Following on from our previous article on "Decibels," we'll address another frequently asked question from FFT analyzer users.
What are the RMS value and power of a signal? What is the difference between them? How are these values expressed in decibels?
The effective value of a time signal, also called RMS (Root Mean Square), is obtained by squaring the signal, taking the average, and then taking the square root. In other words, if the effective value of a time signal x(t) with period T is x rms, it is defined by equation (1).
.................................（1）
The RMS value is often used to represent the magnitude (strength) of an AC time signal. For example, voltage values such as 200 V or 100 V for an AC power supply are RMS values. It is also used in frequency analyzers of AC time signals, such as spectrum analyzers (for high frequencies) and FFT analyzers (for low frequencies), to represent the magnitude (strength) of each frequency component.
First of all, the effective value is a quantity related to electricity; please refer to [Reference Material 1] for details.
The reason why the RMS value in equation (1) is often used is, firstly, that it can be applied to a variety of complex signals, including random signals, and secondly, that it is a quantity that can be given a physically clear meaning in relation to the power of the signal.
Next, let's consider the power of the signal. In equation (1), the value inside the square root is called the mean square, and we define this as the power of the time signal.
.................................（2）
The physical unit of power is given by equation (2) x (t If ) is a voltage signal, then V 2 If a certain physical quantity is EU, then EU 2 This is how it works. Here, "power" is not necessarily directly related to power in the physical sense (energy per unit time), but rather refers to a quantity with dimensions equal to the square of the signal amount. Of course, if the load resistance is known, it can also be correlated with its power consumption.
The power spectrum in an FFT analyzer is the result of calculating the power values by decomposing the power of the signal (total power) in equation (2) into frequency units using FFT calculation.
Next, let's look at the decibel representation of these values.
As noted in [Reference Material 2] "What is a Decibel?", a decibel is defined as 10 times the logarithm of the power ratio. Now, if we define the power spectrum at a certain frequency f as P (f) and the reference value as 1 V 2, then:
| Logarithmic notation of power spectrum | 10log(P(f)) |
| Linear representation of the power spectrum (effective value) |  |
This is how it is expressed.
If we consider it as a logarithmic transformation of the effective value;
This is equivalent to a logarithmic transformation of power.
Specific numerical examples
x(t) = 5sin(2 πft) in the case of
Power value = 12.5 (V 2)
Logarithmic notation = 11.0 dBV
Linear value = 3.54(V)
[Note 1] The numerical examples above represent values at frequency f (single sine wave).
[Note 2] The linear value above is expressed as an RMS value, but in peak value notation it would be expressed as 5 V.
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The following link will take you to our company's website.
Fundamentals of Digital Measurement - Part 3: "Time Waveforms and RMS Values"
Technical document: "What is dB (Decibel)?"
(Excerpt from the email newsletter issued on September 22, 2011)
