Maximum power spectral density. Energy characteristics of signals

Formal definition

Let be a signal considered over a period of time. Then the signal energy in this interval is equal to:

= = = ,

where is the spectral function of the signal. At , average power (variance)

.

Power spectral density (power spectrum density function).

The signal power density spectrum stores information only about the amplitudes of the spectral components. Phase information is lost. Therefore, all signals with the same amplitude spectrum and different phase spectra have the same power density spectra.

Assessment methods

Estimation of the PSD can be performed by the Fourier transform method, which involves obtaining the spectrum in the frequency domain through a fast Fourier transform (FFT). Before the invention of FFT algorithms, this method was practically not used due to the cumbersome nature of direct calculation of the discrete Fourier transform (DFT). Preference was given to other methods, in particular, the correlation function method (Blackman-Tukey) and the periodogram method.

see also

Literature

• Digital signal processing: Handbook. Goldenberg L.M., Matyushkin B.D., Polyak M.N. - M.: Radio and communications, .
• Applied time series analysis. Basic methods. Otnes R., Enokson L. - M.: Mir, .

Wikimedia Foundation. 2010.

• Spectral series
• Spectral series of hydrogen

See what “Power Spectral Density” is in other dictionaries:

Noise power spectral density of a microwave device- 221. Spectral power density of noise of a microwave device Spectral power density of noise Noise spectral power density Psh Noise power of a microwave device in the 1 Hz band Source: GOST 23769 79: Electronic devices and microwave protective devices. Terms... ...

Noise Diode Power Spectral Density- 140. Power spectral density of a noise diode G Ratio of the root mean square value of the power of a noise diode to a given frequency range Source: GOST 25529 82: Semiconductor diodes. Terms, definitions and letter designations... ... Dictionary-reference book of terms of normative and technical documentation

noise power spectral density- spektrinis triukšmo galios tankis statusas T sritis radioelektronika atitikmenys: engl. noise spectral power density vok. Spektralleistungsdichte des Rauschens, f rus. noise power spectral density, f pranc. densité spectrale de puissance… … Radioelektronikos terminų žodynas

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radiation power spectral density- spektrinis spinduliuotės galios tankis statusas T sritis fizika atitikmenys: engl. radiation power spectral density vok. spektrale Strahlungsleistungsdichte, f rus. spectral radiation power density, f pranc. densité spectrale de… … Fizikos terminų žodynas

relative spectral noise power density of a microwave device- NDP. energy spectrum of noise energy spectrum of fluctuations spectral density of noise ΔPsh The ratio of the spectral power density of noise of a microwave device to the output power in a 1 Hz band. [GOST 23769 79] Inadmissible, not recommended... ... Technical Translator's Guide

Relative spectral noise power density of a microwave device- 222. Relative spectral noise power density of the microwave device Ndp. Energy spectrum of noise Energy spectrum of fluctuations Spectral noise density Relative noise spectral power density ΔPш Ratio of power spectral density... ... Dictionary-reference book of terms of normative and technical documentation

Spectral Density- In statistical radio engineering and physics, when studying deterministic signals and random processes, their spectral representation in the form of spectral density, which is based on the Fourier transform, is widely used. If the process has... ... Wikipedia

Spectral radiation density- characteristic of the radiation spectrum, equal to the ratio of the intensity (flux density) of radiation in a narrow frequency interval to the value of this interval. Is the application of the concept of power spectral density to electromagnetic radiation.... ... Wikipedia

Spectral energy (power) density of laser radiation- 5. Spectral energy (power) density of laser radiation* Spectral energy (power) density of SPE (SPM) Wλ, Wv, Pλ, Pv Source ... Dictionary-reference book of terms of normative and technical documentation

Below is a brief description of 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.

	Signal s(t)			Spectral Density

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 an 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 :

The amplitude and phase spectra of the function 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 for = 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 halved, 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.

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 obtain the average power of the kth implementation on the segment T

As T increases, the energy increases, but the ratio tends to a certain limit. Having passed to the limit we obtain

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

International Educational Corporation

Faculty of Applied Sciences

Essay

on the topic"Power density spectrum and its relationship with the correlation function"

By discipline"Theory of electrical communication »

Performed: group student

FPN-REiT(z)-4S *

Dzhumageldin D

Checked: Glukhova N.V.

Almaty, 2015

I Introduction

II Main part

1. Power spectral density

1.1 Random variables

1.2 Probability density of a function of a random variable

2. Random process

3. Method for determining power spectral density using the correlation function

III Conclusion

IV List of used literature

Introduction

Probability theory considers random variables and their characteristics in “statics”. The problem of describing and studying random signals “in dynamics”, as a reflection of random phenomena developing over time or according to any other variable, is solved by the theory of random processes.

As a rule, we will use the variable “t” as a universal coordinate for the distribution of random variables over an independent variable and treat it, purely for convenience, as a time coordinate. Distributions of random variables over time, as well as signals that display them in any mathematical form, are usually called random processes. In technical literature, the terms “random signal” and “random process” are used interchangeably.

In the process of processing and analyzing physical and technical data, one usually has to deal with three types of signals described by statistical methods. Firstly, these are information signals that reflect physical processes that are probabilistic in nature, such as, for example, acts of registration of particles of ionizing radiation during the decay of radionuclides. Secondly, information signals that depend on certain parameters of physical processes or objects, the values ​​of which are unknown in advance, and which usually can be determined from these information signals. And thirdly, this is noise and interference, chaotically varying in time, which accompany information signals, but, as a rule, are statistically independent of them both in their values ​​and in changes over time.

Power Spectral Density

The power spectral density allows one to judge the frequency properties of a random process. It characterizes its intensity at different frequencies or, in other words, the average power per unit frequency band.

The distribution of average power across frequencies is called the power spectrum. The device that measures the power spectrum is called a spectrum analyzer. The spectrum found as a result of measurements is called the hardware spectrum.

The spectrum analyzer operates based on the following measurement methods:

· filtering method;

· transformation method according to the Wiener-Hinchen theorem;

· Fourier transform method;

· method using sign functions;

· method of hardware application of orthogonal functions.

The peculiarity of measuring the power spectrum is the significant duration of the experiment. Often it exceeds the duration of existence of the implementation, or the time during which the stationarity of the process under study remains. Power spectrum estimates obtained from one implementation of a stationary ergodic process are not always acceptable. Often it is necessary to perform numerous measurements, since it is necessary to average the realizations both over time and over the ensemble. In many cases, the implementations of the random processes under study are pre-memorized, which makes it possible to repeat the experiment many times, changing the duration of the analysis, using different processing algorithms and equipment.

In the case of preliminary recording of implementations of a random process, hardware errors can be reduced to values ​​due to the finite duration of the implementation and non-stationarity.

Memorizing the analyzed implementations allows you to speed up hardware analysis and automate it.

Random variables

A random variable is described by probabilistic laws. The probability that a continuous quantity X when measured will fall into any interval x 1<х <х 2 , is determined by the expression:

, Where p(x)- probability density, and . For a discrete random variable x i P(x = x i)=P i, Where P i- probability corresponding to the i-th level of quantity X.

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, defined by (1.47), over all functions leads to a zero spectrum of the process (at M[x(t)]=0 ) 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 under the random function x(t) implies electrical voltage or current, then the mean square of this function can be considered as the average power released in a resistance of 1 ohm. 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 ω . Function dimension W(ω) , 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 task will come later. Here we will limit ourselves to a few general definitions.

By selecting any implementation from the ensemble xk(t) and limiting its duration to a finite interval T, you can apply the usual Fourier transform to it and find the spectral density X kT (ω). Then the energy of the considered segment of implementation can be calculated using the formula:

(1.152)

Dividing this energy into T, we get the average power k-th implementation on the segment T

(1.153)

When increasing T energy ECT increases, but the ratio tends to some limit. Having made the passage to the limit, we get:

G
de

represents average power spectral density the one under consideration k-th implementation.

In general, the value W k (ω) must be averaged over many implementations. Limiting ourselves in this case to consideration of a stationary and ergodic process, we can assume that the function found by averaging over one realization W k (ω) characterizes the entire process as a whole. Omitting the index k, we obtain the final expression for the average power of the random process

For a process with zero mean

(1.156)

From the definition of spectral density (1.155) it is obvious that W X (ω) is an even and non-negative function ω.

1.5.3 Relationship between spectral density and covariance function of a random process

On the one hand, the rate of change X(t) determines the width of the spectrum over time. On the other side, rate of change x(t) determines the course of the covariance function. It is obvious that betweenW X (ω) and K X(τ) there is a close connection.

The Wiener–Khinchin theorem states that TO X (τ) And W x (ω) are related by Fourier transforms:

(1.157)

(1.158)

For random processes with zero mean, similar expressions have the form:

These expressions imply a property similar to the properties of Fourier transforms for deterministic signals: the wider the spectrum of a random process, the smaller the correlation interval, and accordingly, the larger the correlation interval, the narrower the spectrum of the process (see Fig. 1.20).

Fig.1.20. Broadband and narrowband spectra of a random process; borders of the central strip: ±F 1

White noise is of great interest when the spectrum is uniform at all frequencies.

If we substitute in expression 1.158 Wx(ω) = W 0 = const, then we get

where δ(τ) is the delta function.

For white noise with an infinite and uniform spectrum, the correlation function is equal to zero for all values ​​of τ except τ = 0 , at which R x (0) turns to infinity. This kind of noise, which has a needle-like structure with infinitely fine random spikes, is sometimes called a delta-correlated process. The dispersion of white noise is infinitely large.

Self-test questions

Name the main characteristics of a random signal.

How are the correlation function and the energy spectrum of a random signal related mathematically?

Which random process is called stationary.

Which random process is called ergodic.

How the envelope, phase and frequency of a narrowband signal are determined

Which signal is called analytical.