Measurement Uncertainty Evaluation of Digital Modulation Quality Parameters : Magnitude Error and Phase Error

In digital modulation quality parameters traceability, the Error Vector Magnitude, Magnitude Error and Phase Error must be traced, and the measurement uncertainty of the above parameters needs to be assessed. Although the calibration specification JJF1128-2004 Calibration Specification for Vector Signal Analyzers is published domestically, the measurement uncertainty evaluation is unreasonable, the parameters selected is incorrect, and not all error terms are selected in measurement uncertainty evaluation. This article lists formula about magnitude error and phase error, than presents the measurement uncertainty evaluation processes for magnitude error and phase errors.


Measuring method
According to JJF1128-2004 Calibration Specification for Vector Signal Analyzers, digital modulation quality parameters are calibrated by direct comparison method.Schematic diagram is shown in Figure 1.
Connect calibrated Digital Signal Generator (DSG) to a Vector Signal Analyzer (VSA) or a Spectrum Analyzer with the analyzing function of vector signal.The time base output of Vector Signal Generator (VSG) should be connected to the time base input of VSA, and the time base of VSA should be set as external time base.Set the output frequency of the calibrated DSG, and generally the output level is set to -10 dBm.Select the required standard format or the customized modulation parameters, such as modulation, filter type, filter factor Į and symbol rate.Set the input frequency of the VSA equal to the one of calibrated DSG, and input level is -10 dBm or select Level Auto Range button to regulate the input level of VSA automatically.Select the required modulation standard format or set the modulation system, filter type, filter factor Į and symbol rate which is consistent with the VSG.Read the digital modulation quality parameters from VSA.

Definition of digital modulation quality parameter
For various vector modulation, the signal magnitude and phase can both be represented by the points on the constellation diagram.However, due to the effects such as non-ideal performance of vector modem hardware and the precision of vector modulation algorithm, there are errors occurring between signal vector in real and ideal situation.Vector modulation error defines the error between real constellation points and corresponding  As Eq. ( 1)~(3), the subscript rms is expressed as the root mean square value.MagErr rms is expressed as: PhaseErr rms is expressed as: EVM rms is expressed as: In Eq. ( 1)~(3), S is real signal vector, and R is reference signal vector.

Evaluation of measurement uncertainty of magnitude error
EVM will be first introduced before the evaluation of measurement uncertainty of digital modulation quality parameters are introduced.
An ideal vector modulation signal expression is shown in the Eq. ( 4).
Assume I/Q gain imbalance degree g (g=1 means complete balance), phase error ĳ, and interference signal Dsin(2ʌf i t) are exist.Phase noise ș(t) is a random variable which variance is ı P .n(t) is Gauss white noise with a variance of ı G .
Then Eq. ( 4) turns to Eq. ( 5): It is concluded that EVM and the error factor can be obtained by the derivation ([3]).
In Eq. ( 7): I/Q gain imbalance degree g is a linear value, not a logarithm value.
Unit of phase error ĳ is radian.ISR is the power ratio of interfering signals and useful signals, and is also a linear value.
If ĳ=0, EVM in Eq. ( 6) equals MagErr rms . 2 Eq. ( 8) is the expression of measurement uncertainty of magnitude error in digital modulation quality parameters traceability.

Uncertainty source
In JJF1128-2004 Calibration Specification for Vector Signal Analyzers, an example of measurement uncertainty evaluation is also given.For magnitude error and phase error, its uncertainty source is the error of vector signal generated by the VSG, resolution of VSA and measurement repeatability.When the VSG is to be calibrated, the VSA is the standard.The connection graph is the same as Figure 2, which means calibration of digital modulation quality parameters of VSG and digital modulation quality parameters of VSA is mutual cycle.
From Eq. ( 8) it can be concluded, when using the DSG to calibrate the digital modulation magnitude error of the VSA, the magnitude error is related to the following factors: I/Q gain imbalance (including inband flatness) g, signal-noise ratio ISR, phase noise, I/Q origin offset, resolution and measurement repeatability ( [2, 4, 5, 6]).

Uncertainty analysis
The evaluation process of the measurement uncertainty component is introduced below.For the Eq. ( 8) is complex, in the calculation of the uncertainty of the component, consider the impact of individual components, and finally combine the components.

Standard measurement uncertainty components produced by I/Q gain imbalance g
I/Q gain imbalance g includes inband flatness, which means g is related to signal bandwidth.Bandwidth of the signal becomes wider, then I/Q gain imbalance increaser, and magnitude error also increaser.
Only consider the effect of I/Q gain imbalance on magnitude error, it can be obtained from Eq. ( 8).According to the technical specification, I/Q gain imbalance is 1.01.It can be obtained from Eq. (9). a 1 =0.5%,Assuming it's a uniform distribution, the coverage factor is 3 .

Standard measurement uncertainty components produced by signal-noise ratio
Signal-noise ratio is the ratio between signal power and noise power.The greater the SNR is, the stronger the receiver's receptivity.
Only consider the effect of SNR on magnitude error, it can be obtained from Eq. ( 7).

Standard measurement uncertainty components produced by I/Q origin offset
Demodulator imbalance will cause the carrier leakage, which generates an interference signal.It interferences signal vector and deviates it from the correct position, then origin drift.Therefore, origin offset is carrier leakage signal components of demodulated signal.
Ignoring other factors, it can be obtained by the effect of interference signal on magnitude error.It can be obtained from Eq. ( 8).ISR is I/Q origin offset.According to the technical specification of VSA, I/Q origin offset is -60dB.It can be obtained from Eq. ( 14). a 4 = 0.1% Assuming it's a uniform distribution, the coverage factor is 3 .u 4 =0.06%

Standard measurement uncertainty components produced by resolution
When the vector signal is demodulated by the VSA, the minimum resolution is 0.01.a 5 =0.005%Assuming it's a uniform distribution, the coverage factor is 3 .u 5 =0.003%

Standard measurement uncertainty components produced by measurement repeatability
Measure the magnitude error for 10 times when standard value is 4.5%.

The synthesis of the standard uncertainty
Each Standard uncertainty component is independent from each other with no correlation, then consider the larger value of resolution and measurement repeatability involved in the synthesis of the standard uncertainty.

Uncertainty source
Phase error is related to the following factors: I/Q gain imbalance, I/Q origin offset, resolution and measurement repeatability.

Standard measurement uncertainty components produced by I/Q gain imbalance
The formula of the effect of I/Q gain imbalance on phase error is:  3 as an example, the effect of I/Q origin offset on the measurement results is introduced.According to the analysis, the magnitude of the error vector is perpendicular to X axis, and the I/Q origin is shifted in X axis, which has the greatest influence on the phase error.Assuming the origin offset is -60dB, which is 0.1%.According to Figure 3, the influence of the phase error is approximately equal to the origin offset value, which is in radian.It needs to multiply 180 S q ,then turns to degree.a 2 =0.06°Assuming it's a uniform distribution, the coverage factor is 3 .u 2 =0.04°

Standard measurement uncertainty components produced by resolution
Resolution is 0.01°.a 3 =0.005°,Assuming it's a uniform distribution, the coverage factor is 3 .

The synthesis of the standard uncertainty
Each Standard uncertainty component is independent from each other with no correlation, then consider the larger value of resolution and measurement repeatability involved in the synthesis of the standard uncertainty.0.174 u u u u q

Conclusion
In JJF1128-2004 Calibration Specification for Vector Signal Analyzers, the measurement uncertainty evaluation of magnitude error is unreasonable and the measurement uncertainty evaluation formula of phase error is not perfect.In this article, the uncertainty source of the magnitude error and phase error are comprehensively analyzed.

Figure 1 .
Figure 1.Schematic diagram of digital modulation quality parameter calibration in every symbol period.It is also called Digital Modulation Quality Parameter.The major parameters are magnitude error (MagErr), phase error (PhaseErr) and EVM, shown in Figure 2.Those parameters are usually represented by root mean square (RMS) ([2-3]).

Figure 3 .
Figure 3. Phase error with origin offset

Table 2 .
Phase error comparison result (1800MHz, 16QAM modulation, symbol rate 4Mbit/s, filter RRC, filter coefficient 0.22) From table 1 and table 2, the evaluation of measurement uncertainty is basically same as the result assessed by Comparison leading laboratory (NIM).