CN1709202A - Digital demodulation mode mixing biological impedance measuring method - Google Patents

Abstract

The present invention relates to a method for measuring bioelectrical impedance by adopting digital demodulation mode under the mode of frequency mixing excitation. Said invention adopts frequency mixing current of high and low two frequencies to make excitation, and provides a measuring method based on virtual reference vector, and combines said measuring method with digital demodulation mode so as to implement bioelectrical impedance measurement. Said method can provide more complete and accurate information for clinical application.

Description

The mixing bio-impedance measurement method of digital demodulation mode

Technical field

The invention belongs to the bio information detection range, be specifically related under the mixing frequency excitation mode pattern, adopt digital demodulation mode to measure the method for bio-electrical impedance.

Background technology

The bio-electrical impedance measuring technology is to utilize the electrical characteristics of biological tissue and organ (impedance, admittance, dielectric constant etc.) and change, a kind of not damaged detection technique of extraction and Human Physiology, biomedical information that pathological condition is relevant.In early days, the main unifrequency motivation model that adopts, according to biological tissue's frequency impedance characteristic, in β frequency dispersion section, cell membrane electric capacity is basicly stable, increase along with frequency, the capacitive reactance of membrane capacitance reduces, impressed current is walked around the cell membrane extracellular fluid of flowing through during by low frequency and is passed the cell membrane fluid of inside and outside cell of flowing through during to high frequency, in order to obtain the cell internal information, must utilize the flow through characteristic of fluid of inside and outside cell of high frequency electric, therefore the biological tissue's electrical impedance information that records under the single driving frequency can not reflect the organism situation comprehensively, and the multifrequency motivation model of many at present employings promptly adopts the signal of different frequency to encourage respectively, and measure the bio-impedance of this frequency, but human body is dynamic, and this method can not be analyzed the information of same vital movement under different driving frequencies, and the different measuring frequency is when switching, newly the Time Created of bio-electrical impedance information measurement is longer under the frequency, so the data that method provided that this timesharing is measured can not accurately reflect certain electrical impedance information of organism constantly.Because the exciting current by human body must meet safety criterion, often adopt alternating current less than 1mA, therefore the signal of measuring is very faint, mainly carries out the measurement of signal by the method for phase demodulation, and demodulation method commonly used at present has switch demodulation, digital demodulation and multiplication demodulation.The switch demodulation method inevitably will introduce interference, and reference signal is not ideal square wave in amplifier gain handoff procedure, when driving frequency improved, its influence was increasing.The multiplication demodulation method generally realizes simulating phase demodulation with multiplier and low pass filter.Low pass filter is long Time Created, and after exponent number and cut-off frequency were determined, also determined that speed is uncontrolled Time Created thereupon, also is difficult to improve.

It is strong that digital demodulation is compared harmonic wave inhibition ability, no dc shift, and carrying out digital processing has good advantages such as motility.Can adjust sampling rate as required at any time, improve processing speed, the certainty of measurement height.Conventional switch demodulation, multiplication demodulation and digital demodulation is primarily aimed at the processing of single frequency signal.

Summary of the invention

The mixing bio-impedance measurement method that the purpose of this invention is to provide a kind of digital demodulation mode can be measured the bio-impedance under the synchronization different frequency simultaneously.

The present invention adopts the mixing electric current of two kinds of frequencies of height to encourage; At the phase shift problem in the signals transmission,, the measuring method based on empty reference vector has been proposed from the vector space angle.In order to improve the noiseproof feature of system, the mixing frequency digital demodulation scheme is adopted in the date processing motility, has realized synchronization is being measured in the bio-electrical impedance under to different driving frequencies.

Different electrical characteristics embody in biological tissue under different electric field frequencies, in α frequency dispersion section (10Hz-10kHz), and the pericellular ionic environment characteristic of measurement result dominant response; In β frequency dispersion section (10kHz-10MHz), cell membrane electric capacity is basicly stable, and along with the increase of frequency, the capacitive reactance of membrane capacitance reduces, and impressed current is walked around the cell membrane extracellular fluid of flowing through during by low frequency and passed the cell membrane fluid of inside and outside cell of flowing through during to high frequency; In γ frequency dispersion section, measurement result is main relevant with hydrone.In medical application, because most physiopathologic variations all are embodied in α and the β frequency dispersion section, therefore, these two frequency ranges are paid close attention to by people.The present invention is directed to this two frequency dispersion sections, adopt two kinds of frequencies to reach mixing frequency excitation mode mode mode, use the mixing frequency digital demodulation mode, obtain the real part and the imaginary part of the electrical impedance information under the different frequency simultaneously.

In order to obtain the value of biological complex impedance under the high and low frequency simultaneously, high and low different frequency sinusoidal signal is mixed the back as pumping signal, realize mixing frequency excitation mode.The empty reference vector method of impedance information extraction and application is finished in conjunction with the mixing frequency digital demodulation method under the mixing frequency excitation mode.

The present invention adopts high and low two kinds of mixing frequency digital demodulation modes, obtains the real part and the imaginary part of electrical impedance information under the different frequency simultaneously, and the amplitude of the measuring-signal of high and low two frequency content correspondences is respectively A ₁And A ₂, frequency is respectively f ₁=n ₁* f, f ₂=n ₂* f (n ₁≠ n ₂), sample frequency is f _s=N * f (N＞2max (n ₁, n ₂)), the sampling period is q, and the sampling time is the integral multiple of two kinds of excitation signal cycles, and corresponding total sampling number is M=N * q,

If input signal Vs (k) is:

$\mathrm{Vs}\left(k\right)=A_1\sin (\frac{2n_1\mathrm{\&pi;}}{N}k+{\mathrm{\&phi;}}_1)+{\mathrm{<figure-callout\; id="2"\; label="A"\; filenames="A20051001430800091.png"\; state="\{\{state\}\}">A</figure-callout>}}_2\sin (\frac{2{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="A20051001430800091.png"\; state="\{\{state\}\}">n</figure-callout>}}_2\mathrm{\&pi;}}{N}k+{\mathrm{\&phi;}}_2),k=\mathrm{0,1},\&CenterDot;\&CenterDot;\&CenterDot;,M-1$

At f ₁, f ₂Two different frequencies, construct two homophases and orthogonal reference signal respectively:

${\mathrm{Vrs}}_1\left(k\right)=\sin \left(\frac{2n_1\mathrm{\&pi;}}{N}k\right)$

${\mathrm{Vrc}}_1\left(k\right)=\cos \left(\frac{2n_1\mathrm{\&pi;}}{N}k\right)$

${\mathrm{Vrs}}_2\left(k\right)=\sin \left(\frac{{2\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="A20051001430800091.png"\; state="\{\{state\}\}">n</figure-callout>}}_2\mathrm{\&pi;}}{N}k\right)$

${\mathrm{Vrc}}_2\left(k\right)=\cos \left(\frac{{2\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="A20051001430800091.png"\; state="\{\{state\}\}">n</figure-callout>}}_2\mathrm{\&pi;}}{N}k\right)$

Then Vs (k) and Vrs ₁(k), Vrs ₂(k) cross-correlation R _Xrs1, R _Xrs2, Vs (k) and Vrc ₁(k), Vrc ₂(k) cross-correlation R _Xrc1, R _Xrc2Be respectively

$R_{\mathrm{xrs}1}=\frac{1}{M}\underset{k=0}{\overset{M-1}{\mathrm{\&Sigma;}}}\mathrm{Vs}\left(k\right)\&CenterDot;{\mathrm{Vrs}}_1\left(k\right)$

$=\frac{1}{M}\underset{k=0}{\overset{M-1}{\mathrm{\&Sigma;}}}[A_1\sin (\frac{{2n}_1\mathrm{\&pi;}}{N}k+{\mathrm{\&phi;}}_1)+{\mathrm{<figure-callout\; id="2"\; label="A"\; filenames="A20051001430800091.png"\; state="\{\{state\}\}">A</figure-callout>}}_2\sin (\frac{{2\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="A20051001430800091.png"\; state="\{\{state\}\}">n</figure-callout>}}_2\mathrm{\&pi;}}{N}k+{\mathrm{\&phi;}}_2)]\&CenterDot;\sin \left(\frac{{2n}_1\mathrm{\&pi;}}{N}k\right)=\frac{A_1}{2}\cos \left({\mathrm{\&phi;}}_1\right)$

In like manner

$R_{\mathrm{xrc}1}=\frac{1}{M}\underset{k=0}{\overset{M-1}{\mathrm{\&Sigma;}}}\mathrm{Vs}\left(k\right)\&CenterDot;{\mathrm{Vrc}}_1\left(k\right)=\frac{A_1}{2}\sin \left({\mathrm{\&phi;}}_1\right)$

${\mathrm{<figure-callout\; id="2"\; label="R\; xrs"\; filenames="A20051001430800091.png"\; state="\{\{state\}\}">R</figure-callout>}}_{\mathrm{xrs}}=\frac{1}{M}\underset{k=0}{\overset{M-1}{\mathrm{\&Sigma;}}}\mathrm{Vs}\left(k\right)\&CenterDot;{\mathrm{Vrs}}_2\left(k\right)=\frac{{\mathrm{<figure-callout\; id="2"\; label="A"\; filenames="A20051001430800091.png"\; state="\{\{state\}\}">A</figure-callout>}}_2}{2}\cos \left({\mathrm{\&phi;}}_2\right)$

${\mathrm{<figure-callout\; id="2"\; label="R\; xrc"\; filenames="A20051001430800091.png"\; state="\{\{state\}\}">R</figure-callout>}}_{\mathrm{xrc}}=\frac{1}{M}\underset{k=0}{\overset{M-1}{\mathrm{\&Sigma;}}}\mathrm{Vs}\left(k\right)\&CenterDot;{\mathrm{Vrc}}_2\left(k\right)=\frac{{\mathrm{<figure-callout\; id="2"\; label="A"\; filenames="A20051001430800091.png"\; state="\{\{state\}\}">A</figure-callout>}}_2}{2}\sin \left({\mathrm{\&phi;}}_2\right)$

R _Xrs1, R _Xrs2And R _Xrs1, R _Xrs2Represent the output of homophase output and quadrature respectively, can get that measuring voltage is the amplitude and the phase angle of reference to construct signal under two kinds of frequencies:

$A_1=2\sqrt{{R_{\mathrm{xrs}1}}^2+{R_{\mathrm{xrc}1}}^2}$

${\mathrm{\&phi;}}_1=t{g}^{-1}\left(\frac{R_{\mathrm{xrc}1}}{R_{\mathrm{xrs}1}}\right)$

$A_2=2\sqrt{{{\mathrm{<figure-callout\; id="2"\; label="R\; xrs"\; filenames="A20051001430800091.png"\; state="\{\{state\}\}">R</figure-callout>}}_{\mathrm{xrs}}}^2+{{\mathrm{<figure-callout\; id="2"\; label="R\; xrc"\; filenames="A20051001430800091.png"\; state="\{\{state\}\}">R</figure-callout>}}_{\mathrm{xrc}}}^2}$

${\mathrm{\&phi;}}_2=t{g}^{-1}\left(\frac{R_{\mathrm{xrc}2}}{R_{\mathrm{xrs}2}}\right)$

The impedance information of high and low two frequencies measured adopts empty reference vector measurement structure, make up two homophases and the empty reference signal of quadrature empty reference vector as all measurements with digital form,

In Fig. 1, reference resistance r is for electing pure resistance as, and the pressure drop on the reference resistance is V _r, V _rWith the exciting current homophase, the pressure drop in the tested impedance is V _x, vectogram as shown in Figure 2.For the ease of direct visual comparison, having set up is the actual coordinates (among Fig. 2 shown in the dotted line) of benchmark with the exciting current.Be in the actual coordinates of benchmark with the exciting current,

A _r＝r×|I|，A _x＝|Z _x|×|I|

Because precision resistance r is known, but adopts above-mentioned digital demodulation method measuring voltage V _xAnd V _rAmplitude and phase place, i.e. A _xAnd A _rKnown, so Z _xAmplitude can get

$\left|Z_x\right|=\frac{A_x}{A_r}\&times;r$

Z _xThe phase angle correspondence be the actual coordinates of benchmark with the exciting current, as shown in Figure 2,

∠Z _x＝φ _x－φ _r

In the present invention, signal source produces sine voltage signal, and (VCCS) is converted to electric current by Voltage-controlled Current Source, and exciting current puts on tissue by leading.Because VCCS can produce phase shift with voltage transitions during for electric current, and the phase shift that produces for the voltage signal of different frequency is also different, directly adopts " signal source " output as empty reference signal, measures amplitude in the tested impedance and phase place with the drawing-in system error.

The present invention proposes empty reference vector method, measures at the impedance information of each frequency and adopts measurement structure shown in Figure 1, and two homophases of former surface construction and orthogonal reference signal are as the empty reference vector of all measurements.Utilize this empty reference vector method, the phase shift of exciting current signal I and the phase contrast between exciting current and the reference signal can be tried to achieve by known reference resistance r in the real system, therefore, and φ _x-φ _rEliminated the phase error of introducing thus.

The invention has the beneficial effects as follows: cooperate empty reference vector method by the mixing frequency excitation mode mode mode that adopts two kinds of frequencies, the Applied Digital demodulation mode, obtain the real part and the imaginary part of two kinds of electrical impedance information under the frequency simultaneously, and eliminated the error that phase shift causes in current conversion and the transmission course, adopt digital demodulation effectively to eliminate the interference that a large amount of analog devices do not match and noise is introduced simultaneously, thereby provide complete more information accurately for clinical practice.

Description of drawings

Accompanying drawing 1 is empty reference vector measuring method structure chart.

Accompanying drawing 2 is empty reference vector method vectogram.

The specific embodiment

Below by specific embodiment and in conjunction with the accompanying drawings 1～2, the present invention is further illustrated.

Present embodiment mixing frequency excitation mode signal contains high and low two frequency contents, is respectively f ₁And f ₂, wherein,

f ₁＝10kHz，f ₂＝100kHz

Sample frequency f _s=1MHz, 10000 points of sampling, the voltage signal Vs (k) in the tested impedance is:

$\mathrm{Vs}\left(k\right)=A_1\sin (\frac{\mathrm{\&pi;}}{50}k+{\mathrm{\&phi;}}_1)+{\mathrm{<figure-callout\; id="2"\; label="A"\; filenames="A20051001430800091.png"\; state="\{\{state\}\}">A</figure-callout>}}_2\sin (\frac{\mathrm{\&pi;}}{20}k+{\mathrm{\&phi;}}_2),k=\mathrm{0,1},\&CenterDot;\&CenterDot;\&CenterDot;,9999$

Wherein, A ₁, A ₂And φ ₁, φ ₂For to be measured, at f ₁, f ₂Two different frequencies, construct two homophases and orthogonal reference signal respectively:

${\mathrm{Vrs}}_1\left(k\right)=\sin \left(\frac{\mathrm{\&pi;}}{50}k\right)$

${\mathrm{Vrc}}_1\left(k\right)=\cos \left(\frac{\mathrm{\&pi;}}{50}k\right)$

${\mathrm{Vrs}}_2\left(k\right)=\sin \left(\frac{\mathrm{\&pi;}}{20}k\right)$

${\mathrm{Vrc}}_2\left(k\right)=\cos \left(\frac{\mathrm{\&pi;}}{20}k\right)$

Then Vs (k) and Vrs ₁(k) cross-correlation R _Xrs1For

$R_{\mathrm{xrs}1}=\frac{1}{10000}\underset{k=0}{\overset{9999}{\mathrm{\&Sigma;}}}\mathrm{Vs}\left(k\right)\&CenterDot;{\mathrm{Vrs}}_1\left(k\right)$

$=\frac{1}{10000}\underset{k=0}{\overset{9999}{\mathrm{\&Sigma;}}}[A_1\sin (\frac{\mathrm{\&pi;}}{50}k+{\mathrm{\&phi;}}_1)+{\mathrm{<figure-callout\; id="2"\; label="A"\; filenames="A20051001430800091.png"\; state="\{\{state\}\}">A</figure-callout>}}_2\sin (\frac{\mathrm{\&pi;}}{20}k+{\mathrm{\&phi;}}_2)]\&CenterDot;\sin \left(\frac{\mathrm{\&pi;}}{50}k\right)=\frac{A_1}{2}\cos \left({\mathrm{\&phi;}}_1\right)$

In like manner

$R_{\mathrm{xrc}1}=\frac{1}{10000}\underset{k=0}{\overset{9999}{\mathrm{\&Sigma;}}}\mathrm{Vs}\left(k\right)\&CenterDot;{\mathrm{Vrc}}_1\left(k\right)=\frac{A_1}{2}\sin \left({\mathrm{\&phi;}}_1\right)$

${\mathrm{<figure-callout\; id="2"\; label="R\; xrs"\; filenames="A20051001430800091.png"\; state="\{\{state\}\}">R</figure-callout>}}_{\mathrm{xrs}}=\frac{1}{10000}\underset{k=0}{\overset{9999}{\mathrm{\&Sigma;}}}\mathrm{Vs}\left(k\right)\&CenterDot;{\mathrm{Vrs}}_2\left(k\right)=\frac{A_1}{2}\cos \left({\mathrm{\&phi;}}_2\right)$

${\mathrm{<figure-callout\; id="2"\; label="R\; xrc"\; filenames="A20051001430800091.png"\; state="\{\{state\}\}">R</figure-callout>}}_{\mathrm{xrc}}=\frac{1}{10000}\underset{k=0}{\overset{9999}{\mathrm{\&Sigma;}}}\mathrm{Vs}\left(k\right)\&CenterDot;{\mathrm{Vrc}}_2\left(k\right)=\frac{{\mathrm{<figure-callout\; id="2"\; label="A"\; filenames="A20051001430800091.png"\; state="\{\{state\}\}">A</figure-callout>}}_2}{2}\sin \left({\mathrm{\&phi;}}_2\right)$

R _Xrs1, R _Xrs2And R _Xrc1, R _Xrc2Represent the output of homophase output and quadrature respectively, what can get measuring voltage under two kinds of frequencies is the amplitude and the phase angle of reference to construct signal:

$A_1=2\sqrt{{R_{\mathrm{xrs}1}}^2+{R_{\mathrm{xrc}1}}^2}$

${\mathrm{\&phi;}}_1=t{g}^{-1}\left(\frac{R_{\mathrm{xrc}1}}{R_{\mathrm{xrs}1}}\right)$

$A_2=2\sqrt{{{\mathrm{<figure-callout\; id="2"\; label="R\; xrs"\; filenames="A20051001430800091.png"\; state="\{\{state\}\}">R</figure-callout>}}_{\mathrm{xrs}}}^2+{{\mathrm{<figure-callout\; id="2"\; label="R\; xrc"\; filenames="A20051001430800091.png"\; state="\{\{state\}\}">R</figure-callout>}}_{\mathrm{xrc}}}^2}$

${\mathrm{\&phi;}}_2=t{g}^{-1}\left(\frac{R_{\mathrm{xrc}2}}{R_{\mathrm{xrs}2}}\right)$

Because precision resistance r is known, be reference vector with the structure signal, can adopt said method to measure its corresponding driving frequency f ₁And f ₂Amplitude be respectively Ar _lAnd Ar ₂, phase angle is respectively φ _R1, φ _R2, for driving frequency f ₁The amplitude that can get tested impedance is

Phase angle is φ ₁-φ _R1, for driving frequency f ₂The amplitude that can get tested impedance is Phase angle is φ ₂-φ _R2

Claims (1)

1. the mixing bio-impedance measurement method of digital demodulation mode, it is characterized in that: adopt high and low two kinds of mixing frequency digital demodulation modes, obtain the real part and the imaginary part of electrical impedance information under the different frequency simultaneously, the amplitude of the measuring-signal of high and low two frequency content correspondences is respectively A ₁And A ₂, frequency is respectively f ₁=n ₁* f, f ₂=n ₂* f (n ₁≠ n ₂), sample frequency is f _s=N * f (N＞2max (n ₁, n ₂)), the sampling period is q, total sampling number is M=N * q,

Input signal Vs (k) is:

${\mathrm{Vs}\left(k\right)=A}_1\sin (\frac{{2n}_1\mathrm{\&pi;}}{N}k+{\mathrm{\&phi;}}_1)+A_2\sin (\frac{2n_2\mathrm{\&pi;}}{N}k+{\mathrm{\&phi;}}_2)k=0,1,M-1$

At f ₁, f ₂Two different frequencies, construct two homophases and orthogonal reference signal respectively:

${\mathrm{Vrs}}_1\left(k\right)=\sin \left(\frac{{2n}_1\mathrm{\&pi;}}{N}k\right)$

${\mathrm{Vrc}}_1\left(k\right)=\cos \left(\frac{{2n}_1\mathrm{\&pi;}}{N}k\right)$

${\mathrm{Vrc}}_2\left(k\right)=\sin \left(\frac{{2n}_2\mathrm{\&pi;}}{N}k\right)$

${\mathrm{Vrc}}_2\left(k\right)=\cos \left(\frac{{2n}_2\mathrm{\&pi;}}{N}k\right)$

Vs (k) and Vrs ₁(k), Vrs ₂(k) cross-correlation R _Xrs1, R _Xrs2, Vs (k) and Vrc ₁(k), Vrc ₂(k) cross-correlation R _Xrc1, R _Xrc2Be respectively:

$R_{\mathrm{xrs}1}=\frac{1}{M}\underset{k=0}{\overset{M-1}{\mathrm{\&Sigma;}}}\mathrm{Vs}\left(k\right)\&CenterDot;{\mathrm{Vrs}}_1\left(k\right)$

$=\frac{1}{M}\underset{k=0}{\overset{M-1}{\mathrm{\&Sigma;}}}[A_1\sin (\frac{{2n}_1\mathrm{\&pi;}}{N}k+{\mathrm{\&phi;}}_1)+A_2\sin (\frac{2n_2\mathrm{\&pi;}}{N}k+{\mathrm{\&phi;}}_2)]\&CenterDot;\sin \left(\frac{{2n}_1\mathrm{\&pi;}}{N}k\right)$

$=\frac{A_1}{2}\cos \left({\mathrm{\&phi;}}_1\right)$

$R_{\mathrm{xrc}1}=\frac{1}{M}\underset{k=0}{\overset{M-1}{\mathrm{\&Sigma;}}}\mathrm{Vs}\left(k\right)\&CenterDot;{\mathrm{Vrc}}_1\left(k\right)=\frac{A_1}{2}\sin \left({\mathrm{\&phi;}}_1\right)$

$R_{\mathrm{xrs}2}=\frac{1}{M}\underset{k=0}{\overset{M-1}{\mathrm{\&Sigma;}}}\mathrm{Vs}\left(k\right)\&CenterDot;{\mathrm{Vrs}}_2\left(k\right)=\frac{A_2}{2}\cos \left({\mathrm{\&phi;}}_2\right)$

$R_{\mathrm{xrc}2}=\frac{1}{M}\underset{k=0}{\overset{M-1}{\mathrm{\&Sigma;}}}\mathrm{Vs}\left(k\right)\&CenterDot;{\mathrm{Vrc}}_2\left(k\right)=\frac{A_2}{2}\sin \left({\mathrm{\&phi;}}_2\right)$

R _Xrs1, R _Xrs2And R _Xrc1, R _Xrc2Represent that respectively homophase output and quadrature export, the amplitude and the phase angle of measuring voltage under two kinds of frequencies:

$A_1=2\sqrt{{R_{\mathrm{xrsl}}}^2+{R_{\mathrm{xrc}1}}^2}$

${\mathrm{\&phi;}}_1={\mathrm{tg}}^{-1}\left(\frac{R_{\mathrm{xrc}1}}{R_{\mathrm{xrs}1}}\right)$

$A_2=2\sqrt{{R_{\mathrm{xrs}2}}^2+{R_{\mathrm{xrc}2}}^2}$

${\mathrm{\&phi;}}_2={\mathrm{tg}}^{-1}\left(\frac{R_{\mathrm{xrc}2}}{R_{\mathrm{xrs}2}}\right)$

The impedance information of high and low two frequencies measured adopts empty reference vector measurement structure, make up two homophases and the empty reference signal of quadrature empty reference vector as all measurements with digital form,

Reference resistance r is for electing pure resistance as, and the pressure drop on the reference resistance is V _r, V _rWith the exciting current homophase, the pressure drop in the tested impedance is V _x, be in the actual coordinates of benchmark with the exciting current,

A _r＝r×|I|，A _x＝|Z _x|×|I|

Z _xAmplitude be:

$\left|Z_x\right|=\frac{A_x}{A_r}\&times;r$

Z _xThe phase angle correspondence be the actual coordinates of benchmark with the exciting current,

∠Z _x＝φ _x-φ _r。

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