Spectrum, amplitude spectrum, power spectrum, and Energy Spectrum (reproduced)

Spectrum, amplitude spectrum, power spectrum, and Energy Spectrum

 

      In the learning of signal processing, there are some concepts related to spectrum, such as spectrum, amplitude spectrum, power spectrum, and energy spectrum, which are often confusing and unclear. Here we will mainly clarify the differences in terms of concept.

      Perform Fourier transformation on a time-domain signal to obtain the signal spectrum. The signal spectrum consists of two parts: Amplitude Spectrum and phase spectrum. This relationship is simple. So what is power spectrum? What is the energy spectrum? What is the relationship between the power spectrum or energy spectrum and the signal spectrum?

      To distinguish power spectrum and energy spectrum, we must first identify two different types of signals: power signal and energy signal. The concept of energy and power signals is derived from a specific physical system. The voltage and current on the resistance with the known resistance R are respectively V (T) and I (t), then the instantaneous power of this electrical signal is: p (t) = v2 (t) /r = I2 (t) R. For the convenience of qualitative analysis, the normolized power value is obtained by assuming that the resistance R is 1 ohm. During quantitative calculation, the actual power value can be obtained by de-normalization, that is, the actual resistance value can be substituted into it. Abstract The above concepts to define the instantaneous power of the signal x (t) as | f (t) | 2 at the time interval (-T/2T/2) the energy is:

 

                                                         E = int (| f (t) | 2,-T/2, T/2)                    (1)

| F (t) | 2 points, points limited to (-T/2)T/2 ).

The average power within the interval is:

                                                                     P = E/T                             (2)

      If f (t)'s energy at all times is not 0 and there is a time limit, the signal is an energy signal, that is, (1) E is limited when T in the formula tends to be infinite. Typical energy signals include square wave signals and triangular wave signals. However, some signals do not meet the energy signal conditions, such as periodic signals and random signals with unlimited energy. In this case, we need power to describe these signals. When and only when the power of x (t) at all times is not 0 and there is a time limit, the signal is a power signal, that is, (2) when T in the formula tends to be infinite, P is limited. The waveforms in the system either have an energy value or a power value, because the signal power with limited energy is 0, and the signal energy with limited power is infinite. Generally, periodic and random signals are power signals, rather than Periodic Fixed signals are energy signals. Distinguishing signals into energy signals and power signals can simplify mathematical analysis of various signals and noise. Another type of signal has infinite power and energy, such as f (t) = T, which is rarely used.

      The concept of power spectrum and energy spectrum can be well understood after understanding the signal may be an energy signal or a power signal. The energy signal is often described by the energy spectrum. The so-called energy spectrum, also known as the energy spectrum density, refers to the distribution of the signal energy at each frequency point using the density concept. That is to say, the signal energy can be obtained by integrating the energy spectrum in the frequency domain. The energy spectrum is the square of the modulus of the signal amplitude spectrum, and its dimension is focal/hertz. Power signals are often described by power spectrum. The so-called power spectrum, also known as power spectral density, refers to the distribution of signal power at various frequency points with the concept of density. That is to say, the power of the signal can be obtained by integrating the power spectrum in the frequency domain.Theoretically, the power spectrum is the Fourier transformation of the signal Auto-correlation function.Because the power signal does not meet the conditions of Fourier transformation, its spectrum usually does not exist. The ing between the self-correlation function and Fourier transformation is proved by the Gini theory. In engineering practice, even for power signals, due to the limited duration, Fourier transformation can be directly performed on the signals, and then the calculated square of the obtained amplitude spectrum is obtained, divide By the duration to estimate the power spectrum of the signal.

      Deterministic signals, especially non-cyclic deterministic signals, are often described by energy spectrum. However, for random signals, due to the infinite duration, they do not meet the conditions of absolute product and energy product. Therefore, there is no Fourier transformation, so the power spectrum is usually used to describe them. Periodic Signals also do not meet the conditions of Fourier transformation, which are often described by power spectrum. Only special signals such as single-frequency sine signals can be used to solve the Fourier transformation of signals after the delta function is introduced.

      White noise is a special case for random signals described by power spectrum. According to definition, white noise refers to the noise in which the power spectral density is evenly distributed throughout the frequency domain. Strictly speaking, white noise is just an idealized model, because the actual noise power spectrum density cannot have an infinite bandwidth. Otherwise, its power will be infinitely large, which is physically impossible. However, white noise is more convenient in mathematical processing, so it is a powerful tool for system analysis. Generally, as long as the spectrum width of a noise process is much greater than the bandwidth of the system to which it is applied, and the spectrum density of the noise process can be considered as a constant, it can be processed as white noise. For example, the thermal and elastic noise have a uniform power spectral density in a wide frequency range, which can be considered as white noise.

Source: http://blog.sina.com.cn/s/blog_3df0d7f10100hqdy.html>  

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