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The parallel analogue to digital converter (ADC) structure with mixers such as digital bandwidth-interleaving ADC (DBI-ADC) is a practical structureforrealisingthewidebanddataacquisitionsystem.Thesystemcalibrationreliesonaccuratesystemfrequencyresponsemeasurement. However,existingmeasurementmethodsrequirecomplexsynchronouscircuitsoraresusceptibletonoise.Thispaperproposesanewfrequency responsemeasurementmethod,whichcanberealisedbyasimplecir-cuitwithoutanysynchronisationmeasures.Atthesametime,frequency responsemeasurementcanensuresufﬁcientaccuracy.Simulationandexperimentresultsshowthatthemethodiseffectiveandfeasible.

✉ Email: yangkuojun@uestc.edu.cn The parallel analogue to digital converter (ADC) structure with mixers such as digital bandwidth-interleaving ADC (DBI-ADC) is a practical structure for realising the wideband data acquisition system. The system calibration relies on accurate system frequency response measurement. However, existing measurement methods require complex synchronous circuits or are susceptible to noise. This paper proposes a new frequency response measurement method, which can be realised by a simple circuit without any synchronisation measures. At the same time, frequency response measurement can ensure sufficient accuracy. Simulation and experiment results show that the method is effective and feasible.
Introduction: In high-speed data acquisition systems, various parallel ADC structures are usually used to overcome the performance limitations of a single ADC [1,2]. Among these commonly used structures, DBI-ADC is easy to implement and does not require too high component and circuit performance. It is an attractive method to implement high-speed wideband data acquisition systems [3].
The accurate correction of the DBI-ADC system depends on the accurate measurement. References [4] and other DBI-ADC system calibration methods developed based on FI-ADC require accurate measurement of each part of the system, which is unrealistic in engineering. In contrast, references [5,6] that only require measurement of subchannels or the entire system are more practical in engineering.
However, the current wideband ADC system measurement method is not enough to achieve a high-precision measurement of the DBI-ADC system, especially the phase frequency (group delay) response. Since the phase itself is a relative value, the measurement phase must have a reference value, so the input signal used for phase-frequency measurement must be multi-tone. There are two main phase-frequency response methods. One is to measure the device under test (DUT) with single-tone or dual-tone signal [7], and the other is to measure a multi-harmonic signal with a particular phase relationship [8].
During the entire measurement process, sweep frequency measurement requires a strict synchronisation relationship between the two sinusoidal signals (dual-tone) or a strict synchronisation relationship between the DUT and the reference instrument (single-tone an dual-tone). It is not easy to ensure a strict synchronisation relationship, and the measurement accuracy is greatly affected by the synchronisation skew mismatch. Multi-harmonic signal measurement usually uses periodic narrow pulse, fast edge signal, or square wave signal, which do not need to be synchronised. However, because these signals' low-frequency part has too much energy, the high-frequency part may be too small to be accurately measured.
For example, use a Fluke 9500B oscilloscope calibrator and 9550 active probe, which is the highest performance calibration instrument on the market, as a signal source to output 1 MHz, 25.6 ps fast edge signal. Lecroy's WaveMaster 813Zi-B oscilloscope samples the fast-edge signal in 40 GSa/s, 13 GHz, 10,000 times average mode. The measurement result of the amplitude-frequency response result is shown in Figure 1, which has a significant effect on the high frequency.
Besides, multi-harmonic signal measurement has another disadvantage. For DUTs such as the DBI-ADC system that are likely to have spurious signals, due to the multi-harmonic signal's excessive frequency components, it is difficult to distinguish whether the measurement result contains spurious signals that should not be included. When the spurious signal happens to appear at the frequency point where the multiharmonic signal itself has lower energy, this effect may be serious.
In response to the above problems, this paper proposes a new system frequency response measurement method. It can be applied to the Finally, we use a 2-subchannel DBI-ADC system simulation model to verify the effectiveness of this method by comparing the measurement results of different methods. An actual 2-sub-channel DBI-ADC experimental system is used to verify the feasibility of this method, and a comparison with the fast-edge signal measurement results shown in Figure 1 is given. Figure 2, the test frequency resolution is ω 0 . x 1 (t ) and x 2 (t; k) the sine signal output by the high-precision radio frequency source. Assuming that the bandwidth of the DUT is [0, ω c ]. The measurement frequency resolution is ω 0 = ω c /N. N is the number of measurement frequency points.

Measurement methods: As shown in
where M is constant, it determines the interval between the two spectra analysed at one time.
There is no synchronisation between x 1 (t ) and x 2 (t; k), so during measurement, ϕ 1 and ϕ 2 are independent.
Assuming that the sampling time of the DUT is t s . The spectrum of the measured signal input to the DUT is: where H mix (ω) is the frequency response of mixer at ω, H add (ω) is the frequency response of adder at ω.
Because there is no synchronised, the difference of sampling time between the reference instrument and DUT is t sk . The spectrum of the measured signal input to the reference instrument is X REF (ω; k) = X DUT (ω; k)e jω ts k Define Z DUT (ω) is the frequency response of DUT, Y DUT (ω) is the frequency spectrum of DUT. Y DUT (ω) can be described as: Assuming that the reference instrument is an ideal instrument, its frequency response is e − jωτR . Define U REF (ω) is output signal spectrum of the reference instrument. Similar to Equation (3), we can get: Equations (3) and (4) show that for any one measurement, only three frequency points (ω k−M , ω k+M , ω M ) in the outputs are valid. Using these points for calculations can avoid the interference of spurious signals.
Solved by Equation (3) and (4), the amplitude frequency response of points ω k−M can be calculate as: ). Similarly, the phase frequency response at points ω k−M can be calculated as Removing the time variables between Equations (6), it results: In Equation (7), ϕ ZDUT (ω M ) is the phase frequency response at ω M . For a linear time invariant system, it should be the same for all different k.
After completing the test of all frequency points in the entire frequency band, sweeping all k values, according to Group Delay definition, GD = − dϕ dω , we can get its estimation using the difference method incrementing the k value in (7): When using (7) in (8), a constant term − ϕZ (ωM ) Mω0 , independent of k, appears. According to the FI-ADC perfect reconstruction (PR) requirements, we expect the DUT to satisfy that the group delay is a constant. Rewrite the group delay GD ZDUT (ω) as the sum of the average group delay GD zDUT (ω) and the group delay deviation GD zDUT (ω): It can be seen that PR conditions are only concerned with GD zDUT (ω). So GD ZDUT (ω) can be replaced by GD ZDUT (ω). That means − ϕZ (ωM ) Mω0 can be ignored. Therefore, ϕ Z (ω M ) can be considered as a null reference phase. Correspondingly, Equations (7) and (8) can be rewritten as: The spurious signals in the system are mainly divided into two categories, one is the spurious signal with fixed frequency; the other is the frequency change with the input signal. The spurious signal with a fixed frequency has the same influence on all measurement methods and can be corrected by measuring the spurious signal.
For spurious signals whose frequency changes with the input signal, because the frequency of the input signal x(t; k) entering the ADC in this method is controlled by the parameters k and M, the spurious signals can be avoided by changing the values of k and M. The influence of scattered signals. Specifically, the steps are as follows: 1. Determine the parameters M 1 and sequence κ 1 , κ 1 is the value set of k to make k − M and k + M traverse all frequency points. Calculate the frequency point set 1 where the input signal and the spurious signal spectrum overlap. If 1 = ∅, the measurement results using M 1 and k 1 do not need to be corrected and can be used directly. 2. Calculate the parameters M 2 and the sequence κ 2 , the overlapping frequency point sequence 2 satisfies 1 ∩ 2 = ∅. 3. Measure the system according to M 1 , κ 1 and M 2 , κ 2 respectively, and record the measurement results as Z s1 , Z s2 . 4. Z is the average value of Z S1 and ZS2 (0 for unmeasured frequency points) As what has been discussed above, we can get DUT frequency response measurement steps of this method are: 1. Determine the appropriate ω 0 , M 1 , κ 1 , M 2 , κ 2 , to reduce the influence of the second term in (7) or (10). This way, M 1 and M 2 should be as large as possible. Establish also the number of times, I, that the measurements have to be repeated. 2. Connect the measurement system according to Figure 2 Simulation: In order to verify the validity and feasibility of the measurement method proposed in this paper, simulations and experiments were carried out in this section and the next section. Simulation mainly analyzes the effectiveness of the measurement method, and compares the theoretical performance with other methods.
In the simulation, a 2-subchannel DBI-ADC system has been designed in MATLAB, and the sampling rate of each subchannel is 10 GS/s. System sampling rate is f s = 20GS/s, bandwidth is c = 5 GHz. The simulation results are compared with the other three methods under different measurement errors. The measurement result of the sine frequency sweep method without noise is used as the reference's true value, that means this result considered as true (ref Z DUT . The frequency response of the mixer and adder used in this method is regarded as an equivalent 5th-order elliptical low-pass filter. Set frequency are f 0 = 10 MHz, M 1 = 501, κ 1 = 1, 2, . . . , M 1 − 1, M 2 = 511, κ 2 = 11, 12, . . . , M 2 − 11. When the system adds Gaussian white noise with σ 2 = 10 −3 and σ 2 = 10 −5 . Table 1 is the mean square error (MSE) of amplitude and group delay deviation with different Gaussian white noise by different measurement methods, σ 2 is noise variance.
As shown in Table 1, the mean square error of the group delay deviation measured by this method is much smaller than the square wave and the periodic pulse when the noise is high. Table 2 is the MSE of group delay deviation with skew mismatch by different measurement methods. Assuming that skew mismatch obeys Gaussian distribution, the confidence probability is 99%, and μ is the    Table 1 and 2, it can be seen that the accuracy of sine sweep method is obviously affected by the tilt mismatch. Other methods are almost unaffected by skew mismatch. Skew mismatch <0.01T s can only be achieved by TI-ADC now, and in other ADC structures will be larger. This means that in practice, skew mismatch has a greater impact on the sine sweep method. Table 3 is the MSE of amplitude and group delay deviation with spurious by different measurement methods. Assuming that the spurious signals existing in the system have the same amplitude, α is their attenuation relative to the input signal. It can be seen from Table 3 that our method after correction (in column E) is obviously superior to square wave and periodic pulse, and achieves the accuracy similar to sine sweep.
Experiment: The experiment system is a 2-subchannel DBI-ADC system without correction. It has a sampling rate of 20 GSa/s and a bandwidth of 5.5 GHz. The frequency response measurement methods are the fast edge signal shown in Figure 1, and this method.
The system amplitude-frequency response measurement value is shown in Figure 3, the group delay deviation measurement value is shown in Figure 4.
According to Figures 3 and 4, the measurement errors of this method are significantly smaller than that of the fast-edge signal. One reason is that the fast edge signal is greatly affected by noise, especially the high frequency part. The other reason is that the energy of the fast side signal between the low frequency lobes drops sharply, which leads to the increase of group delay measurement error.
Conclusion: This paper proposes a new frequency response measurement method for wideband ADC acquisition systems such as DBI-ADC. This method combines the advantages of the multi-harmonic signal method and the sine frequency sweep method. The accuracy of the measurement results can reach the level of the sine frequency sweep method. Simulation and experiment verify the effectiveness of this method.