This time, we will discuss how the spectral values obtained from FFT analysis change depending on the frequency resolution, and how to interpret the analyzed spectral values based on the time waveform.
As explained in the previous section on "Time Window Length and Spectral Resolution," the frequency spectrum obtained from an FFT analyzer is the power (mean square) of an AC signal that has passed through a group of filters with a certain width called a bin. We will call this spectrum the power spectrum. The spectrum of a periodic signal, such as a sine wave, is a line spectrum, so if that line spectrum fits within the width of the bin, its power value does not change regardless of the width. The physical unit of the power spectrum is V² (or EU²).
In contrast, the spectrum of an irregular time waveform (random signal) without periodicity, such as amplifier noise, is a continuous function of frequency and therefore a continuous spectrum. The power value of such a random signal spectrum depends on the bin width Δf of the power value passing through it, and therefore changes depending on the number of analysis lines (number of bins) of the FFT analyzer.
For example, if we analyze with 800 lines and 1600 lines and compare the power values of the same frequency band in the two power spectra, the 800-line analysis will have approximately twice the power value on average. To reduce this difference in power values due to differences in analysis bandwidth, the power value is normalized by the analysis bandwidth Δf. This spectrum per unit frequency (1 Hz) is called the power spectral density function (hereinafter referred to as PSD). That is, if the power spectrum is P(f), then PSD can be calculated as P(f)/Δf. The physical unit of PSD is V²/Hz (or EU²/Hz).
Power Spectrum
Power spectral density function (PSD)
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Figure 1: Examples of pink noise analysis using 800 and 1600 lines.
The two types of time waveforms mentioned above (periodic signals and random signals) are so-called stationary signals, so their spectra are independent of the time averaging by the FFT time window. However, in the case of transient signals such as shock waves, the energy value is finite, so the power spectrum depends greatly on the time window length. For example, if a transient signal lasting 0.5 seconds is analyzed with time windows of 1 second and 2 seconds, the power value will be twice as large in the 1-second analysis. Of course, in this case as well, since it is a non-periodic signal, the spectrum will be continuous.
For transient signals, the spectral quantity is determined in units of energy, not power. This is called the energy spectral density function (ESD), and it can be calculated as ESD = PSD・T (where T is the time window length of the FFT). The physical unit of ESD is V 2 s/Hz (or EU 2 s/Hz).
(Note)
The relationship between PSD and ESD is similar to that between time-averaged sound pressure level (LT) and acoustic exposure level (LE) used in noise measurement;
This is the relationship between the two.
Power Spectrum
Power spectral density function (PSD)
Energy spectral density function (ESD)
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Figure 2 shows examples of transient sound analysis using 400 lines, 800 lines, and 1600 lines.
As mentioned earlier, time waveforms analyzed by an FFT analyzer can be broadly classified into periodic signals, random signals, and transient signals. Table 1 summarizes how to read the spectra.
| periodic signal | Random signal | transient signal | |
| Duration of time signal | infinite | infinite | limited |
| Power | limited | limited | limited |
| Energy | infinite | infinite | limited |
| Shape of the spectrum | Line spectrum | Continuous spectrum | Continuous spectrum |
| Spectral evaluation function | Power Spectrum | Power spectral density | Energy spectral density |
| The above units | EU2 | EU2/Hz | EU2・s/Hz |
| Calculation method | P (f) | P (F)/Δf | （P (F)/Δf）・T |
(Note)
EU stands for Engineering Unit, and it represents any unit of physical quantity.
(Excerpt from the email newsletter issued on June 26, 2008)
