FFT Analysis Basic Terminology Glossary (Starting with "S")
(1) Mean (MEAN)
(2) RMS value
(3) Standard deviation (S.D.)
The second moment around the mean is called the variance, and the square root of the variance is called the standard deviation. The RMS value and standard deviation of a signal excluding the DC component are the same. They are calculated using the following formula.
The relationship between (1), (2), and (3) is as follows:
(4) Skewness
This is the third moment around the mean value, normalized by σ³, and is used as an indicator of asymmetry around the mean value. It is calculated using the following formula:
(5) Kurtosis
It is an index representing the sharpness of a waveform, normalized by σ⁴ (the fourth moment around the mean). It is calculated using the following formula. Since the crustosis value for a time signal with a normal distribution (Gaussian distribution) is 3, sometimes the crustosis is calculated by subtracting 3 from the following formula.
This is the square root of the mean square of the signal. In English, it's called root mean square (rms).
The formula is
For a sine wave, the RMS value is 1/√2 of the peak value. The data for each line in the power spectrum is the square of the RMS value of the signal in that bandwidth, i.e., the mean squared value.
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Figure 1
When considering the energy (heat) generated when the voltage is 100 V and the resistance is 100 Ω, the current flowing through the resistor is given by Ohm's law;
Therefore, I = E/R = 100 / 100 = 1 (A), and the power W is W = E × I = 100 × 1 = 100 (W). Since calculating the current every time is troublesome, generally, from I = E/R:
This is a commonly used method. It has been found that applying 100V to a 100Ω electric heater generates 100W of heat. However, household power supplies have an effective value of 100V, but they are not DC; in the Kanto region, they are 50Hz AC (sine wave). If a 100Ω electric heater is connected to this household power supply, how many watts of heat will be generated?
The answer is 100 W. In other words, "the RMS value is the equivalent DC voltage that performs the same work." So, how many watts would the same electric heater generate if used with AC 200 V? (In reality, doing this would likely break the heater.)
AC 200 V has the same power as DC 200 V, so if we consider it as DC 200 V, the power becomes 2002/100 = 400 W, not 200 W. This is because "when the voltage doubles, the current also doubles, and the power becomes 4 times 22." So, what will the RMS value be in Figure 2?
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Figure 2
Simply averaging the absolute values gives 1.5 V, but let's consider the actual power generated. Since the calculation is complicated with a resistance of 100 Ω, we'll use 1 Ω. The power generated at 1 V and 2 V is given by the following formulas, but the result remains the same even if we omit the 1 Ω resistor.
The square of the voltage is the power (electrical power) generated across a 1 Ω resistor. Whenever you see the square of voltage or current, you should think of it as power. (In the case of 1 Ω, voltage and current are equal, so from equation 2, the square of the current is also power.)
The average power is 2.5 W, since the 1 W interval is halved and the 4 W interval is also halved. In other words, this waveform has the same power as a DC voltage that generates 2.5 W of power through a 1 Ω resistor.
Therefore, we can see that the effective value is 1.58 V.
The RMS value is the square root of the average power. In other words, the RMS value is the average of the square roots, which in English is called "root mean square." When you want to emphasize that it is the RMS value, you can use the abbreviation "rms."
The RMS value of a sine wave is 1/√2 of the peak value = 0.707. Therefore, the waveform of a household 100V power supply has a peak value of 141V, which is √2 times the RMS value, as shown in Figure 3 below. (Incidentally, the average value is 2/π = 0.637.)
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Figure 3
Let's connect a 100 Ω electric heater to this power supply again. Then the maximum power will be:
Therefore, it becomes 200 W. Since the minimum power is clearly 0 W, it is easy to imagine that the generated power will fluctuate between 0 W and 200 W. This is illustrated in Figure 4 below. It is clear that the average value at this time is 100 W.
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Figure 4
The frequency response function (transfer function) represents the relationship between the input and output of an electrical system or a vibration transmission system in a structure, and is expressed as the ratio of the Fourier spectrum A(ƒ) of the input to the Fourier spectrum B(ƒ) of the output.
In other words, the frequency response function H(ƒ) is
In this instrument, the numerator and denominator of the right-hand side of the above equation are multiplied by the complex conjugate A*(ƒ) and the calculation is performed as shown in the following equation.
The denominator, A(ƒ) × A*(ƒ), is the power spectrum of A(ƒ), and the numerator, B(ƒ) × A*(ƒ), is the cross spectrum of A(ƒ) and B(ƒ). Therefore, the frequency response function H(ƒ) can be obtained by dividing the input-output cross spectrum by the input power spectrum.
The frequency response function can also be estimated using the following calculation method.
When the output signal b(t) has a lot of external noise, averaging can minimize random errors.
In the case of nonlinear systems, linearization is possible by averaging using random signals (least squares approximation).
When the input signal a(t) has a lot of external noise, averaging can minimize random errors.
If leakage error is suspected at the resonance point, it is possible to reduce the bias error.
Now, if we let the true transfer function be Ht(ƒ), then when there is a lot of noise on both the input and output,
The relationship is as follows: (However, we assume the system is a linear system.) Also, regarding the phase, it is equal to the phase of the cross spectrum, just like with .
The relationship with the coherence function is
This is the result. γ 2 is the ratio of H2 to H1. If we denote the ratio of the input and output power spectra (transfer characteristics) as |Ha(ƒ)| 2, then
Therefore,
Alternatively, take the logarithm.
This means that the average of the logarithms of the gains of H1 and H2 is equal to the logarithm of the frequency response characteristic Ha.
The frequency response function is represented by gain characteristics and phase characteristics. Gain characteristics represent how the amplitude changes as a signal passes through the system, with the X axis representing frequency and the Y axis representing 20log 10 H(ƒ) in decibels (the ratio of the output amplitude to the input amplitude). Phase characteristics represent the phase lead or lag between the input and output signals, with the X axis representing frequency and the Y axis representing degrees or radians.
Differentiation along the frequency axis is performed by multiplying by (ω)n in the power spectrum and by (jω)n in the frequency response function.
(j is the imaginary unit, ω = 2πf)
(Note 1) In the power spectrum, the first derivative is ω² and the second derivative is ω⁴ because it represents the power value.
(Note 2) The frequency response function is calculated using complex numbers, so the imaginary unit j is also multiplied.
Integration along the frequency axis is performed by dividing by (ω)n in the power spectrum and by (jω)n in the frequency response function.
(j is the imaginary unit, ω = 2πf)
(Note 1) In the power spectrum, the single integral is ω² and the double integral is ω⁴ because they represent power values.
(Note 2) Because the frequency response function involves complex number calculations, the imaginary unit j is also divided.
The autocorrelation function is a function of the quantity τ, obtained by using a waveform x(t) and a waveform x(t+τ) shifted by τ, and is defined as follows:
The autocorrelation function is useful for determining the period of a waveform. The autocorrelation function reaches its maximum value when τ = 0, that is, when the product of itself is taken. If the waveform is periodic, the autocorrelation function will also show a peak at the same period. Furthermore, for irregular signals, if the fluctuation is slow, the value will be high when τ is large, and if the fluctuation is rapid, the value will be high when τ is small, so τ can be used as a temporal indicator of the fluctuation.
The autocorrelation function is obtained by the inverse Fourier transform of the power spectrum.
The cross-correlation function is a function of the amount of shift τ when one of the waveforms of two signals is delayed by τ, and is defined as follows:
The cross-correlation function is used to measure the similarity and time delay between two signals. If the two signals are completely different, the cross-correlation function approaches 0 regardless of τ. When two signals correspond to the input and output of a system, it is used to estimate the time delay of that system, detect the presence of signals buried in external noise, and determine the signal propagation path.
This is obtained by the inverse Fourier transform of the cross spectrum.