CN102325106B - Gaussian filtered minimum shift keying (GSMK) digital demodulation error metering method - Google Patents

Abstract

The invention relates to a Gaussian filtered minimum shift keying (GSMK) digital demodulation error metering method and a system. The metering method comprises the steps of: 1) taking a phase modulation (PM) signal as a standard GSMK signal; 2) setting 0.5Rs difference value delta f by setting the modulation frequency and standard of the PM signal, and setting the capture length N of a vector signal analyzer at the same time, wherein the N and the delta f satisfy a certain constraint relation; 3) establishing an analytic relation between the frequency parameter delta f and a modulation error parameter; and 4) comparing the calculable digital modulation error parameter with a value measured by the vector signal analyzer to realize the metering and calibration of a GMSK demodulation error parameter. The metering system provided by the invention comprises an analog signal generator, a power divider, a tunable receiver and a measured vector signal analyzer. The method and the system provided by the invention can be used for metering and calibrating the digital demodulation errors of vector signal analyzers and mobile communication comprehensive testers with a GMSK analysis and demodulation function, and can be widely used.

Description

A kind of metering method of GMSK digital demodulating error

Technical field

The present invention relates to radio communication and broadcast and television information transmission technology, especially design the metering of the GMSK signal modulation error of second generation mobile communication system use.

Background technology

Abbreviation and Key Term definition

PhaseErrRms, phase error root-mean-square value

PhaseErrPeak, phase error peak value

EvmRms, Error Vector Magnitude root-mean-square value

EvmPeak Error Vector Magnitude peak value

MagErr range error

PM signal.Phase-modulated signal

Digital modulation signals has become the important foundation stone of information-intensive society, is the main carriers of communication.Most typical is exactly application in radio communication and radio data system, and such as the GMSK signal that second generation mobile communication system is used, the cellphone subscriber's number based on this signal surpasses 1,000,000,000 in the whole world, and relevant metered quantity is also quite huge.

The main parameters of weighing digital modulation signals modulation error has: error vector magnitude (EVM), phase error (Phase Err) etc.The same with all physical quantitys, these values also must through credible, can trace to the source, metering accurately, guarantee value accurately, unified, the normal operation of guarantee information system.But due to technology and theory, the traced to the source metering of Digital Modulation parameter is a worldwide difficult problem at present, can trace to the source, believable method and apparatus is not all set up always.

With regard to digital modulation signals, main gage work concentrates on the calibration aspect of digital modulation signal generator (VSG) and VSA (VSA).And as test instrumentation, remove other modulation-demodulation devices in test production, scientific research through VSG and the VSA of calibration.

Although the gage work of the digital modulation and demodulation error parameter that VSG is relevant with VSA is being carried out always,, still there is two large problems in the basic demand according to metering:

One, be difficult to trace to the source, mainly refer to: current standard method is to use VSA calibration signal source, use digital modulation signal generator as standard calibration analyte instrument, this is a closed loop, this closed loop and other standard be not contact almost, being the closed loop that can not trace to the source, is " castles in the air " in whole measuring system.

Gage work operation with closed ring, is easy to occur error circulation and accumulation, is difficult to guarantee accuracy and confidence level.Fig. 1 figuratively understands this problem.

Two, use standard digital modulation signal, lack the setting to error parameter, in general, measuring parameter setting will be contained the main scope of application of the utensil that is measured, such as the range of certain slide calliper rule is 10 centimetres, and only using the scale of 1 centimetre of gauge block calibration, is inadequate.In calibration, during VSA, the modulation signal of digital modulation signals source emission standard, does not carry out error setting to signal, is only actually near 0 of error and tests, and this is disengaging real requirement.Because VSA is in real work, there is larger error in the signal receiving, the effect of VSA is exactly that modulation error parameter is measured, and should to signal, carry out different error settings within the specific limits often, obtain different measured values, metering result is more realistic demand.But why not problem solves, be because error setting and and concrete modulation error parameter between theory relation not yet set up, method of operation lacks especially.

Summary of the invention

This patent is intended to regard to this use of GMSK modulation system very widely, solve gage work operation with closed ring, be easy to occur error circulation and accumulation, be difficult to guarantee the problem of accuracy and confidence level, and within the specific limits signal is carried out to different error settings, obtain different measured values, make to measure the more realistic demand of result.

In order to realize object of the present invention, a kind of metering method of GMSK digital demodulating error is proposed, comprise step:

1) by modulating frequency, be 0.5R _s, modulation index is 0.2431, carrier frequency is f _cpM signal as the GMSK signal of a standard, R _sfor character rate, its Gaussian filter BT=0.3;

2) by modulating frequency and the standard configuration 0.5R of PM signal are set _sdifference DELTA f, what VSA was set simultaneously catches length Result Length N, wherein N and Δ f meet the restriction relation of following formula:

$N\left|\mathrm{\&Delta;f}\right|<\frac{{\mathrm{<figure-callout\; id="2"\; label="R\; s"\; filenames="HDA0000076575730000012.png,HDA0000076575730000031.png"\; state="\{\{state\}\}">R</figure-callout>}}_s}{2};$

3) set up frequency parameter Δ f and modulation error parameter PhaseErrRms, PhaseErrPeak, EvmRms, the analytic relationship between EvmPeak is as follows:

PhaseErrPeak＝β ₀[sin2π(f _p+Δf)t ₀-sin2f _pt ₀]

$\mathrm{EvmPeak}=2\sin \left(\frac{\mathrm{PhaseErrPeak}}{2}\right)$

$\mathrm{PhaseErrRms}=\sqrt{\frac{{\&Integral;}_0^{t_0}{\left\{{\mathrm{\&beta;}}_0\right[\sin 2\mathrm{\&pi;}{f}_{p}t-\sin 2\mathrm{\&pi;}(f_p+\mathrm{\&Delta;f})t\left]\right\}}^2\mathrm{<figure-callout\; id="0"\; label="dt\; t"\; filenames="HDA0000076575730000012.png,HDA0000076575730000022.png"\; state="\{\{state\}\}">dt</figure-callout>}}{t_{}}}$

$=\frac{{\mathrm{\&beta;}}_0}{\sqrt{2{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="HDA0000076575730000012.png,HDA0000076575730000022.png"\; state="\{\{state\}\}">t</figure-callout>}}_0}}\sqrt{2t_0-\sin 4\mathrm{\&pi;}{f}_p{t}_0-\sin 4\mathrm{\&pi;}(f_p+\mathrm{\&Delta;f}){\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="HDA0000076575730000012.png,HDA0000076575730000022.png"\; state="\{\{state\}\}">t</figure-callout>}}_0+\sin 2\mathrm{\&pi;}({2f}_p+\mathrm{\&Delta;f})t_0-\sin 2\mathrm{\&pi;\&Delta;}{\mathrm{ft}}_0}$

$\mathrm{EvmRms}=\sqrt{\frac{\underset{0}{\overset{t_0}{\&Integral;}}{\left\{2\sin \right\{\frac{{\mathrm{\&beta;}}_0[\sin 2\mathrm{\&pi;}{f}_{p}t-\sin 2\mathrm{\&pi;}(f_p+\mathrm{\&Delta;f})t]}{2}\left\}\right\}}^2\mathrm{<figure-callout\; id="0"\; label="dt\; t"\; filenames="HDA0000076575730000012.png,HDA0000076575730000022.png"\; state="\{\{state\}\}">dt</figure-callout>}}{t_{}}};$

4) by this computable Digital Modulation error parameter and the contrast of VSA measured value, realize the measurement and calibration to GMSK demodulating error parameter.

The present invention also proposes a kind of metering system of GMSK digital demodulating error, and this system comprises:

Analog signal generator, generation PM signal;

Power splitter, receives the PM signal that analog signal generator produces, and it is distributed downwards;

Tuned receiver, receives the PM signal that described power splitter is distributed, monitoring PM parameter;

Tested VSA is worked under GMSK mode of operation, receives the PM signal of described power splitter distribution.

Described tuned receiver is R & S FSMR receiver, guarantees that the PM signal modulating frequency of output is f _p+ Δ f, f _p=135.417kHz, modulation index is 0.2431, measuring accuracy is ± 0.0005.

The frequency, demodulation frequency of described tested VSA is set to 1GHz, character rate R _s=270.834kBaud, BT=0.3.

Accompanying drawing explanation

Fig. 1 is the closed-loop digital modulation parameter metering schematic diagram that can not trace to the source;

Fig. 2 is that proof GMSK signal is the flow chart of a sinusoidal phase-modulation signal;

Fig. 3 is experimental provision schematic diagram of the present invention;

Fig. 4 is the contrast of PhaseErrRms measurements and calculations in first group of experimental result;

Fig. 5 is the contrast of EvmPeak measurements and calculations in first group of experimental result;

Fig. 6 is the contrast of PhaseErrPeak measurements and calculations in first group of experimental result;

Fig. 7 is the contrast of PhaseErrRms measurements and calculations in second group of experimental result;

Fig. 8 is the contrast of EvmPeak measurements and calculations in second group of experimental result;

Fig. 9 is the contrast of PhaseErrPeak measurements and calculations in second group of experimental result.

Embodiment

For making the object, technical solutions and advantages of the present invention clearer, below in conjunction with specific embodiment and coordinate accompanying drawing, the present invention is described in more detail.

The character rate of GMSK signal is R _s, and symbol sebolic addressing is 101010 ..., when Gaussian filter BT=0.3, this GMSK signal is a sinusoidal phase-modulation signal so, modulating frequency is 0.5R _s, and modulation index is 0.2431, this conclusion is the follow-up basis of carrying out error structure and tracing to the source, and will prove this conclusion below.Concise and to the point proof flow chart as shown in Figure 2.

As shown in Figure 2, sequence is that baseband signal g (t) waveform of 10101 correspondences is bipolar square wave pulses, the symbol period T that pulse duration is GMSK _s, the Fourier series form of this waveform is:

$g\left(t\right)=\frac{R_s}{\mathrm{\&pi;}}\left\{\sin \right[2\mathrm{\&pi;}\left(\frac{R_s}{2}\right)t]+\frac{1}{3}\sin [2\mathrm{\&pi;}\left(\frac{3R_s}{2}\right)t]+\frac{1}{5}\sin [2\mathrm{\&pi;}\left(\frac{5R_s}{2}\right)t]+\&CenterDot;\&CenterDot;\&CenterDot;\}---\left(1\right)$

The frequency domain of Gaussian filter is corresponding:

$H\left(f\right)=\exp \left[{-\mathrm{\&alpha;}}^2{\left(\frac{f}{R_s}\right)}^2\right]---\left(2\right)$

By BT=0.3, obtained:

$\exp \left({-\mathrm{\&alpha;}}^2{0.3}^2\right)=\frac{1}{2}---\left(3\right)$

By (3), obtain α=0.27752, by its substitution (2), calculating g (t) by gaussian filtering H (f) is afterwards:

Obviously harmonic amplitude is too little, can ignore.G ₀(t) signal after frequency modulation is required GMSK signal:

$S_0\left(t\right)=\sin [2\mathrm{\&pi;}{f}_{c}t+2\mathrm{\&pi;}\underset{0}{\overset{t}{\&Integral;}}{g}_0\left(\mathrm{\&tau;}\right)\mathrm{d\&tau;}]=\sin [2\mathrm{\&pi;}{f}_{c}t-0.2431\cos \left(\frac{2\mathrm{\&pi;}{R}_{s}t}{2}\right)]---\left(5\right)$

The canonical parameter that current mobile communication system that above analytic process is selected is used is as example, and when parameter changes, above analysis process is still effective.The knowledge innovation that this analysis process is explained for this patent, also should be subject to patent protection.

Frequency is set

modulation index β ₀=0.2431, by being set, the frequency error of frequency modulation controls vector modulation error, and PM signal frequency is set to f _p+ Δ f, the vector V after demodulation _e(t) be:

V _e(t)＝exp[jβ ₀sin2π(f _p+Δf)t] (6)

V so _e(t) have phase error and EVM, MagErr is 0 in theory, establishes:

${\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="HDA0000076575730000012.png,HDA0000076575730000022.png"\; state="\{\{state\}\}">t</figure-callout>}}_0=0.5T_c=\frac{N}{{2R}_s}---\left(7\right)$

If N is that the symbol numbers (Result Lengt) of GMSK demodulated sequence in VSA is through the mathematical derivation of series of complex, obtain meeting under the constraints of (8) formula, the actual value of 4 modulation error parameters can be calculated according to (9) formula～(12) formula.So just will

$N\left|\mathrm{\&Delta;f}\right|<\frac{R_s}{2}---\left(8\right)$

PhaseErrPeak＝β ₀[sin 2π(f _p+Δf)t ₀-sin 2πf _pt ₀] (9)

$\mathrm{EvmPeak}=2\sin \left(\frac{\mathrm{PhaseErrPeak}}{2}\right)---\left(10\right)$

$\mathrm{PhaseErrRms}=\sqrt{\frac{{\&Integral;}_0^{t_0}{\left\{{\mathrm{\&beta;}}_0\right[\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000076575730000012.png,HDA0000076575730000031.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&pi;}{f}_{p}t-\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000076575730000012.png,HDA0000076575730000031.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&pi;}(f_p+\mathrm{\&Delta;f})t\left]\right\}}^2\mathrm{dt}}{t_0}}---\left(11\right)$

$=\frac{{\mathrm{\&beta;}}_0}{\sqrt{2{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="HDA0000076575730000012.png,HDA0000076575730000022.png"\; state="\{\{state\}\}">t</figure-callout>}}_0}}\sqrt{2t_0-\mathrm{<figure-callout\; id="4"\; label="sin"\; filenames="HDA0000076575730000012.png"\; state="\{\{state\}\}">sin</figure-callout>}4\mathrm{\&pi;}{f}_p{t}_0-\mathrm{<figure-callout\; id="4"\; label="sin"\; filenames="HDA0000076575730000012.png"\; state="\{\{state\}\}">sin</figure-callout>}4\mathrm{\&pi;}(f_p+\mathrm{\&Delta;f}){\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="HDA0000076575730000012.png,HDA0000076575730000022.png"\; state="\{\{state\}\}">t</figure-callout>}}_0+\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000076575730000012.png,HDA0000076575730000031.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&pi;}({2f}_p+\mathrm{\&Delta;f})t_0-\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000076575730000012.png,HDA0000076575730000031.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&pi;\&Delta;}{\mathrm{<figure-callout\; id="0"\; label="ft"\; filenames="HDA0000076575730000012.png,HDA0000076575730000022.png"\; state="\{\{state\}\}">ft</figure-callout>}}_0}$

$\mathrm{EvmRms}=\sqrt{\frac{\underset{0}{\overset{t_0}{\&Integral;}}{\left\{2\sin \right\{\frac{{\mathrm{\&beta;}}_0[\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000076575730000012.png,HDA0000076575730000031.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&pi;}{f}_{p}t-\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000076575730000012.png,HDA0000076575730000031.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&pi;}(f_p+\mathrm{\&Delta;f})t]}{2}\left\}\right\}}^2\mathrm{dt}}{t_0}}---\left(12\right)$

By above derivation, sum up briefly conclusion as follows: modulating frequency is 0.5R _s, modulation index is 0.2431, carrier frequency is f _cpM signal, be the GMSK signal of a standard, its Gaussian filter BT=0.3.By modulating frequency and the standard configuration 0.5R of PM signal are set _sdifference DELTA f, what VSA (VSA) was set simultaneously catches length Result Length N, wherein N and Δ f meet the restriction relation of (8) formula.Now just can set up frequency parameter Δ f and modulation error parameter PhaseErrRms, PhaseErrPeak, EvmRms, analytic relationship between EvmPeak, these analytic relationships are as shown in (9) formula～(12) formula, and obviously this is just traceable to frequency parameter by the modulation error of GMSK signal.Solved tracing to the source of Digital Modulation parameter simultaneously, with error, problem has been set.We think " actual value " by this computable Digital Modulation error parameter, by calculated value and the contrast of VSA measured value, have just realized the measurement and calibration to GMSK demodulating error parameter.

Experimental verification, we use the experimental provision shown in Fig. 3, and as shown in the figure, simulation PM signal generator is used for simulating phase-modulated signal, and the parameters such as its carrier frequency, phase modulation frequency, modulation index should be adjustable continuously, and possess good stability.The current good signal generator of quality, respective model instrument and module as Agilent company, Rohde & Schwarz Representative Office etc. produce, can meet the demands.The effect Shi Jiang mono-road PM signal of power splitter is distributed into two-way, and a road is for monitored, and a road is for calibration vector signal analyzer.Tuned receiver is used for monitoring PM parameter, also can say for calibrating the modulating frequency of PM signal, two parameters of modulation index, this instrument must be through effectively verifying checking, measurement and calibration, receiver PM demodulation parameter will effectively be traceable to corresponding upper level measurement standard, the accuracy that tuned receiver is measured is higher, just higher to the accuracy of VSA metering, the present invention uses R & S FSMR receiver monitoring PM signal in experiment, modulation index is ± 0.0005 for measuring the limits of error, certainly other similar receiver can use.Measured piece is the GMSK demodulation module of VSA, and at present communication is conventional arranges is-symbol speed R _s=270.834kBaud, BT=0.3.

Use R & S FSMR receiver monitoring PM signal, guarantee that the PM signal modulating frequency of output is f _p+ Δ f, f _p=135.417kHz, modulation index is 0.2431 (measuring accuracy is ± 0.0005).The frequency, demodulation frequency of PM carrier frequency and VSA is all set to 1GHz.VSA is set to the GMSK pattern of standard, character rate R _s=270.834kBaud, BT=0.3.

When frequency error not being set, all modulation error parameters approach 0, and its vector solution is changed the line map as shown in Figure 4:

Carried out altogether two groups of experiments, setting and result are as follows:

1. Result Length N=140 is set, Δ f is set: from-900Hz stepping 100Hz, be incremented to 900Hz.Measurement result is as shown in the table.

First group of experimental result of table 1 (Result Length N=140, Δ f changes, N and Δ f meet (8) formula constraints)

Δf(Hz)	-900	-800	-700	-600	-500	-400	-300	-200	-100	0
											PhaseRms(deg)	7.862	7.067	6.244	5.404	4.534	3.651	2.754	1.848	0.929	0.106
EvmPeak(％)	24.490	23.617	22.050	20.496	18.054	14.836	11.415	7.876	4.056	0.469
											PhasePeak(deg)	-14.068	-13.557	-12.660	-11.763	10.359	-8.511	6.545	4.515	-2.323	-0.268
Δf(Hz)	100	200	300	400	500	600	700	800	900
											PhaseRms(deg)	0.928	1.841	2.756	3.654	4.541	5.409	6.258	7.081	7.872
Evm Peak(％)	4.010	7.873	11.516	14.829	17.593	20.219	22.064	23.576	24.343
											PhasePeak(deg)	2.297	-4.512	6.602	8.504	10.093	-11.606	12.668	-13.541	-13.983

Shown in following Fig. 4～Fig. 6 of contrast of first group of experimental result and calculating: wherein measure error Meas Err refers to that measured value deducts calculated value.

2. Δ f=100Hz is set, and Result Length N arranges: from 50, be incremented to 1300.Measurement result is as shown in table 2.

(Δ f=100Hz, Result Length N changes second group of experimental result of table 2.N and Δ f meet (8) formula constraints)

N	50	100	200	300	400	500	600
								PhaseRms(deg)	0.341	0.661	1.321	1.971	2.620	3.264	3.902
EvmPeak(％)	1.545	2.953	5.934	8.420	11.221	13.369	15.763
								PhasePeak(deg)	0.885	1.692	-3.401	4.827	6.432	-7.662	-9.041
N	700	800	900	1000	1100	1200	1300
								PhaseRms(deg)	4.533	5.154	5.763	6.364	6.950	7.523	8.081
EvmPeak(％)	17.834	19.566	21.248	22.766	23.357	24.266	24.668
								PhasePeak(deg)	10.233	-11.227	12.200	13.072	-13.411	13.933	14.169

Shown in following Fig. 7～Fig. 9 of contrast of second group of experimental result and calculating: wherein measure error Meas Err refers to that measured value deducts calculated value.

First this patent has proved modulating frequency is f _p=0.5R _s, and modulation index is 0.2431, carrier wave is f _cpM signal be that the carrier wave of a standard is f _cgMSK signal, its character rate is R _s, Gaussian filter BT=0.3.By to PM signal modulating frequency and standard value 0.5R are set _sbetween deviation delta f, result length (Result Length) N is caught in the analysis demodulation that VSA is set simultaneously, just can obtain four Digital Modulation error parameter value: PhaseErrRms that can resolve clearing, PhaseErrPeak, EvmRms, EvmPeak, the theoretical value of these four modulation errors is only relevant with Δ f and N, has so just set up the GMSK modulation error parameter that can trace to the source, can calculate, can arrange.

This metering method and system that has just provided a kind of GMSK digital demodulating error parameter realizes.This method and system can carry out measurement and calibration to the digital demodulating error of the VSA with GMSK analytic solution conditioning function, mobile communication comprehensive tester, has a wide range of applications.

Above-described specific embodiment; object of the present invention, technical scheme and beneficial effect have been carried out to further detailed description; institute is understood that; the foregoing is only specific embodiments of the invention; be not limited to the present invention; within the spirit and principles in the present invention all, any modification of making, be equal to replacement, improvement etc., within all should being included in protection scope of the present invention.

Claims (1)

1. a metering method for GMSK digital demodulating error, is characterized in that comprising step:

1) by modulating frequency, be 0.5R _s, modulation index is 0.2431, carrier frequency is f _cphase-modulated signal as the GMSK signal of a standard, R _sfor character rate, its Gaussian filter BT=0.3;

2) by modulating frequency and the standard configuration 0.5R of phase-modulated signal are set _sdifference DELTA f, what VSA was set simultaneously catches length Result Length N, wherein N and Δ f meet the restriction relation of following formula:

$N\left|\mathrm{\&Delta;f}\right|<\frac{R_s}{2};$

3) set up frequency parameter Δ f and modulation error parameter PhaseErrRms, PhaseErrPeak, EvmRms, the analytic relationship between EvmPeak is as follows:

PhaseErrPeak＝β ₀[sin2π(f _p+Δf)t ₀-sin2πf _pt ₀]

$\mathrm{EvmPeak}=2\sin \left(\frac{\mathrm{PhaseErrPeak}}{2}\right)$

$\begin{array}{c}\mathrm{PhaseErrRms}=\sqrt{\frac{{\&Integral;}_0^{t_0}{\left\{{\mathrm{\&beta;}}_0\right[\sin 2\mathrm{\&pi;}{f}_{p}t-\sin 2\mathrm{\&pi;}(f_p+\mathrm{\&Delta;f})t\left]\right\}}^2\mathrm{dt}}{t_0}}\\ =\frac{{\mathrm{\&beta;}}_0}{\sqrt{{2t}_0}}\sqrt{{2t}_0-\sin 4\mathrm{\&pi;}{f}_p{t}_0-\sin 4\mathrm{\&pi;}(f_p+\mathrm{\&Delta;f})t_0+\sin 2\mathrm{\&pi;}({2f}_p+\mathrm{\&Delta;f})t_0-\sin 2\mathrm{\&pi;\&Delta;}{\mathrm{ft}}_0}\end{array}$

$\mathrm{EvmRms}=\sqrt{\frac{\underset{0}{\overset{t_0}{\&Integral;}}\{2\sin {\left\{\frac{{\mathrm{\&beta;}}_0[\sin 2\mathrm{\&pi;}{f}_{p}t-\sin 2\mathrm{\&pi;}(f_p+\mathrm{\&Delta;f})t]}{2}\right\}\}}^2\mathrm{dt}}{t_0}}$

In formula, β ₀for phase-modulation index, f _pfor phase modulation frequency, T ₀for time-parameters;

4) by this computable Digital Modulation error parameter and the contrast of VSA measured value, realize the measurement and calibration to GMSK demodulating error parameter.

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