CN102121975A - Method for reconstructing precise phase response of sampling oscilloscope based on NTN calibration and K-K conversion - Google Patents

Abstract

The invention discloses a method for reconstructing precise phase response of a sampling oscilloscope based on nose-to-nose (NTN) calibration and Kramers-Kronig (K-K) conversion, belongs to the technical field of high-speed electronics metering and solves the problem that the frequency resolution of phase response of the sampling oscilloscope, acquired on the basis of NTN calibration, is low. The method comprises the following steps of: converting an amplitude response function h(f) of the oscilloscope theoretically to obtain a phase response function Phi(f); converting the amplitude response function of the oscilloscope with a truncation error in a practical project to obtain a phase response function PhiOmega(f); obtaining a phase error function delta(f); performing approximate expansion and orthogonalization on the phase error function delta(f); and calculating to obtain a coefficient value of a basis function according to a difference between a phase obtained by the NTN calibration method and a phase on a corresponding frequency point of the PhiOmega(f), and correcting the phase response function PhiOmega(f) of the oscilloscope in the practical project to obtain a precise phase response function of the sampling oscilloscope in the practical project. The method is used for acquiring the precise phase response of the sampling oscilloscope.

Description

Based on the sampling oscilloscope meticulous phase response reconstructing method of NTN calibration with the K-K conversion

Technical field

The present invention relates to a kind of meticulous phase response reconstructing method of sampling oscilloscope, belong to high-velocity electrons and learn field of measuring techniques based on NTN calibration and K-K conversion.

Background technology

The nineties in last century,, directly measure the complicated large-signal response of the circuit, device and the system that contain nonlinear element and simplified greatly along with the progress of radio frequency, microwave theory and technique.Non-linear vector network analyzer is a up-to-date utility of directly finishing the large-signal network analysis.Present stage non-linear vector network analyzer be faced with two hang-ups: the one, push present highest frequency scope (50GHz) to higher frequency limitation; The 2nd, the frequency resolution of raising phase alignment.

The phase alignment spare of non-linear vector network analyzer is by the sampling oscilloscope calibration, and therefore the phase response that obtains sampling oscilloscope seem particularly important.The phase response that obtains sampling oscilloscope at present mainly relies on the NTN calibration.NTN (Nose toNose) calibration steps is to utilize two oscillographs directly to dock, and wherein oscillograph produces the kick-out pulse as exciting signal source, and another oscillograph is as receiver, and the method by deconvolution can obtain oscillographic response function.Because the restriction of NTN self-technique, its frequency resolution can only reach 250MHz, and this has influenced the precision of follow-up measurement greatly, obtains the meticulous phase response of sampling oscilloscope so be badly in need of a kind of new method.

Summary of the invention

The objective of the invention is to have the low problem of frequency resolution, provide a kind of based on the meticulous phase response reconstructing method of sampling oscilloscope of NTN calibration with the K-K conversion in order to solve the phase response that relies on the NTN calibration to obtain sampling oscilloscope.

The present invention includes following steps:

Step 1: in theory oscillographic amplitude response function h (f) is carried out the Kramers-Krong conversion, obtain oscillographic in theory phase response function φ (f);

Step 2: the oscillographic amplitude response function process Kramers-Krong conversion that has truncation error in the engineering reality is obtained oscillographic phase response function φ in the engineering reality _Ω(f);

Step 3: obtain to block in the engineering reality phase error function Δ (f) afterwards by step 1 and step 2;

Step 4: (f) does approximate expansion to the phase error function Δ, and the basis function that launches the back acquisition is carried out orthogonalization;

Step 5: according to phase place that obtains by the NTN calibration method and φ _Ω(f) difference of the phase place on the respective frequencies point is calculated, and obtains the coefficient value of described orthogonalized basis function, in conjunction with sampled value to oscillographic phase response function φ in the engineering reality _Ω(f) revise, obtain the meticulous phase response function of sampling oscilloscope in the engineering reality.

Advantage of the present invention is: the present invention utilizes Kramers-Krong (K-K) conversion to reconstruct the meticulous phase response of sampling oscilloscope in conjunction with NTN calibration method and frequency sweep method.Experiment shows: the oscillograph phase response that minimum phase response and the NTN calibration method that reconstructs measured is very approaching on corresponding frequency, thinks that herein the result that NTN obtains is the actual value of oscillograph phase response.

Description of drawings

Fig. 1 is orthogonalized basis function { Ψ ₁, Ψ ₂, Ψ ₃Curve map;

Fig. 2 is the comparison diagram of the sampling of NTN calibration method phase place that obtains and the phase place that adopts the inventive method match to obtain;

Fig. 3 is an oscillograph 1MHz-40GHz phase response fitting result curve map in the embodiment six;

Fig. 4 is the uncertainty curve map of minimum phase response in the embodiment six.

Embodiment

Embodiment one: present embodiment may further comprise the steps:

Step 1: in theory oscillographic amplitude response function h (f) is carried out the Kramers-Krong conversion, obtain oscillographic in theory phase response function φ (f);

Step 2: the oscillographic amplitude response function process Kramers-Krong conversion that has truncation error in the engineering reality is obtained oscillographic phase response function φ in the engineering reality _Ω(f);

Step 3: obtain to block in the engineering reality phase error function Δ (f) afterwards by step 1 and step 2;

Step 4: (f) does approximate expansion to the phase error function Δ, and the basis function that launches the back acquisition is carried out orthogonalization;

Step 5: according to phase place that obtains by the NTN calibration method and φ _Ω(f) difference of the phase place on the respective frequencies point is calculated, and obtains the coefficient value of described orthogonalized basis function, in conjunction with sampled value to oscillographic phase response function φ in the engineering reality _Ω(f) revise, obtain the meticulous phase response function of sampling oscilloscope in the engineering reality.

Embodiment two: present embodiment is for to the further specifying of embodiment one, and the expression formula of oscillographic in theory phase response function φ (f) is in the described step 1:

$\mathrm{\φ}\left(f\right)=\frac{2f}{\mathrm{\π}}\underset{\mathrm{\π}}{\overset{\∞}{\∫}}\frac{1}{f^2-s^2}\ln \left(\right|\tilde{h}\left(s\right)\left|\right)\mathrm{ds};$

F represents frequency in the formula;

Oscillographic phase response function φ in the engineering reality in the described step 2 _Ω(f) expression formula is:

${\mathrm{\φ}}_{\mathrm{\Ω}}\left(f\right)=\frac{2f}{\mathrm{\π}}\underset{0}{\overset{\mathrm{\Ω}}{\∫}}\frac{1}{f^2-s^2}\ln \left(\right|\tilde{h}\left(s\right)\left|\right)\mathrm{ds};$

The expression formula of the phase error function Δ (f) after blocking in the engineering reality in the described step 3 is:

$\mathrm{\Δ}\left(f\right)=\mathrm{\φ}\left(f\right)-{\mathrm{\φ}}_{\mathrm{\Ω}}\left(f\right)=\frac{2\mathrm{\π}}{f}\underset{\mathrm{\Ω}}{\overset{\∞}{\∫}}\frac{1}{f^2-s^2}\ln \left(\right|h\left(s\right)\left|\right)\mathrm{ds},$

Ω is the maximum frequency values that adopts sweep measurement method and NTN calibration method to reach in the engineering reality in the formula.

Embodiment three: present embodiment is for to the further specifying of embodiment two, and in the described step 4 phase error function Δ (f) done the expression formula that approximate expansion obtains to be:

$\mathrm{\Δ}\left(f\right)\≈c_{1}f+c_{2}f\underset{\mathrm{\Ω}}{\overset{\∞}{\∫}}\frac{1}{f^2-s^2}\mathrm{ds}+c_{3}f\underset{\mathrm{\Ω}}{\overset{\∞}{\∫}}\frac{1}{f^2-s^2}\ln \left(s\right)\mathrm{ds}$

$=C_1{\mathrm{\ψ}}_1\left(f\right)+C_2{\mathrm{\ψ}}_2\left(f\right)+C_3{\mathrm{\ψ}}_3\left(f\right),$

Basis function ψ wherein ₁(f)=f, ${\mathrm{\ψ}}_2\left(f\right)=\ln \left(\frac{\mathrm{\Ω}+f}{\mathrm{\Ω}-f}\right),$ ${\mathrm{\ψ}}_3\left(f\right)=\mathrm{f\Φ}(\frac{f^2}{{\mathrm{\Ω}}^2},2,\frac{1}{2});$

ψ wherein ₃(f) in the expression formula,

It is the Lerch transcendental function;

The basis function that launches the back acquisition is carried out orthogonalized method is:

Adopt the Gram-Schmidt method to launching the basis function orthogonalization that the back obtains, the function after the definition orthogonalization is { Ψ ₁, Ψ ₂, Ψ ₃, obtain

${\mathrm{\Ψ}}_1\left(f\right)=\frac{{\mathrm{\ψ}}_1\left(f\right)}{\sqrt{E_1}}=\sqrt{3}f,$

${\mathrm{\Ψ}}_2\left(f\right)=\frac{{\mathrm{\θ}}_2\left(f\right)}{\sqrt{E_{{\mathrm{\θ}}_2\left(f\right)}}}=\frac{1}{\sqrt{\frac{{\mathrm{\π}}^2}{3}-3}}\·(\ln \left(\frac{1+f}{1-f}\right)-3f),$

${\mathrm{\Ψ}}_3\left(f\right)=\frac{{\mathrm{\θ}}_3\left(f\right)}{\sqrt{E_{{\mathrm{\θ}}_3\left(f\right)}}}\≈18.0102\·\underset{n=0}{\overset{\∞}{\mathrm{\Σ}}}\frac{4}{{(2n+1)}^2}{f}^{2n+1}-4.1224\·\ln \left(\frac{1+f}{1-f}\right)-66.9176\·f.$

Oscillographic in theory phase response function φ (f) in the present embodiment, be that Hilbert conversion by its amplitude logarithm recovers out, the Kramers-Krong that adopts is transformed to a kind of concrete form in the Hilbert conversion, owing in engineering reality, can cause very large absolute error on the phase calculation if simply use this theory, but described absolute error has low order in essence, therefore utilize the Hilbert conversion to have the character of localization, construct described basis function and estimate this class error.

Basis function

In,

Be that Lerch surmounts the water hole number.

The process of in the present embodiment phase error function Δ (f) being done approximate expansion is mathematical operation, and the first step is to make Taylor's technology to launch, and second one is to the integration as a result after launching, and obtains the expression formula of three basis functions then.

Embodiment four: present embodiment is for to the further specifying of embodiment three, and directly sampling oscilloscope sampled in the described step 5 and the method that finally obtains the meticulous phase response function of sampling oscilloscope in the engineering reality is:

Utilize frequency sweep method that oscillograph is carried out the amplitude-frequency calibration, the frequency of its amplitude response function is { f _n| n=1 ..., N}; Utilize the NTN calibration method that oscillograph is carried out complex frequency calibration, the frequency of its phase response function be f ' _m| m=1 ..., M}, obtain thus M * N rank dense matrix K (f ' _m, f _n), wherein M is that the frequency of phase response function is counted, N is that the frequency of amplitude response function is counted; Described dense matrix K (f ' _m, f _n) m, the solution formula of the element of n item is:

K _m.n＝KB _n(f)，

K is an operator in the formula, and

$K=\frac{1}{\mathrm{\π}}\underset{\mathrm{\α}}{\overset{\mathrm{\β}}{\∫}}\frac{2f}{f^2-s^2}(\mathrm{ms}+b)\mathrm{ds}=\frac{m}{\mathrm{\π}}f\ln \left|\frac{{\mathrm{\α}}^2-f^2}{{\mathrm{\β}}^2-f^2}\right|-\frac{b}{\mathrm{\π}}\ln |\frac{\mathrm{\α}+f}{\mathrm{\α}-f}\·\frac{\mathrm{\β}-f}{\mathrm{\β}+f}|,$

B _n(f) the linear difference function for constructing,

Then

${\mathrm{KB}}_n\left(f\right)=\frac{1}{\mathrm{\π}(f_n-f_{n-1})}\·\left(f\ln \right|\frac{f_{n-1}^2-f^2}{f_n^2-f^2}|+f_{n-1}\ln |\frac{f_{n-1}+f}{f_{n-1}-f}\·\frac{f_n-f}{f_n+f}\left|\right)$

$-\frac{1}{\mathrm{\π}(f_{n+1}-f_n)}\·\left(f\ln \right|\frac{f_n^2-f^2}{f_{n+1}^2-f^2}|+f_{n+1}\ln |\frac{f_n+f}{f_n-f}\·\frac{f_{n+1}-f}{f_{n+1}+f}\left|\right),$

The oscillographic phase response function that makes the NTN calibration method obtain is φ _NTN(f ' _m), according to

Formula, the phase sample value that is obtained by described frequency sweep calibration method can be calculated as follows:

φ _Ω(f′ _m)＝K(f′ _m，f _n)h，

H is the vector that logarithm value constituted by the amplitude response function of frequency sweep method acquisition in the formula,

The oscillographic phase response function that is obtained by the NTN calibration is φ _NTN(f ' _m) and calculate the φ that obtains _Ω(f ' _m) work difference acquisition Δ (f ' _m), by the approximate expansion of described phase error function Δ (f) to Δ (f ' _m) carry out match, obtain:

Δ(f′ _m)＝α ₁Ψ ₁(f′ _m)+α ₂Ψ ₂(f′ _m)+α ₃Ψ ₃(f′ _m)，

Utilize least square method to find the solution following formula, obtain factor alpha ₁, α ₂, α ₃,

Suppose that residual error is fully little in least square fitting, and be the sign of minimum phase hypothesis correctness, obtain that the meticulous phase response function of sampling oscilloscope is in the engineering reality:

φ _mp(f)＝K(f，f _n)h+α ₁Ψ ₁(f)+α ₂Ψ ₂(f)+α ₃Ψ ₃(f)，

Obtain the phase response function of the sampling oscilloscope under the arbitrary frequency point after the reconstruct thus.

The frequency sweep method that adopts in the present embodiment is with respect to the NTN collimation technique, and the amplitude information of frequency response that can measuring oscilloscope in the attainable scope of existing microwave power standard, is generally 1MHz-50GHz, and frequency sweep method can be measured the amplitude of any frequency.But because frequency sweep method can not provide phase information, so use this method can not carry out oscillograph time domain measurement and calibration separately.

Use the Gram-Schmidt method of mathematics original substrate { ψ in the present embodiment _jBecome { Ψ at the bottom of the orthogonal basis _jBe to make matrix A because do like this _Mj=Ψ _j(f ' _m) state be that conditional number has diminished.The status number of a matrix A be defined as cond (A)=|| A||||A ^-1||, it is to measure a kind of index number of a solution of equations for the coefficient data changing sensitivity, especially plays an important role in the analysis of linear algebraic number value.In the present invention its expression when Δ (f ' _m) α={ α when subtle change takes place ₁, α ₂, α ₃Degree of stability, the more little α of status number={ α ₁, α ₂, α ₃With respect to Δ (f ' _m) variation stable more.This status number plays an important role when comparison algorithm good and bad, is difficult to calculate but be actually in engineering.Usually use an analytical approach qualitatively in the engineering actual analysis: the status number of the more little then matrix of the angle between the matrix column vector is big more; If the orthogonal then matrix status number of rectangular array vector levels off to 1.So original substrate { ψ _jBecome { Ψ at the bottom of the orthogonal basis _jAfterwards result calculated is more stable with respect to the variation meeting of the test data that collects, thereby increased the robustness of algorithm, the reliability that obtains the result is strengthened.

Δ in the present embodiment (f ' _m) with respect to the φ of actual measurement _NTN(f ' _m) very big, the expression formula that the application phase error function is done approximate expansion obtains.

The method that employing frequency sweep method described in the present embodiment is sampled to oscillograph is: use power meter and calibrated power splitter to come the output power in real-time measuring-signal source, can calculate the amplitude response value of sampling oscilloscope under different frequency point by the reading on sampling oscilloscope and the power meter, and then obtain the amplitude response function in the certain bandwidth of sampling oscilloscope.

Described NTN calibration method: the scheme of nineteen ninety U.S. Hewlett-Packard Corporation proposition: do not adopt any tool master, only use the hypervelocity waveform digitization system of two performance unanimities, their signal input part is directly docked, carry out the calibration of system's complex transfer function.Find by testing their repeatedly: when system during to a direct current voltage sample, sample circuit will produce extremely narrow kickout pulse, and to the input end projection, this extremely narrow kickout pulse contains hypervelocity waveform digitization system linearity characteristic information.Therefore, they propose important guess: when certain dc offset voltage is set, can produce little " kickout " pulse at oscillographic input port.This pulse almost shape with the time domain impulse response of oscillograph self is identical.This guess after experiment in obtained confirmation.The method is called as NTN (Nose-to-Nose) calibration method.

Embodiment five: present embodiment is for further specifying the frequency { f of described amplitude response function to embodiment four _n| n=1 ..., the frequency of N} and phase response function f ' _m| m=1 ..., the pass between the M} is:

Min{f _n}≤Min{f′ _m}＜Max{f′ _m}＜Max{f _n}。

Embodiment six: below in conjunction with Fig. 1 to Fig. 4 present embodiment is described, utilizes Agilent 86100C sampling oscilloscope to experimentize.At first utilize the NTN collimation technique, two 86100C sampling oscilloscope butt joints.Obtained the complex frequency response of the 1-50GH of 86100C, resolution is 250MHz.

Utilize frequency sweep method to obtain the amplitude response information of the 82484A module of sampling oscilloscope then.Use AgilentE8257D 250KHz-40GHz PSGAnalog Signal Generator as the triggering source; The synthetic sweep generator 0.01-40GHz of AV1487 ultra broadband is as signal source.The triggering source uses identical 10MHz benchmark (benchmark of AV1487) to carry out synchronously with signal source.Calibrating device is HP 437B Power Meter and Agilent 8487APower Sensor50MHz-50GHz 1 μ W-100mW.The background noise of oscillograph module 83484A is: Mean:566.031 μ V, Std dev:10.74 μ V have obtained utilizing frequency sweep method amplitude response information that obtains and the phase response information of utilizing the NTN calibration to obtain thus.

The frequency of the phase information of measuring in the present embodiment f ' _m| m=1 ..., M} can not with the frequency { f of the amplitude information of measuring _n| n=1 ..., N} is equally dense.Be that the amplitude information frequency range should cover the phase information frequency range, otherwise can cause dispersing on the data processing, need to satisfy:

Min{f _n}≤Min{f′ _m}＜Max{f′ _m}＜Max{f _n}。

The meticulous phase response function of sampling oscilloscope in engineering reality:

φ _Mp(f)=K (f, f _n) h+ α ₁Ψ ₁(f)+α ₂Ψ ₂(f)+α ₃Ψ ₃(f) in, the Kramers-Kronig operator by with amplitude measurement frequency { f _nIdentical set forms, target f can be an optional frequency.Experimental result as shown in Figures 2 and 3.The α that obtain this moment ₁=-0.4175; α ₂=-1.1156; α ₃=0.4798.

The uncertainty of the phase place that obtains by above-mentioned K-K conversion mainly is made up of three parts: the uncertainty u that NTN calibrates ²[h]; The uncertainty u of frequency sweep method ²[φ _NTN] and the uncertainty brought of transformation matrix and match.Because phase place is the Hilbert conversion by the amplitude information logarithm Obtain.Can obtain after using certain mathematic(al) manipulation:

K wherein _M.n=KB _n(f ' _m) element that lists for the capable n of m of matrix K.

The average of supposing stochastic variable X is E (X)=X ₀, variance is var (X)=u ²[X], variable Y=F (X), wherein F is fully can be little, then can obtain:

E[Y]＝F ₀[X ₀]， $u^2\left[Y\right]={\left(\frac{\∂F}{\∂s}{|}_{s=X_0}\right)}^2{u}^2\left[X\right],$

So can obtain

Wherein

Be the diagonal matrix that comprises vectorial h uncertainty, described uncertainty comprises system and at random.Can obtain the standard uncertainty of frequency sweep method data in addition between 0.005-0.17dB.

The u that obtains ²Element on [φ Ω] principal diagonal for our needed φ (f ' _m) not authenticity.Because the measurement of NTN calibration and frequency sweep method is separate, so uncertainty u ²[φ _NTN] and u ²[φ _Ω] be the quadrature addition.

Because match basis function { ψ ₁(f), ψ ₂(f), ψ ₃(f) } quadrature turns to substrate { Ψ ₁, Ψ ₂, Ψ ₃, make matrix A like this _Mj=Ψ _j(f ' _m) state be that conditional number has diminished, this makes the vectorial α={ α calculate ₁, α ₂, α ₃Error reduce greatly.And suppose φ here _Mp(f) stochastic variable h and { α in the final expression formula ₁, α ₂, α ₃Be separate.Follow their linear propagation forms separately by the propagation of error of integral operator and three pointwise multiplication, its uncertainty result is that quadrature can add.

The phase place uncertainty of the 0-30GHz that obtains at last as shown in Figure 4.

Claims (5)

1. one kind based on the NTN calibration meticulous phase response reconstructing method of sampling oscilloscope with the K-K conversion, and it is characterized in that: it may further comprise the steps:

Step 1: in theory oscillographic amplitude response function h (f) is carried out the Kramers-Krong conversion, obtain oscillographic in theory phase response function φ (f);

Step 2: the oscillographic amplitude response function process Kramers-Krong conversion that has truncation error in the engineering reality is obtained oscillographic phase response function φ in the engineering reality _Ω(f);

Step 3: obtain to block in the engineering reality phase error function Δ (f) afterwards by step 1 and step 2;

Step 4: (f) does approximate expansion to the phase error function Δ, and the basis function that launches the back acquisition is carried out orthogonalization;

Step 5: according to phase place that obtains by the NTN calibration method and φ _Ω(f) difference of the phase place on the respective frequencies point is calculated, and obtains the coefficient value of described orthogonalized basis function, in conjunction with sampled value to oscillographic phase response function φ in the engineering reality _Ω(f) revise, obtain the meticulous phase response function of sampling oscilloscope in the engineering reality.

2. according to claim 1 based on the meticulous phase response reconstructing method of sampling oscilloscope of NTN calibration with the K-K conversion, it is characterized in that:

The expression formula of oscillographic in theory phase response function φ (f) is in the described step 1:

$\mathrm{\φ}\left(f\right)=\frac{2f}{\mathrm{\π}}\underset{\mathrm{\π}}{\overset{\∞}{\∫}}\frac{1}{f^2-s^2}\ln \left(\right|\tilde{h}\left(s\right)\left|\right)\mathrm{ds};$

F represents frequency in the formula;

Oscillographic phase response function φ in the engineering reality in the described step 2 _Ω(f) expression formula is:

${\mathrm{\φ}}_{\mathrm{\Ω}}\left(f\right)=\frac{2f}{\mathrm{\π}}\underset{0}{\overset{\mathrm{\Ω}}{\∫}}\frac{1}{f^2-s^2}\ln \left(\right|\tilde{h}\left(s\right)\left|\right)\mathrm{ds};$

The expression formula of the phase error function Δ (f) after blocking in the engineering reality in the described step 3 is:

$\mathrm{\Δ}\left(f\right)=\mathrm{\φ}\left(f\right)-{\mathrm{\φ}}_{\mathrm{\Ω}}\left(f\right)=\frac{2\mathrm{\π}}{f}\underset{\mathrm{\Ω}}{\overset{\∞}{\∫}}\frac{1}{f^2-s^2}\ln \left(\right|h\left(s\right)\left|\right)\mathrm{ds},$

Ω is the maximum frequency values that adopts sweep measurement method and NTN calibration method to reach in the engineering reality in the formula.

3. according to claim 2 based on the meticulous phase response reconstructing method of sampling oscilloscope of NTN calibration with the K-K conversion, it is characterized in that:

In the described step 4 phase error function Δ (f) being done the expression formula that approximate expansion obtains is:

$\mathrm{\Δ}\left(f\right)\≈c_{1}f+c_{2}f\underset{\mathrm{\Ω}}{\overset{\∞}{\∫}}\frac{1}{f^2-s^2}\mathrm{ds}+c_{3}f\underset{\mathrm{\Ω}}{\overset{\∞}{\∫}}\frac{1}{f^2-s^2}\ln \left(s\right)\mathrm{ds}$

$=C_1{\mathrm{\ψ}}_1\left(f\right)+C_2{\mathrm{\ψ}}_2\left(f\right)+C_3{\mathrm{\ψ}}_3\left(f\right),$

Basis function ψ wherein ₁(f)=f, ${\mathrm{\ψ}}_2\left(f\right)=\ln \left(\frac{\mathrm{\Ω}+f}{\mathrm{\Ω}-f}\right),$ ${\mathrm{\ψ}}_3\left(f\right)=\mathrm{f\Φ}(\frac{f^2}{{\mathrm{\Ω}}^2},2,\frac{1}{2});$

ψ wherein ₃(f) in the expression formula,

It is the Lerch transcendental function;

The basis function that launches the back acquisition is carried out orthogonalized method is:

Adopt the Gram-Schmidt method to launching the basis function orthogonalization that the back obtains, the function after the definition orthogonalization is { Ψ ₁, Ψ ₂, Ψ ₃, obtain

${\mathrm{\Ψ}}_1\left(f\right)=\frac{{\mathrm{\ψ}}_1\left(f\right)}{\sqrt{E_1}}=\sqrt{3}f,$

${\mathrm{\Ψ}}_2\left(f\right)=\frac{{\mathrm{\θ}}_2\left(f\right)}{\sqrt{E_{{\mathrm{\θ}}_2\left(f\right)}}}=\frac{1}{\sqrt{\frac{{\mathrm{\π}}^2}{3}-3}}\·(\ln \left(\frac{1+f}{1-f}\right)-3f),$

${\mathrm{\Ψ}}_3\left(f\right)=\frac{{\mathrm{\θ}}_3\left(f\right)}{\sqrt{E_{{\mathrm{\θ}}_3\left(f\right)}}}\≈18.0102\·\underset{n=0}{\overset{\∞}{\mathrm{\Σ}}}\frac{4}{{(2n+1)}^2}{f}^{2n+1}-4.1224\·\ln \left(\frac{1+f}{1-f}\right)-66.9176\·f.$

4. according to claim 3 based on the meticulous phase response reconstructing method of sampling oscilloscope of NTN calibration with the K-K conversion, it is characterized in that:

The method of directly sampling oscilloscope being sampled in the described step 5 and finally obtaining the meticulous phase response function of sampling oscilloscope in the engineering reality is:

Utilize frequency sweep method that oscillograph is carried out the amplitude-frequency calibration, the frequency of its amplitude response function is { f _n| n=1 ..., N}; Utilize the NTN calibration method that oscillograph is carried out complex frequency calibration, the frequency of its phase response function be f ' _m| m=1 ..., M}, obtain thus M * N rank dense matrix K (f ' _m, f _n), wherein M is that the frequency of phase response function is counted, N is that the frequency of amplitude response function is counted; Described dense matrix K (f ' _m, f _n) m, the solution formula of the element of n item is:

K _m.n＝KB _n(f)，

K is an operator in the formula, and

$K=\frac{1}{\mathrm{\π}}\underset{\mathrm{\α}}{\overset{\mathrm{\β}}{\∫}}\frac{2f}{f^2-s^2}(\mathrm{ms}+b)\mathrm{ds}=\frac{m}{\mathrm{\π}}f\ln \left|\frac{{\mathrm{\α}}^2-f^2}{{\mathrm{\β}}^2-f^2}\right|-\frac{b}{\mathrm{\π}}\ln |\frac{\mathrm{\α}+f}{\mathrm{\α}-f}\·\frac{\mathrm{\β}-f}{\mathrm{\β}+f}|,$

B _n(f) the linear difference function for constructing,

Then

${\mathrm{KB}}_n\left(f\right)=\frac{1}{\mathrm{\π}(f_n-f_{n-1})}\·\left(f\ln \right|\frac{f_{n-1}^2-f^2}{f_n^2-f^2}|+f_{n-1}\ln |\frac{f_{n-1}+f}{f_{n-1}-f}\·\frac{f_n-f}{f_n+f}\left|\right)$

$-\frac{1}{\mathrm{\π}(f_{n+1}-f_n)}\·\left(f\ln \right|\frac{f_n^2-f^2}{f_{n+1}^2-f^2}|+f_{n+1}\ln |\frac{f_n+f}{f_n-f}\·\frac{f_{n+1}-f}{f_{n+1}+f}\left|\right),$

The oscillographic phase response function that makes the NTN calibration method obtain is φ _NTN(f ' _m), according to

Formula, the phase sample value that is obtained by described frequency sweep calibration method can be calculated as follows:

φ _Ω(f′ _m)＝K(f′ _m，f _n)h，

H is the vector that logarithm value constituted by the amplitude response function of frequency sweep method acquisition in the formula,

The oscillographic phase response function that is obtained by the NTN calibration is φ _NTN(f ' _m) and calculate the φ that obtains _Ω(f ' _m) work difference acquisition Δ (f ' _m), by the approximate expansion of described phase error function Δ (f) to Δ (f ' _m) carry out match, obtain:

Δ(f′ _m)＝α ₁Ψ ₁(f′ _m)+α ₂Ψ ₂(f′ _m)+α ₃Ψ ₃(f′ _m)，

Utilize least square method to find the solution following formula, obtain factor alpha ₁, α ₂, α ₃,

Suppose that residual error is fully little in least square fitting, and be the sign of minimum phase hypothesis correctness, obtain that the meticulous phase response function of sampling oscilloscope is in the engineering reality:

φ _mp(f)＝K(f，f _n)h+α ₁Ψ ₁(f)+α ₂Ψ ₂(f)+α ₃Ψ ₃(f)，

Obtain the phase response function of the sampling oscilloscope under the arbitrary frequency point after the reconstruct thus.

5. according to claim 4 based on the meticulous phase response reconstructing method of sampling oscilloscope of NTN calibration with the K-K conversion, it is characterized in that: the frequency { f of described amplitude response function _n| n=1 ..., the frequency of N} and phase response function f ' _m| m=1 ..., the pass between the M} is:

Min{f _n}≤Min{f′ _m}＜Max{f′ _m}＜Max{f _n}。

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