Linear and non-linear Model-based Testing (MbT) has been developed in the past for mixed-signal devices. For Digital- to-Analog Converters (DACs) with... Computational modeling - Predictive models - Low pass filters - Radio frequency - Digital signal processing - Testing - Converters - digital-analogue conversion - low-pass filters - system-in-package - testing - model based testing - digital-to-analog converters - on-chip error source - device under test - low-pass filter - system-in-package

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Institute for Traffic Communications Engineering Dresden University of Technology, Dresden, Germany

Infineon Technologies AG Munich, Germany

Email: {Reik.Mueller, Hans-Joachim.Jentschel}@tu-dresden.de, [email protected]

Abstract—Linear and non-linear Model-based Testing (MbT) has been developed in the past for mixed-signal devices. For Digitalto-Analog Converters (DACs) with a static nonlinear transfer characteristic, linear MbT has even been adopted in production test. The core principle of MbT is to parameterize a model of the on-chip error sources based on the measurements that are performed on the Device Under Test (DUT). The model parameters are then used in order to determine whether the DUT passes or fails the test. Using a linear model reduces the computational effort in applying MbT to multiplying matrices during production test. For non-linear models, the computational effort is larger. In this paper, we report a generalization of MbT applied to combinations of static nonlinearities and dynamic linear systems. The specific example of a DAC followed by a low-pass filter is found, for example, at the boundary between the digital baseband and the RF front-end of a typical transceiver. One technology of integrating the digital baseband and the RF front-end is System-in-Package (SiP). Limited access to internal signals, such as the DAC output in our example, poses particular problems for parameterizing a linear error source model. With our generalization, we provide a reference MbT which allows us to evaluate and compare various linear and non-linear MbT approaches with respect to a specific test application.

I. I NTRODUCTION An RF-transceiver, as shown in Fig. 1, is a typical example of a mixed-signal circuit used in wireless communication systems. The drive for increased system integration leads to transceiver realizations as Systems-on-Chip (SoC) or Systemsin-Package (SiP). The advantage of SiP is the option of mixing fabrication technologies on a block-by-block basis for optimizing system performance and cost. In [1], a test approach targeting mixed-signal blocks on SoC platforms is reported. In this approach, which standardizes the access to the tested module with a digital wrapper, the analog inputs and outputs are assumed to be accessible from the SoC pins. However, for SiPs, we might not have direct access to the analog signals at the block inputs and outputs. Hence, we need an approach where test measurements at the system level allow us to identify performance degradations at the block level without explicit access to internal analog nodes. Block level diagnosis is a requirement in order to identify weaknesses of the system during product characterization and production ramp-up. After this characterization step, our strategy for high-volume production testing of a SiP can

1-4244-2396-5/08/$20.00 ©2008 IEEE

RF Transceiver

analog RF

Tx

I Q

Rx

I Q

DAC

I

digital baseband

Fig. 1. Typical RF-transceiver architecture with digital I/Q modulated signals being transmitted to or received from the antenna via the analog/RF blocks labeled Tx and Rx, respectively; encircled are the I-channel DAC, amplifier and filter stages in the transmit path.

focus on measuring previously identified weaknesses (or error sources) rather than on verifying the complete set of system specifications. Even if efficient test and diagnosis strategies are available for individual blocks of the system, the limited access to internal nodes poses a significant test problem. For the example of the Digital-to-Analog Converter (DAC) followed by a Programmable Gain Amplifier (PGA) and a low-pass filter stage as shown in Fig. 1, efficient test and diagnosis methods have been reported for stand-alone data converters [2], [3] and for linear filters [4]. However, when testing the completely assembled system, we have no access to the analog DAC output signal nor the filter input signal. In fact, we might only have access to the transmitter (Tx) output signal. Due to such access limitations, any attempt of finding the root cause for DAC performance degradations becomes significantly more difficult than applying the techniques described in [2] for stand-alone data converters with direct access to input and output signals. One alternative to block-level test and diagnosis is to perform system-level tests without regard to the system architecture and block structure. This approach would require a full functional test of the transceiver system properties. This specification-oriented test approach has lead to the perception of ever-increasing test costs [5] due to the increasing number of specifications and the complexity of their measurement. Therefore, we aim for a structural test strategy [6] that exploits the knowledge of the system structure. When this structure is represented by a parameterizable model, we call this strategy “Model-based Test” [7], [8]. The parameters of the model

are determined based on key measurements carried out on the DUT. From these parameters, the system performance is computed and can be tested against the system specifications in order to make pass/fail decisions. Moreover, the model parameters can be used for model-based fault diagnosis [3], [9]. Various flavors of MbT have been developed by a number of research groups. Linear MbT has originally been proposed in [10] by the National Institute of Standards and Technologies (NIST). The model is represented by a system of simultaneous linear equations. Determining the model parameters from the key measurements is computationally very efficient [11] as it only involves a matrix multiplication. For the example of an 11-bit video-DAC architecture, linear MbT has been introduced in high-volume production [12] in order to reduce the cost of testing a video encoder SoC containing six DACs. A comparative study [13] considers the NIST method and the wavelet transform as alternatives for modeling a PGA. When testing the static transfer characteristic of the PGA, the authors conclude that applying the NIST method yields more accurate test results. Applications of non-linear MbT to RF transceiver circuits has also been reported. A recent publication [8] reports MbT for a direct conversion receiver where the model is based on an Artificial Neural Network (ANN). The main drawback of using ANNs for modeling is that the simulations required for finding the ANN response, i.e. the pass/fail decision for the DUT, is computationally expensive. In [14], an RF subsystem is modeled by applying multivariate adaptive regression splines. The authors report an interesting approach to automatic test stimulus selection. An optimized multi-tone test stimulus is derived by maximizing the sensitivity of the regression parameters to changes in any of the system parameters. In this paper, we present a generalization of the linear and non-linear MbT approaches that can deal with an arbitrarily complex system model. The parameters of this model are determined by minimizing the difference between the responses of the system model and the DUT. With this approach we can evaluate the accuracy of MbT with respect to the structure of the system model, the test input signal and measurement noise. The paper starts in Sec. II with a description of the general model-based approach to test and diagnosis as applied to our target device type. In particular, we introduce the concept of the prediction error which is defined with respect to the target specification for which we are testing the DUT. In Sec. III, we present the results of model-based parameter identification as a basis for comparing two model-based test approaches. Finally, the conclusions are drawn in Sec. IV. II. T EST AND DIAGNOSIS TASK A. Definition of prediction error For the purposes of test and diagnosis of a system, we assume to have a system model with n parameters. These parameters are written as a parameter vector p˜ ∈ Rn . This parameter

x(t)

DUT

y(t, p)

DSP

−

+ system y˜(t, p˜) DSP model p˜

E(·, p, p˜)

Fig. 2. General Model-based Test: for an input signal x, the responses of the DUT and parameterized system model, y and y˜, are post-processed by a DSP block in order to determine performance values; the difference between these performance values defines the prediction error E.

vector is determined such that, for a given test signal at the input, the responses of the DUT and model match. This is illustrated in Fig. 2 for a DUT with an input signal x(t) and an output signal y(t, p), where t ∈ R denotes time and p ∈ Rk denotes the (unknown) parameter vector associated with the DUT. Depending on the test specification, we might be interested in matching post-processed versions of the output signals y(t, p) and y˜(t, p˜). For example, we might want to test specifications in the frequency domain. Therefore, in Fig. 2, the output signals are fed through a Digital Signal Processing (DSP) block before the difference is computed and denoted as the prediction error signal E(·, p, p˜). The prediction error characterizes how well the system model parameterized by p˜ approximates the DUT’s input/output behavior with respect to the target specifications. This error depends on the model parameter vector p˜, on the DUT parameter vector p and on another independent variable, e.g. time t or frequency f , according to operation performed by the DSP. Ideally, the prediction error is zero for all significant values of the independent variables. Therefore, we consider the worst-case value of the prediction error signal over a range or set I of (significant) values for the independent variable. This leads us to the definition of Emax (p, p˜) = max E(i, p, p˜). ∀i∈I

(1)

B. Model-based Test Let us assume that we have a method of determining the model parameter vector p˜ from a set of DUT measurements such that the prediction error is bounded by an acceptable value for Emax for all test signals x(t). Then we do not need to measure the specified DUT performance explicitly, instead, we can compute the response y˜(t, p˜) of the system model, infer the system performance by applying the DSP operations, and compare this computed performance with the test specifications in order to make a pass/fail decision. For the example of DAC testing, the specification of Integral Non-Linearity (INL [15]) requires a measurement of the DAC output for all possible digital input values. For an 11-bit DAC this requires 211 = 2048 measurements per DUT. In [12], the application of MbT reduced the number of measurements to 72 per DUT. From these measurements, the vector p˜ ∈ R36 , i.e. 36 model parameters are determined and the full set of

TABLE I D ESIGN VALUES FOR DUT.

DUT

x(t)

NL

sNL (t)

LTI

y(t) Parameter Value

Fig. 3. Continuous-time representation of a signal chain comprising DAC, gain and filter: a static nonlinearity (NL) followed by an LTI block; no access to the internal signal sNL is available during test.

2048 DAC output values, and hence the INL performance, are calculated from these parameters. It is shown for the example in [12] that the prediction error Emax is less than 5% of the test limits. C. Model-based Diagnosis The model-based approach can also be applied to diagnosing the root-cause of (tested) performance degradations. For this application, the system model in Fig. 2 is comprised of individual blocks with respective parameters p˜1 , p˜2 , . . . , p˜n forming the model parameter vector p˜ ∈ Rn . During the application of MbT to a DUT with unknown parameter vector p ∈ Rk we determine the parameter vector p˜ ∈ Rn . If we can map the individual parameters p˜i to the blocks of the system model, we can identify the location and the amount of performance degradations associated with the individual parameters, and therefore system blocks. For linear MbT applied to data converters, the methodology of diagnosing parametric deviations and hard-faults is described in detail in [9]. However, the model-based test approach requires a sufficiently rich and accurate model. Assuming that we can obtain such a model, the advantage of this approach is that the test and diagnosis task reduces to a model parameter identification problem. D. DUT description and specifications For the RF-transceiver shown in Fig. 1, the Tx-path takes the digital I- and Q-signals from the baseband and converts these with a Digital-to-Analog Converter (DAC) to the analog domain. The analog I- and Q-signals are amplified and lowpass filtered before they are up-converted into the RF domain. As a demonstrator for MbT, we extract from the full RF transceiver the DAC, gain and filter blocks as depicted in Fig. 1. We represent this DUT by the block diagram shown in Fig. 3, i.e. the DUT is comprised of a static nonlinearity (NL) followed by a dynamic Linear Time-Invariant (LTI) filter stage. For the purposes of this paper, we assume that this DUT representation is adequate, because we want to focus on the model-based test and diagnosis procedures, rather than on the development of an adequate model. Thus, for simplicity, we describe the nonlinear block by a polynomial sNL (t) = p0 + p1 x(t) + p2 x2 (t) + p3 x3 (t)

(2)

for its input/output relationship in the time domain. The LTI block is described in the frequency domain by its transfer function 1 Y (s) = (3) SNL (s), p4 s + 1

p0

p1

p2

p3

1/(2πp4 )

0.50

10.00

0.05

-0.01

100 Hz

where SNL (s) and Y (s) denote the Laplace transforms of the input and output signals of the LTI block. Thus, the LTI block represents a first-order low-pass filter with a cut-off frequency of fcut-off = 1/(2πp4 ). The above DUT description contains five parameters with the design values listed in Table I. The DUT input signal is a multi-tone signal x(t) =

7

Ak sin(2πfk + φk )

(4)

k=1

with seven frequency components spread around the filter cutoff frequency of 100 Hz as detailed in Table II. The amplitudes of the sinewaves comprising this test signal are set all equal to A = 100 mV. The phases φk can be chosen such as to minimize the crest-factor, i.e. the ratio between the maximum and the rms values of the input signal. The DUT response to this test signal can be simulated. We determine the steady-state response of the DUT which is sampled at a rate of 16.384 kHz for the duration of one second. After applying an FFT to the DUT output signal we obtain the amplitude spectrum shown in Fig. 4. In this plot one can clearly identify the linearly transferred input signal with its seven components—the higher frequency components are attenuated more strongly due to the low-pass filter action. Intermodulation products are four and six orders of magnitude below the linearly transferred input signal components. About fifteen orders of magnitude below the linearly transferred signals lies the computational noise which is shown in gray in Fig. 4. For such a device, we can specify certain performance parameters based on the output spectrum, e.g. Total Harmonic Distortion, Spurious-Free Dynamic Range, Third-order Intercept Point, etc. For example, we determine the TotalHarmonic-Distortion (THD) for a seven-tone test signal as follows: ⎞ ⎛ ⎜ Y 2 (ωk , p) ⎟ ⎟ dB. THD(p) = min ⎜ 10 log 10 ⎠ 3 k=1,...,7 ⎝ 2 Y (i · ωk , p)

(5)

i=2

TABLE II M ULTI - TONE TEST SIGNAL DEFINITION WITH COMPONENT FREQUENCY f , PHASE φ AND AMPLITUDE A = 100 mV. f / Hz

70

80

91

103

116

149

200

φ / rad

2.21

0.7

0.75

4.49

4.73

0.69

5.99

0

10

−5

Amplitude

mainly linear transferred signals

intermodulation products and harmonics

10

E. Calculation of prediction error For a target specification, such as THD, we need an approximation y˜(t, p˜) of the DUT output signal y(t, p) in Fig. 2 such that (in the ideal case) the Power Spectral Ratios Y˜ 2 (ω, p˜)/Y 2 (ω, p) = 1 (7) for all frequencies ω of interest with Y 2 (ω, p) 0. Therefore, the DSP operation in Fig. 2 includes an FFT and a conversion of the amplitude spectrum Y (ω, p) to the power spectrum

−10

10

lower signal bound dividing wanted and unwanted signal contributions

−15

P (ω, p) = 10 log10 Y 2 (ω, p) in dB.

noise caused by computational limits

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−20

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0

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1

2

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10 f / Hz + 1

3

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4

Fig. 4. Amplitude spectrum |Y versus frequency f = ω/2π for the response y(t, p0 ) of the DUT with design parameter vector p0 given in Table I. 2

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Power in dB

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|E(ω, p, p0 )|

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|E(ωI , p, p0 )|

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10 f / Hz + 1

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Fig. 5. Prediction error E(ω, p, p0 ) versus frequency f = ω/2π; shown in dark is the error for the set of frequencies Ω with significant signal content as defined in (10), shown in gray is the error for all other frequencies; the color scheme matches that in Fig. 4.

This requires the test signal frequencies in Table II to be chosen such that none of the harmonic components share the same frequency, i.e. we require that i · ωk = j · ωm

∀(i, k) = (j, m).

Based on this definition of the DSP operation, the prediction error is defined as follows E(ω, p, p˜) = P˜ (ω, p˜) − P (ω, p). (9)

10

(ω, p0 )|

10

(8)

(6)

The THD caused by the non-linear block in (2) alone is of course frequency-independent and evaluates to 92.0412 dB for the design parameters in Table I. For the overall DUT, we expect better THD performance over the range of test frequencies due to the attenuation of the harmonics. For the lowest test frequency of 70 Hz, the THD evaluates its worstcase value of 95.0227 dB; for the highest test frequency of 200 Hz, the THD evaluates to its best-case value of 97.3602 dB.

This has the effect that any error in predicting harmonics and intermodulation products becomes as important as an error in the linearly transferred signals. We illustrate this by the example of the DUT having a parameter vector p that deviates (due to assumed manufacturing tolerances) from the design parameter vector p0 . We assume for simplicity that the system model has the same structure as the DUT, and that we parameterize the system model with the design parameters by setting p˜ = p0 in Fig. 2. Then we obtain the prediction error E(ω, p, p0 ) which is shown versus frequency in Fig. 5. Note that the prediction error should only be evaluated at the frequencies of interest, i.e. those frequencies that are required for determining any of the DUT specifications. For the example of THD defined in (5), we are only interested in second- and third-order distortions, i.e. the frequency locations fk of the input signal x(t) (as defined in (4)) and the their integer multiples 2fk and 3fk . III. M ODEL PARAMETER IDENTIFICATION A. Optimization approach A standard approach to identifying the model parameter vector p˜ in the block diagram in Fig. 2 is to use nonlinear optimization [16]. With nonlinear optimization we implement the feedback loop shown in Fig. 6. For other specifications, intermodulation products might also be of interest, and therefore, we define the set of significant frequencies Ω = {ω : |Y (ω, p0 )| > 10−12 }

(10)

where the threshold of 10−12 is application specific and determined in our case from the amplitude spectrum shown Fig. 4. The set of significant frequencies should include all frequencies of interest, but exclude all frequencies, where E(ω, p, p˜) would be dominated by noise as marked in Fig. 5 by gray dots. Starting from an initial guess p˜0 , the system model parameter vector p˜ is iteratively updated while searching for a minimum of the cost function f (˜ p). With our definition of the prediction error in (9) we use the cost function f (˜ p) = E 2 (ω, p, p˜) (11) ω∈Ω

x(t)

DUT

y(t, p)

system y˜(t, p˜) model

Fig. 6.

FFT

FFT

Y (ω, p)

Y˜ (ω, p˜)

+ 10 log10 (·)2

E(ω, p, p˜)

arg min · 2 p˜

P˜ (ω, p˜)

p˜

0

0

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−1

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10% parameter variation 1% parameter variation

−1

10 in dB

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max

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Prediction error Emax in dB

Nonlinear Optimization

Block diagram of model parameter identification by minimizing prediction error in the least-squares sense.

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Fig. 7. Expected prediction error Emax (p, p˜) versus number of iterations for optimizing p˜ starting with the design values p0 ; for the experiment twenty DUTs were chosen randomly by drawing the parameter vector p from a Gaussian distribution with mean p0 and standard deviations 0.01 (marked by circles) and 0.1 (marked by dots).

Fig. 8. Expected prediction error Emax (p, p˜) versus model order l for linearly predicting the model parameter vector p˜ as in (12).

which restricts the optimization to the significant frequencies ω ∈ Ω and minimizes the prediction error in the least-squares sense. In order to estimate the maximum prediction error that we have to expect when applying MbT, we need to evaluate Emax (p, p˜) as defined in (1) over an ensemble of devices. In order to carry out this experiment in simulation, we assume that the elements pi in the parameter vector p are Gaussian distributed around the design values given in Table I. For the standard deviation of this multivariate distribution we assume two cases: 0.01 and 0.1. In this experiment, we deal with a special case where the system model is structurally identical to the DUT, and for p˜ = p, the prediction error becomes zero. For this particular case, we show in Fig. 7 the prediction error that we obtain for a given number of iterations in the optimization loop. By inspection, we can observe that the prediction error decreases by an order of magnitude for every ten iterations until computational inaccuracies limit the further decrease at Emax ≈ 10−7 . These observations are independent of the variation assumed for the DUT parameters. However, larger variation in the DUT parameters p causes the initial guess p0 to be further away from p, and therefore, requires more iterations.

B. Linear MbT approach Linear MbT can be used for estimating the parameter vector p˜ of the system model in Fig. 2. The procedures are described in detail in [9] for data converters. The model parameters are predicted as follows: ˜ + p˜0 p˜ = Apred x

(12)

l

where the vector x ˜ ∈ R is the solution of a linear system of equations b = Aest x ˜ + b0 . (13) The vector b represents measurements performed on the DUT, e.g. bi = Y (ωi , p), and b0i = Y (ωi , p0 ), for all ωi ∈ Ω, the set of significant frequencies defined in (10). The matrices Aest and Apred represent the linear model with its own set of l parameters. The higher the number of model parameters, the lower the expected prediction error. By performing the same simulations as previously for the optimization approach (see Sec. IIIA) we obtain the prediction error Emax versus the order l for the linear model as shown in Fig. 8. By comparison of the plots in Figs. 7 and 8 we can infer that 1) the prediction error achievable with linear modeling is by three orders of magnitude worse than the error

0

10

10% parameter variation 1% parameter variation

−1

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parameters of the system model identify all sources of performance degradation and enable an in-depth diagnosis of the root-cause of performance degradations.

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ACKNOWLEDGMENT

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This work has been supported by the German government (BMBF) in project Dionysys within the research promotion program IKT 2020 under grant No. 01 M 3084.

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R EFERENCES

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Fig. 9. Expected prediction error Emax (p, p˜) versus model order l for linearly predicting the model parameter for the special case of keeping p4 = p04 constant in (3).

achievable with nonlinear parameter estimation; 2) larger variation in the DUT parameters cause larger prediction errors, and linear MbT cannot fully compensate for this increased error by using higher model orders, while the nonlinear optimization-based parameter estimation can. These results suggest that linear MbT as described in [9] is not adequate for our DUT topology. In contrast to the case shown in Fig. 8, a linear model of order l = 4 can correctly estimate the parameters of the non-linear block in the DUT in Fig. 3 when the LTI-block description in (3) has its parameter fixed at p4 = p04 . In this case, we obtain the plots shown in Fig. 9. Our simulations suggest that a prediction error of Emax = 1.1 · 10−8 is achievable. How to extend linear MbT in order to deal with variations in the parameter(s) of the LTI-block is currently on-going work and will be reported elsewhere. IV. C ONCLUSION We presented a generalization of Model-based Test and Diagnosis that abstracts from any particular model parameter estimation technique (such as linear MbT) and can serve as a basis of comparison between competing modeling techniques. For the DUT architecture that we considered, i.e. a static nonlinearity followed by a low-pass filter, we have shown that linear MbT is not adequate; it lacks significantly behind the reference MbT that uses an optimization-based parameter estimation algorithm. However, we have shown how to obtain a reference MbT and how to determine a lower bound for the best-possible prediction error using MbT. Furthermore, we have shown that MbT in general is able to determine the parameters of the system model despite measurement access is limited to the input and output signals of the DUT. The such determined

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