CN114244457A - Method for rapidly predicting multi-platform electromagnetic interference problem - Google Patents
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Abstract
The invention discloses a method for quickly predicting the problem of multi-platform electromagnetic interference, which establishes mathematical models of a transmitter, a receiver and an antenna of multiple platforms through a mathematical analytic expression and establishes an electric wave propagation model among the multiple platforms according to a combat environment. And the electromagnetic interference problem possibly existing among the multiple platforms is quickly predicted through quick amplitude-frequency screening. According to the rapid prediction method for the multi-platform electromagnetic interference problem, the electromagnetic compatibility problem possibly existing in multi-platform development can be predicted in the design stage, and the electromagnetic compatibility design scheme is optimized; in the development stage, the cost of the electromagnetic compatibility test can be reduced, a large amount of manpower, material resources and time are saved, the test device is fast and efficient, and the risk of burning the test device can be reduced to a large extent.
Description
Technical Field
The invention relates to the field of electromagnetic compatibility design, in particular to a method for quickly predicting the problem of electromagnetic interference of multiple platforms.
Background
With the development of the times, the cooperation and coordination among multiple platforms are more emphasized, and a large amount of various frequency-using equipment, namely high-sensitivity receiving equipment and high-power/broadband working interference equipment, is required to be intensively used. In a limited space, the possibility of mutual interference between the frequency-using devices is greatly increased. The rapid prediction method is a low-cost and effective method for analyzing the electromagnetic interference problem in the complex electromagnetic environment.
In order to effectively improve the electromagnetic compatibility of the platform, the spectrum overall planning and predictive analysis should be developed at the model demonstration stage and the initial development stage to find out potential mutual interference equipment and take effective measures to avoid as early as possible. At present, the problem of electromagnetic interference in complex electromagnetic environments is caused by the simultaneous operation of many multi-frequency devices. The multiple platforms are high in power and sensitivity, and the broadband devices are densely arranged, so that more electromagnetic compatibility problems are possibly caused. Therefore, analysis of electromagnetic interference problems in complex electromagnetic environments is very difficult.
Disclosure of Invention
In view of this, the present invention provides a fast prediction method for solving the problem of electromagnetic interference, which is used to analyze the electromagnetic compatibility hidden danger existing in multiple platforms and predict the electromagnetic compatibility problem that may occur.
Therefore, the invention provides a method for rapidly predicting the multi-platform electromagnetic interference problem, which comprises the following steps: a transmitter, a receiver, an antenna mathematical model, a radio wave propagation model and a rapid amplitude-frequency screening method; wherein:
the transmitter, the receiver and the antenna mathematical model are established with the electric wave propagation model according to the electromagnetic parameters of the transmitter and the receiver and are used for developing the rapid amplitude-frequency screening method.
The rapid amplitude-frequency screening method is used for predicting the multi-platform electromagnetic interference problem.
In a possible implementation manner, in the method for rapidly predicting the multi-platform electromagnetic interference problem provided by the present invention, the transmitter, the receiver, and the antenna mathematical model specifically include a transmitter mathematical model, a receiver mathematical model, and an antenna mathematical model;
the transmitter mathematical model specifically comprises the following steps of establishing a relevant nonlinear model for an amplifier of a transmitter by using a power series, wherein the power series is in the form of:
the power series can simulate the higher harmonic component generated by the amplifier, and the order of the power series can be adjusted according to the actual situation in the modeling process, for example, the third-order power series model is as follows:
the receiver mathematical model comprises the specific steps of establishing a simple receiver mathematical model based on a signal-to-interference ratio (SIR), wherein the receiver sensitivity mathematical model can be expressed as follows:
S＝-174+NF+10lg(B)+10lg(SNR)
in the formula, NF is a noise factor (generally 10dB), B is a signal bandwidth, and SNR is a demodulation signal-to-noise ratio, which is also referred to as a demodulation threshold or a minimum output signal-to-noise ratio.
The antenna mathematical model specifically comprises the step of calculating the gain of the antenna in a certain specific direction according to the angle. A mathematical model for a single antenna can be built using a directional pattern function. The typical directional pattern function is:
from the above equation, the gain to any one of the three-dimensional space directions of the angle parameter can be passed according to the mathematical model of the single antenna. For example, the specific pattern function may be:
for the establishment of the array antenna mathematical model, 5 factors of the number of array elements, the array element spacing, the distribution form, the excitation phase and the amplitude need to be considered, if one antenna array consists of M array elements, the directional diagram of the mth array element in the array is
The directional pattern function of the entire array is then:
if each array element
Similarly, the above equation can be simplified as:
s is an array factor that is related to the position of the antenna elements in the array. According to the directional diagram product, the directional diagram of the array antenna can be specifically developed as follows:
wherein A ism、αmAs array element amplitude and phase weighting coefficients, Δ RmThe distance from the m-th array element to the observation point. Taking a planar array in which M × N antenna elements located in the yoz plane form a rectangular grid array as an example, the directional diagram function can be expressed as:
the radio wave propagation model specifically comprises the step of establishing a radio wave propagation mathematical model by using a Fourier transmission formula and a radar equation. Pr is the received power, Pt is the transmitted power, At is the effective aperture of the transmitting antenna, Ar is the effective aperture of the receiving antenna, f is the signal carrier frequency, c is the speed of light, σ is the radar cross-section, Sr is the power density of the backscatter At distance r, and Sin is the power density of the incident on the object.
The radio wave propagation model further takes into account the earth occlusion problem. On smooth ground, even if there are no significant obstacles between the transmit and receive antennas' line of sight, additional ground scattering losses need to be introduced if there is a blockage in its first fresnel zone.
The loss of free space can be expressed by
Lfs＝32.45+20log10dkm+201og10fMHz
The radius of the first Fresnel zone can be calculated according to the following formula
In a possible implementation manner, in the method for quickly predicting the multi-platform electromagnetic interference problem provided by the present invention, the quick amplitude-frequency screening method specifically includes a quick screening method and an amplitude screening method;
the quick screening method comprises the specific steps of carrying out preliminary judgment from the frequency perspective, deleting the transmitting/receiving pairs which cannot interfere with each other, predicting the possible interference types of the transmitting/receiving pairs, and providing a basis for selecting an interference margin calculation model in the amplitude screening method.
The amplitude screening method specifically comprises the steps of judging from the amplitude of an interference signal, and realizing accurate prediction of the multi-platform electromagnetic interference problem by calculating and analyzing the interference margins of a transmitter and a receiver. The interference allowance comprises a same frequency interference allowance (A), an adjacent frequency interference allowance (B), a second harmonic interference allowance (C) and a third harmonic interference allowance (D).
And when the possibility of interference is judged according to the quick screening method, corresponding interference allowance calculation is carried out according to the existing interference type. And calculating the same-frequency interference margin (A-IM), the adjacent-frequency interference margin (B-IM), the second harmonic interference margin (C-IM) and the third harmonic interference margin (D-IM) item by item.
If this interference margin is less than the receiver sensitivity (screening level), no significant interference is generated.
The model for calculating the interference margin is as follows:
IM＝PT-PS+GT-LTR+GR-PR+CF
wherein, PTFor transmitters at transmission frequency fTThe time-dependent transmission power, which needs to be confirmed according to the design index of the transmitter; gain G of transmitting antennaTFor transmitting antenna at transmitting frequency fTA gain in time; degree of isolation LTRFor the isolation between the transmitting antenna and the receiving antenna, the isolation can be obtained by modeling calculation by adopting electromagnetic field calculation software and can also be simplified into free space attenuation; pRFor a receiving device RnAt a receiving frequency fTA sensitivity threshold level of time; pSIs an out-of-band inhibition capability; the interference margin correction coefficient CF is set to 0 in the amplitude screening.
The method of the invention has the benefits that:
(1) in the design stage, predicting the possible electromagnetic compatibility problem in the development stage according to the multi-platform functional electromagnetic parameters, and optimizing an electromagnetic compatibility design scheme;
(2) in the development stage, the boundary detection of multi-platform electromagnetic compatibility can be realized through limited simulation, and the consumption of manpower and material resources generated by large-scale tests is avoided.
Drawings
FIG. 1 is an overall block diagram of the method of the present invention.
FIG. 2 is a schematic view of earth occlusion considered by the present invention.
Fig. 3 is a schematic view of the blocking of the first fresnel region of the transceiving antenna considered by the present invention.
FIG. 4 is a flow chart of the rapid screening method of the present invention.
Fig. 5 is a flow chart of amplitude screening according to the present invention.
FIG. 6 is a simplified diagram of a simulated external field test arrangement of the present invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only illustrative and are not intended to limit the present invention.
According to the method for rapidly predicting the multi-platform electromagnetic interference problem, the electromagnetic compatibility problem possibly existing in multi-platform development can be predicted in the design stage, and the design scheme of the electromagnetic compatibility is optimized; in the development stage, the cost of the electromagnetic compatibility test can be reduced, and the method is fast and efficient.
Referring to fig. 1, a method for rapidly predicting a multi-platform electromagnetic interference problem includes: transmitter, receiver, antenna mathematical model, electric wave propagation model, and fast amplitude-frequency screening method. Wherein:
the transmitter, the receiver and the antenna mathematical model are established with the electric wave propagation model according to the electromagnetic parameters of the transmitter and the receiver and are used for developing the rapid amplitude-frequency screening method.
The rapid amplitude-frequency screening method is used for predicting the multi-platform electromagnetic interference problem.
In a possible implementation manner, in the method for quickly predicting the multi-platform electromagnetic interference problem provided by the present invention, the transmitter, the receiver, and the antenna mathematical model specifically include: transmitter mathematical model, receiver mathematical model, antenna mathematical model.
The transmitter mathematical model specifically comprises the following steps of establishing a relevant nonlinear model for an amplifier of a transmitter by using a power series, wherein the power series is in the form of:
the power series can simulate the higher harmonic component generated by the amplifier, and the order of the power series can be adjusted according to the actual situation in the modeling process, for example, the third-order power series model is as follows:
where A is the amplitude of the signal x (t), C1、C2、C3Is a polynomial coefficient.
The receiver mathematical model comprises the specific steps of establishing a simple receiver mathematical model based on a signal-to-interference ratio (SIR), wherein the receiver sensitivity mathematical model can be expressed as follows:
S＝-174+NF+10lg(B)+10lg(SNR)
in the formula, NF is a noise factor (generally 10dB), B is a signal bandwidth, and SNR is a demodulation signal-to-noise ratio, which is also referred to as a demodulation threshold or a minimum output signal-to-noise ratio.
The antenna mathematical model can obtain the gain of the antenna in a specific direction according to the angle. The specific steps are that gains corresponding to different azimuth angles of the antenna are packaged into a matrix form in MATLAB, and the gain value of the angle can be called after a specific angle is input. A mathematical model for a single antenna can be built using a directional pattern function. The typical directional pattern function is:
from the above formula, according to the mathematical model of the single antenna, the gain of any one direction in the three-dimensional space can be obtained through the angle parameter. For example, the specific pattern function may be:
for the establishment of the array antenna mathematical model, 5 factors of the number of array elements, the array element spacing, the distribution form, the excitation phase and the amplitude need to be considered, if one antenna array consists of M array elements, the directional diagram of the mth array element in the array is
The directional pattern function of the entire array is then:
if each array element
Similarly, the above equation can be simplified as:
s is an array factor that is related to the position of the antenna elements in the array. According to the directional diagram product, the directional diagram of the array antenna can be specifically developed as follows:
wherein A ism、αmAs array element amplitude and phase weighting coefficients, Δ RmThe distance from the m-th array element to the observation point. Taking a planar array in which M × N antenna elements located in the yoz plane form a rectangular grid array as an example, the directional diagram function can be expressed as:
referring to fig. 2 and 3, the strength of the interference signal picked up by the receiving antenna can be obtained by fries' formula.
Where Pr is the receive power, Pt is the transmit power, At is the effective aperture of the transmit antenna, Ar is the effective aperture of the receive antenna, f is the signal carrier frequency, and c is the speed of light.
Because the effective aperture of the antenna and the gain of the antenna have the following relationship:
therefore, the following Fries formula can be rewritten as:
where Pr is the received power, Pt is the transmitted power, Gt is the transmit antenna gain, Gr is the receive antenna gain, and f is the analysis frequency. It should be noted that the maximum radiation directions of the transmitting antenna and the receiving antenna are not always aligned, so the specific values of Gt and Gr are modified according to the specific paths. That is, the propagation direction of the radio wave (θ 0, φ 0) is obtained, and the antenna gain pattern is substituted to obtain Gt (θ 0, φ 0) and Gr (pi- θ 0,2 × pi- φ 0). Based on the above equation, a specific skew-received power curve can be obtained. However, due mainly to the influence of the surface structure, the actual received power of the radar is usually smaller than the theoretical calculation result. Therefore, the influence of the earth occlusion problem needs to be considered.
The earth shielding problem considered by the invention takes the isolation of a transmitting antenna and a receiving antenna as an example, the heights of the transmitting antenna and the receiving antenna from the ground are h1 and h2 respectively, the height difference is h, the slant distance between the transmitting antenna and the receiving antenna is d, and the horizontal distance between a radar and a target is d0. d and d0The relationship of (c) can be expressed as:
d2＝h2+d0 2
assuming that the position of the transmitting antenna is A, the position of the receiving antenna is B, and a tangent AC is made through the point A and the earth is tangent to the point C. Wherein the heights of the transmitting antenna and the receiving antenna are h1And h2The earth radius is ae, which is numerically about 6370 km.
When the point B is on the left side of the OC, the distance between the point AB and the point AB can be directly measured, and the problem of earth shielding does not exist; when the point B is on the right side of the OC, the earth occlusion problem can exist along with the change of the height of the point B, and the critical value can be obtained through simple calculation.
On smooth ground, even if there are no significant obstacles between the transmit and receive antennas' line of sight, additional ground scattering losses need to be introduced if there is a blockage in its first fresnel zone.
The loss of free space can be expressed by
Lfs＝32.45+20log10dkm+20log10fMHz
The radius of the first Fresnel zone can be calculated according to the following formula
According to calculation, when f is 600MHz, and d1 is d2 is 5km, R is 35.3 m. When the antenna is not sufficiently elevated, the ground scattering effect, i.e. L, will occurdf＝-F(X)-G(Y1)-G(Y2)。
In a possible implementation manner, in the method for quickly predicting the multi-platform electromagnetic interference problem provided by the present invention, the method for quickly screening amplitude and frequency specifically includes: a rapid screening method and an amplitude screening method.
Referring to fig. 4, the fast screening method specifically includes reading electromagnetic parameters of a transmitter and a receiver, primarily judging multiple platforms from the aspects of co-frequency interference, adjacent frequency interference, second harmonic interference and third harmonic interference from the aspect of frequency, deleting transmission-reception pairs which do not interfere with each other, predicting interference types which may exist in the transmission-reception pairs, and providing a basis for model selection of interference margin calculation in amplitude screening. Setting the operating frequency of the transmitter to { (f)T)min，(fT)maxBandwidth of (f)T)bw(ii) a The working frequency range of the receiver is { (f)R)min，(fR)max}. According to the electromagnetic compatibility engineering design experience, when rapid screening is carried out:
1) co-channel interference (a): the transceiver devices are at the same frequency;
2) adjacent channel interference (B): the transceiver device is at the transmitter near frequency (transmitter 20% bandwidth);
3) second harmonic interference (C): the second harmonic of the transmitting device falls within the receiver band;
4) third harmonic interference (D): the third harmonic of the transmitting device falls within the receiver band.
Referring to fig. 5, the amplitude screening method specifically includes, when it is determined that there is interference possibility according to the fast screening method, performing corresponding interference margin calculation according to the type of the interference. And calculating the same-frequency interference margin (A-IM), the adjacent-frequency interference margin (B-IM), the second harmonic interference margin (C-IM) and the third harmonic interference margin (D-IM) item by item.
If this interference margin is less than the receiver sensitivity (screening level), no significant interference is generated.
The model for calculating the interference margin is as follows:
IM＝PT-PS+GT-LTR+GR-PR+CF
wherein, PTFor transmitters at transmission frequency fTThe time-dependent transmission power, which needs to be confirmed according to the design index of the transmitter; gain G of transmitting antennaTFor transmitting antenna at transmitting frequency fTA gain in time; degree of isolation LTRFor the isolation between the transmitting antenna and the receiving antenna, the isolation can be obtained by modeling calculation by adopting electromagnetic field calculation software and can also be simplified into free space attenuation; pRFor a receiving device RnAt a receiving frequency fTA sensitivity threshold level of time; pSIs an out-of-band inhibition capability; the interference margin correction coefficient CF is set to 0 in the amplitude screening.
Example (b): a platform consisting of 5 pieces of electronic information equipment is set under a certain scene, the electronic information equipment is respectively set as A1 sets and B1 sets of transmitters, and C2 receivers and D1 sets of receivers, and specific electromagnetic parameters are as follows:
a: frequency: 500-600 MHz; emission power: 75 dBm; a directional antenna pointing in the + x direction at maximum; coordinates are as follows: (0m, 0m, 10 m);
b: frequency: 700 and 1100 MHz; emission power: 50 dBm; omnidirectional antenna, vertical polarization; coordinates are as follows: (-1000m, 0m, 5 m);
c1: frequency: 450-550 MHz; reception sensitivity: -80 dBm; omnidirectional antenna, vertical polarization; coordinates are as follows: (0m, 2000m, 5 m);
c2: frequency: 650-750 MHz; reception sensitivity: -80 dBm; omnidirectional antenna, vertical polarization; coordinates are as follows: (2000m, 2000m, 5 m);
d: frequency: 900-3000 MHz; reception sensitivity: -100 dBm; omnidirectional antennas, monopole antennas; coordinates are as follows: (0m, 3000m, 5 m);
according to the provided electromagnetic parameters such as frequency, power, sensitivity, antenna and coordinate information of the platform transmitter and the receiver, a transmitter, a receiver, an antenna mathematical model and a radio wave propagation model are established, rapid screening is carried out according to the model, and the screening results are as follows:
co-channel interference: a and C1 have same-frequency overlapping in the range of 500-550 MHz; b and C1 have same-frequency overlapping in the range of 700-750 MHz; b and D have same frequency overlap in the range of 900-1100 MHz.
And (3) adjacent channel interference: a and C1 have adjacent frequency overlapping in the range of 480-; b and C1 have adjacent frequency overlapping in the range of 650-700 MHz; b and D have adjacent frequency overlapping in the range of 1100-1180 MHz.
Second harmonic interference: second harmonic overlapping exists between A and D within the range of 1000-1200 MHz; b and D have second harmonic overlapping in the range of 1400-2200 MHz.
Third harmonic interference: the A and the D have third harmonic overlapping in the range of 1500-1800 MHz; b and D have third harmonic overlapping in the range of 2100-3000 MHz.
And according to the quick screening result, carrying out corresponding interference margin calculation on the transmitting/receiving pairs possibly having interference according to the existing interference types. And calculating the same frequency interference margin, the adjacent frequency interference margin, the second harmonic interference margin and the third harmonic interference margin item by item. The results are shown in the following table:
Claims (4)
1. a method for predicting a multi-platform electromagnetic interference problem is characterized by comprising the following steps: a transmitter, a receiver, an antenna mathematical model, a radio wave propagation model and a rapid amplitude-frequency screening method; wherein:
the transmitter, the receiver and the antenna mathematical model are established with the electric wave propagation model according to the electromagnetic parameters of the transmitter and the receiver and are used for developing the rapid amplitude-frequency screening method.
The rapid amplitude-frequency screening method is used for predicting the multi-platform electromagnetic interference problem.
2. The method for predicting the multi-platform electromagnetic interference problem of claim 1, wherein the transmitter, the receiver and the antenna mathematical model specifically comprise: a transmitter mathematical model, a receiver mathematical model and an antenna mathematical model; wherein:
the transmitter mathematical model specifically comprises the following steps of establishing a relevant nonlinear model for an amplifier of a transmitter by using a power series, wherein the power series is in the form of:
the power series can simulate the higher harmonic component generated by the amplifier, and the order of the power series can be adjusted according to the actual situation in the modeling process, for example, the third-order power series model is as follows:
the receiver mathematical model comprises the specific steps of establishing a simple receiver mathematical model based on a signal-to-interference ratio (SIR), wherein the receiver sensitivity mathematical model can be expressed as follows:
S＝-174+NF+10lg(B)+10lg(SNR)
in the formula, NF is a noise factor (generally 10dB), B is a signal bandwidth, and SNR is a demodulation signal-to-noise ratio, which is also referred to as a demodulation threshold or a minimum output signal-to-noise ratio.
The antenna mathematical model specifically comprises the step of calculating the gain of the antenna in a certain specific direction according to the angle. A mathematical model for a single antenna can be built using a directional pattern function. The typical directional pattern function is:
from the above equation, the gain to any one of the three-dimensional space directions of the angle parameter can be passed according to the mathematical model of the single antenna. For example, the specific pattern function may be:
for the establishment of the array antenna mathematical model, 5 factors of the number of array elements, the array element spacing, the distribution form, the excitation phase and the amplitude need to be considered, if one antenna array consists of M array elements, the directional diagram of the mth array element in the array is
The directional pattern function of the entire array is then:
if each array element
Similarly, the above equation can be simplified as:
s is an array factor that is related to the position of the antenna elements in the array. According to the directional diagram product, the directional diagram of the array antenna can be specifically developed as follows:
wherein A ism、αmAs array element amplitude and phase weighting coefficients, Δ RmThe distance from the m-th array element to the observation point. Taking a planar array in which M × N antenna elements located in the yoz plane form a rectangular grid array as an example, the directional diagram function can be expressed as:
3. the method of predicting the multi-platform electromagnetic interference problem of claim 2, wherein the electrical wave propagation model;
the radio wave propagation model specifically comprises the step of establishing a radio wave propagation mathematical model by using a Fourier transmission formula and a radar equation. Pr is the received power, Pt is the transmitted power, At is the effective aperture of the transmitting antenna, Ar is the effective aperture of the receiving antenna, f is the signal carrier frequency, c is the speed of light, σ is the radar cross-section, Sr is the power density of the backscatter At distance r, and Sin is the power density of the incident on the object.
The radio wave propagation model further takes into account the earth occlusion problem. On smooth ground, even if there are no significant obstacles between the transmit and receive antennas' line of sight, additional ground scattering losses need to be introduced if there is a blockage in its first fresnel zone.
The loss of free space can be expressed by
Lfs＝32.45+20log10dkm+20log10fMHz
The radius of the first Fresnel zone can be calculated according to the following formula
4. The method for predicting the multi-platform electromagnetic interference problem of claim 3, wherein said fast amplitude-frequency screening method; the method specifically comprises the following steps: a rapid screening method and an amplitude screening method; wherein:
the quick screening method comprises the specific steps of carrying out preliminary judgment from the frequency perspective, deleting the transmitting/receiving pairs which cannot interfere with each other, predicting the possible interference types of the transmitting/receiving pairs, and providing a basis for selecting an interference margin calculation model in the amplitude screening method.
The amplitude screening method specifically comprises the steps of judging from the amplitude of an interference signal, and realizing accurate prediction of the multi-platform electromagnetic interference problem by calculating and analyzing the interference margins of a transmitter and a receiver. The interference allowance comprises a same frequency interference allowance (A), an adjacent frequency interference allowance (B), a second harmonic interference allowance (C) and a third harmonic interference allowance (D).
And when the possibility of interference is judged according to the quick screening method, corresponding interference allowance calculation is carried out according to the existing interference type. And calculating the same-frequency interference margin (A-IM), the adjacent-frequency interference margin (B-IM), the second harmonic interference margin (C-IM) and the third harmonic interference margin (D-IM) item by item.
If this interference margin is less than the receiver sensitivity (screening level), no significant interference is generated.
The model for calculating the interference margin is as follows:
IM＝PT-PS+GT-LTR+GR-PR+CF
wherein, PTFor transmitters at transmission frequency fTThe time-dependent transmission power, which needs to be confirmed according to the design index of the transmitter; gain G of transmitting antennaTFor transmitting antenna at transmitting frequency fTA gain in time; degree of isolation LTRFor the isolation between the transmitting antenna and the receiving antenna, the isolation can be obtained by modeling calculation by adopting electromagnetic field calculation software and can also be simplified into free space attenuation; pRFor a receiving device RnAt a receiving frequency fTA sensitivity threshold level of time; pSIs an out-of-band inhibition capability; the interference margin correction coefficient CF is set to 0 in the amplitude screening.
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Cited By (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN115643136A (en) * | 2022-09-29 | 2023-01-24 | 北京航空航天大学 | Multi-domain cooperative spectrum interference method and system based on evaluation indexes |
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