

To measure the transfer system characteristics of a system using an FFT analyzer, it is necessary to excite the system (i.e., input a specific signal to the transfer system). FFT analyzers with 2 or more channels usually have a signal output function to easily measure the transfer function. This time, we will discuss methods for exciting the transfer system, the signals to input to the system, and precautions to take when using them.

When broadly classified by the combination of the output signal and the method for measuring the transfer function, they can be seen in Table 1 below. (1) involves sweeping a single-frequency sinusoidal signal to determine the transfer function in a certain bandwidth. In contrast, (2) is a special signal containing frequency components over a wide bandwidth, and it effectively utilizes the FFT's ability to simultaneously analyze that wide frequency bandwidth.

Table 1 Combinations of methods for measuring output signals and transfer functions

The following explains the properties of these signals and the corresponding analysis methods.

The sinusoidal sweep method is a method for determining the transfer function over a specified frequency range by exciting the system at a single frequency, finding the transfer function at only one point, and then repeating the measurement while changing that single frequency (sweeping) (Figure 1). This method is intuitive and easy to understand because the excitation time waveform is a sine wave, and it is a classic method that has been used for a long time.

Furthermore, since this method determines the transfer function of only one frequency point in a single measurement, it does not require a multi-point FFT, and a single-point DFT method called FRA (Frequency Response Analysis) is often used. An analyzer that uses this calculation method is called a servo analyzer. In either method, it is possible to remove noise and harmonic components from the input signal to the system and the output signal components from the system for analysis, effectively providing the function of a tracking filter.

This method has the following characteristics:

1. Because it can excite with a signal of much higher energy compared to other waveforms, it can be used for vibration testing of large structures such as ships, bridges, and large buildings.
2. Since only one frequency is measured at a time, the analyzer's auto-ranging function can be used, enabling measurements with a high dynamic range.
3. The amplitude can be freely controlled, allowing you to investigate the nonlinearity of the system or control the system's inputs and outputs to achieve a specific amplitude characteristic.
4. Since this method determines the frequency of a single point in a single measurement, the sweep frequency bandwidth, speed, and frequency resolution can be freely controlled. Especially for wide frequency bandwidths, logarithmic sweep (logarithmic resolution) functionality is also possible. (Note: This function is usually performed using FRA, not FFT.)
5. The biggest drawback is that the measurement takes a long time.

Table 1 (2) uses a special broadband signal linked to the frequency band analyzed by the FFT, allowing simultaneous measurement of the entire bandwidth using FFT technology (Figure 2). The features of this method are as follows:

1. Because it simultaneously measures a wide bandwidth, it can perform measurements at high speed.
2. It is also possible to simultaneously determine the coherence function, which allows us to check the reliability of the transfer function.
3. Due to the principles of FFT, it can only measure linear resolution.

The signals listed in Table 2 are explained below.

1. Random waves (irregular waves)

This signal is completely non-periodic, with irregular amplitude and phase. Its amplitude characteristics are approximately normal (Gaussian), and its frequency characteristics become flat after sufficient averaging. It is commonly known as white noise. (Note: Gaussian noise and white noise are completely different in nature.) The greatest advantage of using this signal is that by applying it to amplitude-dependent or nonlinear systems and performing sufficient averaging, a linearly approximated transfer function can be estimated.
Furthermore, since this signal is a continuous signal without periodicity, a Hanning window is usually used in FFT analyzers, but the biggest drawback is that leakage errors cannot be avoided. In particular, these errors become large near the resonance point and can be confirmed by the coherence function. A way to compensate for this drawback is to use a burst random wave (short-duration irregular wave), which generates the signal only in a portion of the FFT analyzer's time window and converges the response signal within that time window.
This burst random wave is a highly practical signal with a wide range of applications, retaining the advantages of the original random wave while overcoming its drawbacks, and is frequently used in experimental modal analysis. However, it's important to note that its relatively low signal power can easily lead to a decrease in the signal-to-noise ratio (SNR).

2. Pseudorandom waves

This signal is a repeating signal obtained by performing an inverse FFT on a Fourier spectrum with constant amplitude and randomized phase, within a time window. Because this signal is a periodic signal repeating within a time window, it has a discrete spectrum with power only at the frequencies (i.e., bins) that match the frequency resolution of the FFT, synchronized with the FFT. Therefore, for example, a time window of 2048 points results in a spectrum with 800 lines; this can be considered a time waveform obtained by adding 800 sine waves with constant amplitude and randomized phases. In other words, multiSine wave (multiple sine wave) It is also said that...
The advantage of this signal is that it is a signal synchronized with a time window. Rectangular window It can be used and does not produce leakage errors.
In linear systems such as electrical circuits, the transfer function can be determined quickly with little to no averaging. However, in nonlinear systems, the averaging effect is not obtained, making it practically useless.
Furthermore, inverse FFT can be used to generate pseudo-random waves with arbitrary amplitude characteristics.
Therefore, it is used in random vibration controllers and other applications.

3. Swept sine wave (fast sweep wave, chirp wave)

This signal is typically a continuous sine wave across the frequency range of an FFT analyzer, from the lower limit to the upper limit.
This waveform is swept and then repeated within that time window.
The properties of this signal are almost the same as the pseudo-random wave described above. Because a square wave window can be used, there is no leakage error, and the transfer function can be determined quickly in linear systems, but it cannot be used in nonlinear systems.
The difference between pseudorandom waves and swept sine waves lies in the time localization of frequencies; in systems with relatively low attenuation, pseudorandom waves are more suitable.

4. Impulse wave

This signal is one in which the signal power is localized to a single point on the time waveform. Like pseudo-random waves and swept sine waves, it is a waveform that is repeated in the time window of an FFT analyzer. In vibration testing, it is rarely used with typical vibrators, but is used as a simple test with an impulse hammer.
Its basic properties are the same as the two signals mentioned above, but its crest factor is relatively large, making it more susceptible to disturbances, which is a drawback.

This is a summary of the signals used for transfer function measurement that we have explained so far.

Table 2 Summary of Comparison of the Properties of Various Signals

Generally, in vibration testing, the excitation method is determined by the measurement method and the shape of the signal, and the use of an exciter is required.
There are three types of excitation: sinusoidal excitation, random excitation, and impulse excitation using an impulse hammer.
It is classified as follows (Figure 8).

A comparison of these excitation methods is shown in Table 3, based on the signal properties described above.
Table 3 Comparison of three types of vibration excitation methods

Excitation source	Waveform and spectrum	Vibrator	time	Features
Sine wave sweep		Electromagnetic and hydraulic vibrators	long	Large excitation force Steady-state response Vibration is possible at a constant G.
Random wave (Burst Random)		Electromagnetic and hydraulic vibrators	Medium	Close to actual working waveform Optimal transfer function measurement is possible even in nonlinear systems. Steady-state response
impact (impact)		Impulse Hammer	short	Easy Unable to control the excitation force Free response

Finally, here's a summary.

1. To estimate the transfer function with an FFT analyzer, an optimal signal is required, which can be broadly classified into sinusoidal sweeps and broadband signals optimized for FFT processing.
2. The sinusoidal sweep method measures only one frequency point per measurement, allowing for the determination of a transfer function with very high accuracy and a wide dynamic range. However, its drawback is that it is time-consuming to perform.
3. Random waves can linearly approximate highly nonlinear systems through sufficient averaging, but they have the drawback of introducing leakage errors.
4. The drawbacks of the random waves mentioned above can be overcome by using burst random waves.
5. Pseudorandom waves and swept sine waves are signals with similar properties and are synchronized with the FFT window, so there is no leakage error, and the transfer function can be determined quickly for linear systems.
6. Currently, the main excitation methods used in vibration testing are sinusoidal sweep testing, random testing, and hammering testing.

【keyword】
Transfer function, FRA, FFT, sinusoidal sweep, DFT, servo analyzer, tracking filter, auto-ranging function, dynamic range, coherence function, log sweep, linear resolution, random wave, irregular wave, normal distribution, Gaussian distribution, white noise, linear approximation, Hanning window, leakage error, burst random wave, short-time irregular wave, experimental modal analysis, S/N ratio, pseudo-random wave, multi-sine wave, multiple sine wave, square window, swept sine wave, fast-swept wave, chirp wave, impulse wave, crest factor, crest factor, impulse hammer

【reference】

1. "Introduction to Modal Analysis," by Akio Nagamatsu, Corona Publishing Co., Ltd. (1994)
2. "Handbook of Mechanical Dynamics," edited by Naruhiko Kaneko and Masaaki Okuma, Asakura Shoten (2015).
   <12. Vibration Testing Methods (pp. 383-400)>

(Excerpt from the email newsletter issued on March 17, 2016)