Comments about spectrum, energy spectrum, and Power Spectrum

1. Energy and Power Signals

If the energy of the Signal Integration can be obtained, instead of infinity, that is, the energy is limited, and the average power in all time is 0, the signal is an energy signal. If the energy is infinite, the power is used to describe the energy of the signal, which is the power signal. Any signal is either an energy signal or a power signal, because the power of the signal can never be infinite.

2. spectrum, energy spectrum, and Power Spectrum

In the North Science Edition signal and system, signals can be divided into energy signals and power signals. The non-periodic energy signal has the energy spectral density, which is the square of Fourier transformation. The power signal has the power spectral density, and its self-correlation function is a pair of Fourier transformation pairs, which is equal to the square/Interval Length of Fourier transformation. The power signal has no power spectrum.

In his book, Mr. Hu guangshu found that "random signals are infinite in time and there are infinite samples, so the random signal energy is infinite, it should be a power signal. The power signal does not meet the absolute product conditions of Fourier transformation, so its Fourier transformation does not exist. If the Fourier transformation of a deterministic sine function does not exist, only the Impulse Function is introduced to obtain the Fourier transformation. Therefore, the spectrum analysis of random signals is not simply a spectrum, but a power spectrum ."

Periodic Signals are power signals, so periodic signals have power spectral density. However, periodic signals may be deterministic or random signals. There is also a power spectrum density for random signals with an infinite duration.

For a deterministic signal, it can be an energy signal (without a power spectral density) or a power signal (with a power spectral density ), therefore, whether the signal has a power spectrum is not necessarily related to whether the signal is a deterministic signal.

The following arguments come from the Research Forum, which can be helpful, but may have some minor issues:

Spectrum is the Fourier transformation of signals, which describes the distribution of signals at various frequencies. The square of the spectrum (called the energy spectrum when the energy is limited and the average power is 0) describes the distribution of signal energy at various frequencies.

The power spectrum is a discrete Fourier transformation of the auto-correlation function of random signals (note that the auto-correlation function is a deterministic sequence, and the discrete signal itself does not have a discrete Fourier transformation ). It describes the distribution of random signal power at various frequencies, rather than the energy distribution.

During the computation process, the sample data is computed using the fast Fourier transformation. However, the difference is that the signal spectrum is a complex number, including the amplitude and phase frequencies. The results of repeated calculation are basically the same. The random signal power spectrum can also perform FFT on the data, but the square of the modulus must be calculated because the power spectrum is a real number. After a group of samples is changed, the calculation result is slightly different because the sample values of random signals are different. To obtain a real power spectrum, you must perform multiple average times. The more times, the better.

The power spectrum can be defined in two aspects. One is the Fourier transformation of the Self-correlation function mentioned by the author, and the other is the Fourier transformation modulus square of the time domain signal, which is divided by the time length. The first definition is often referred to as the inner theory, while the second is actually from the energy spectrum density. According to the parseval theorem, the wavelet transform modulus square is defined as the energy spectrum, that is, the signal energy contained within the Unit frequency range. Naturally, there is a time average relationship between energy and power. Therefore, the time average of the energy spectral density gets the power spectrum. (This statement is not accurate)

Conversion from: Reference 1. Reference 2