Source: Original article by CSDN blogger "Fang Keming".
Many things in life rely on the propagation of signals, and the propagation of signals is invisible, but it exists in the form of waves, and this kind of signal will produce power, and the signal power per unit frequency band is called the power spectrum. It shows the distribution of signal power over frequency over a certain area. In the field of science, the power spectrum and the spectrum have a certain relationship, but they are still different and different.
Power spectral density
In physics, signals are usually represented in the form of waves, such as electromagnetic waves, random vibrations, or sound waves. When the power spectral density of the wave is multiplied by an appropriate factor, the power carried per unit frequency of the wave is obtained, which is called the power spectral density (PSD) of the signal; not to be confused with spectral power distribution (SPD). The unit of power spectral density is usually expressed in watts per hertz (W/Hz), which is expressed in wavelength rather than frequency, i.e., watts per nanometer (W/nm).
Power spectrum related definitions
Power spectral density spectroscopy is a probabilistic statistical method that is a measure of the mean square value of a random variable. Generally used in random vibration analysis, the continuous transient response can only be described by the probability distribution function, that is, the probability of a certain horizontal response. The definition of power spectral density is the "power" (mean square value) in the unit frequency band, and the power spectral density is the statistical result of the response of the structure under the excitation of random dynamic loads, and is a relationship curve between the power spectral density value and the frequency value, where the power spectral density can be in the form of displacement power spectral density, velocity power spectral density, acceleration power spectral density, force power spectral density, etc. Mathematically, the area under the power spectral density value-frequency value curve is the mean square value, and when the mean value is zero, the mean square value is equal to the variance, that is, the square value of the standard deviation of the response.
Although it is not necessary to assign a certain physical dimension to a signal or its variables, the following discussion assumes that the signal varies in the time domain. The definition of spectral density above requires that the Fourier transform of the signal must exist, i.e. the signal squares can be product or squared. An often more useful alternative representation is the power spectral density (PSD), which defines how the power of a signal or time series is distributed with frequency. Here the power may be the actual physical power, or more often it is convenient to represent that an abstract signal is defined as the square of the signal value, i.e., the actual power when the load of the signal is 1 ohm (ohm). This instantaneous power (the median of the average power) can be expressed as:
Since the signal with a non-zero mean value is not squarely integrable, there is no Fourier transform in this case. Fortunately, the Wiener-Khinchin theorem provides a simple alternative to the Fourier transform of the signal's autocorrelation function if the signal can be seen as a stationary random process.
The difference between the power spectrum and the spectrum
1. Calculations
The calculation of the power spectrum requires the signal to be autocorrelated first, and then to perform the FFT operation.
The spectrum is calculated by FFT the signal directly.
2. Formula
The power spectrum is the study of the signal, but it is the study of the signal in terms of energy.
The spectrum is also used to describe a signal, but the way of representation has changed, from time domain to frequency domain representation, that is, a signal is represented in a different way.
The difference between the power spectrum and the spectrum sum boils down to the correlation between signal, power, and energy.
3. Definitions
The definition of power spectrum is the transformation of signal power in the unit frequency band in the case of a finite signal, and the power changes with frequency, thus becoming the power spectrum, which is the energy expressed by the analysis of the available finite signal of power energy. It contains some amplitude information about the spectrum, but the phase information is discarded.
In contrast, the spectrum is extremely less rigorous, mainly reflecting the average transformation of the signal, and only the average amount over a period of time is required.
Therefore, it is often said that in the case of different spectral signals, its power spectrum is likely to be the same.
4. Nature
Although the process of the power spectrum is random, because it is an average concept, it is equivalent to a stationary random process, and the power spectrum of this process is a deterministic function.
The sample of the spectrum undergoes a Fourier transformation, although the process is also random, but for this random change process, the spectrum is formed as a random frequency domain sequence, and the function is uncertain.
5. Requests
There is also a difference between the concept of power extremes and amplitudes of the power spectrum and the spectrum, and their existential requirements are also different. The existence of the power spectrum requires the convergence of changes, while the existence of the spectrum only requires convergence or not.
The power spectrum and the spectrum have similarities and are related, but these differences are the important points that determine the usefulness of both of them. Although the power spectrum and the spectrum are both the study of the signal, but the research direction is different, the angle is not the same, and there are differences in their properties, the randomness of the power spectrum is a little worse, more rigorous, and there is a definite function support, and the spectrum has fewer requirements, and the randomness is quite strong, which leads to its signal change, but this is also where its research value lies.
Original link:
https://blog.csdn.net/godlovechux/article/details/77030793
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