CN101900900A - Liquid crystal phased array wave control data determination method based on wave surface iteration - Google Patents

Abstract

The invention relates to a liquid crystal phased array wave control data determination method based on wave surface iteration, which belongs to the technical field of laser phased array radar and relates to liquid crystal phased array wave beam control. The method mainly comprises the following steps of: (1) calculating an emergency stage wave surface by using a traditional wave control method according to a deflexion angle required to be obtained; (2) calculating an average slope of the stage wave surface and an ideal linear wave surface to obtain an average slope error; (3) recording a step edge position included in the stage wave surface and calculating the change of the average slope of each step edge after translation; and (4) judging whether the change of the average slope after the translation of the step reduces the average slope error or not; and if so, judging whether the step edges are moved leftwards or rightwards and carrying out iteration; and if not, stopping the iteration. By utilizing the relation of the average slope of the emergency stage wave surface of the liquid crystal phased array and the angle error, the invention obtains high-precision wave control data through the translation of the step edges, and can realize wave beam scanning with higher precision and more angles in comparison with the traditional wave control method.

Description

LCD phased array ripple control data based on the wave front iteration are determined method

Technical field

The invention belongs to the phased-array laser radar technical field, relate to the control of LCD phased array wave beam, be specifically related to definite method of a kind of LCD phased array wave beam control data (being voltage identification code).

Background technology

The LCD phased array technology is a kind of novel phased-array technique, and it utilizes the electrically conerolled birefringence characteristic of liquid crystal, promptly passes through to load the birefraction of different magnitudes of voltage with the control liquid crystal, thereby realizes the phase modulation (PM) of light wave is reached the deflection of light beam.Comparing with traditional microwave phased array has that cost is low, in light weight, driving voltage low and characteristics such as pointing accuracy height, can be used for fields such as wave beam control, beam scanning, laser communication.In addition, in that deflection angle is required small but excellent especially application scenario, also more suitable.When carrying out laser communication such as satellite and ground receiving station, the beam director on the satellite makes very small angle of the every deflection of emergent light, will cause the span of very big distance on the ground.Thereby the LCD phased array that is used for laser radar and free space optical communication has two important characteristic parameter: ripple control precision and ripple control efficient, the two is directly relevant with the PHASE DISTRIBUTION of LCD phased array outgoing light wave.

Traditional ripple control method is that the family curve by the phase delay of liquid crystal and on-load voltage obtains the voltage identification code of loading, thereby realizes the deflection of light beam.But because the discrete feature of on-load voltage, attainable on-load voltage value is limited, thereby also can't realize for some phase delay, thereby brings the error of outgoing wave front, so ripple control efficient and ripple control precision also can reduce, and quantitatively also can be restricted in attainable scanning angle.

For ripple control efficient, people such as Scott Harris and Charies M.Titus have carried out many relevant researchs, and wherein ScottHarris has proposed a kind of optimization voltage loaded value to obtain high efficiency method.But then less relatively for ripple control Study on precision, David

Proposed a kind of Deng the people by optimizing phase pushing figure to obtain the method for high-precision beam scanning.This method can reduce the trueness error of deflection angle to a certain extent, but can realize quantitatively not significantly improving of deflection angle.Therefore need a kind of better method can obtain higher ripple control precision.

Summary of the invention

The invention provides a kind of LCD phased array ripple control data and determine method based on the wave front iteration, be modified to the basis with iteration, utilize the average gradient of liquid crystal outgoing wave front and the relation of ripple control trueness error, expression formula by the derivation average gradient obtains the average gradient variation that all possible phase place step edge translation is brought, judge that these average gradients change the error that whether can remedy step edge average gradient and desirable average gradient, when error can't reduce by translation phase place step, stop iteration again, write down the position and the corresponding voltage identification code value (being ripple control data) that obtains loading of step edge at last.

The present invention utilizes iteration correction LCD phased array outgoing wave front phase place step edge to obtain to be carried in the voltage identification code on the LCD phased array, reaches the purpose of accurate control far field wave beam deflection, and can receive the more deflection angle realized.By the check stopping criterion for iteration, judge whether iterative process is finished.Change error greater than step edge average gradient and desirable average gradient if set stopping criterion for iteration and be average gradient after the edge translation of phase place step, represent that promptly iteration finishes, the far field beam distribution that is obtained can satisfy high-precision requirement.

In order to solve the problem of obtaining the high precision beam scanning, technical solution of the present invention is as follows:

A kind of LCD phased array ripple control data based on the wave front iteration are determined method, as shown in Figure 2, may further comprise the steps:

Step 1:, utilize the conventional wave control method to calculate LCD phased array outgoing staircase waveform front according to the desirable deflection angle θ that the LCD phased array needs are realized As shown in Figure 1, wherein the phase value of the realization of n phased array unit is:

Wherein M is a phase place

Interior quantification number of steps,

λ is an optical wavelength, d phased array unit interval, " round " expression get with bracket in the immediate round values of value, N is a phased array unit sum.

Step 2: calculate average gradient error k _Error=k _I-k, wherein k is a LCD phased array outgoing staircase waveform front average gradient, k _lBe ideal linearity outgoing wave front average gradient and satisfied:

$k=\frac{2\mathrm{\πd}}{\mathrm{\λ}}\sin \left({\mathrm{\θ}}_{\mathrm{stair}}\right)---\left(3\right)$

Wherein d is the phased array unit interval, and θ is that the deflection angle of ideal linearity wave front correspondence is the desirable deflection angle that LCD phased array need be realized, θ _StairDeflection angle for staircase waveform front correspondence.

Step 3: array is also used in the position, step edge that is comprised in the statistics LCD phased array outgoing staircase waveform front $\stackrel{\→}{Q}=\{a_1,a_2,...,a_{p-1}\}$ Record, wherein position, step edge a _qBe defined as the sequence number of q step edge first left phase control unit, 1≤q≤p and 1≤a _q≤ N-1, p are the step number of staircase waveform front.

Step 4: by formula

And

Calculate any q step edge respectively, its position is a _q(1≤a _q≤ N-1), the step edge is to the right or the average gradient conversion that produces behind the phased array unit that moves to left

Or

Wherein $C=\frac{6}{N^2-1},$ C is a constant under the total constant situation of phase control unit,

M is the quantification number of steps in phase place 0～2 π,

Represent a _qThe weights of individual phased array unit, and $A_{a_q}=a_q(1-\frac{a_q}{N}).$

Step 5 definition $\stackrel{\→}{{\mathrm{\Δk}}_R}=\{{\mathrm{\Δk}}_{a_1\_R},{\mathrm{\Δk}}_{a_2\_R},...,{\mathrm{\Δk}}_{a_{p-1}\_R}\},$ $\stackrel{\→}{{\mathrm{\Δk}}_L}=\{{\mathrm{\Δk}}_{a_1\_L},{\mathrm{\Δk}}_{a_2\_L},...,{\mathrm{\Δk}}_{a_{p-1}\_L}\},$ Calculate by step 4

With

In comprised the average gradient that the edge translation of all possible step brought and changed, and in array

In search all positions, edge, always having a position is q _R(1≤q _R≤ p) step edge makes:

${\mathrm{\σ}}_R=\min \left(\right|k_{\mathrm{error}}-\stackrel{\→}{{\mathrm{\Δk}}_R}\left[\stackrel{\→}{Q}\right]\left|\right)=|k_{\mathrm{error}}-\stackrel{\→}{{\mathrm{\Δk}}_R}[q_R\left]\right|---\left(4\right)$

Equally also always having a position is q _L(1≤q _L≤ p) step edge makes:

${\mathrm{\σ}}_L=\min \left(\right|k_{\mathrm{error}}-\stackrel{\→}{{\mathrm{\Δk}}_L}\left[\stackrel{\→}{Q}\right]\left|\right)=|k_{\mathrm{error}}-\stackrel{\→}{{\mathrm{\Δk}}_L}[q_L\left]\right|---\left(5\right)$

Wherein minimum value is got in " min " expression, and " || " expression takes absolute value,

Expression is by array

The vector that constitutes, " [q _R] " be q _RVector representation, " [q _L] " be q _LVector representation.

If step 6 σ _R＞| k _Error| and σ _L＞| k _Error|, representative can't reduce the average gradient error again | k _Error|, stop iteration and forward step 7 statistics angular error to; If σ _R≤ | k _Error| or σ _L≤ | k _Error|, then begin following iterative process;

If step 6-1 is σ _R＜σ _L, represent right-shift operation can better reduce the average gradient error | k _Error|, to q _RThe step edge at place carries out after translation is finished, forwarding step 2 to phased array unit of right translation to;

If step 6-2 is σ _R〉=σ _L, represent shift left operation can better reduce the average gradient error | k _Error|, to q _LThe step edge at place carries out after translation is finished, forwarding step 2 to phased array unit of left to.

Step 7 is calculated the staircase waveform front average gradient k after iteration is finished, then according to ideal linearity wave front average gradient k _lWith ${\mathrm{\&theta;}}_{\mathrm{error}}=\arcsin \left(\frac{{\mathrm{\&lambda;<figure-callout\; id="2"\; label="k\; I"\; filenames="DEST\_PATH\_HSB00000253105200011.png,DEST\_PATH\_HSB00000253105200021.png"\; state="\{\{state\}\}">k</figure-callout>}}_I}{2\mathrm{\&pi;d}}\right)-\arcsin \left(\frac{\mathrm{\&lambda;<figure-callout\; id="2"\; label="k"\; filenames="DEST\_PATH\_HSB00000253105200011.png,DEST\_PATH\_HSB00000253105200021.png"\; state="\{\{state\}\}">k</figure-callout>}}{2\mathrm{\&pi;d}}\right)$ Calculate the deflection angle error after iteration is finished, and record LCD phased array outgoing this moment staircase waveform front is to obtain correct on-load voltage code (being ripple control data).

Need to prove:

1, traditional ripple control method is to be based upon on the limited phase quantization thresholding, comes the linear wave front of approximate ideal by the PHASE DISTRIBUTION of ladder, and only realizes realizing quantizing phase place with desirable wave front is hithermost based on each phased array unit of (1) formula;

2, the average gradient of LCD phased array outgoing notch cuttype wave front is for obtaining the slope of straight line behind its linear fit.For staircase waveform front U[n], the straight line that simulates is U _Fit[n]=a+kn, n=1 wherein, 2 ..., N, N is the sum of phase control unit, the straight line U that then simulates _FitThe slope of [n] can be expressed as:

$k=\frac{\stackrel{\&OverBar;}{\mathrm{nU}}-\stackrel{\&OverBar;}{n}\&CenterDot;\stackrel{\&OverBar;}{U}}{\stackrel{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DEST\_PATH\_HSB00000253105200011.png,DEST\_PATH\_HSB00000253105200021.png"\; state="\{\{state\}\}">n</figure-callout>}}^2}-{\left(\stackrel{\&OverBar;}{n}\right)}^2}---\left(6\right)$

Wherein $\stackrel{\&OverBar;}{U}=\frac{1}{N}\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}U\left[n\right],$ $\stackrel{\&OverBar;}{n}=\frac{1}{N}\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}n=\frac{N+1}{2},$ $\stackrel{\&OverBar;}{n^2}=\frac{1}{N}\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}{n}^2=\frac{(N+1)(2N+1)}{6},$ $\stackrel{\&OverBar;}{\mathrm{nU}}=\frac{1}{N}\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}\mathrm{nU}\left[n\right].$ If with straight line U _Fit[n] as the linear wave front, then slope k is staircase waveform front U[n] average gradient;

3, because step edge location definition is the sequence number of step edge first left phase control unit, then for the staircase waveform front that p phase place step arranged, its step edge number is p-1, promptly $\stackrel{\&RightArrow;}{Q}=\{a_1,a_2...a_{p-1}\};$

4, in the step 4, the transformed value of the corresponding average gradient in step edge of every translation, as the step edge step that moves to left, then average gradient changes:

If move right a step, then average gradient changes:

Wherein $C=\frac{6}{N^2-1},$ C is a constant under the total constant situation of phase control unit.

M is the quantification number of steps in phase place 0～2 π.

Represent a _qThe weights of individual phased array unit, and $A_{a_q}=a_q(1-\frac{a_q}{N}),$ N represents the phased array unit number of staircase waveform front.

The invention has the beneficial effects as follows:

The problem that the present invention will control the deflection of LCD phased array wave beam high precision is converted into the problem of the position, phase place step edge of revising LCD phased array outgoing staircase waveform front, expression formula by derivation step wave front average gradient obtains the average gradient variation that phased array cell position of step edge translation brings, be modified to the difference minimum that position that the basis changes the step edge makes the average gradient of the average gradient of outgoing wave front and desirable wave front with iteration, thereby obtain corresponding on-load voltage code.Do not need too complex calculations process, can obtain voltage identification code corresponding on each phase shifter array element yet, obtain the beam quality of high precision scanning simultaneously, and realize more scanning angle quantity.Therefore, method provided by the invention has stronger operability.

Description of drawings

Fig. 1 is the synoptic diagram of conventional wave control method.

Fig. 2 is a process flow diagram of the present invention.

Fig. 3 provides examples of simulation at traditional ripple control method, has provided it at 0～θ _SpotAttainable deflection angle and theoretical deflection angle in the scope.

Fig. 4 is at the examples of simulation that the specific embodiment of the invention provides, compared that the present invention proposes based on the position, step edge of iteration correction outgoing wave front performance with the method for obtaining the on-load voltage code and traditional ripple control method.

Fig. 5 has compared many groups N and M combination and has carried out the revised trueness error of wave front iteration at the examples of simulation that the specific embodiment of the invention provides.

Fig. 6 has provided it at 0～θ at the examples of simulation that the specific embodiment of the invention provides _MaxObtain the required iterations of attainable deflection angle in the scope.

Embodiment

In order to verify feasibility of the present invention, provide the specific embodiment of the present invention below in conjunction with an examples of simulation, and simulation result is compared analysis, and the performance of itself and conventional wave control method relatively.

If the attainable step number of LCD phased array M=4, optical maser wavelength is 1.064 μ m, total number of poles (being LCD phased array unit number) N=64, and electrode separation d is 5 μ m.Calculate at 0～θ according to formula (2) and formula (3) _SpotThe deflection angle and the theoretical deflection angle that realize in the scope, wherein ${\mathrm{\&theta;}}_{\mathrm{spot}}=\frac{\mathrm{\&lambda;}}{\mathrm{Nd}}$ Expression far field beam angle.

Among Fig. 3, deflection angle that traditional ripple control method is realized and theoretical deflection angle have bigger inconsistent, and definition normalization trueness error is staircase waveform front deflection angle θ _StairWith the error of ideal linearity wave front deflection angle θ and the ratio of far field beam angle:

${\mathrm{\&epsiv;}}_{\mathrm{norm}}=\frac{|\mathrm{\&theta;}-{\mathrm{\&theta;}}_{\mathrm{stair}}|}{{\mathrm{\&theta;}}_{\mathrm{spot}}}---\left(9\right)$

Zui Da normalization trueness error ε as can be seen from Fig. 3 _{Norm, max}=0.1519=0.625M ^-1, POS 1 place in the drawings appears, and definition POS 2 places are the bigger positions of another trueness error, the phenomenon of the deflection angle flyback of wave beam also occurred at POS 1 and POS 2 places.

Fig. 4 compared that the present invention proposes based on the position, step edge of iteration correction outgoing wave front performance with the method for obtaining the on-load voltage code and traditional ripple control method.The deflection angle correction of each realization among Fig. 3 is obtained revised attainable deflection angle be close to very much theoretical deflection angle.As shown in Figure 4, at POS 3 places, maximum normalization trueness error ${\mathrm{\&epsiv;}}_{\mathrm{norm},\max }^{\mathrm{opt}}=0.0111,$ Compare with traditional ripple control method and to have reduced 13.7 times.

In order better to verify the correction effect of wave front iteration modification method, carry out the correction of wave front iteration at N of the many groups among Fig. 3 and M combination, revised trueness error is as shown in Figure 5.The maximum normalization trueness error that revised trueness error is compared the conventional wave control method has very large improvement.But every group of trueness error all has bigger peak value when the low-angle of left side, reason is when low-angle, the step of wave front is less in monocycle, and iterative algorithm itself can not increase step, therefore the numbers of steps that is used for the iteration correction is not enough, can not reach certain correction precision, the peak value trueness error of this moment is ${\mathrm{\&epsiv;}}_{\mathrm{norm},\max }^{\mathrm{opt}}=2.6N^{-1}{M}^{-1}.$ When step reaches some, revise trueness error and just can reach stable, as shown in Figure 5:

1, compares with traditional ripple control method, based on the maximum normalization trueness error of the ripple control method ε of iteration correction _{Norm, max}Reduce ${\mathrm{\&epsiv;}}_{\mathrm{norm},\max }/{\mathrm{\&epsiv;}}_{\mathrm{norm},\max }^{\mathrm{opt}}\&ap;{0.625M}^{-1}/2.6N^{-1}{M}^{-1}\&ap;0.25N$ Doubly, array number is only depended in the reduction of visible maximum normalization trueness error;

2, with N=32, M=8 is an example, and traditional ripple control method is at 0～θ _MaxScope in average normalization trueness error ε _Norm≈ 1.395, and through after the ripple control method correction based on the iteration correction ${\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}_{\mathrm{norm}}^{\mathrm{opt}}\&ap;0.029,$ Trueness error reduces ${\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}_{\mathrm{norm}}/{\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}_{\mathrm{norm}}^{\mathrm{opt}}\&ap;257\&ap;M\&times;N$ I.e. reduction value is proportional to N * M;

3, with N=32, M=8 is an example, at 0.2 θ _Max～θ _MaxAngular range, David

Carry ripple control method ε Deng the people _{Norm, max}≈ 0.0117, can reduce maximum normalization trueness error in this scope based on the ripple control method of iteration correction ${\mathrm{\&epsiv;}}_{\mathrm{norm},\max }/{\mathrm{\&epsiv;}}_{\mathrm{norm},\max }^{\mathrm{opt}}\&ap;0.0117/0.00109\&ap;11$ Doubly;

4, along with the increase of M or N, can increase with the number of angles that N * M ratio reduces the normalization trueness error based on realization in the ripple control method of iteration correction thereupon.

In addition, this iterative algorithm is all not high to the requirement of iterations and required iteration time, and with combination N=32, the spatial light modulator of M=8 is an example, obtains 0～θ _MaxThe iterations of angle distributes as shown in Figure 6 in the scope.But less at all deflection angle scope inner iterations time number average, maximum is no more than 10 times.At CPU frequency 1.61GHz, the enterprising line operate of the PC of internal memory 512MB, single iteration required time only are t ≈ 2.19 * 10 ^-4S.

Claims (2)

1. the LCD phased array ripple control data based on the wave front iteration are determined method, may further comprise the steps:

Step 1:, utilize the conventional wave control method to calculate LCD phased array outgoing staircase waveform front according to the desirable deflection angle θ that the LCD phased array needs are realized

Wherein the phase value of the realization of n phased array unit is:

Wherein M is the quantification number of steps in phase place 02 π,

λ is an optical wavelength, d phased array unit interval, " round " expression get with bracket in the immediate round values of value, N is a phased array unit sum;

Step 2: calculate average gradient error k _Error=K _I-k, wherein k is a LCD phased array outgoing staircase waveform front average gradient, k _IBe ideal linearity outgoing wave front average gradient and satisfied:

$k=\frac{2\mathrm{\&pi;d}}{\mathrm{\&lambda;}}\sin \left({\mathrm{\&theta;}}_{\mathrm{stair}}\right)$

Wherein d is the phased array unit interval, and θ is that the deflection angle of ideal linearity wave front correspondence is the desirable deflection angle that LCD phased array need be realized, θ _StairDeflection angle for staircase waveform front correspondence;

Step 3: array is also used in the position, step edge that is comprised in the statistics LCD phased array outgoing staircase waveform front $\stackrel{\&RightArrow;}{Q}=\{a_1,a_2,\&CenterDot;\&CenterDot;\&CenterDot;,a_{p-1}\}$ Record, wherein position, step edge a _qBe defined as the sequence number of q step edge first left phase control unit, 1≤q≤p and 1≤a _q≤ N-1, p are the step number of staircase waveform front;

Step 4: by formula

And

Calculate any q step edge respectively, its position is a _q, 1≤a _q≤ N-1, the step edge to the right or the average gradient conversion that behind the phased array unit that moves to left, produces Or

Wherein $C=\frac{6}{N^2-1},$ C is a constant under the total constant situation of phase control unit,

M is the quantification number of steps in phase place 0～2 π,

Represent a _qThe weights of individual phased array unit, and $A_{a_q}=a_q(1-\frac{a_q}{N});$

Step 5 definition $\stackrel{\&RightArrow;}{{\mathrm{\&Delta;k}}_R}=\{\mathrm{\&Delta;}{k}_{a_1\_R},\mathrm{\&Delta;}{k}_{a_2\_R},\&CenterDot;\&CenterDot;\&CenterDot;,\mathrm{\&Delta;}{k}_{a_{p-1}\_R}\},$ $\stackrel{\&RightArrow;}{\mathrm{\&Delta;}{k}_L}=\{\mathrm{\&Delta;}{k}_{a_1\_L},\mathrm{\&Delta;}{k}_{a_2\_L},\&CenterDot;\&CenterDot;\&CenterDot;,\mathrm{\&Delta;}{k}_{a_{p-1}\_L}\},$ Calculate by step 4

With

In comprised the average gradient that the edge translation of all possible step brought and changed, and in array

In search all positions, edge, always having a position is q _R, 1≤q _R≤ p, the step edge, make:

${\mathrm{\&sigma;}}_R=\min \left(\right|k_{\mathrm{error}}-\stackrel{\&RightArrow;}{{\mathrm{\&Delta;k}}_R}\left[\stackrel{\&RightArrow;}{Q}\right]\left|\right)=|k_{\mathrm{error}}-\stackrel{\&RightArrow;}{\mathrm{\&Delta;}{k}_R}[q_R\left]\right|$

Equally also always having a position is q _L, 1≤q _L≤ p, the step edge, make:

${\mathrm{\&sigma;}}_L=\min \left(\right|k_{\mathrm{error}}-\stackrel{\&RightArrow;}{{\mathrm{\&Delta;k}}_L}\left[\stackrel{\&RightArrow;}{Q}\right]\left|\right)=|k_{\mathrm{error}}-\stackrel{\&RightArrow;}{\mathrm{\&Delta;}{k}_L}[q_L\left]\right|$

Wherein minimum value is got in " min " expression, and " || " expression takes absolute value,

Expression is by array The vector that constitutes, " [q _R] " be q _RVector representation, " [q _L] " be q _LVector representation;

If step 6 σ _R＞| k _Error| and σ _L＞| k _Error|, representative can't reduce the average gradient error again | k _Error|, stop iteration and forward step 7 statistics angular error to; If σ _R≤ | k _Error| or σ _L≤ | k _Error|, then begin following iterative process;

If step 6-1 is σ _R＜σ _L, represent right-shift operation can better reduce the average gradient error | k _Error|, to q _RThe step edge at place carries out after translation is finished, forwarding step 2 to phased array unit of right translation to;

If step 6-2 is σ _R〉=σ _L, represent shift left operation can better reduce the average gradient error | k _Error|, to q _LThe step edge at place carries out after translation is finished, forwarding step 2 to phased array unit of left to;

Step 7 is calculated the staircase waveform front average gradient k after iteration is finished, then according to ideal linearity wave front average gradient k _IWith ${\mathrm{\&theta;}}_{\mathrm{error}}=\arcsin \left(\frac{\mathrm{\&lambda;}{k}_I}{2\mathrm{\&pi;d}}\right)-\arcsin \left(\frac{\mathrm{\&lambda;k}}{2\mathrm{\&pi;d}}\right)$ Calculate the deflection angle error after iteration is finished, and record LCD phased array outgoing this moment staircase waveform front is to obtain correct on-load voltage code, i.e. ripple control data.

2. the LCD phased array ripple control data based on the wave front iteration according to claim 1 are determined method, it is characterized in that, LCD phased array outgoing staircase waveform front average gradient is for obtaining the slope of straight line in the step 2 behind its linear fit; For LCD phased array outgoing staircase waveform front U[n], the straight line that simulates is U _Fit[n]=a+kn, n=1 wherein, 2 ..., N, N is the sum of phase control unit, the straight line U that then simulates _FitThe slope of [n], i.e. LCD phased array outgoing staircase waveform front U[n] average gradient be expressed as:

$k=\frac{\stackrel{\&OverBar;}{\mathrm{nU}}-\&CenterDot;\stackrel{\&OverBar;}{n}\&CenterDot;\stackrel{\&OverBar;}{U}}{\stackrel{\&OverBar;}{n^2}-{\left(\stackrel{\&OverBar;}{n}\right)}^2}$

Wherein $\stackrel{\&OverBar;}{U}=\frac{1}{N}\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}U\left[n\right],\stackrel{\&OverBar;}{n}=\frac{1}{N}\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}n=\frac{N+1}{2},\stackrel{\&OverBar;}{n^2}=\frac{1}{N}\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}{n}^2=\frac{(N+1)(2N+1)}{6},\stackrel{\&OverBar;}{\mathrm{nU}}=\frac{1}{N}\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}\mathrm{nU}\left[n\right].$

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