CN113030888A - Axial correction method for measurement error of polarized phased array radar - Google Patents

Abstract

The invention relates to the technical field of meteorological radar axial correction, and discloses an axial correction method for a polarized phased array radar measurement error. The invention corrects the polarization measurement error through the non-orthogonality of the radiation electric field in the direction of the polarized phased array antenna wave beam, and is suitable for the phased array radar, so thatZ _DRIf the measurement error is less than 0.1dB, the relative error of the antenna directional diagram measurement is less than 1%, and the measurement is fast and accurateThe polarization measurement performance is analyzed, the measurement precision is high, and the polarization phased array radar antenna measurement is of great practical value.

Description

Axial correction method for measurement error of polarized phased array radar

Technical Field

The invention relates to the technical field of meteorological radar axial correction, in particular to an axial correction method for a measurement error of a polarization phased array radar.

Background

At present, meteorological observation usually adopts meteorological radar observation, the meteorological radar has certain range resolution and Doppler resolution, and parameter measurement of a target is respectively carried out in different resolution units. Generally, the echo sampled signals of each range bin are corrected by correlation and the difference reflectivity Z is used_DRAnd linear depolarization ratio L_DRAs an index to measure the performance of the polarization error correction method. The polarization measurement error of the general mode is large because the general mode introduces not only a second-order error but also a first-order error of the antenna cross-polarization component.

In antenna measurements, the antennas are usually placed in the xy plane, and it is well known that the radiation pattern of a single antenna is very different from its pattern in a finite array. This is because, in a finite array, when the antenna element is excited, the radiated electromagnetic field is received, reflected and re-radiated by other elements in the array. This electromagnetic interaction between the antenna elements is called Mutual Coupling. In analyzing the array antenna Pattern, an Active Element Pattern (AEP) is generally used. AEP is defined as the antenna pattern when a certain element is excited and all other elements are loaded with matching loads.

First, for a real radar antenna, an electric field is radiated

And

it needs to be obtained by measurement, and the antenna measurement inevitably introduces measurement errors, so the measurement errors of the antenna pattern will have an influence on the correction based on equation (2.41). Secondly, in meteorological observation, the observed objects are all distributed objects, such as rainfall, snowfall, cloud layers and the like, and therefore the received voltage equation should be expressed in an integral form. Then, for a practical array antenna, the mutual coupling between the array elements can have a significant impact on the characteristics of the antenna pattern. Again, the disparity of the H and V channels can also have an effect on the accuracy of the polarization correction. To make Z_DRThe measurement error of the antenna directional diagram is less than 0.1dB, the relative error of the antenna directional diagram measurement is less than 1%, and the high measurement precision provides a serious challenge for the measurement of the antenna directional diagram of the polarized phased array radar.

Disclosure of Invention

In order to overcome the defects of the prior art, the invention aims to provide an axial correction method for the measurement error of the polarized phased array radar.

In order to achieve the purpose, the invention adopts the technical scheme that:

an axial correction method for a polarized phased array radar measurement error comprises the following steps:

firstly, establishing an array model

1) Active cell direction, the active direction of the cell in the mth row and nth column is denoted as

2) Array direction, transmission and reception pattern of the array F_T(θ,φ；θ_S,φ_S) And F_R(θ,φ；θ_S,φ_S) Is shown as

3) A receive voltage equation, and in ATSR mode, when only the H port is excited, the electric field radiated by the phased array antenna

Is composed of

When the incident field is

The scattered field of a single raindrop is

Horizontal channel receiving voltage dV_hhAnd a vertical channel receiving voltage dV_vhAnd scattering electric field

In a relationship of

Secondly, deriving a receiving voltage equation,

1) axial correction method-ATSR mode, using projection matrix as reference, based on (2.31), the corrected receiving voltage equation is expressed as

Wherein

Representing the corrected received voltage matrix, C_TAnd C_RTo correct the matrix, expressed as

2.1 theorem, setting the direction of a real dual-polarized antenna unit as f, the absolute error of the antenna direction measurement as e, and | e_ij|＝|f_ijIf there is

(f^-1+e^-1)^-1≈e (2_.62)

Prove that the matrix inversion theorem is utilized, have

Due to | e_ij|＝|f_ijI, have

e+f≈f (2.64)

According to theorem 2.1, get δ_ijAnother expression of upper bound estimation

Is provided with

As the relationship with gamma becomes larger (cross polarization becomes larger),

and then becomes larger;

(2) calibration performance analysis under ideal H/V channel conditions,

1) in the case of a single spherical raindrop, if the raindrop polarization scattering matrix is a unit matrix, (2.31) is simplified

2) In case of a large amount of spherical raindrops, when the parameter α is_ij、β_ijAnd delta_ijUnknown, a Monte Carlo simulation-based method is adopted to analyze the polarization measurement performance of the phased array radar;

(3) calibration performance analysis under non-ideal H/V channel conditions

1) The non-ideal H/V channel is modeled,

the non-idealities of the transmit and receive channels are represented by two 2 x 2 matrices, a and B, which are expressed as follows

2) In the case of a single spherical raindrop, the inclusion of H/V channel imperfections

And

is shown as

In the STSR mode, for a distributed target composed of a large number of raindrops, the received signal is represented as

It s_h(t) and s_v(t) waveform divided into H and V port transmissions, set S_hv(θ,φ)＝S_vhWhen (θ, Φ)' is 0, (2.114) is

In practice, s_h(t) and s_v(t) cannot be perfectly orthogonal, so define

Then

Thereby obtaining

When the waveform s_h(t) and s_vWhen (t) is known, Q is a constant matrix, and (2.120) is also equivalent to (2.31).

Due to the adoption of the technical scheme, the invention has the following advantages:

an axial correction method for measurement errors of a polarized phased array radar is characterized in that a complete dual-polarized unit directional diagram, a dual-polarized array transmitting and receiving directional diagram model and a distributed rainfall particle receiving voltage equation are established from a basic array antenna of a meteorological radar. Then, a projection matrix method and an axial correction method are expanded, polarization measurement errors caused by the non-orthogonality of the radiation electric field in the direction of the polarized phased array antenna beam are corrected through the axial correction method, the polarization error correction performance is realized under two modes of ATSR and STSR, and the engineering design is carried out by utilizing the coupling relation among the axial correction errors, the polarization measurement errors, the antenna directional diagram measurement relative errors, the channel amplitude-phase inconsistency and the polarization isolation degree.

The invention simulates and analyzes the polarization measurement performance of the phased array radar, and limits the wave beam width to L_DRThe effect of the measurement is significant and on Z_DRThe effect of the measurement was not significant. L is the beam width of the phased array, which varies with beam pointing_DRThe measurement performance of (c) also varies with beam pointing. The wider the beam, L_DRThe poorer the accuracy of the measurement. In contrast, Z_DRThe measurement performance of the beam is not obviously changed along with the beam pointing.

The invention is suitable for phased array radar, makes the transmitting directional diagram and the receiving directional diagram reciprocal, corrects the polarization measurement error by the non-orthogonality of the radiation electric field in the direction pointed by the polarized phased array antenna wave beam, and quickly and accurately analyzes the polarization measurement performance, so that Z is enabled to be in a Z shape_DRThe measurement error is less than 0.1dB, the relative error of the antenna directional diagram measurement is less than 1%, the measurement precision is high, and the method has great practical value for the polarized phased array radar antenna measurement.

Drawings

FIG. 1 is a diagram of a spherical coordinate system and a planar array antenna structure;

FIG. 2. epsilon_ijA real part diagram of (a);

FIG. 3 ε_ijAn imaginary part diagram of (a);

FIG. 4 delta_ijA simulation result graph;

FIG. 5 matrix f (θ)_S,φ_S) An infinity norm based condition number versus γ plot;

FIG. 6

A graph of variation with Δ;

FIG. 7

A graph of variation with Δ;

FIG. 8 is a schematic diagram of a T/R assembly of a polarized phased array radar;

FIG. 9 is a diagram of a non-ideal H/V channel model;

FIG. 10 shows a single raindrop and a large number of raindrops

A simulation result graph;

FIG. 11 shows a single raindrop and a large number of raindrops

A simulation result graph;

FIG. 12 Ideal and nonideal H/V channel conditions

A simulation result graph;

FIG. 13 conditions of ideal and non-ideal H/V channels

And (5) a simulation result graph.

Detailed Description

As shown in fig. 1 to 13, an axial calibration method for polarization phased array radar measurement errors analyzes the polarization measurement performance of the phased array radar, firstly, a complete array model is established, and a receiving voltage equation is derived from the complete array model; a spherical coordinate system and planar array antenna configuration is shown in figure 1,

and

is a unit vector under a spherical coordinate system. The planar array antenna is located on the yz plane and comprises N_rowRow and N_colAnd (4) columns.

Representing the electric field radiated by the H-port in the beam axial Direction (Boresight Direction),

representing the electric field radiated by the V-port in the beam axis direction. The array scans over-45 to 45 in azimuth and 0 to 30 in pitch. For an array antenna, its directional diagram should be symmetric about phi, which is limited to [0 °,45 ° ]]Theta is in the range of [60 DEG, 90 DEG ]]. The antenna is placed on the yz plane, and the definition of the horizontal and vertical polarization bases and the unit vector in the spherical coordinate system

And

are consistent.

The active cell pattern, in fig. 1, the active pattern of the cell located in the mth row and nth column can be represented as

Wherein f is_hh(θ, φ) is the radiated electric field component in the H direction when the H port is excited; f. of_hv(θ, φ) is the radiated electric field component in the H direction when the V port is excited; f. of_vh(θ, φ) is the radiated electric field component in the V direction when the H port is excited; f. of_vv(θ, φ) is the radiated electric field component in the V direction when the V port is excited.

Array directional diagram, namely a transmitting directional diagram and a receiving directional diagram F of the array based on the active unit directional diagram_T(θ,φ；θ_S,φ_S) And F_R(θ,φ；θ_S,φ_S) Can be expressed as

Where the subscripts "T" and "R" denote Transmission (Transmission) and Reception (Reception), respectively. X_mn(θ_S,φ_S) And Y_mn(θ_S,φ_S) Respectively, the complex weighting coefficients of the units mn.

Since each dual polarized antenna element has two ports, H and V, strictly speaking, Xmn (θ)_S,φ_S) And Y_mn(θ_S,φ_S) Should be represented as a 2 x 2 matrix. For example, the transmit pattern weighting coefficient matrix should be expressed as

Where columns 1 and 2 represent the excitation of the H and V ports, respectively. Is provided with

Thus, it is possible to provide

Can be expressed as a scalar coefficient

For receive weighting, the H and V ports are typically weighted separately for optimal measurement performance. Here the same weighting is applied to the H and V beams. Thus also using a scalar coefficient

To represent the weighting of the received beams.

(2.15) and (2.16) give a general form of the array transmit and receive pattern. Since the active cell patterns of the individual cells are not exactly the same, it is inconvenient to analyze the polarization measurement error and correct the problem with (2.15) and (2.16). On the other hand, for a large array, since most cells are in very similar array environments, it is assumed here thatThe active patterns of the cells in a large array are all the same. On the basis of this, the method is suitable for the production,

and

can be simplified into

Wherein f (theta; phi) represents the active directional diagram of each cell, AF_TAnd AF_RIs an array factor expressed as follows

Wherein

(as shown in FIG. 1) is the position vector of the unit mn, k 2 π/λ is the wavenumber, λ is the wavelength, and the exponential term

The phase difference of the cell mn and the reference cell with respect to the target point is shown.

Receive voltage equation in ATSR mode, when only the H-port is excited, the phased array antenna radiates the electric field

Is composed of

Assuming that there is a single raindrop in the beam direction, the polarization scattering matrix is S', when the incident field is

The scattered field of a single raindrop is

Scattered electric field

The voltage is generated after being received by the antenna, and according to the antenna theory, the horizontal channel receives the voltage dV_hhAnd a vertical channel receiving voltage dV_vhAnd scattering electric field

In a relationship of

Wherein

(2.24) can be expressed in a matrix form as follows

To simplify the analysis, terms relating to distance and gain are omitted (2.27). It should be noted that S' (θ; φ) includes the effects of attenuation and phase shift during propagation of electromagnetic waves, and the expression is shown below

Where T represents a one-way path transmission matrix describing the attenuation and phase shift in the propagation of the electromagnetic wave. S (θ; Φ) represents a polarization scattering matrix inherent to a single raindrop.

When only the V port is excited, the voltage component dV is received_hvAnd dV_vvCan be expressed as

(2.27) and (2.29) may be combined into

For a large number of raindrops distributed in space, the received voltage V may be expressed as

Where Ω denotes a solid angle, d Ω is sin θ d θ d Φ. It should be noted that equation (2.31) is only a mathematical processing method, and two columns in the matrix V respectively represent voltage components measured at different times. In the following analysis and derivation, the identity matrix in (2.31) will be omitted.

Since the echoes of a large number of raindrops are incoherent, the total received power P_ij(i, j ═ h, v) can be represented by

P_ij＝∫_Ω<|dV_ij|²>dΩ (2.32)

Wherein<·>Representing the ensemble average. Based on (2.32), differential reflectivity Z_DRIs defined as

If S'_hv＝S′_vhThe actual measured linear depolarization ratio then represents the error in the linear depolarization ratio, which is 0.

In S'_hv＝S′_vhLinear depolarization ratio error under the assumption of 0

Is defined as

The "Projection Matrix" method is generally used to correct polarization measurement errors caused by the non-orthogonality of the H and V port radiated electric fields. When the dual-polarized antenna unit only radiates electromagnetic waves at the H port, an electric field is radiated

Can be expressed as

When only the V port radiates electromagnetic waves, an electric field is radiated

Can be expressed as

And

are respectively as

And

unit vector in direction. Under the H/V polarization base, the electric field radiated by the dual-polarized antenna unit can be expressed as

Wherein the projection matrix P can be expressed as

The physical meaning of the projection matrix P is "to polarize the basis

And

the characterized radiation electric field is transformed into a radiation electric field characterized by H and V polarization radicals. Mathematically, P represents a unit vector that would be two non-orthogonal

And

projected to orthogonal unit vectors

And

the above. Based on the projection matrix P, the received voltage equation of a single point target can be expressed as

Wherein S is a target polarization scattering matrix, and superscript t represents matrix transposition. To simplify the analysis, the term relating to the distance r and the gain is omitted from equation (2.39).

In the ATSR mode, when the H and V ports alternately transmit the unit signal (amplitude of 1, phase of 0), the reception voltage equation can be expressed as

Thus, the target polarization scattering matrix S can be expressed as

S＝C^t·V·C (2.41)

Wherein C ═ P^-1Referred to as the projection correction matrix.

As can be seen from (2.41), the projection matrix correction method implies the following assumptions:

electric field radiated by H-port and V-port

And

the accuracy is known; 2. the target is a point target; 3. the radiation characteristics of all the antenna units are completely consistent; the amplitude-phase characteristics of the H and V channels are ideal; 5. the radar transmission and reception patterns are the same, i.e. the radar is reciprocal.

The axial correction method, namely ATSR mode, is based on (2.31) by using a projection matrix method for reference, and the corrected receiving voltage equation is expressed as

Wherein

Representing the corrected received voltage matrix, C_TAnd C_RTo correct the matrix, can be expressed as

And

indicating measured array transmit and receive patterns in the beam direction (theta)_S,φ_S) The value of (c) above.

The "projection matrix correction method" can be regarded as a "array element level" correction method, and the correction method shown in (2.42) can be regarded as an "array level" correction method. Strictly speaking, the active patterns of the array elements are different, so the correction method given by (2.42) is more general. Since the correction matrices (2.43) and (2.44) are based only on the information of the array transmission and reception direction axes, the correction method given by (2.42) is called an axial correction method (Boresight correction).

Projection matrix correction methods assume that the correction matrix is precisely known, so in axial correction methods, the correction matrix is defined based on the measured array transmit and receive patterns. In addition, the received electric field in the axial correction method is expressed as an integral of a space domain, which represents the characteristics of a distributed target such as raindrops. In conclusion, (2.42) extends the "projection matrix correction method".

As can be seen from (2.43) and (2.44), the axial correction includes two layers: one is to

And

correcting for non-orthogonality; the second is to compensate for the non-uniformity of the transmit and receive beam gains. Based on (2.18) and (2.19), C_TAnd C_RCan be simplified into

Wherein f is_m(θ_S,φ_S) Representing the measured antenna element pattern.

(1) Linear model

(3.42) can be expressed as

Wherein

And

referred to as the corrected array antenna pattern, which is defined as

Note that (2.47) is mathematically similar to (2.31), so the corrected received power

Can still be calculated as (2.32) except that F_TAnd F_RTo be replaced by

And

it should be noted that (2.48) and (2.49) are only mathematical processes, and that in practice the correction matrix only acts on the received voltage V.

In the in-depth analysis of (2.47), first, the corrected cross-polarization pattern pair will be discussed

And

the influence of (c). In order to eliminate the influence of the scattering matrix of the target, assuming that S' is a unit matrix, the (2.47) is expanded into a scalar form, and

in the case of (2.50), the,

and

contains the second order term of the cross-polarization pattern, and

and

the expression (c) contains the first order term of the cross-polarization pattern. Therefore, the temperature of the molten metal is controlled,

is insensitive to cross-polarization patterns, and

is sensitive to cross-polarization patterns. In the case of the STSR mode,

containing the first order term of the cross-polarization pattern. Therefore, the cross-polarization pattern requirements for the ATSR mode are lower than for the STSR mode.

For the purpose of deep analysis of (2.47), the active patterns of the individual antenna elements are assumed to be identical. Based on (2.45) and (2.46),

and

can be expressed as

If measured cell pattern f_m(θ_S,φ_S) Exactly without any error, then f (θ, φ) · f_m(θ_S,φ_S)^-1In the beam direction (theta)_S,φ_S) Above would be an identity matrix. In practice, however, f (θ, φ) f due to the effect of antenna pattern measurement errors_m(θ_S,φ_S)^-1In the beam direction (theta)_S,φ_S) Not an identity matrix.

Therefore, will

And

is modeled as

Wherein ε ij (θ, φ; θ)_S,φ_S) (i, j ═ h, v) is called the axial correction error, and describes the effect of finite antenna beamwidth on polarization measurement error.²AF_TAnd²AF_Ris a normalized array factor, i.e.

And

through the actually measured and simulated directional diagram of the dual-polarized microstrip patch antenna unit, the directional diagram of the antenna unit can be approximated to a linear function of theta and phi. To this end epsilon_ijApproximated as a linear function of theta and phi, as follows

Wherein alpha is_ij，β_ijAnd delta_ijAre complex parameters. The linear model of the axial correction error is called (2.55), the model describes the variation relation of the correction error of the axial correction method along with the spatial angles theta and phi, and the rationality of the linear model is that the directional diagram of the electrically small-sized dual-polarized antenna unit is emptyThe fact that is a slowly varying function of the angles theta and phi.

The rationality of the linear model shown in (2.55) is verified, and the rationality of (2.55) is demonstrated by using a simulated dual-polarized microstrip patch antenna element pattern. Corrected microstrip patch antenna directional pattern

Is shown as

Selecting (theta)_S,φ_S) Based on the simulated antenna element pattern, e is calculated (60 °,45 °)_ijThe real and imaginary parts are shown in fig. 2 and 3, respectively. As can be seen from FIGS. 2 and 3,. epsilon_ijThe linear relationship between theta and phi in the neighborhood of (60 degrees, 45 degrees) illustrates the rationality of the linear approximation shown in (2.55), indicating the axial correction error epsilon_ijIt is indeed a slowly varying function of the spatial angles theta and phi.

To further verify the linear approximation in FIGS. 2 and 3, a Matlab Curve Fitting kit (Current Fitting Toolbox) was used, with θ and φ as parameters, for ε_ijA linear fit was performed and the results are shown in table 2.1. In Table 2.1, R²(R-square) is called Coefficient of Determination (Coefficient of Determination) and is between 0 and 1. It describes how well the model fits to the given data. As can be seen from the table, R²Are all very close to 1, which further verifies the rationality of the linear model (2.55) and also shows that the axial correction error epsilon is approximated by a linear function_ijWith high accuracy.

TABLE 2.1 Linear model parameter fitting results

	α	β	δ	R2
					εhh	0.4266+0.0558j	-0.8391+0.0069j	-0.0015-0.0001j	0.97
εhv	0.3152+0.0236j	-0.2178+0.0279j	0.0003+0.0002j	0.98
					εvh	-0.5297+0.0236j	0.4412-0.0307j	-0.0014+0.0000j	0.98
εvv	0.4187-0.0198j	-0.6536-0.0823j	-0.0014+0.0001j	0.96

Linear model (2.55) parameter α_ij,β_ijAnd delta_ijHas profound physical significance. From (2.55), α_ijAnd beta_ijRepresents epsilon_ijThe rate of change in theta and phi. In addition, if the measured cell pattern is completely accurate with no error, then there is ε_ij(θ_S,φ_S；θ_S,φ_S) 0, i.e. delta _ij0. Whereas in practice due to errors in antenna pattern measurements delta_ijNot equal to 0. Hence the term δ_ijTo correct the error axially, it characterizes the accuracy of the axial correction. In view of δ_ijOf the measured dual-polarized antenna element pattern f_m(θ_S,φ_S) Can be expressed as the sum of the true direction diagram and the absolute measurement error, i.e.

Wherein f (theta)_S,φ_S) Representing the true cell pattern, e (θ)_S,φ_S) Representing the absolute error of the antenna pattern measurement. Corrected cell directivity pattern

Can be expressed as

For the convenience of derivation, in (2.59) (2.81),

and

are respectively abbreviated as

f and f_m. Using momentsInverse lemnism of the array to obtain

(f_m)^-1＝(f+e)^-1＝f^-1-f^-1(f^-1+e^-1)^-1f^-1 (2_.59)

It is assumed here that e is invertible. Substitution of (2.59) into (2.58) can give

Where I denotes an identity matrix. Further can obtain

(2.61) indicates. delta_ijAs well as the absolute measurement error e and the antenna element pattern f itself. Theorem 2.1 gives the relation between the true antenna element pattern f and the measured absolute error e.

2.1 theorem, setting a real dual-polarized antenna unit directional diagram as f, an antenna directional diagram measurement absolute error as e, and | e_ij|＝|f_ijIf there is

(f^-1+e^-1)^-1≈e (2_.62)

Prove that the matrix inversion theorem is utilized, have

Due to | e_ij|＝|f_ijI, have

e+f≈f (2.64)

Therefore (2.63) can be approximated as

(e^-1+f^-1)^-1-e≈-e·f^-1·e (2_.65)

Based on the matrix norm theory, there are

Wherein | · | purple_∞Representing the matrix ∞ -norm. Further, (2.66) can be expressed as

(2.67) can be understood as meaning in the matrix ∞ -norm with (e)^-1+f^-1)^-1To approximate the relative error of e. It is noted that

Wherein | f | purple_∞·||f^-1||_∞The condition number based on the ∞ -norm for the matrix f. Definition of κ (f) ═ f | | non-conducting phosphor_∞·||f^-1||_∞Then (2.68) can be expressed as

According to the definition of infinity-norm of matrix, | | e | | ventilation_∞Can be expressed as

||e||_∞＝max{|e_hh|+|e_hv|，|e_vh|+|e_vv|} (2.70)

Since the relative error is more reflective of the measurement accuracy, we define the upper bound of the relative error of the antenna element pattern measurement

Strictly speaking, the absolute measurement error e_ijIs random and is based onDepending on the spatial distribution of the antenna element main polarization and cross polarization patterns. Thus, E^f _ijAlso depending on the spatial distribution of the antenna element main polarization and cross polarization patterns. To simplify the analysis, assume E^f _ij＝E_fAnd E_fIs a constant within the beam sweep area.

Thereby can obtain

Based on (2.72) have

Without loss of generality, assume | f_hh|+|f_hv|≥|f_vh|+|f_vvIf (2.73) can be expressed as

According to (2.70), a

||e||_∞≤E_f·||f||_∞ (2.75)

Substituting (2.75) into (2.69) there are

||-e·f^-1||_∞≤E_f·κ(f) (2.76)

Therefore, (2.67) can be expressed as

For a designed dual polarized antenna element, if the antenna cross polarization is below-10 dB, then there is κ (f)<2. Meanwhile, in the antenna measurement, E_f<5% is also easily satisfied. Thus, E_fκ (f) is a very small amount.

Then there are

(e^-1+f^-1)^-1≈e (2_.78)

Bring (2.78) into (2.61), have

Then delta_hhCan be estimated as

Can similarly obtain

Since the derivation (2.81) uses the approximate relation (f)^-1+e^-1)^-1E, therefore (2)_.81) Is only delta_ijAn approximate estimate of the upper bound. To verify the estimated performance of (2.81), the following numerical simulation example is given. Suppose that

And E_fBased on (2.81), we calculate δ as 1%_ijAs shown in (2.83)

On the other hand, the absolute measurement error e is generated by a random number generator such that e_ijSatisfy the requirement of

Generation of e_ijThen, δ was calculated directly using (2.61)_ijThe simulation results are shown in fig. 4. | δ in fig. 4_hh∣,∣δ_hv∣,∣δ_vh| and | δ_vv| the maximum values are 0.0160, 0.0133, 0.0129 and 0.0162, respectively, which is in good agreement with the estimation result of (2.83). Thus, δ is estimated using equation (2.83)_ijThe upper bound of (c) is reasonable.

Further numerical simulation analysis shows that when k (f (theta))_S,φ_S) (matrix f (θ))_S,φ_S) Infinity-norm based condition number) less than 2, while E _f5% or less, and the formula (2.81) has good estimation accuracy, and these conditions (k (f (theta))_S,φ_S) < 2 and E)_f≦ 5%) may be satisfied in antenna design and measurement. Therefore, δ was analyzed by (2.81)_ijThe upper bound of (c) is reasonable. As can be seen from the simulation results of FIG. 4 and the approximate estimation based on (2.83), δ_ijUpper bound of (E) and relative error E_fIn the same order of magnitude.

According to theorem 2.1, δ can be obtained_ijAnother more general expression for upper bound estimation

From (2.85), δ_ijUpper bound and relative measurement error E_fAnd an antenna element pattern f (theta)_S,φ_S) There is a relationship. And κ (f (θ)_S,φ_S) Is characterized by f (theta)_S,φ_S) The magnitude of the cross-polarization. This is explained below by way of an example. Suppose that

The relationship with γ is shown in fig. 5. As can be seen from fig. 5, when γ becomes large (cross polarization becomes large),

and therewith becomes larger.

(2.81) and (2.85) give 2 different estimates of δ_ijThe method of the upper bound, it should be noted that these 2 methods are not equivalent. The estimation of (2.81) facilitates numerical calculation, while (2.85) is suitable for theoretical analysis. As mentioned above, α_ijAnd beta_ijRepresents epsilon_ijThe rate of change in theta and phi. It is therefore necessary to discuss the antenna pattern measurement error vs_ijThe effect of the spatial rate of change. To study epsilon_ijSpatial rate of change of (2), definition

Wherein

And

the corrected spatial rate of change of the antenna element pattern is described,

i.e. epsilon_ijThe spatial rate of change of (c). According to (2.58), the compound (I) can be obtained

Since the antenna element pattern measurement error is generally very small, there is f_m(θ_S,φ_S)≈f(θ_S,φ_S) From this can be obtained

Wherein

And

the spatial rate of change of the corrected element pattern is described when the antenna element pattern measurements are error free. As can be seen from (2.90) and (2.91), when the antenna element pattern measurement error is small, the measurement error is epsilon_ijThe spatial rate of change effect is also small.

(2) And (4) performing correction performance analysis under an ideal H/V channel condition, and only considering the influence of antenna element directional diagram measurement errors and limited beam width on polarization measurement errors under the condition that the amplitude-phase characteristics of the H channel and the V channel are consistent.

In the case of a single spherical raindrop, first, we assume that there is only one spherical raindrop in the beam pointing direction, and no raindrop at other angles. In this case, there is Z_DR0dB and L_DRInfinity dB. Since only a single raindrop is considered,

the integral over the entire spatial domain in (2.31) can be removed. In the case of a single spherical raindrop, assuming that the raindrop polarization scattering matrix is a unit matrix, (2.31) can be simplified to

From (2.92) may be

Further assume | δ_ij∣＝Δ，δ_ijIn the phase of [0,2 π]Are uniformly distributed. Taylor expansion is performed on (2.93) and the higher-order terms are ignored, then | Z_DRThe mathematical expectation of | may be approximated as

Wherein

The representation takes a mathematical expectation.

At | δ_ijΔ | ═ and Arg (δ)_ij):[0,2π]Under the assumption that the first and second images are different,

symmetrically distributed around 0, then

Hence we calculate here

Is a mathematical expectation of

FIG. 6 shows

The variation with Δ, wherein the red line is calculated based on (2.94) and the blue line is simulated based on Monte Carlo. In Monte Carlo simulation, | δ is set_ijΔ | ═ Δ, and δ_ijIs generated by a random number generator,

calculated according to (2.93), and then obtained by multiple times of simulation

Are averaged to obtain the final

As can be seen from fig. 6, the approximation based on equation (2.94) agrees well with the Monte Carlo simulation.

Using a similar derivation, one can obtain

FIG. 7 shows

Variation with Δ, wherein the red line is calculated using (2.95), the blue line is simulated using Monte Carlo, the Monte Carlo simulation process is aligned with that of FIG. 6

The simulation is similar. As can be seen from FIG. 7, the (2.95) -based approximation matches well with the simulation results using Monte Carlo.

As can be seen from fig. 6 and 7, the error δ is corrected in the axial direction_ijTo pair

And

the effect of (a) is significant. For a single spherical raindrop, it is sufficient

Δ is less than 0.01. This means that the relative error of the antenna element pattern measurements is of the order of 1%. Such high pattern measurement accuracy requirements present significant challenges to antenna measurements. On the other hand, FIGS. 6 and 7 show

Increases approximately linearly with increasing delta, and

the increase in (c) increases approximately logarithmically with increasing Δ.

In the case of a large number of spherical raindrops, it can be seen from the previous analysis that if the parameter α of the linear model (2.55) is a_ij、β_ijAnd delta_ijAs is known, the polarization measurement performance of the phased array radar can be calculated from (2.53), (2.54), and (2.47). However, since the actual array pattern is not yet available, the parameter α is_ij、β_ijAnd delta_ijIs unknown. Therefore, a Monte Carlo simulation-based method is provided for analyzing the polarization measurement performance of the phased array radar, and the method comprises the following steps: step 1: pointing (θ) for a given beam_S,φ_S)，α_ij、β_ijAnd delta_ijGenerated by a random number generator, in turn e_ijCalculated from (2.55);

Step 2:

and

calculated from (2.53) and (2.54);

step 3 corrected received power

Calculated from (2.32);

Step 4:

and

calculated from (2.33) and (2.34), where n represents the nth simulation.

In the beam pointing direction (theta)_S,φ_S) Repeating the above simulation, and then comparing the obtained results

And

averaging to obtain the direction of a given beam

And

calculated by the above method

And

is represented by the parameter alpha_ij、β_ijAnd delta_ijThe average value under a certain distribution of (a) is a statistical description of the polarization measurement performance.

Table 2.2 shows the corresponding simulation parameters, where U (a, b) denotes the position [ a, b ]]Arg (z) denotes the phase of complex number z. From table 2.1, it can be seen that | α is for a well-designed microstrip patch antenna element_ij|p1，|β_ij| p 1. By further analysis, | α is known for infinitesimal electric dipoles_ij|≤2，|β_ijI.ltoreq.2, so we assume here. | α_ij|≤2，|β_ijLess than or equal to 2. As can be seen from Table 2.1, α_ij、β_ijIs much larger than the imaginary part, indicating a_ij、β_ijIs close to 0, so its phase is set in table 2.2 to

Are uniformly distributed.

TABLE 2.2 Monte Carlo simulation parameters

First, an example is analyzed, namely a correction matrix C_TAnd C_RIs completely accurate and contains no errors. Under this condition, there is delta _ij0. In this chapter, if not otherwise specified, Z is assumed to be_DR＝0dB，L_DRInfinity dB. Obtained by Monte Carlo simulation

And

within the entire beam scanning range,

less than 3X 10^-3dB，

Below-36 dB. The result is a hypothetical correction matrix C_TAnd C_RAre fully accurate and therefore these results can be considered as the best results under the given conditions and can be used as a reference for other simulation results. In addition, the first and second substrates are,

the increase from the array normal to the beam pointing (60, 45) is about 2.8dB, which indicates that

The measured performance of (a) is varied with beam pointing.

Still set delta _ij0 while increasing | α_ij| and | β_ijThe distribution range of | is, within the whole beam scanning range,

less than 8 x 10^-3dB，

Increasing from-33.34 dB to-30.4 dB, an increase of approximately 2.9 dB.

The effect of antenna element pattern measurement errors on polarization correction performance is analyzed below. Setting | δ_ij0.01 and will | α_ij| and | β_ijThe range of | varies from U (0,1) to U (0,2), over the entire beam sweep,

there was a slight fluctuation in the vicinity of 0.1dB, which indicates that

Hardly depending on the beam scanning direction, and is given by | α_ij| and | β_ijThe amplitude of | is not sensitive. On the contrary

As the beam is pointed. From

Increased from-35.4 dB to-32.8 dB (increased by about 1.6dB), and

increasing from-31.78 dB to-29.55 dB (an increase of about 2.2 dB). This means that

Also with | α_ij| and | β_ijThe amplitude distribution range of l is relevant.

Axial correction error delta_ijHas a great influence on the performance of polarization correction, and determines

And

the lower limit that can be reached.

In addition, the fluctuation speed of the spatial domain polarization of the antenna unit is |, alpha_ij| and | β_ijThe magnitude of the l is such that,

to L_DRThe measurement of (2) also has a large influence. Therefore, in practice, an antenna unit pair with small fluctuation of space domain polarization characteristics is designed to improve L_DRThe accuracy of the measurement is beneficial.

Simulation results show that the aim is to achieve

The relative error of antenna element pattern measurements is up to 1%, which is consistent with previous analysis based on a single spherical raindrop. The requirement of 1% of antenna unit directional diagram measurement accuracy is difficult to achieve in practice, and the requirement of 5% of antenna unit directional diagram measurement accuracy is easy to meet in practice.

Based on | δ_ijThe simulation result of 0.05 is |,

there is little fluctuation around 0.5 dB.

Increasing from-25.4 dB to-24.4 dB,

increasing from 24.4dB to-23.2 dB.

(3) Calibration performance analysis under non-ideal H/V channel conditions

And (4) non-ideal H/V channel modeling, and FIG. 8 is a schematic diagram of T/R components of the polarized phased array radar in an ATSR mode and an STSR mode. Non-idealities in the T/R components can cause coupling between the H and V channels and amplitude phase inconsistencies, thus affecting the accuracy of the polarization measurement.

Channel Isolation (CIS) is used to indicate coupling between H and V channels, and Channel Imbalance (CIM) is used to indicate non-uniformity of amplitude and phase of H and V channels. The non-ideal H/V channel model is shown in FIG. 9, where a_ijAnd b_ijTo indicate the coupling and amplitude phase inconsistency of the H and V channels. Two 2 x 2 matrices A and B are used to represent the non-idealities of the transmit and receive channelsThe expression is shown below

Wherein a is_hh、a_vv、b_hhAnd b_vvDescribe the imbalance of the H/V channel, and a_hv、a_vh、b_hvAnd b_vhCoupling between H/V channels is described. To simplify the analysis, assume a_hh＝b _hh1, CIM is thus defined as

Note that if | a_vvIf | is greater than 1, then there is

For CIM to be a non-negative value, use is made of

Similarly, CIS is defined as

Based on the above analysis, the transmit and receive patterns of an array containing H/V channel non-idealities can be represented as

Assuming that the active patterns of all antenna elements are the same, then

Wherein

The case of a single spherical raindrop, according to previous analysis, contains H/V channel non-idealities

And

is shown as

Further, A and B may be represented as

Assuming that there is only a single in the beam directionA spherical raindrop, so that only calculation is needed

And

due to the beam pointing

And

can compensate

Induced phase change, therefore

And

can be expressed as

According to

And

knowing that the double summation in (2.109) and (2.110) approximates the mathematical term values of A and B;

suppose gamma_hv,γ_vh,ψ_hv,ψ_vhU (0,2 π), then

And

thus, can obtain

If delta_ijWhen the value is 0, (2.111) can be simplified to 0

(2.112) it was shown that even if the antenna element pattern measurements were completely accurate, the inconsistencies in the H and V channels would be aligned

An influence is produced. Further assume | a_vv|＝|b_vvI < 1, according to CIM definition, can be obtained

From (2.113), the conditions were satisfied

The requirement of (3) is that CIM is less than 0.05dB, namely that the amplitude phase imbalance of the H/V channel is less than 0.05 dB.

Of a single raindrop

Simulation result of η wherein_vv＝τ_vv＝0.99，γ_vv,ψ_vv～U(-10°，10°)，η_hv＝η_vh＝τ_hv＝τ_vh＝0，|δ_ij|＝Δ，Arg(δ_ij) U (0,2 π). When delta<At the time of 0.01, the alloy is,

almost constant at 0.26dB, when Δ>0.02，

Increasing linearly. This indicates that when the antenna pattern measurement error is small, channel coupling is the main source of polarization measurement error; when the antenna pattern measurement error is large, the polarization measurement error mainly originates from the antenna pattern measurement error.

In the case of a large amount of spherical raindrops, a Monte Carlo simulation-based method is also used here to analyze polarization measurement errors in the case of a large amount of spherical raindrops, and the simulation flow is as follows:

step 1: given | Delta_ij|

Step 2. Generation of alpha by a random number Generator_ij，β_ij，δ_ij

Step 3 Generation A by a random number Generator_mnAnd B_mn

Step 4 calculation

And

step 5 calculation

And

shown in figures 10 and 11 are

And

the simulation result of (1), wherein

|α_ij|＝|β_ij|＝2,η_hv＝η_vh＝τ_hv＝τ_vh＝0,η_vv,τ_vv～N(0.99,0.01²) And gamma_vv,ψ_vvU (-10, 10). In fig. 10, single raindrop and heavy raindrop conditions

The fit is very good. In fig. 11, only when Δ is large, under the single raindrop and large raindrop conditions

The agreement is compared, and when Δ is smaller, the two conditions are

The difference is very large. This illustrates a finite beamwidth pair

The measurement influence is large.

Setting τ_hv,η_vh,τ_hv,τ_vh～N(0.01,0.01²)，γ_hv,γ_vh,ψ_hv,ψ_vhU (-10 °,10 °) and keeping the other parameters unchanged. The simulation results are shown in fig. 12 and 13. As can be seen from fig. 12, when CIS is 40dB, the H/V channel couples

The effect of (a) was not significant. In FIG. 13, when Δ is small, the H/V channel coupling pair

The influence of (a) is significant.

Axial calibration method-STSR mode, for distributed targets consisting of a large number of raindrops, the received signal is represented as

Wherein s is_h(t) and s_v(t) waveforms transmitted for the H and V ports, respectively. Suppose S_hv(θ,φ)＝S_vhWhen (θ, Φ) — 0, (2.114) is represented as

From (2.115), V_h(t) and V_v(t) contains the 1 st and 2 nd order terms of the cross-polarization pattern, while in ATSR mode the received electric field component contains only the 2 nd order term of the cross-polarization pattern. Therefore, the system accuracy requirement is higher in the STSR mode than in the ATSR mode.

To overcome the effect of the 1 st order term of the cross-polarization pattern, orthogonal waveforms may be employed. Received signal V_h(t) and V_v(t) after passing through a matched filter, can be expressed as

Wherein

And

matched filters for the H and V channels, respectively.

Representing the convolution of the signal.

If s is_h(t) and s_v(t) is completely orthogonal, then

In this case, (2.116) is equivalent to (2.31). The foregoing analytical methods and conclusions regarding the ATSR mode can therefore be directly applied.

In practice, s_h(t) and s_v(t) cannot be perfectly orthogonal, so define

Then

Thus, can obtain

Wherein C is_TAnd C_RSee (2.43) and (2.44). When the waveform s_h(t) and s_vWhen (t) is known, Q is a constant matrix, and (2.120) is also equivalent to (2.31).

Claims (1)

1. An axial correction method for a polarized phased array radar measurement error is characterized by comprising the following steps: the method comprises the following steps:

firstly, establishing an array model

1) Active cell direction, the active direction of the cell in the mth row and nth column is denoted as

Wherein

·f_hh(θ, φ) is the radiated electric field component in the H direction when the H port is excited;

·f_hv(θ, φ) is the radiated electric field component in the H direction when the V port is excited;

·f_vh(θ, φ) is the radiated electric field component in the V direction when the H port is excited;

·f_vv(θ, φ) is the V-square when the V port is energizedAn upward radiated electric field component;

2) array direction, transmission and reception pattern of the array F_T(θ,φ；θ_S,φ_S) And F_R(θ,φ；θ_S,φ_S) Is shown as

Wherein the subscripts "T" and "R" denote Transmission (Transmission) and Reception (Reception), X, respectively_mn(θ_S,φ_S) And Y_mn(θ_S,φ_S) Complex weighting coefficients respectively representing the units mn;

since each dual polarized antenna element has two ports, H and V, Xmn (theta)_S,φ_S) And Y_mn(θ_S,φ_S) Expressed as a 2 x 2 matrix, the transmit pattern weighting coefficient matrix should be expressed as

Where columns 1 and 2 represent the excitation of the H and V ports, respectively, assuming

Thus, it is possible to provide

Expressed as a scalar coefficient

For receive weighting, the same weighting is used for the H and V beams, and one is also usedScalar coefficient

To represent the weighting of the received beams;

formulae (2.15) and (2.16),

and

simplified to

Wherein

Showing the active pattern, AF, of each cell_TAnd AF_RIs an array factor expressed as follows

Wherein

Is the position vector of unit mn, k 2 pi/lambda is wave number, lambda is wavelength, exponential term

The phase difference of the cell mn and the reference cell with respect to the target point is shown;

3) a receive voltage equation, and in ATSR mode, when only the H port is excited, the electric field radiated by the phased array antenna

Is composed of

Assuming that there is a single raindrop in the beam direction, the polarization scattering matrix is S', when the incident field is

The scattered field of a single raindrop is

Scattered electric field

The voltage is generated after being received by the antenna, and according to the antenna theory, the horizontal channel receives the voltage dV_hhAnd a vertical channel receiving voltage dV_vhAnd scattering electric field

In a relationship of

Wherein

The formula (2.24) is expressed in a matrix form as follows

For simplicity, (2.27) omits the terms of distance and gain, it is noted that S' (θ; φ) includes the effects of attenuation and phase shift during propagation of electromagnetic waves, and the expression is shown below

Wherein T represents a one-way path transmission matrix and describes attenuation and phase shift in the electromagnetic wave propagation process, and S (theta; phi) represents a polarization scattering matrix inherent to a single raindrop;

when only the V port is excited, the voltage component dV is received_hvAnd dV_vvIs shown as

(2.27) and (2.29) are combined to

For a large number of raindrops distributed in space, the received voltage V is expressed as

Where Ω denotes a solid angle, d Ω is sin θ d θ d Φ,

when the echoes of a large number of raindrops are incoherent, the total received power P is therefore_ij(i, j ═ h, v) is represented by

P_ij＝∫_Ω〈|dV_ij|²>dΩ (2.32)

Based on (2.32), differential reflectivity Z_DRIs defined as

If S'_hv＝S′_vhWhen the linear depolarization ratio is 0, the actually measured linear depolarization ratio represents the error of the linear depolarization ratio;

in S'_hv＝S′_vhLinear depolarization ratio error under the assumption of 0

Is defined as

The "Projection Matrix" method is used to correct polarization measurement errors caused by the non-orthogonality of the H and V port radiated electric fields; when the dual-polarized antenna unit only radiates electromagnetic waves at the H port, an electric field is radiated

Is shown as

When only the V port radiates electromagnetic waves, an electric field is radiated

Is shown as

And

are respectively as

And

the unit vector in the direction, under the H/V polarization base, the electric field radiated by the dual-polarized antenna unit is expressed as

Wherein the projection matrix P is represented as

Projecting the matrix P as "to be based on polarization

And

the characterized radiation electric field is transformed into a radiation electric field characterized by H and V polarization radicals, P denotes the transformation of two non-orthogonal unit vectors

And

projected to orthogonal unit vectors

And

the above step (1); based on the projection matrix P, the equation of the received voltage of the single point target is expressed as

Wherein S is a target polarization scattering matrix, superscript t represents matrix transposition, and the terms related to distance r and gain are omitted in formula (2.39);

in the ATSR mode, when the H and V ports alternately transmit the unit signal: when the amplitude is 1 and the phase is 0, the receiving voltage equation is expressed as

Thus, the target polarization scattering matrix S is represented as

S＝C^t·V·C (2.41)

Wherein C ═ P^-1Referred to as a projection correction matrix;

from (2.41), the projection matrix correction method implies the following settings:

1) electric field radiated by H-port and V-port

And

the accuracy is known; 2) the target is a point target; 3) the radiation characteristics of the antenna elements are completely consistent; 4) the amplitude phase characteristics of the H and V channels are ideal; 5) the radar transmission and reception patterns are the same, i.e. the radar is reciprocal;

secondly, deriving a receiving voltage equation,

1) axial correction method-ATSR mode, using projection matrix as reference, based on (2.31), the corrected receiving voltage equation is expressed as

Wherein

Representing the corrected received voltage matrix, C_TAnd C_RTo correct the matrix, expressed as

And

indicating measured array transmit and receive directions in the beam direction (theta)_S,φ_S) A value of (d) above; (2.42) is an array-level correction method, the active directions of the array elements are different, and the correction matrixes (2.43) and (2.44) are based on the information of the array transmitting direction axis and the array receiving direction axis direction only, so that the correction matrixes are called as(2.42) the correction method given is axial correction (Boresight correction);

when the transmitting and receiving directions of the array are not reciprocal, two correction matrixes are defined to correct the transmitting and receiving directions of the array respectively, so that (2.42) the projection matrix correction method is expanded;

from (2.43) and (2.44), the axial correction includes: one is to

And

correcting for non-orthogonality; secondly, the non-uniformity of the gain of the transmitting beam and the receiving beam is compensated; based on (2.18) and (2.19), C_TAnd C_RIs simplified into

Wherein f is_m(θ_S,φ_S) Indicating the measured antenna element direction;

(1) linear model, (3.42) can be expressed as

Wherein

And

referred to as corrected array antenna patternIs defined as

(2.47) is mathematically similar to (2.31), so that the corrected received power

Can still be calculated as (2.32) except that F_TAnd F_RTo be replaced by

And

(2.48) and (2.49) are mathematical processes only, and the correction matrix in practice acts only on the received voltage V;

first, the corrected cross polarization direction pair is analyzed

And

assuming that S' is a unit matrix, the (2.47) is developed into a scalar form, including

In the case of (2.50), the,

and

contains the second order term of the cross-polarization pattern, and

and

the expression (c) contains the first order term of the cross-polarization direction,

is not sensitive to cross-polarization direction, and

the method is sensitive to the cross polarization direction; in the case of the STSR mode,

a first order term comprising the cross-polarization direction; therefore, the cross-polarization pattern requirements for the ATSR mode are lower than for the STSR mode;

analyzing the formula (2.47), and assuming that the active directions of the antenna units are the same; based on (2.45) and (2.46),

and

is shown as

If measured cell direction f_m(θ_S,φ_S) Exactly without any error, then f (θ, φ) · f_m(θ_S,φ_S)^-1In the beam direction (theta)_S,φ_S) Above would be an identity matrix; and in practice, f (theta, phi) · f_m(θ_S,φ_S)^-1In the beam direction (theta)_S,φ_S) Is not an identity matrix;

therefore, will

And

is modeled as

Wherein ε ij (θ, φ; θ)_S,φ_S) (i, j ═ h, v) called axial correction error, describes the effect of finite antenna beamwidth on polarization measurement error;

and

is a normalized array factor, i.e.

And

ε_ijthe analytic expression of (2) knows that the direction of the antenna unit is approximate to a linear function of theta and phi through the direction of the actually measured and simulated dual-polarized microstrip patch antenna unit; to this end epsilon_ijApproximated as a linear function of theta and phi, as shown below

Wherein alpha is_ij，β_ijAnd delta_ijIs a plurality of parameters;

the formula (2.55) is a linear model of the axial correction error, describes the change relation of the correction error of the axial correction method along with the airspace angles theta and phi, verifies the rationality of the linear model of the formula (2.55), and explains the rationality of the formula (2.55) by utilizing the direction of the simulated dual-polarized microstrip patch antenna unit; corrected microstrip patch antenna directional pattern

Is shown as

Selecting (theta)_S,φ_S) Equal to (60 °,45 °), e is calculated based on the simulated antenna element directions_ij，ε_ijThe linear relationship between theta and phi in the neighborhood of (60 deg., 45 deg.) is approximately linear, and the linear approximation of (2.55) indicates the axial correction error epsilon_ijIs a slow-varying function of the spatial angles theta and phi;

using Matlab's Curve Fitting toolkit (Curve Fitting Toolbox), with theta and phi as parameters, for epsilon_ijPerforming a linear fit, R²(R-square), called Coefficient of Determination (coeffient of Determination), between 0 and 1, describes how well the model fits to a given datum;

linear model (2.55) known as α_ijAnd beta_ijRepresents epsilon_ijThe rate of change in theta and phi, if the measured cell orientation is error free, then there is epsilon_ij(θ_S,φ_S；θ_S,φ_S) 0, i.e. delta_ij0; measured dual-polarized antenna unit direction f_m(θ_S,φ_S) Expressed as the sum of true directional diagram and absolute measurement error, i.e.

Wherein f (theta)_S,φ_S) Representing the true cell direction, e (θ)_S,φ_S) Corrected element directional diagram representing absolute error of antenna direction measurement

Is shown as

In (2.59) (2.81),

and

are respectively abbreviated as

f and f_mUsing matrix inversion theorem to obtain

(f_m)^-1＝(f+e)^-1＝f^-1-f^-1(f^-1+e^-1)^-1f^-1 (2.59)

Assuming that e is reversible, substituting (2.59) into (2.58) yields

Wherein I represents an identity matrix, and further obtaining

(2.61) indicates. delta_ijMeanwhile, the relation between the real antenna unit direction f and the measurement absolute error e is given by theorem 2.1, which relates to the absolute measurement error e and the antenna unit direction f;

2.1 theorem, setting the direction of a real dual-polarized antenna unit as f, the absolute error of the antenna direction measurement as e, and | e_ij|＝|f_ijIf there is

(f^-1+e^-1)^-1≈e (2.62)

Prove that the matrix inversion theorem is utilized, have

Due to | e_ij|＝|f_ijI, have

e+f≈f (2.64)

Therefore (2.63) can be approximated as

(e^-1+f^-1)^-1-e≈-e·f^-1·e (2.65)

Based on the matrix norm theory, there are

Wherein | · | purple_∞Denotes the matrix ∞ -norm, and further (2.66) denotes

(2.67) in the sense of the matrix ∞ -norm with (e)^-1+f^-1)^-1To approximate the relative error of e, note that

Wherein | f | purple_∞·||f^-1||_∞Define κ (f) | | f | | survival as a condition number based on an infinity norm for the matrix f_∞·||f^-1||_∞Then (2.68) is expressed as

According to the definition of infinity-norm of matrix, | | e | | ventilation_∞Is shown as

||e||_∞＝max{|e_hh|+|e_hv|,|e_vh|+|e_vv|} (2.70)

Defining an upper bound on relative error of antenna element directional measurements

Absolute measurement error e_ijIs random and depends on the spatial distribution of the main and cross-polarization directions of the antenna elements, E^f _ijAlso dependent on the spatial distribution of the main and cross-polarization patterns of the antenna elements, let E^f _ij＝E_fAnd E_fIs a constant in the beam scanning area, thereby obtaining

Based on (2.72) have

Without loss of generality, assume | f_hh|+|f_hv|≥|f_vh|+|f_vvIf (2.73) is expressed as

According to (2.70), obtaining

||e||_∞≤E_f·||f||_∞ (2.75)

Substituting (2.75) into (2.69) there are

||-e·f^-1||_∞≤E_f·κ(f) (2.76)

Therefore, (2.67) is expressed as

If the antenna cross polarization is below-10 dB, then there is k (f)<In antenna measurement, E_f<5% is also easily satisfied, therefore, E_fκ (f) is a very small quantity and thus has

(e^-1+f^-1)^-1≈e (2.78)

Bring (2.78) into (2.61), have

Then delta_hhIs estimated as

Can similarly obtain

Derivation (2.81) uses the approximate relationship (f)^-1+e^-1)^-1E, so (2.81) is only δ_ijAn approximate estimate of the upper bound, to verify the estimated performance of (2.81), the following numerical simulations are given: suppose that

And E_fCalculated δ based on (2.81) ═ 1%_ijThe upper bound of (a) is,

(2.83) generating the absolute measurement error e by means of a random number generator such that e_ijSatisfy the requirement of

Generation of e_ijThen, δ was calculated directly using (2.61)_ij，

When k (f (θ)_S,φ_S) (matrix f (θ))_S,φ_S) Infinity-norm based condition number) less than 2, while E_f5% or less, and the formula (2.81) has good estimation accuracy, and these conditions (k (f (theta))_S,φ_S) < 2 and E)_f5%) are satisfied in antenna design and measurement; delta_ijUpper bound of (E) and relative error E_fIn a phaseThe same magnitude;

according to theorem 2.1, get δ_ijAnother expression of upper bound estimation

From (2.85), δ_ijUpper bound and relative measurement error E_fAnd an antenna unit direction f (theta)_S,φ_S) All have a relationship; and κ (f (θ)_S,φ_S) Is characterized by f (theta)_S,φ_S) The size of the cross-polarization;

is provided with

As the relationship with gamma becomes larger (cross polarization becomes larger),

and then becomes larger;

(2.81) and (2.85) give 2 different estimates of δ_ijThe method of the upper bound, (2.81) estimation facilitates numerical calculation, while (2.85) is suitable for theoretical analysis; alpha is alpha_ijAnd beta_ijRepresents epsilon_ijThe rate of change in θ and φ;

antenna directional diagram measurement error pair epsilon_ijInfluence of the spatial rate of change, ε_ijSpatial rate of change of (2), definition

Wherein

And

describing the spatial rate of change of the antenna element orientation after correction, i.e. epsilon_ijAccording to (2.58)

Since the direction measurement error of the antenna unit is generally very small, f is_m(θ_S,φ_S)≈f(θ_S,φ_S) Thereby obtaining

Wherein

And

describing the spatial rate of change of the corrected cell direction when the antenna cell direction measurement is error free;

as known from (2.90) and (2.91), when the antenna element pattern measurement error is small, the measurement error is in the pair ε_ijThe spatial change rate of (a) is also less affected;

(2) the correction performance analysis under the ideal H/V channel condition is carried out, the amplitude phase characteristics of the H channel and the V channel are consistent, and the influence of antenna unit directional diagram measurement errors and limited beam width on polarization measurement errors is set;

1) in the case of a single spherical raindrop, first, it is assumed that there is only one spherical raindrop in the beam pointing direction, in this case Z_DR0dB and L_DRInfinity dB; only single raindrop is considered, integral of the whole space domain is removed in (2.31), and in the case of single spherical raindrop, if a raindrop polarization scattering matrix is taken as a unit matrix, the (2.31) is simplified into

From (2.92)

Further assume | δ_ij∣＝Δ，δ_ijIn the phase of [0,2 π]Are uniformly distributed; taylor expansion is performed on (2.93), then | Z_DR| the mathematical expectation is approximated as

Wherein

Expressing a mathematical expectation;

at | δ_ijΔ | ═ and Arg (δ)_ij):[0,2π]Under the assumption that the first and second images are different,

symmetrically distributed around 0, then

Thus in the calculation of

Is a mathematical expectation of

As a function of the change in delta,

in Monte Carlo simulation, | δ is set_ijΔ | ═ Δ, and δ_ijIs generated by a random number generator,

calculated according to (2.93), and then obtained by multiple times of simulation

Are averaged to obtain the final

The approximation based on equation (2.94) is consistent with the results of Monte Carlo simulations;

derived using a derivation method

Monte Carlo simulation procedure and pair for variation with delta

Similar to the simulation of (2.95), the approximation results based on (2.95) are consistent with the results using the Monte Carlo simulation;

for this purpose, error delta is corrected axially_ijTo pair

And

the effect of (a) is significant, and for a single spherical raindrop, it is sufficient

The measurement accuracy of (1), Δ is less than 0.01; this means that the relative error in the measurement of the antenna element orientation is of the order of 1%, indicating that

Increases approximately linearly with increasing delta, and

increases in (c) increase approximately logarithmically with increasing Δ;

2) in case of a large number of spherical raindrops, if the parameter α of the linear model (2.55)_ij、β_ijAnd delta_ijAs known, the polarization measurement performance of the phased array radar is calculated by (2.53), (2.54) and (2.47);

when the parameter α is_ij、β_ijAnd delta_ijUnknown, a Monte Carlo simulation-based method is adopted to analyze the polarization measurement performance of the phased array radar, and the method comprises the following steps:

step 1: pointing (θ) for a given beam_S,φ_S)，α_ij、β_ijAnd delta_ijGenerated by a random number generator, in turn e_ijCalculated from (2.55);

Step 2:

and

calculated from (2.53) and (2.54);

step 3 corrected received power

Calculated from (2.32);

Step 4:

and

calculated from (2.33) and (2.34), where n represents the nth simulation;

in the beam pointing direction (theta)_S,φ_S) Repeating the above simulation, and then comparing the obtained results

And

averaging to obtain the direction of a given beam

And

calculated by the above method

And

is represented by the parameter alpha_ij、β_ijAnd delta_ijThe average value under a certain distribution of (a) is a statistic of polarization measurement performance;

for a designed microstrip patch antenna element, | α_ij|p1，|β_ijI p1 byFurther analysis shows that the | alpha is provided for infinitesimal electric dipoles_ij|≤2，|β_ijI.ltoreq.2, therefore assume; | α_ij|≤2，|β_ij|≤2，α_ijThe real part of β ij is much larger than the imaginary part, indicating α_ij、β_ijIs close to 0, and thus is set to be in phase

Are uniformly distributed;

first, the matrix C is corrected_TAnd C_RIs completely accurate and contains no error, under which condition, there is delta_ij0, if not stated, Z is assumed_DR＝0dB，L_DRInfinity dB, obtained by Monte Carlo simulation

And

within the entire beam scanning range,

less than 3X 10^-3dB，

Lower than-36 dB; the result is a hypothetical correction matrix C_TAnd C_RAre completely accurate, so that these results are considered to be the best results under given conditions and can be used as a reference for other simulation results, in addition to

The increase from the array normal to the beam pointing (60, 45) is about 2.8dB, which indicates that

The measured performance of (a) is a function of beam pointing; still set delta_ij0 while increasing | α_ij| and | β_ijThe distribution range of | is, within the whole beam scanning range,

less than 8 x 10^-3dB，

Increased from-33.34 dB to-30.4 dB, an increase of approximately 2.9 dB;

the influence of antenna unit directional diagram measurement error on polarization correction performance is set to be delta_ij0.01 and will | α_ij| and | β_ijThe range of | varies from U (0,1) to U (0,2), over the entire beam sweep,

there was a slight fluctuation in the vicinity of 0.1dB, which indicates that

Hardly depending on the beam scanning direction, and is given by | α_ij| and | β_ijThe amplitude of | is not sensitive, but rather

As the beam is directed, from

Increasing from-35.4 dB to-32.8 dB (an increase of about 1.6dB),

increasing from-31.78 dB to-29.55 dB (an increase of approximately 2.2dB) indicates that

Also with | α_ij| and | β_ijThe amplitude distribution range of | is related;

axial correction error delta_ijThe influence on the performance of the polarization correction, which determines

And

the lower limit can be reached, and the fluctuation speed of the spatial domain polarization of the antenna unit is the | alpha_ij| and | β_ijAmplitude of L, for L_DRThe measurement of (2) also has a large influence, so that in practice, an antenna unit with small fluctuation of space domain polarization characteristics is designed to improve L_DRThe accuracy of the measurement is beneficial;

simulation results show that the aim is to achieve

The relative error of the antenna unit directional diagram measurement is 1%, and the measurement precision of the antenna unit directional diagram of 5% is relatively easy to meet in practice based on the | delta%_ijThe simulation result of 0.05 is |,

there is little fluctuation around 0.5dB,

increasing from-25.4 dB to-24.4 dB,

increased from 24.4dB to-23.2 dB;

(3) calibration performance analysis under non-ideal H/V channel conditions

1) Non-ideal H/V Channel modeling, namely adopting T/R components of a polarized phased array radar in an ATSR mode and an STSR mode, using Channel Isolation (CIS) to represent coupling between H and V channels, and using Channel Imbalance (CIM) to represent inconsistency of amplitude phases of the H and V channels;

non-ideal HA of the/V channel model_ijAnd b_ijTo represent the coupling and amplitude phase inconsistency of the H and V channels, two 2 x 2 matrices a and B to represent the non-idealities of the transmit and receive channels,

the expression is as follows

Wherein a is_hh、a_vv、b_hhAnd b_vvDescribe the imbalance of the H/V channel, and a_hv、a_vh、b_hvAnd b_vhCoupling between H/V channels is described, assuming a_hh＝b_hh1, CIM is thus defined as

Note that if | a_vvIf | is greater than 1, then there is

For CIM to be a non-negative value, use is made of

Similarly, CIS is defined as

From the above analysis, the array transmit and receive directions containing H/V channel non-idealities are expressed as

Assuming that the active directions of all antenna elements are the same, then

Wherein

2) In the case of a single spherical raindrop, the inclusion of H/V channel imperfections

And

is shown as

Further, A and B are represented by

Assuming that there is only a single spherical raindrop in the beam pointing direction, so only one calculation is needed

And

due to the beam pointing

And

can compensate

Induced phase change, therefore

And

is shown as

According to

And

knowing that the double summation in (2.109) and (2.110) approximates the mathematical term values of A and B;

let gamma_hv，γ_vh，ψ_hv，ψ_vhU (0,2 π), then

And

thereby obtaining

If delta_ijWhen the value is 0, (2.111) is simplified to

(2.112) it was shown that even if the antenna element pattern measurements were completely accurate, the inconsistencies in the H and V channels would be aligned

Has an influence of_vv|＝|b_vv< 1, according to CIM

From (2.113), it is satisfied

The CIM is less than 0.05dB, namely the amplitude phase imbalance of the H/V channel is less than 0.05 dB;

of a single raindrop

Simulation result of η wherein_vv＝τ_vv＝0.99，γ_vv,ψ_vv～U(-10°，10°)，η_hv＝η_vh＝τ_hv＝τ_vh＝0，|δ_ij|＝Δ，Arg(δ_ij) U (0,2 π), when Δ<At the time of 0.01, the alloy is,

almost constant at 0.26dB, when Δ>0.02，

The linearity increases, indicating that channel coupling is the main source of polarization measurement error when the antenna pattern measurement error is small; when the antenna directional diagram measurement error is large, the polarization measurement error mainly comes from the antenna directional diagram measurement error;

in the case of a large amount of spherical raindrops, polarization measurement errors in the case of a large amount of spherical raindrops are analyzed by a Monte Carlo simulation method, and the simulation flow is as follows:

step 1: given | Delta_ij|

Step 2. Generation of alpha by a random number Generator_ij，β_ij，δ_ij

Step 3 Generation A by a random number Generator_mnAnd B_mn

Step 4 calculation

And

step 5 calculation

And

and

as a result of the simulation of (a),

|α_ij|＝|β_ij|＝2,η_hv＝η_vh＝τ_hv＝τ_vh＝0,η_vv,τ_vv～N(0.99,0.01²) And gamma_vv,ψ_vvU (-10, 10), single raindrop and heavy raindrop conditions

Quite coincident, only when the delta is larger, under the conditions of single raindrop and a large number of raindrops

The agreement is compared, and when Δ is smaller, the two conditions are

The difference is very large, which shows the finite beamwidth pair

The measurement influence is large;

setting eta_hv,η_vh,τ_hv,τ_vh～N(0.01,0.01²)，γ_hv,γ_vh,ψ_hv,ψ_vhU (-10 °,10 °) and keeping the other parameters constant, when CIS is 40dB, the H/V channel coupling pair

Has insignificant influence, and when the delta is small, the H/V channel coupling pair

The influence of (2) is more obvious;

in the STSR mode, for a distributed target composed of a large number of raindrops, the received signal is represented as

Wherein s is_h(t) and s_v(t) waveforms transmitted for the H and V ports, respectively, assuming S_hv(θ,φ)＝S_vhWhen (θ, Φ) — 0, (2.114) is represented as

Known from (2.115), V_h(t) and V_v(t) contains the 1 st and 2 nd order terms of the cross-polarization pattern, while the received field component in the ATSR mode contains only the 2 nd order term of the cross-polarization pattern, and therefore the system accuracy requirement in the STSR mode is higher than that in the ATSR mode;

in order to overcome the influence of the 1 st order term of the cross-polarization directional diagram, the orthogonal waveform is adopted, and a receiving signal V is received_h(t) and V_v(t) after passing through a matched filter, is represented as

Wherein

And

matched filters for the H and V channels respectively,

representing a signal convolution; if s is_h(t) and s_v(t) is completely orthogonal, then

In this case, (2.116) is equivalent to (2.31);

in practice, s_h(t) and s_v(t) cannot be perfectly orthogonal, so define

Then

Thereby obtaining

Wherein C is_TAnd C_RSee (2.43) and (2.44); when the waveform s_h(t) and s_vWhen (t) is known, Q is a constant matrix, and (2.120) is also equivalent to (2.31).

Abstract

The invention relates to the technical field of meteorological radar axial correction, and discloses an axial correction method for measurement errors of a polarized phased array radarPolarization measurement errors caused by non-orthogonality of the radiation electric field in the beam pointing direction are analyzed, and polarization error correction performance under ATSR and STSR modes is analyzed. The invention corrects the polarization measurement error through the non-orthogonality of the radiation electric field in the direction of the polarized phased array antenna wave beam, is suitable for the phased array radar, and ensures that the Z-shaped antenna wave beam is not orthogonal_DRThe measurement error is less than 0.1dB, the relative error of the antenna directional diagram measurement is less than 1%, the polarization measurement performance is analyzed quickly and accurately, the measurement precision is high, and the method has great practical value for the polarization phased array radar antenna measurement.

Images