Last time, we discussed the power spectrum (Part 1), which is the most basic and important function of an FFT analyzer as an electrical measuring instrument. This time, we will continue with the power spectrum (Part 2).
As mentioned previously, the power spectrum is a function of frequency and represents the power (energy per unit time) at each frequency contained in the original time signal. A conceptual diagram of how to obtain the power spectrum using an FFT analyzer is shown in Figure 1. As explained using filters as shown in this diagram, obtaining the power spectrum with an FFT analyzer is equivalent to passing the time signal x(t) through a group of L filters with a very steep bandwidth Δf (L is the number of analysis lines on the frequency axis as explained previously, for example L = 800) and calculating the mean square (power) result.
(Note: This type of signal processing is not actually performed in real FFT analyzers.)
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Figure 1: Conceptual diagram illustrating the acquisition of the power spectrum using an FFT analyzer.
The issue here is the bandwidth Δf of the bandpass filter (which affects the power value). As mentioned in "Frequency Analysis from the Basics (8) "Discrete Fourier Transform (DFT)", periodic time signals have a discrete frequency distribution (line spectrum). Therefore, for periodic time signals, their power spectral value is hardly affected (see Figure 2).
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Figure 2 Power values of line spectra unaffected by differences in bandwidth Δf.
In contrast, the spectrum of a non-periodic random signal (irregular signal) is a continuous spectrum, so its power value changes significantly depending on the bandwidth Δf being analyzed (see the illustrative diagram in Figure 3). For example, comparing the case with a bandwidth of 10 Hz and the case with a bandwidth of 5 Hz, the power value at 10 Hz will be about twice as large, depending on the shape of the spectrum. To minimize this effect, the power value is usually normalized per unit frequency (1 Hz bandwidth). This spectral function normalized per unit frequency (1 Hz bandwidth) is called the Power Spectral Density Function (PSD). Specifically, if the frequency resolution of the power spectrum P(k) explained in the previous power spectrum (Part 1) is Δf, then the PSD G(k) is:
.................................(1)
It is required as such.
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Figure 3: Differences in power due to analytical bandwidth of continuous spectrum.
The definition of PSD using the Fourier spectrum X(k) is:
.................................(2)
This is the result. Below, for simplicity, we will omit the limit in the explanation. As explained previously, X(k) is T times the complex Fourier coefficient ck;
.................................(3)
As shown in equation (4) below, the frequency resolution Δf is the reciprocal of the time window length T of the FFT;
.................................(4)
We can see that equation (3) is equal to equation (1).
Similar to the power spectrum P(k), the relationship between the mean squared value (total power) of the time signal x(t) and the PSD G(k) is:
.................................(5)
Therefore, by integrating the PSD, we can obtain the mean square of the time signal x(t).
PSD is used for evaluating the self-noise of amplifiers and for determining the spectral shape in random vibration tests. Its physical unit is V²/Hz, or EU²/Hz if EU is any physical quantity. In particular, when the physical quantity is vibration acceleration, it is sometimes called Acceleration Spectral Density (ASD). If the unit of acceleration is m/s², then the unit of ASD is m²/s³.
In summary, the power spectral density function (PSD) expresses the power (energy per unit time) per unit frequency (1 Hz width) contained in the original time signal as a function of frequency.
Next, let's consider the case where the time signal x(t) is not an infinitely continuing waveform, but a finite-time waveform such as an impulse waveform.
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Figure 4 Finite impulse waveform and the time window length for analyzing it.
As explained in "Frequency Analysis from the Basics (6) - Fourier Transform and Convolution", in equation (2), ( ) 2 X k is the energy spectrum, and PSD is its energy (power) per unit time, so it is in the form of being divided by the FFT time window length T. Therefore, when calculating the PSD of an impulse waveform like the one in Figure 4, for example, there will be a difference in the power value between the time windows T1 and T2, so the calculated PSD is further multiplied by the time window length T.
The frequency function obtained in this way is called the Energy Spectral Density Function (ESD). If E(k) is ESD, then:
E (k ) = T G (k ) .................................(6)
ESD is used to determine the energy distribution of transient signals such as shock waveforms. Similar to PSD, integrating ESD over the entire frequency band yields the total energy of the original time signal (mean square multiplied by T). Its physical units are V²s/Hz, or EU²s /Hz, where EU is an arbitrary physical quantity.
As shown in Table 1 below, FFT analyzers utilize three types of spectra: power spectrum, PSD, and ESD, each within a given bandwidth.
Table 1 Comparison of three types of spectra
|
Types of spectra |
Physical meaning |
Target signal |
Physical units |
|
Power Spectrum |
Power distribution by frequency bandwidth |
periodic signal |
EU |
|
PSD |
Power distribution per unit frequency |
Continuous random signal |
EU |
|
ESD |
Energy distribution per unit frequency |
transient signal |
EU |
Finally, here's a summary.
- Obtaining a power spectrum with an FFT analyzer is equivalent to passing a time signal x(t) through a group of filters with a very steep analysis bandwidth of Δf and simultaneously calculating the mean squared power.
- The spectrum of a periodic signal becomes a line spectrum, and its power value is not significantly affected by the analysis bandwidth, so the usual power spectrum can be applied.
- The spectrum of a time-sustaining random signal is a continuous spectrum, and in a typical power spectrum, the power value depends on the analysis width, so the power spectral density function (PSD) is applied.
- Transient signals, such as shock pulses, are finite in nature, and their power values depend on the analysis time window length, so the energy density function (ESD) is applicable.
【keyword】
Power spectrum, power, line spectrum, random signal, continuous spectrum, power spectral density function, PSD, acceleration spectral density, ASD, energy spectrum, energy spectral density function, ESD
[Reference materials]
- "Spectral Analysis," by Mikio Hino, Asakura Shoten (1977)
- "Digital Fourier Analysis (I) -Fundamentals-" by Kenichi Kido, Corona Publishing Co., Ltd. (2007)
- "Easy Guide to Using an FFT Analyzer," edited by Yamaguchi and Ono, Ohmsha (1994)
(Excerpt from the email newsletter issued on January 23, 2014)
