This time, we'll talk about decibels (dB), which are commonly used in FFT.
(1) decibel
In the fields of electronics, acoustics, and vibration, the unit dB (decibel) is used as a practical expression of energy and power, defined as 10 times the common logarithm of the ratio of the measured value to a reference value. In the fields of acoustics and vibration, the magnitude of a physical quantity expressed on a decibel scale is called a level.
Now, if we express the power level as a physical quantity using an equation, it will be as follows:
Lw=10Log(W/Wo) dB...(1)
Lw: Power level
Wo: Power set as a standard value (unit: W: watts)
W: Measured power value (unit: Watts)
Next, let's consider how to represent the voltage level. In this case, by applying the relationship that power W is proportional to the square of voltage V (W∝V^2) to equation (1), the voltage level L of voltage V can be expressed as shown in equation (2).
L=10Log(V^2/Vo^2)=20Log(V/Vo)... (2)
Vo: Reference value of voltage level (unit: V: volts)
V: Measured voltage value (unit: V: volts)
Similarly, the levels of sound pressure P (unit μPa: micropascals) and vibration acceleration A (unit m/s^2: meters per second squared) are expressed in equations (3) and (4).
Lp＝20Log（P／Po）dB ・・・ （ 3 ）
La＝20Log（A／Ao）dB ・・・ （ 4 ）
Here, Po is the reference sound pressure of 20 μPa, and Ao is the reference vibration acceleration, which is defined as 10⁻⁵ m/s² in JIS (10⁻⁶ m/s² in ISO).
According to JIS 1502-1990, sound pressure level is defined as "10 times the common logarithm of the value obtained by dividing the square of the effective value of the sound pressure by the square of the reference sound pressure (20 μPa). The unit is decibels, and the unit symbol is dB." Vibration level meters are also defined in JIS 1510-1995. This effective value is quite tricky. The effective value is used to represent AC signals, but if the time constant is different when calculating the effective value, the value will be different. Also, in the case of sound level meters, it is said that the value is converted to a level rather than calculated as an effective value. It is not possible to consider the signal processing of this sound level meter and the signal processing of FFT from the same perspective, but please understand that we will deliberately overlook the theoretical aspects here and explain it with the aim of understanding the concept.
(2) FFT and Decibels
In an FFT analyzer, the input signal is a voltage. This signal is sampled, and for example, from 2048 sample points, the power spectrum of the frequency components is obtained and displayed using FFT calculation. The voltage (amplitude) of each frequency component is displayed in dB using equation (2).
Now, the effective value is calculated from the mean square of the sampled values.
RMS = √{1/N * Σ(xi)^2} ... (5)
{ }: Indicates what is inside the square root.
Σ: Indicates the sum of N elements.
xi: i-th sample value
This value is calculated using a formula derived from the time-domain waveform, but each frequency component of the FFT is also derived from the same sampled values. Therefore, we will arbitrarily consider the time required for sampling to be equivalent to the time constant used when determining the RMS value in sound level meters, etc.
The power spectrum is displayed as a decomposition of frequencies, and the power is represented by the amplitude of each frequency component. In the case of sine and cosine waves, the RMS value and the single-amplitude value have a √2 multiplier relationship, and can be converted with a simple calculation.
The amplitude of a sine wave = √2 × the RMS value of the sine wave ... (6)
In power spectra, it is sometimes more convenient to display the amplitude as a single unit. In this case, the reference value Vo can be switched to 1Vrms (effective value 1V), and 1Vo-p (single amplitude 1V) can be used as the reference for display. Since √2 times equals 3dB, this can be calculated mentally by adding 3dB to the effective value.
(supplement)
The RMS value obtained from the time waveform sample (time domain) using equation (5) is essentially the same as the overall RMS value obtained from each frequency component of the power spectrum. This should be understandable as it represents the same phenomenon in both the time domain and the frequency domain.
(3) Decibels of the sound level meter
Noise levels are measured using a sound level meter, with a baseline of 20 μPa and displayed in dB. The terms "sound pressure level" and "noise level" are commonly used when referring to sound level meters. Until recently, the frequency response corresponding to human hearing was called A-weighting. "Noise level" is calculated by frequency-weighting the raw sound captured by a microphone through this A-weighting, and then converting it to a level. "Sound pressure level" is determined without passing through this A-weighting. Therefore, "sound pressure level" is thought to represent the loudness of the sound source, while "noise level" represents the loudness of that sound as heard by a human ear. Currently, A-weighting no longer considers the relationship with auditory perception, and "sound pressure level" and "noise level" are now referred to as "A-weighted sound pressure level" (abbreviated as "sound level").
When leveling, there are two time constants (time weights): FAST (0.125s) and SLOW (1s). The time constant represents the dynamic characteristic (responsiveness). For example, if the sound suddenly becomes louder, it is defined as the time it takes to reach 63% of that increased sound volume. When considering the leveling process of a sound level meter, if you look at the FFT from the sound level meter, the FFT is analyzing the instantaneous sound. To match the FFT display to the sound level meter display, considerations such as exponential averaging or averaging measurements are necessary.
(4) Decibels of the vibration level meter
Since vibration level meters use the same signal processing as sound level meters, it's easier to understand them by comparing them. The terms "vibration acceleration level" and "vibration level" are also used for vibration level meters.
Just as sound pressure level is considered to represent the sound source, vibration acceleration level represents the magnitude of acceleration at the vibration measurement point. Similarly, just as "vibration level" is considered to be the noise level after passing through an A-weighting filter, the magnitude of vibration is represented through the frequency weights (sensory correction values) of the vertical and horizontal vibrations that the body perceives.
The time constant for leveling is set to 0.63s.
Vibration level meters simultaneously measure vibrations in three axes: front-to-back (X-axis), left-to-right (Y-axis), and up-and-down (Z-axis), taking into account differences in vibration perception.
Furthermore, when directly inputting the signals from the AC output terminals of a sound level meter or vibration level meter into an FFT analyzer, or when directly inputting signals from Accelerometer or microphone into an FFT analyzer for power spectrum analysis, the approach described in section (2) applies. Recently, models with real-time octave analysis capabilities that incorporate the time constant of noise and vibration have also become available.
(5) Unit calibration between dB for FFT and dB for noise and vibration
When inputting the AC output signal from a sound level meter into an FFT analyzer for power spectrum analysis, the FFT unit calibration is performed so that the signal can be directly read as dB from the sound level meter.
By calibrating, the dB values of the FFT can be adjusted to match the dB values of the sound pressure level.
Sometimes, subscripts such as dBspl (spl: sound pressure level) are used to indicate sound pressure levels.
If you know the CAL (calibration signal output) of the sound level meter in volts and the corresponding dBspl, you can use equation (3) to calculate the Pa per volt. Let this be the coefficient K, and express it as P = kV, then you can perform the unit calibration calculation as shown in equation (6).
Lp=20Log(kV/Po) (dBspl)...(6)
Po: 20 μPa
V: Measured voltage level
K: Coefficient for converting voltage to Pa.
The same conversion can be made using equation (7) when using the AC signal from a vibration level meter.
Lv=20Log(K'V/Ao) (dB Z-axis)...(7)
(6) Regarding the overall value of the FFT and the sound level meter reading
The FFT analyzer has a function that automatically calculates the conversion factor K when the calibration signal from the sound level meter is input. The display value of the sound level meter corresponds to the overall value when measured in the 20kHz or 10kHz range on the FFT analyzer. Therefore, the calibration procedure in section (5) is performed using the overall value.
To reiterate, there are differences in signal processing between a sound level meter and an FFT.
In calibration signals, the signal is constant, so the reading on the sound level meter and the overall value of the FFT will match. However, if the sound being measured changes significantly, there will be a difference between the sound level meter value and the overall value of the FFT. This is because the FFT analyzes the sound instantaneously, and therefore, by averaging the results, it will match the equivalent sound level of the sound level meter. Please understand this as a difference in signal processing between the sound level meter and the FFT.
For a detailed explanation of sound level meters, please refer to the technical report "What is a Sound Level Meter?".
(Excerpt from the email newsletter issued on December 24, 2004)
