1) In its physical meaning, the power spectrum is real and non-negative:
Therefore, it is fundamentally impossible to reconstruct any individual realization of a random process from the power spectrum.
2) Since the argument is an even function, the corresponding power spectrum is an even function of frequency. It follows that the pair of Fourier transforms (6.14), (6.15) can be written using integrals in semi-infinite limits:
(6.17)
(6.18)
3. It is advisable to introduce the so-called one-sided power spectrum of a random process, defining it as follows:
(6.19)
The function allows you to calculate the variance of a stationary random process by integrating over positive (physical frequencies):
(6.20)
4. In technical calculations, a one-sided power spectrum N(f) is often introduced, which is the average power of a random process per frequency interval 1 Hz wide:
(6.21)
At the same time, how easy it is to see
A very important parameter of random processes is the correlation interval. Random processes typically have the following properties: their correlation function tends to zero as the time shift increases. The faster the function decreases, the smaller the statistical relationship between the instantaneous values of a random signal at two divergent points in time turns out to be.
The numerical characteristic used to evaluate the “rate of change” of the implementation of a random process is the correlation interval determined by the expression:
(6.22)
If information about the behavior of any implementation “in the past” is known, then a probabilistic forecast of a random process for a time of order is possible.
Another essential parameter for a random process is the effective spectrum width. Let the random process under study be characterized by a function - a one-sided power spectrum, and - the extreme value of this function. Let us mentally replace this random process with another process whose power spectral density is constant and equal within the effective frequency band , selected from the condition that the average powers of both processes are equal:
This gives the formula for the effective spectral width:
(6.23)
Outside the specified band, the spectral density of the random process is considered equal to 0.
This numerical characteristic is often used for engineering calculations of noise signal dispersion: .
If realizations of a random process have the dimension of voltage (V), then the relative power spectrum N has the dimension of .
White noise and its properties. Gaussian random process.
A) White noise.
a stationary random process with a constant power spectral density at all frequencies is called white noise.
(7.1)
According to the Wiener-Khinchin theorem, the white noise correlation function is:
is equal to zero everywhere except the point. The average power (dispersion) of white noise is unlimited.
White noise is a delta-correlated process. The uncorrelated nature of the instantaneous values of such a random signal means an infinitely high rate of change over time - no matter how small the interval is, the signal during this time can change by any predetermined value.
White noise is abstract mathematical model and the corresponding physical process certainly does not exist in nature. However, this does not prevent us from approximately replacing real fairly broadband random processes with white noise in cases where the bandwidth of the circuit affected by the random signal turns out to be significantly narrower than the effective width of the noise spectrum.
When we mean a random process as a set (ensemble) of functions of time, it is necessary to keep in mind that functions with different shapes correspond to different spectral characteristics. Averaging the complex spectral density introduced in § 2.6 or 2.1 over all functions leads to a zero spectrum of the process (at ) due to the randomness and independence of the phases of the spectral components in different implementations.
It is possible, however, to introduce the concept of the spectral density of the mean square of a random function, since the value of the mean square does not depend on the phase relationship of the summed harmonics. If a random function means electrical voltage or current, then the mean square of this function can be considered as the average power released in a 1 ohm resistance. This power is distributed over frequencies in a certain band, depending on the mechanism of formation of the random process. Average power spectral density is the average power per Hz at a given frequency. The dimension of the function, which is the ratio of power to frequency band, is
The spectral density of a random process can be found if the mechanism of formation of the random process is known. In relation to noise associated with the atomic structure of matter and electricity, this problem will be considered in § 7.3. Here we will limit ourselves to a few general definitions.
By selecting any realization from the ensemble and limiting its duration to a finite interval T, you can apply the usual Fourier transform to it and find the spectral density (ω). Then the energy of the considered segment of realization can be calculated using formula (2.66):
Dividing this energy by we get the average power k-th implementation on segment T
As T increases, the energy increases, but the ratio tends to a certain limit. Having passed to the limit we get
represents the average power spectral density of the implementation in question.
In general, the value must be averaged over many implementations. Limiting ourselves in this case to considering a stationary and ergodic process, we can assume that the function found by averaging over one implementation characterizes the entire process as a whole.
Omitting the index k, we obtain the final expression for the average power of the random process
If a random process with a non-zero mean value is considered, then the spectral density should be represented in the form
The most important characteristic of stationary random processes is the power spectral density, which describes the distribution of noise power over the frequency spectrum. Let us consider a stationary random process, which can be represented by a random sequence of voltage or current pulses following each other at random time intervals. A process with a random sequence of pulses is non-periodic. Nevertheless, we can talk about the spectrum of such a process, meaning in this case by spectrum the distribution of power over frequencies.
To describe noise, the concept of power spectral density (PSD) of noise is introduced, also called in the general case spectral density (SP) of noise, which is determined by the relation:
where P(f) - time-averaged noise power in the frequency band f at the measurement frequency f.
As follows from relation (2.10), the noise frequency has the dimension W/Hz. In general, SP is a function of frequency. The dependence of SP noise on frequency is called energy spectrum, which carries information about the dynamic characteristics of the system.
If a random process is ergodic, then the energy spectrum of such a process can be found from its single implementation, which is widely used in practice.
When considering the spectral characteristics of a stationary random process, it often turns out to be necessary to use the concept of noise spectrum width. The area under the curve of the energy spectrum of a random process, related to the noise frequency at some characteristic frequency f 0 is called effective spectrum width, which is determined by the formula:
(2.11)
This quantity can be interpreted as the width of the uniform energy spectrum of a random process in the band 
, equivalent in average power to the process under consideration.
Noise power P, contained in the frequency band f 1 …f 2 is equal to
(2.12)
If the SP noise in the frequency band f 1 ...f 2 is constant and equal S 0, then for the noise power in a given frequency band we have: 
where f=f 2 -f 1 – frequency band passed by the circuit or measuring device.
An important case of a stationary random process is white noise, for which the spectral density does not depend on frequency over a wide frequency range (theoretically, over an infinite frequency range). Energy spectrum of white noise in the frequency range -∞< f < +∞ is given by:
= 2S 0
= const, (2.13)
The white noise model describes a random process without memory (without aftereffect). White noise occurs in systems with a large number of simple homogeneous elements and is characterized by a normal distribution of the amplitude of fluctuations. The properties of white noise are determined by the statistics of independent single events (for example, the thermal movement of charge carriers in a conductor or semiconductor). However, true white noise with an infinite frequency band does not exist, since it has infinite power.
In Fig. 2.3. shows a typical oscillogram of white noise (dependence of instantaneous voltage values on time) (Fig. 2.3a) and the probability distribution function of instantaneous voltage values e, which is a normal distribution (Fig. 2.3b). The shaded area under the curve corresponds to the probability of occurrence of instantaneous voltage values e, exceeding the value e 1 .
Rice. 2.3. Typical white noise oscillogram (a) and probability density distribution function of instantaneous noise voltage amplitude values (b).
In practice, when assessing the noise level of any element or sub-device, the rms noise voltage is usually measured 
in units of B 2 or rms current in units of A 2. In this case, the SP noise is expressed in units of V 2 / Hz or A 2 / Hz, and the spectral densities of voltage fluctuations S u
(f) or current S I
(f) are calculated using the following formulas:
(2.14)
Where 
and – time-averaged noise voltage and current in the frequency band f respectively. The bar above means averaging over time.
In practical problems, when considering fluctuations of various physical quantities, the concept of generalized spectral density of fluctuations is introduced. In this case, the SP of fluctuations, for example, for resistance R expressed in units of Ohm 2 /Hz; Magnetic induction fluctuations are measured in units of T 2 /Hz, and fluctuations in the frequency of the self-oscillator are measured in units of Hz 2 /Hz = Hz.
When comparing noise levels in linear two-terminal networks of the same type, it is convenient to use the relative spectral noise density, which is defined as
=
,
(2.15)
Where u– DC voltage drop across a linear two-terminal network.
As can be seen from expression (2.15), the relative spectral density of noise S(f) is expressed in units of Hz -1.
When studying automatic control systems, it is convenient to use another characteristic of a stationary random process, called spectral density. In many cases, especially when studying the transformation of stationary random processes by linear control systems, spectral density turns out to be a more convenient characteristic than the correlation function. The spectral density of a random process is defined as the Fourier transform of the correlation function, i.e.
If we use Euler’s formula, then (9.52) can be represented as
Since the function is odd, in the last expression the second integral is equal to zero. Taking into account that the function is even, we get
Since it follows from (9.53) that
Thus, the spectral density is a real and even function of frequency o). Therefore, on the graph, the spectral density is always symmetrical about the ordinate axis.
If the spectral density is known, then using the inverse Fourier transform formula you can find the corresponding correlation function:
Using (9.55) and (9.38), we can establish an important relationship between the dispersion and spectral density of a random process:
The term "spectral density" owes its origin to the theory of electrical oscillations. The physical meaning of spectral density can be explained as follows.
Let be the voltage applied to an ohmic resistance of 1 Ohm, then the average power dissipated across this resistance over time is equal to
If we increase the observation interval to infinite limits and use (9.30), (9.38) and (9.55), then we can write the formula for the average power as follows:
Equality (9.57) shows that the average signal power can be represented as an infinite sum of infinitesimal terms, which extends to all frequencies from 0 to
Each elementary term of this sum plays the role of power corresponding to an infinitesimal portion of the spectrum, contained in the range from to. Each elementary power is proportional to the value of the function for a given frequency. Therefore, the physical meaning of spectral density is that it characterizes the distribution of signal power over the frequency spectrum.
The spectral density can be found experimentally through the average value of the squared amplitude of the harmonics of the implementation of a random process. Instruments used for this purpose and consisting of a spectrum analyzer and a calculator for the average value of the squared harmonic amplitude are called spectrometers. It is more difficult to find the spectral density experimentally than the correlation function, therefore, in practice, the spectral density is most often calculated using a known correlation function using formula (9.52) or (9.53).
The mutual spectral density of two stationary random processes is defined as the Fourier transform of the cross correlation function, i.e.
Using the mutual spectral density, by applying the inverse Fourier transform to (9.58), we can find an expression for the cross correlation function:
Cross spectral density is a measure of the statistical relationship between two stationary random processes: If the processes are uncorrelated and have zero average values, then the cross spectral density is zero, i.e.
Unlike the spectral density, the cross spectral density is not an even function of o and is not a real, but a complex function.
Let's consider some properties of spectral densities
1 The spectral density of a pure random process, or white noise, is constant over the entire frequency range (see Fig. 9.5, d):
Indeed, substituting expression (9.47) for the white noise correlation function into (9.52), we obtain
The constancy of the spectral density of white noise over the entire infinite frequency range, obtained in the last expression, means that the energy of white noise is distributed evenly across the entire spectrum, and the total energy of the process is equal to infinity. This indicates the physical impossibility of a random process such as white noise. White noise is a mathematical idealization of a real process. In fact, the frequency spectrum falls off at very high frequencies (as shown by the dotted line in Fig. 9.5, d). If, however, these frequencies are so high that when considering any specific device Since they do not play a role (because they lie outside the frequency band transmitted by this device), then idealizing the signal in the form of white noise simplifies the consideration and is therefore quite appropriate.
The origin of the term “white noise” is explained by the analogy of such a process with white light, which has the same intensities of all components, and by the fact that random processes such as white noise were first identified in the study of thermal fluctuation noise in radio engineering devices.
2. The spectral density of a constant signal is a -function located at the origin of coordinates (see Fig. 9.5, a), i.e.
To prove this, let us assume that the spectral density has the form (9.62), and from (9.55) the corresponding correlation function. Because
then when we get
This (in accordance with property 5 of the correlation functions) means that the signal corresponding to the spectral density defined by (9.62) is a constant signal equal to
The fact that the spectral density is a -function of means that all the power of the DC signal is concentrated at zero frequency, as would be expected.
3. The spectral density of a periodic signal represents two -functions located symmetrically relative to the origin of coordinates at (see Fig. 9.5, e), i.e.
To prove this, let us assume that the spectral density has the form (9.63), and using (9.55) we find the corresponding correlation function:
This (in accordance with the property of 6 correlation functions) means that the signal corresponding to the spectral density determined by (9.63) is a periodic signal equal to
The fact that the spectral density represents two -functions located at means that all the power of the periodic signal is concentrated at two frequencies: If we consider the spectral density only in the region of positive frequencies, we obtain,
that all the power of a periodic signal will be concentrated at one frequency.
4. Based on the above, the spectral density of the time function expanded in a Fourier series has the form
This spectral density corresponds to a line spectrum (Fig. 9.9) with -functions located at positive and negative harmonic frequencies. In Fig. 9.9 -functions are conventionally depicted in such a way that their heights are shown proportional to the coefficients of the unit -function, i.e., the values and
Note that the spectral density, as follows from (9.64), does not contain, just like the correlation function defined by (9.44), any information about the phase shifts of individual harmonic components. and vice versa. This corresponds to the physical essence of the process: the wider the spectral density graph, i.e., the more high frequencies presented in spectral density, the higher the degree of variability of the random process and the same graphs of the correlation function. In other words, the relationship between the type of spectral density and the type of time function is inverse compared to the relationship between the correlation function and the type of time function. This is especially true when considering a constant signal and white noise. In the first case, the correlation function has the form of a horizontal straight line, and the spectral density has the form of an -function (see Fig. 9.5, a). In the second case (see Fig. 9.5, d) the opposite picture occurs.
6. The spectral density of a random process, on which periodic components are superimposed, contains a continuous part and separate functions, corresponding to the frequencies of periodic components.
Individual peaks in the spectral density plot indicate that the random process is mixed with hidden periodic components that may not be detected at first glance at individual records of the process. If, for example, one is superimposed on a random process periodic signal with frequency then graph; spectral density has the form shown in Fig. 9.10,
Sometimes a normalized
spectral density which is the Fourier image of the normalized correlation function (9.48):
The normalized spectral density has the dimension of time.
Below is short description some signals and their spectral densities are determined. When determining the spectral densities of signals that satisfy the condition of absolute integrability, we directly use formula (4.41).
The spectral densities of a number of signals are given in Table. 4.2.
1) Rectangular pulse (Table 4.2, item 4). The oscillation shown in Fig. (4.28, a) can be written in the form
Its spectral density
The spectral density graph (Fig. 4.28, a) is built on the basis of a previously performed analysis of the spectrum of a periodic sequence of unipolar, rectangular pulses (4.14). As can be seen from (Fig. 4.28, b), the function goes to zero at the values of the argument = n, Where P - 1, 2, 3, ... - any integer. In this case, the angular frequencies are equal to = .
Rice. 4.28. Rectangular pulse (a) and its spectral density (b)
The spectral density of a pulse at is numerically equal to its area, i.e. G(0)=A. This position is valid for the impulse s(t) free form. Indeed, assuming in the general expression (4.41) = 0, we obtain
i.e. pulse area s(t).
Table 4.3.
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When the pulse is stretched, the distance between the zeros of the function is reduced, i.e., the spectrum is compressed. At the same time, the value increases. On the contrary, when the pulse is compressed, its spectrum expands and the value decreases. Figure 4.29, a, b) shows graphs of the amplitude and phase spectra of a rectangular pulse.
Rice. 4.29. Amplitude graphs (a) Fig. 4.30. A pulse of rectangular shape, and phase (b) spectra shifted by time
When the pulse is shifted to the right (delay) for a time (Fig. 4.30), the phase spectrum changes by the amount determined by the multiplier argument exp() (Table 4.2, item 9). The resulting phase spectrum of the delayed pulse is shown in Fig. 4.29, b with a dotted line.
2) Delta function (Table 4.3, item 9). We find the spectral density function using formula (4.41), using the filtering property δ -functions:
Thus, the amplitude spectrum is uniform and is determined by the area δ -function [= 1], and the phase spectrum is zero [= 0].
The inverse Fourier transform of the function = 1 is used as one of the definitions δ -functions:
Using the property of time shift (Table 4.2, item 9), we determine the spectral density of the function , delayed by time relative to :
Amplitude and phase spectra functions are shown in table. 4.3, pos. 10. The inverse Fourier transform of a function has the form
3) Harmonic oscillation 
(Table 4.3, item 12). A harmonic oscillation is not a completely integrable signal. Nevertheless, to determine its spectral density, a direct Fourier transform is used, writing formula (4.41) in the form:
Then, taking (4.47) into account, we obtain
δ(ω) – delta functions, shifted along the frequency axis by frequency , respectively to the right and left relative. As can be seen from (4.48), the spectral density of a harmonic vibration with a finite amplitude takes on an infinitely large value at discrete frequencies.
By performing similar transformations, one can obtain the spectral density of vibration 
(Table 4.3, item 13)
4) Type function 
(Table 4.3, item 11)
Signal spectral density as a constant level A is determined by formula (4.48), setting = 0:
5) Unit function (or unit jump) (Table 4.3, item 8). The function is not completely integrable. If represented as the limit of exponential momentum , i.e.
then the spectral density of the function can be defined as the limit of the spectral density of the exponential pulse (Table 4.3, item 1) at:
The first term on the right side of this expression is equal to zero at all frequencies except = 0, where it goes to infinity, and the area under the function is equal to a constant value
Therefore, the function can be considered the limit of the first term. The limit of the second term is the function. Finally we get
The presence of two terms in expression (4.51) is consistent with the representation of the function in the form 1/2+1/2sign( t). According to (4.50), the constant component 1/2 corresponds to the spectral density, and the odd function - imaginary value of spectral density.
When analyzing the effect of a single step on circuits whose transfer function at = 0 is zero (i.e., on circuits that do not pass direct current), in formula (4.51) only the second term can be taken into account, representing the spectral density of a single step in the form
6) Complex exponential signal (Table 4.3, item 16). If we represent the function as
then, based on the linearity of the Fourier transform and taking into account expressions (4.48) and (4.49), the spectral density of the complex exponential signal
Consequently, a complex signal has an asymmetric spectrum, represented by a single delta function, shifted by frequency to the right relative to it.
7) Arbitrary periodic function. Let us represent an arbitrary periodic function (Fig. 4.31, a) by a complex Fourier series
where is the pulse repetition rate.
Fourier series coefficients
expressed through the spectral density of a single pulse s(t) at frequencies ( n=0, ±1, ±2, ...). Substituting (4.55) into (4.54) and using relation (4.53), we determine the spectral density of the periodic function:
According to (4.56), the spectral density of an arbitrary periodic function has the form of a sequence of functions shifted relative to each other by frequency (Fig. 4.31, b). Coefficients at δ -functions change in accordance with the spectral density of a single pulse s(t) (dashed curve in Fig. 4.31, b).
8) Periodic sequence of δ-functions (Table 4.3, item 17). Spectral density of a periodic sequence of functions
is determined by formula (4.56) as a special case of the spectral density of a periodic function at = 1:
Fig.4.31. Arbitrary sequence of pulses (a) and its spectral density (b)
Rice. 4.32. Radio signal (a), spectral densities of the radio signal (c) and its envelope (b)
and has the form of a periodic sequence δ -functions multiplied by the coefficient .
9) Radio signal with a rectangular envelope. The radio signal presented in (Fig. 4.32a) can be written as
According to pos. 11 Table 4.2, the spectral density of the radio signal is obtained by shifting the spectral density of the rectangular envelope along the frequency axis to the right and left with the ordinate halving, i.e.
This expression is obtained from (4.42) by replacing the frequency with the frequencies – shift to the right and – shift to the left. The transformation of the envelope spectrum is shown in (Fig. 4.32, b, c).
Examples of calculating the spectra of non-periodic signals are also given in.
