FFT解析基礎用語集「サ行」
Mean (MEAN)
(2) Effective value (RMS)
Standard Deviation (S.D.)
The second moment around the mean is called variance, and the square root of variance is referred to as standard deviation. The effective value of a signal, excluding the DC component, is the same as the standard deviation. These values are calculated using the following formula.
The relationships of (1), (2), and (3) are as follows.
(4) Skewness
It is the third moment around the mean normalized by σ 3, used as an indicator of the asymmetry around the mean. It is calculated by the following formula.
(5) Kurtosis
The fourth moment around the mean normalized by σ 4 is an indicator of the sharpness of the waveform. It is calculated using the following formula. Since the value of kurtosis for a normal distribution (Gaussian distribution) in time signals is 3, it is sometimes referred to as kurtosis by subtracting 3 from the value obtained from the following formula.
It is the square root of the mean of the squares of the signal. In English, it is called root mean square (rms).
The formula is
For a sine wave, the effective value is 1/√2 of the peak value. The data for each line of the power spectrum corresponds to the square of the effective value of the signal in that band, which is the mean square value.
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Figure 1
Considering the energy (heat) generated when the voltage is 100 V and the resistance is 100 Ω, the current flowing through the resistor is determined by Ohm's law.
Thus, I = E/R = 100 / 100 = 1 (A), and the power W is W = E × I = 100 × 1 = 100 (W). Since calculating current each time is cumbersome, it is generally stated that I = E/R.
is commonly used. We found that applying 100 V to a 100 Ω electric heater generates 100 W of heat. Household power supplies have an RMS voltage of 100 V, but rather than direct current, they provide 50 Hz alternating current (a sine wave) in the Kanto region. How many watts of heat will be generated when a 100 Ω electric heater is connected to this household power supply?
The answer is 100 W. In other words, “the RMS value is the DC voltage that does the same amount of work.” So, how many watts would the same heater generate if used with AC 200 V? <In reality, doing this would probably break the heater.>
AC 200 V has the same power as DC 200 V, so if we consider it as DC 200 V, 200 2 /100 = 400 W, not 200 W. This is because “when the voltage doubles, the current also doubles, and the power becomes four times as great, or 2 2.” So, what is the RMS value in the following 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 calculating with a resistance of 100 Ω is cumbersome, we will use 1 Ω instead. The power generated at 1 V and 2 V can be calculated using the following formula, but omitting the resistance of 1 Ω does not change the results.
The square of the voltage corresponds to the power (electricity) generated in a 1 Ω resistor. When you see the square of the voltage or current, 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 also represents power.)
The average power is 2.5 W, as the interval of 1 W is 1/2 and the interval of 4 W is also 1/2. In other words, this waveform has the same power as a DC voltage that generates 2.5 W of power across a 1 Ω resistor.
It can be seen that the effective value is 1.58 V.
The effective value is the square root of the average power. In other words, the effective value is the average of the squares taken square root, which is referred to in English as 'root mean square', and when emphasizing that it is specifically the effective value, it is abbreviated as 'rms'.
The effective value of a sine wave is 1/√2 = 0.707 of the peak value, so the waveform of a household 100 V power supply has a peak value of 141 V, which is √2 times the effective value, as shown in Figure 3 below. (By the way, the average value is 2/π = 0.637.)
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Figure 3
Once again, let's connect a 100 Ω electric heater to this power source. Then, the maximum power is;
Therefore, it will be 200 W. It is clear that the minimum power is 0 W, so it is easy to imagine that the generated power fluctuates between 0 W and 200 W. This situation is illustrated in Figure 4 below. It is evident 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 input and output in electrical systems and structural vibration transmission systems, expressed as the ratio of the Fourier spectrum of the input A(f) to the Fourier spectrum of the output B(f).
That is, the frequency response function H(ƒ) is
In this device, we multiply the complex conjugate A*(ƒ) with the numerator and denominator of the right side of the above equation, and we perform the calculations as shown in the next expression.
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). From this, the frequency response function H(ƒ) can be obtained by dividing the cross spectrum of the input and output by the power spectrum of the input.
The frequency response function can also be estimated using the following calculation method.
When there is a lot of external noise in the output signal b(t), it is possible to minimize random errors through averaging.
In the case of nonlinear systems, it is possible to linearize them using random signals through averaging (least squares approximation).
When there is a lot of external noise in the input signal a(t), it is possible to minimize random errors through averaging.
If leakage errors are considered at resonance points, it is possible to reduce bias errors.
Now, if we denote the true transfer function as Ht(f), in cases where there is a lot of noise in both the input and output,
This will be the relationship. (However, we assume that the system is a linear system.) Also, with respect to the phase, it will be equal to the phase of the cross-spectrum.
The relationship with coherence function is
This means that γ 2 is the ratio of H2 to H1. If the ratio of the input-output power spectrum (transfer characteristics) is | Ha(ƒ) | 2 then
Therefore,
or taking the logarithm
In other words, the average of the logarithmic values of the gains H1 and H2 is equal to the logarithmic value of the frequency response characteristic Ha.
The frequency response function is represented by gain characteristics and phase characteristics. The gain characteristic represents how the amplitude changes as the signal passes through the system, with the X-axis showing frequency and the Y-axis displaying 20log 10 H(ƒ) in decibels (the amplitude ratio of output to input). Additionally, the phase characteristic represents the lead and lag of the phase between the input signal and the output signal, with the X-axis showing frequency and the Y-axis displayed in degrees or radians.
Differentiation operations in the frequency domain are performed by multiplying (ω)n in the power spectrum and (jω)n in the frequency response function.
(j is the imaginary unit, ω=2πf)
(Note 1) In the power spectrum, the first derivative is proportional to ω squared and the second derivative is proportional to ω to the fourth power, because they are power values.
(Note 2) The frequency response function involves complex number calculations, so the imaginary unit j is also multiplied.
The integration operation in the frequency domain 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 proportional to the square of omega (ω^2) and the double integral is proportional to the fourth power of omega (ω^4) because they represent power values.
(Note 2) The frequency response function involves complex number operations, so the imaginary unit j is also involved in division.
The autocorrelation function is defined as a function of the shift amount τ, using the waveform x(t) and the waveform x(t + τ) shifted by τ, as shown in the following equation.
The autocorrelation function is effective for examining the periodicity of waveforms. The autocorrelation function reaches its maximum value when τ=0, which is when it takes its own product, and if the waveform is periodic, the autocorrelation function will also show peaks at the same period. Additionally, in irregular signals, if the fluctuations are slow, it will have high values at larger τ, and when the fluctuations are fine, it will show high values at smaller τ, indicating that τ serves as a temporal reference for the variations.
We are calculating the autocorrelation function using the inverse Fourier transform of the power spectrum.
The cross-correlation function is defined as a function of the shift amount τ when one of the waveforms of the two signals is delayed by τ, as expressed in the following equation.
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 zero regardless of the time lag (τ). It is used to estimate time delays in systems corresponding to input and output signals, detect the presence of signals buried in external noise, and determine signal propagation paths.
It is obtained by the inverse Fourier transform of the cross spectrum.