In the fourth lesson, we explained that the power spectrum is the decomposition of the power (mean squared) of a time-domain signal x(t) into its frequency components. But how exactly is it calculated, and what are its units?
If X(f) is the Fourier transform of a time-domain signal x(t), then X(f) is generally a complex number.
X (f) = X R (f) ＋ j X I (f)
It can be expressed as follows, and X(f) is called the complex Fourier spectrum. Here, XR(f) is the real part and XI(f) is the imaginary part.
It can also be expressed in terms of its absolute value (amplitude) and phase.
Amplitude |X (f)|=√{XR (f) 2 + XI (f) 2}
The contents of { } represent what is inside the square root.
Phase θ (f) = arctan (XI (f) / XR (f))
These are sometimes called the amplitude spectrum and the phase spectrum, respectively. This phase is the phase relative to the beginning of the time window of the time-domain signal. Conversely, given amplitude and phase information, the original time-domain signal can be perfectly reconstructed using the inverse Fourier transform.
We define the power spectrum as the square of the amplitude spectrum.
P (f) ＝｜X (f)｜ 2＝ X R (f) 2 + X I (f) 2
Thus, the power spectrum P(f) represents the power components for each frequency, and the phase information of the original time-domain signal is lost, so it is not possible to reconstruct the original waveform from it. Its unit is V^2 (or the square of the physical quantity), depending on the unit of x(t). In the case of an irregular signal, the infinite signal is cut out for each time window, and P(f) (an estimate of P(f)) is obtained by averaging a large number of times (theoretically infinitely many times).
Strictly speaking, the power spectrum P(f) is the power (mean square) of the time signal passing through the analysis bandwidth Δf. For periodic signals, it is a line spectrum and is independent of the resolution bandwidth Δf. However, in the case of irregular signals, it is a continuous spectrum and the power depends on the resolution bandwidth Δf. In such cases, each band of the power spectrum is divided by the analysis bandwidth Δf and normalized to a unit frequency (i.e., 1 Hz).
This is called power spectral density (PSD), and its unit is V^2/Hz.
Next, let's consider the spectrum of a transient signal, such as an impulse waveform. Since the duration of a transient signal is finite, its value changes depending on the mean-squared time interval. Therefore, we multiply the above PSD by the mean-squared time interval T (the time window length of the FFT) to make it independent of the mean-squared time interval. This is called the Energy Spectrum Density (ESD), and its unit is V² s/Hz. Energy is defined as the square integral of the time-domain signal x(t), and power is defined as that value divided by the integration time (i.e., averaged), so the relationship between PSD and ESD can be easily understood.
(Note)
The total energy of the disorder signal x(t) is
Lim Integral (0 to T) x 2 (t) dt...(1)
T→∞
Integral (0 to T): The first-order integral with one period from 0 to T.
x² (t): Represents x²(t) squared.
It is defined as follows: for continuous signals, it is infinite, but for transient signals, it is finite. Also, since power is the average value over time,
Lim 1/T * Integral (0 toT) x 2 (t) dt ・・・(2)
T→∞
It is defined as follows and has a finite value.
Up until now, we've discussed the spectrum of a single-channel signal. Next time, we'll talk about the frequency function between two channels.
(Excerpt from the email newsletter issued on May 29, 2003)
