The concept of several spectrum-related spectra

Spectrum, amplitude spectrum, power spectrum, and energy spectrum

in the study 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 confusing. This is primarily a conceptual clarification of the differences between them.

For a time domain signal Fourier transform, the spectrum of the signal can be obtained, the spectrum of the signal is composed of two parts: amplitude spectrum and phase spectrum. The relationship is still simple. So, what is a power spectrum? What is energy spectrum? What does the power spectrum or energy spectrum have to do with the spectrum of the signal?

         to differentiate between power spectrum and energy spectrum, it is first clear that there are two different types of signals: power signals and energy signals. We draw the concept of energy signals and power signals from a specific physical system. Voltages and currents on resistors with known resistance R are V (t)   and  i ( T), the instantaneous power of this electrical signal is:  p (t) = V2 (t)/R = i2 (t) r. In the case of qualitative analysis, for convenience, it is generally assumed that the resistance R is 1 ohms and is normalized   (normolized)  . The actual power value can be obtained by substituting the actual resistor value for quantitative calculation. Make an abstraction of the above concept, the signal  x (t)   Define its instantaneous power of  |f (t), at the time interval   (-T/2   T/2)   is:

E=∫|f (t) to DT (1)

The integral range is-T/2 to T/2. The average power in this interval is:

p = e/t (2)

         If and only if f (t) energy is not 0 and limited at all times, the signal is an energy signal, i.e. (1) in the  t  tends to infinity when E is limited. Typical energy signals such as square-wave signals, triangular-wave signals, etc. But some signals do not satisfy the conditions of the energy signal, such as the periodic signal and the energy infinite random signal, at this time need to use power to describe such signals. When and only if X (t) at all times the power is not 0 and limited, the signal is a power signal, namely   (2)    t  tends to infinity when  p   is limited. Waveforms in the system either have an energy value or have a power value because the energy is limited to a signal power of 0, and the power of a limited signal is infinitely large. In general, periodic signals and random signals are power signals, and non-periodic deterministic signals are energy signals. Distinguishing signals into energy and power signals simplifies mathematical analysis of a variety of signals and noises. There is also a kind of signal whose power and energy are infinite, such as  f (t) = t, such signals are seldom used.

Understanding The concept of power spectrum and energy spectrum can be well understood when the signal may be an energy signal or a power signal. For energy signals, the energy spectra are used to describe them. The so-called energy spectrum, also known as energy spectral density, refers to the distribution of signal energy at each frequency point using the concept of density. That is to say, the energy spectrum in the frequency domain of the integration can be obtained signal energy. The energy spectrum is the square of the modulus of the signal amplitude spectrum, and its dimension is kj/khz. For power signals, the common power spectrum is described. The so-called power spectrum, also known as power spectral density, refers to the distribution of signal power at each frequency point by means of the concept of density. In other words, the power spectrum can get the power of the signal by integrating it in the frequency domain. Theoretically, the power spectrum is the Fourier transform of the signal autocorrelation function. Since the power signal does not meet the conditions of Fourier transform, its spectrum is usually absent, and the Wiener-Sinzing theorem proves the correspondence between autocorrelation function and Fourier transform. In engineering practice, even if the power signal, due to the continuous time is limited, the signal can be directly Fourier transform, and then the resulting amplitude spectrum modulo squared, divided by the duration to estimate the power spectrum of the signal.

deterministic signals, especially non-periodic deterministic signals, are described in common energy spectra. But for the stochastic signal, because the duration time is infinite, does not satisfy the absolute integrable and the energy integrable condition, therefore does not have the Fourier transform, therefore usually uses the power spectrum to describe. Periodic signal, also is not satisfied with the Fourier transform conditions, commonly used power spectrum to describe, which has been described in the previous. Only a few special signals, such as single-frequency sinusoidal signals, can solve the Fourier transform of a signal after the introduction of the delta function.

White Noise is a special case for random signals described by the power spectrum. By definition, white noise refers to the uniformly distributed noise of the power spectral density throughout the frequency domain. Strictly speaking, white noise is only an idealized model, because the actual noise of the power spectral density can not have unlimited bandwidth, otherwise its power will be infinitely large, is physically impossible to achieve. However, white noise is more convenient in mathematical processing, so it is a powerful tool for system analysis. Generally, as long as a noise process has a spectral width far greater than the bandwidth of its system, and in this bandwidth, its spectral density can be considered as a constant, it can be treated as white noise. For example, thermal and spurious noise have uniform power spectral densities over a wide frequency range, and they can often be thought of as white noise.

The concept of several spectrum-related spectra