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A Simple Nonparametric Preprocessing Technique to Correct for Nonstationary Effects in Measured Data Kurt Barbé, Wendy Van Moer, Lieve Lauwers, and Niclas Björsell
Abstract—The general approach for modeling systems assumes that the measured signals are (weakly) stationary, i.e., the power spectrum is time invariant. However, the stationarity assumption is violated when: 1) transient effects due to experimental conditions are dominant; 2) data are missing due to, for instance, sensor failure; or 3) the amplitude of the excitation signals smoothly varies over time due to, for instance, actuator problems. Although different methods exist to deal with each of these nonstationary effects specifically, no unified approach is available. In this paper, a new and general technique is presented to handle nonstationary effects, based on processing overlapping subrecords of the measured data. The proposed method is a simple preprocessing step where the user does not need to specify which nonstationary effect is present, nor the time interval where the nonstationary effect appears. The merits of the proposed approach are demonstrated on an operational wireless system suffering from interrupted link effects. Index Terms—Incomplete measurements, missing data, nonstationarity, preprocessing, signal modeling, spectral leakage, system modeling, transients.
I. I NTRODUCTION
I
N MANY measurement applications, one aims at characterizing the dynamic behavior of a device under test (DUT), observing a periodic component buried in the observed signal, studying the coloring of the disturbing noise sources (see [1]). The standard technique to reach this objective is the measurement of the frequency response function (FRF). Hence, the measurement of the FRF can be considered as a fundamental measurement topic. Looking at the fundamentals of FRF measurements, it is clear that the classical technique to obtain the FRF operates under strict conditions. Departures of these conditions may lead to systematic errors in the measured FRF.
Manuscript received October 28, 2011; revised April 23, 2012; accepted April 24, 2012. Date of publication June 1, 2012; date of current version July 13, 2012. This work was supported in part by the Flemish Research Foundation (FWO) under a postdoctoral fellowship and in part by the Flemish Government through the Minister for Innovation under Methusalem (Meth-1) Grant. The Associate Editor coordinating the review process for this paper was Dr. Dario Petri. K. Barbé and L. Lauwers are with the Department of Fundamental Electricity and Instrumentation (ELEC), Research Team Medical Measurements and Signal Analysis (M2ESA), Vrije Universiteit Brussel, 1050 Brussel, Belgium (e-mail:
[email protected]). W. Van Moer is with the Department of Fundamental Electricity and Instrumentation (ELEC), Research Team Medical Measurements and Signal Analysis (M2ESA), Vrije Universiteit Brussel, 1050 Brussel, Belgium, and also with the Department of ITB/Electronics, University of Gävle, 801 76 Gävle, Sweden. N. Björsell is with the Department of ITB/Electronics, University of Gävle, 801 76 Gävle, Sweden. Digital Object Identifier 10.1109/TIM.2012.2198269
This paper studies measurement scenarios in which the measurement of the FRF is difficult and prone to errors, and we propose a general technique to circumvent these problems. The proposed technique is a preprocessing technique that can be safely applied. Indeed, when the measurements reveal no measurement problems, the technique leaves the measurement invariant, implying that the classical techniques to obtain the FRF can be used. Otherwise, the technique will correct the measurement problems prior to using any user-preferred technique to compute the FRF. Classical methodologies available in literature to compute the FRF operate under the assumption that the measured signals are stationary [2], such that the underlying signal’s properties remain invariant over time [3]. This assumption is clearly violated when dealing with time-varying systems. However, also for time-invariant systems, the stationarity assumption can become invalid due to the measurement process/errors. This is, for instance, the case when the following three scenarios occur: I) transient effects due to initial and final experimental conditions; II) missing data due to sensor failure; III) amplitude variations of the excitation signals due to actuator problems. Transient effects typically arise when a finite-measurement time window is used [2]. Missing data occur when the sensor performing the measurements incorrectly registers or simply fails to register the measurements [4]. Actuator problems are present when the actuator fails to apply the intended excitation signal to the system. One way to tackle this problem is by measuring the actual applied input signal [5]. Unfortunately, this is, for many applications, difficult or even infeasible (e.g., for wireless systems). Hence, actuator problems lead to systematic differences between the applied and the intended excitation. In these three scenarios, the measurement engineer needs to take into account the influence of the nonstationary effects since they introduce systematic errors in the measured FRF of the DUT [2], [6]. Different methods exist to deal with each of the aforementioned nonstationary effects specifically. However, most of these techniques require a parametric model (e.g., [4] and [6]–[8]), on top of the use of time windows to eliminate transients [9], [10]. For example, in the time series analysis literature, the problem of missing data is dealt with by representing the missing data as additional parameters or unknowns. As a result, knowledge of the time interval where the nonstationary effect appears is required.
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Assume the system to be stable such that its impulse response function g(t) is absolutely integrable [9] ∞ |g(t)|dt < ∞.
(1)
0
Fig. 1. Scheme of the theoretical measurement setup.
In this paper, we propose a general technique to correct for nonstationary effects in the measured data, based on processing overlapping subrecords. The technique is based on the Welch method [9], [11], [12] for nonparametric power spectrum estimators to suppress transient influences due to nonstationary effects present in the data. The proposed technique has the following advantages. I) The technique presents a unified approach to handle both missing data and transient phenomena. II) The user does not need to specify which nonstationary effect applies, nor the time window when the effect appears. III) The technique is completely nonparametric such that it can be used as a preprocessing step without the need of additional modeling steps. IV) The technique is fully automated without any user interaction. No real disadvantage of the proposed method is present besides the fact that the computation time slightly increases due to the additional preprocessing step. Indeed, throughout the paper, we show that the variability due to the preprocessing step is in general not increased. On top of this, we study the performance of the preprocessing step as a function of the fraction of overlap. This paper shows that the mean squared error (MSE) is a decreasing function of the fraction of overlap such that every decrease in MSE implies an increase in the computational workload. The outline of this paper is as follows. In Section II, the theoretical and experimental framework is introduced. Section III presents a detailed study of the nonstationary effects in the frequency domain. Section IV describes the correction algorithm to compensate for nonstationary effects. In Section V, user interaction is eliminated such that the algorithm becomes fully automated. Section VI shows the performance of the presented technique on simulations, whereas Section VII applies the technique on measurements of a wireless system. II. T HEORETICAL AND E XPERIMENTAL F RAMEWORK This section describes the theoretical measurement setup that is used to derive the results mathematically (see Fig. 1). A. System Properties Let G be a time-invariant, causal, linear, and continuous-time system with transfer function G(ω) at angular frequency ω.
The system is excited by a (weakly) stationary signal u0 (t) for t ≥ 0 with a power spectrum Su0 (ω). Note that u0 (t) consists of a large class of multifrequency signals, e.g., filtered/colored noise, multisine, and pseudorandom binary sequence. Furthermore, it is assumed that power spectrum Su0 (ω) is integrable such that the signal’s energy is finite. The response of the system G to the signal u0 (t) is given by y0 (t). This assumption implies that the Fourier coefficients at frequency ω of both u0 (t) and y0 (t) exist and will be respectively denoted as U0 (ω) and Y0 (ω). Excitation signal u0 (t) is realized by the actuator A(ω) where the signal uref (t) is the computer-stored reference signal. B. Noise Considerations Let signal y(t) be the measured version of the exact response signal y0 (t). As a result of the measurement process, signal y(t) is a noisy version of true response y0 (t) y(t) = y0 (t) + n(t).
(2)
Hence, noise process n(t) is a (weakly) stationary signal with an integrable power spectrum Sn (ω). Furthermore, excitation signal u0 (t) and the noise process n(t) are assumed to be independent. For the sake of generality, the noise process does not need to follow a Gaussian distribution such that signal n(t) may also account for possible quantization noise. C. Sampling and Discretization Reference signal uref (t) is a band-limited signal with maximal frequency ωm . Measured output signal y(t) is sampled at a sampling frequency fs that is larger than twice the highest frequency ωm of reference signal uref (t). As a result, the measured output does not exhibit any aliasing errors. D. Considered Sources of Nonstationary Effects The theory and methodology proposed in this paper correct for three different types of nonstationary effects in measurements. Note that the different types may be simultaneously present. 1) Transient Conditions: In a practical measurement experiment, excitation signal u0 (t) is applied to the system G(ω) during the time interval t ∈ [0, T ] with T = (N − 1)Ts , where Ts is the sampling period. It is well known that a sudden switchon at t = 0 and switch-off at t = T of the excitation signal results in a nonstationary disturbance in the system’s response known as the transient effect [2]. 2) Missing Data: The system’s response y0 (t) is measured at time instants t = 0, Ts , . . . , (N − 1)Ts . Sensor failure
BARBÉ et al.: NONPARAMETRIC TECHNIQUE TO CORRECT FOR NONSTATIONARY EFFECTS IN MEASURED DATA
implies that a number of measured samples t˜ of y(t) were not registered such that y(t˜) = 0. Note that the value zero is chosen for simplicity. In practice, y(t˜) may be equal to any constant equal to the dc level of the measurement device or equal to noise satisfying the conditions in Section II-B. 3) Amplitude Variations: An ideal actuator is able to realize the reference signal such that u0 (t) = uref (t). Actuator problems imply that there exists a time interval [t, t] where u0 (t) = uref (t). In other words, the influence of the actuator problem can be described by a (piecewise) continuous function / [t, t]. h(t) such that uref (t) = h(t)u0 (t) with h(t) = 1 for t ∈ Clearly, ideal actuators do not exist such that at most u0 (t) ≈ uref (t) holds. However, we assume that function h(t) is larger than possible imperfections of the actuator (e.g., actuator noise).
In this section, the influence of the nonstationary effects described in Section II-D is studied in the frequency domain. Hereby, we will ignore measurement noise n(t). A. Transient Conditions In [15], it was shown that transient conditions present themselves as a leakage effect in the frequency domain. We compute the discrete Fourier transform (DFT) of measured output signal y(nTs ), n = 0, . . . , N − 1, at frequency ωk = (2πk/N )fS as N −1
y(nTs )e−jωk nTs .
(3)
n=0
Here, it is shown that missing data (as described in Section II-D) act as a leakage effect in the frequency domain. No transient phenomena are considered since the focus is only put on missing data. Hence, system G is assumed to operate under steady-state conditions. Next, y(t˜) = 0 holds for the samples t˜ = nTs , (n + 1)Ts , . . . , nTs of the missing data interval. The relation between Y (ωk ) and Y0 (ωk ) is given by ˜ k) Y (ωk ) = Y0 (ωk ) ∗ D(ω
−j ˜ k) = e D(ω
(n+n)Ts
N
sin (ωk (n − n − 1)Ts ) sin(ωk Ts )
C. Amplitude Variations Amplitude variations of the excitation signal due to actuator problems, as described in Section II-D, are a more general case of missing data. Piecewise continuous function h(t), as defined in Section II-D, may be arbitrarily well approximated by for t ∈ / [t, t] for m M ≤ t − t ≤ (m + 1) M (5) where ∆M = (t − t)/M and m = 0, . . . , M − 1. Equation (5) shows that actuator problems are closely related to missing data in the sense that every interval m∆M ≤ t − t ≤ (m + 1)∆M implies “missing data.” Without a formal proof, this reasoning motivates the presence of leakage errors due to nonstationarity problems of the data.
n=m
u ˜0 (nTs )e−jωk nTs = U0 (ωk ) ∗ Dt (ωk )
ωk 2
and is proven in Appendix B. The difference between Y (ωk ) and Y0 (ωk ) is a leakage effect. By increasing the measured record (N → ∞), the leakage error due to missing data will be decreased.
where m is an index such that (m − 1)Ts ≤ t ≤ mTs . The leakage errors appear due to N −1
B. Missing Data
˜ M (t) = h
Equation (3) can be further simplified to (see Appendix A) N −1 ∞ −jωk nTs Y (ωk ) = g(t) u ˜0 (nTs )e e−jωk t dt 0
for N → ∞. This is what we expect from linear system’s theory in the absence of transients [1].
where
III. N ONSTATIONARY E FFECTS IN THE F REQUENCY D OMAIN
Y (ωk ) =
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1 h(m M )
(4)
n=m
D. Concluding Remark
where ∗ is the convolution operator and Dt (ωk ) = −
ωk 1 −j ωk (m−1)Ts sin 2 mTs 2 e N sin ω2k Ts
for (m − 1)Ts < t ≤ mTs . Appendix A provides a proof of the calculations. It is well known [13] that Dt (ωk ) converges to Dirac delta function δ(ωk ) [14] for N → ∞ such that Y (ωk ) → G(ωk )U0 (ωk )
All nonstationary effects studied in the frequency domain act as a leakage effect. Hence, the user does not need to specify the nature of these nonstationary effects beforehand.
IV. C ORRECTING FOR L EAKAGE E RRORS D UE TO N ONSTATIONARY E FFECTS This section presents a method to correct leakage errors based on overlapping subrecords of the measured FRF. Hereby, the time intervals in which a nonstationary effect occurs are assumed to be known.
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stationary. Let I be the set of indices of the subrecords of the output signal for which no nonstationary disturbances are observed, and let J be the set of the remaining indices. Hence, (9) represents a nonparametric estimator of the leakage contribution due to the nonstationary effects 1 [i] 1 [i] Xr (k) − Xr (k) T˜x (k) = #J #I i∈J
Fig. 2. Extracted circular overlapping subrecords with a fraction of overlap r = 2/3.
A. Extracting Circular Overlapping Subrecords To extract overlapping subrecords, one needs to partition the measured input/output signals into at least two segments. Hence, for generality, we assume that P segments are measured. The number of segments depends on the frequency resolution desired by the user and the total measurement time window. Let x[i] (nTs ) be the nth measured sample of the ith segment (i = 1, . . . , P ) of the signal x, then x[i] (nTs ) = x(((i − 1)L + n) Ts ) .
(6)
Next, the nth measured sample of the ith (i = 1, . . . , (P/1 − r)) overlapping subrecord is defined as x[i] r (nTs ) = x ([(i − 1)L(1 − r) + n]P L Ts )
(7)
where r denotes the fraction of overlap and [a]P L denotes a modulus P L. Subrecords extracted following the rule in (7) satisfy the scheme in Fig. 2 known as circular overlap [12]. If the fraction of overlap in (7) is equal to zero, definitions (6) and (7) naturally coincide. Note that, in case of circular overlap, some subrecords consist of a part of the last measured period and a part the first measured period. This is in contrast with regular overlap. Although this concatenation of signals imposes a discontinuity, its properties are well known [12], [16]. B. Nonparametric Correction for Nonstationary Effects The nonparametric correction technique is performed in the frequency domain, which is a natural choice as nonstationary disturbances act as leakage in the frequency domain. Hence, we generalize the DFT in (3) to the computation of the discrete Fourier coefficients of the overlapping subrecords. The discrete [m] Fourier coefficient Xr (k) at frequency ωk of the ith overlap[m] ping subrecord Xr (nTs ) is defined as L−1 2πk [m] [m] −j 2πk n Xr (ωk ) = xr (nTs )e L (8) e−j L τi n=0
where τi = (1 − r)(m − 1)L to refer all overlapping subrecords to the same time origin. It is clear that Fourier coef[m] ficients Xr (k) will be heavily influenced by leakage errors if these Fourier coefficients are computed from those subrecords where the nonstationary data are present. The nonstationary behavior can be only observed in the output signal due to the fact that reference signal uref (t) is
(9)
i∈I
where #J and #I are the numbers of indices in sets I and J, respectively. The rationale behind (9) is straightforward. In the ideal case, when there are no nonstationary effects, there should be no [i] significant difference between any Xr (k) due to the stationarity of the input and output signals. If there is a difference between the DFT coefficients belonging to the index set I and J, this difference is due to both nonstationary effects and [i] 1 measurement noise. Computing the average #J l∈I Xr (k) [i] 1 and #I i∈I Xr (k) suppresses the noise if the noise is zero mean, such that the difference (9) is an estimate of the error due to nonstationarity. C. Correction of Measured DFT Coefficients In (9), we quantified the difference between the spectra corresponding to the stationary and nonstationary subrecords. Next, it is described how to apply the correction (9). The Fourier coefficients are modified in the following way: [i] for i ∈ I ˜ [i] (k) = Xr (k) X (10) r [i] ˆ Xr (k) − Tx (k) for I ∈ J. Please note that in (10), only the subrecords with an index belonging to set J are modified since it is assumed that the subrecords belonging to index set I are not influenced by nonstationary disturbances. Furthermore, the Fourier coefficients corresponding to the reference signal x = uref are modified as well, although no nonstationary effects occur in x = uref . However, to keep the phase coherence between the reference and the output Fourier coefficients, the reference and output signals need to be processed in the same way. Since correction Tˆx (k) is data driven, the variance of the ˜ r[i] (k) is altered with respect to corrected Fourier coefficients X [i] the originally measured Fourier coefficients Xr (k). A simplified analysis in which the correlation among the overlapping subrecords is ignored reveals 1 1 ˜ [i] (k) ≈ + 1 − Sx (ωk )2 . Var X r #I #J This simplified analysis suggests that if both groups have approximately the same number of segments and the variance is not increased, a small decrease in variance may be expected when the stationary group holds more segments than the nonstationary group (#I > #J); otherwise, a small increase in variance may be expected. However, in practice, the number of segments in both groups does not sufficiently differ to observe
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a significant in–or decrease in variance. See Section VI for a numerical illustration. V. AUTOMATIC S ENSING OF N ONSTATIONARY E FFECTS In the previous section, we assumed that the subrecords, with nonstationary effects, are a priori known. This is, in reallife examples, not feasible. Hence, to avoid user interaction, an automatic sensing algorithm is required to detect those subrecords where nonstationary disturbances are present. [m] The distribution of the measured power spectra |Yr (k)|2 corresponding to the overlapping subrecords implies that [m] 2 ∆ [m] Yr (k) = Sy (ωk )χ22
(11)
Fig. 3. Pole-zero configuration of transfer function G(jω).
A. Initial and Final Conditions ∆
where = denotes equality in distribution, χ22 is a chi-squared [m] distribution [17] with two degrees of freedom, and Sy (ωk ) is [m] the power spectrum corresponding to signal y (t). Under the condition that all subrecords are stationary and hence have the [m] same power spectrum Sy (ωk ), the null hypothesis of the test is equal to H0 ≡ Sy[i] (ωk ) = Sy[j] (ωk )
for i = j.
(12)
Under the additional condition that measured signal y(t) is a Gaussian process, the z-test can be derived from (12), which is asymptotically (L → ∞) optimal [18]. As a result, for a userspecified probability of making a type-I error, the test provides the user the smallest possible probability of making a type-II error. The test statistic becomes [m] 2 Yr (k) L2 −1 2 − MM−1 k=0 1 M [n] Yr (k) M L n=1
− 1. (13) Zm = 3 M 2 (M −1)2 (M −2)
Asymptotically, for L → ∞, this test statistic follows a Gaussian distribution with zero mean and unit variance. Hence, based on the z-test (13), the different subrecords can be split into two groups, i.e., a stationary group and a nonstationary group. According to (13), the subrecords exhibiting stationary behavior have a Zm value close to 0. As a result, the following discrimination rule is applied. A subrecord m is assessed being stationary if |Zm | ≤ median ([|Z1 |, |Z2 |, . . . , |ZM |]) .
In this example, a continuous-time system was simulated with the following transfer function represented as a partial fraction decomposition: 1+j G(ω) = 2.25 × 104 jω − 2π103 (−1 + 227j)
(14)
Based on rule (14), the different subrecords can be assigned to either the set of indices I if the inequality (14) holds or to set J otherwise, as defined in Section VI-B. VI. S IMULATION E XAMPLES In this section, the proposed correction method is illustrated on two examples: 1) compensation of initial and final conditions; and 2) compensation of missing data.
+ +
1−j jω − 2π103 (−1 − 227j) 1 jω −
2π103 (−10
+ 308 103 j)
1 . + jω − 2π103 (−10 − 308 103 j) The system’s pole-zero configuration is shown in Fig. 3. The system is stable with two complex conjugate zeros captured between the two resonances and one real-valued nonminimumphase zero. The corresponding differential equation of system G(ω) was simulated in a time interval [0, 2]µs in which a sampling frequency of 1 MHz was selected. The initial conditions of the differential equation were randomly chosen. Excitation signal u0 (t) is a random phase multisine with a unit amplitude spectrum u0 (t) =
F
sin(2πkf0 t + φk )
for t ≥ 0
k=0
where f0 = 1 kHz, F = 499, and the phases φk are randomly drawn from the interval [0, 2π]. For simplicity, only output measurement noise is considered. Output noise process ny (t) is a Gaussian process filtered by a second-order Butterworth filter with a cutoff frequency of 400 kHz. The output signal-to-noise ratio (SNR) is varied from 0 to 100 dB in steps of 10 dB. To eliminate the transient contribution, a fraction of overlap of 80% is used for two measured periods. One period consists of 1000 samples. The FRF is obtained by computing GFRF (ωk ) =
Yˆ (ωk ) ˆ (ωk ) U
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Fig. 4. Ten different amplitude spectra of output signal y [3] (t) extracted from 80% overlap. The gray curve represents the nonstationary subrecords; the black curve represents the stationary subrecords.
Fig. 5. MSE for the transient example of the FRFs computed (top plot) without compensation and (bottom plot) with compensation.
ˆ (ωk ) are the average discrete Fourier where Yˆ (ωk ) and U coefficients of the output and input overlapping subrecords. The FRF is computed once by using the proposed transient elimination method and once without the correction. In Fig. 4, the magnitudes of the different overlapping DFT spectra reveal two groups. The black group is significantly less influenced by transient errors than the gray group, as revealed by the automatic sensing method in Section V. Fig. 5 shows the MSE of the obtained FRF and the true transfer function computed over 1000 simulation runs. It is immediately clear that the transient elimination procedure improves the MSE. Indeed, in the bottom plot, the MSE has improved by approximately 20 dB near the resonance of 227 kHz. Moreover, the MSE after compensation no longer reveals the characteristics of the system such that we may conclude that the transient elimination procedure was successful. B. Missing Data The same system was applied as in the example in Section VI-A. Instead of using a multisine signal, we use a noise signal to excite the system. This example shows that random parts of the output signal unobserved due to missing data can be partly reconstructed to improve the error of the measured transfer function. Signal u0 (t) is a Gaussian noise signal with zero mean and a unit power spectrum in the frequency band [0, 500] kHz and
Fig. 6. Time-domain signals of (top plot) the applied excitation and (bottom plot) the measured response.
Fig. 7. Ten different amplitude spectra of the output signal extracted from 80% overlap.
zero outside this band. The response of the system G(ω) to input signal u0 (t) was computed under steady-state conditions. Signal u0 (t) is applied for 3 µs, and this record is partitioned in three segments (P = 3). The sampling frequency is again 1 MHz. Note that no measurements of the output signal are available in the time window t˜ ∈ [0.1, 1]µs, such that y(t˜) = 0. This results in missing data of 90% of the first segment or 30% of the entire measured output signal (see Fig. 6). For simplicity, only output measurement noise is considered. The output noise process ny (t) is a Gaussian process filtered by a second-order Butterworth filter and a cutoff frequency at 400 kHz. The output SNR is varied from 10 to 100 dB. To eliminate the nonstationary effects, a fraction of overlap of 80% is used for three measured segments (P = 3). Every subrecord consists of 1000 samples. Since the applied excitation signal is not a periodic signal, the FRF is obtained by computing GFRF (ωk ) =
Sˆyu (ωk ) Sˆuu (ωk )
where Sˆyu (ωk ) and Sˆuu (ωk ) are the cross-power spectrum of (y, u0 ) and the power spectrum of u0 , respectively, averaged over the different overlapping subrecords. The FRF is computed once by using the proposed missing data correction and once without the correction. Although less
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Fig. 8. MSE for the missing data example of the FRFs computed (top plot) without compensation and (bottom plot) with compensation.
Fig. 10. MSE as a function of (top) the fraction of overlap and as a function of (bottom) the number of overlapping segments.
The optimal choice suggests a fraction of overlap of 0.8 because, from that point on, an increasing fraction of overlap no longer contributes to an improved MSE. Please note that this optimal choice is only valid for two initial segments (P = 2). The optimal choice of 0.8 can be lowered for increasing initial segments (see Fig. 10). Fig. 9. (Black) True output signal y0 (t) and (gray) reconstructed signal. The time interval corresponding to the missing data is shown by the black dashed lines.
visible due to noise, the automatic detection of the nonstationary subrecords was able to partition the different overlapping subrecords in two groups. In Fig. 7, the different amplitude spectra corresponding to the overlapping subrecords are shown. The detection method classified the black subrecords as stationary and the gray subrecords as nonstationary. Fig. 8 shows the MSE of the obtained FRF and the true transfer function computed over 1000 simulation runs. The difference in MSE between the corrected and uncorrected FRFs is approximately 7 dB. This improvement suggests that the correction algorithm was successful in reconstructing at least a part of the missing data in signal y(t). The reconstructed signal in the time domain is shown in Fig. 9 by the gray curve. It is clear that the algorithm is not able to fully reconstruct the time-domain signal but the method serves as a correction to reduce errors in the FRF introduced by missing data. C. Improved MSE as a Function of the Fraction of Overlap In this section, the first example in Section VI-A is revisited. Under the same conditions as in Section VI-A, the MSE is studied as a function of the fraction of overlap and the associated number of overlapping segments. The fraction of overlap ranges from 0.33 to 0.94 such that the number of segments ranges from 3 to 30.
VII. M EASUREMENT E XAMPLE : I NTERRUPTED W IRELESS L INK This measurement example shows the merits of the proposed approach when a sender–receiver setup encounters missing links and fading effects. A. Measurement Setup The wireless setup is shown in Fig. 11 and is a simplified sender–receiver system. The sender consists of a generator that produces a multisine of 5000 tones between 55 and 80 MHz. The multisine is then upconverted around a center frequency of 2.15 GHz and is sent to the antenna. After the receiving antenna, the signal is downconverted by a mixer with an Local Oscillator signal of 2.015 GHz. After low-pass filtering the downconverted signal, one sideband remains between 55 and 80 MHz. This signal is then digitized by an analog-to-digital converter with a sampling frequency of 400 MHz. Two possible interrupted link effects have been mimicked. In a first setup, a switch was inserted directly after the receiving antenna. As a result, the signal is completely cut off for a certain amount of time. In a second setup, a steerable attenuator was inserted after the receiving antenna. By steering the attenuator with a sawtooth, one can mimic the behavior of a fading signal. B. Measurement Results 1) Signal Loss: Fig. 12 shows the measured signal fallout in the time domain. The signal contains 11 periods of
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Fig. 11. Signal fallout.
Fig. 14. Signal fallout: (black) stationary output spectrum; (light gray) corrected output spectrum; and (dark gray) uncorrected output spectrum.
Fig. 15. Signal fallout, no automatic sensing algorithm: (black) stationary output spectrum; (light gray) corrected output spectrum; and (dark gray) uncorrected output spectrum.
C. Corrected Output Spectra Fig. 12. (a) Sender side. (b) Receiver side.
Fig. 13. Signal fading.
80 000 sample points. The gray vertical lines separate the different periods. Note that the signal fallout is different for each period. 2) Fading Signal: Fig. 13 shows the measured fading signal in the time domain. The signal contains 11 periods of 80 000 sample points. The gray vertical lines separate the different periods. Note that fading is different in each period of the original signal.
The measured spectra containing nonstationary effects are compensated by the automatic correction algorithm developed in this paper. 1) Signal Loss: The automatic correction algorithm has been applied on the data in Fig. 12 with an overlap percentage of 80%. Fig. 14 presents the uncorrected output spectrum in dark gray and the corrected output spectrum in light gray. These output spectra are compared with the stationary output spectrum, i.e., no signal fallout is present (black). One can clearly see that the corrected output spectrum lies much closer to the stationary output spectrum than the noncorrected output spectrum. The difference between the uncorrected and corrected output spectra is 3 dB. In the next step, the automatic sensing of nonstationary effect (see Section V) has not been used. Instead, we visually inspected the different overlapping subrecords and selected those segments containing the smallest fractions of signal fallout. An overlap percentage of 80% has been used. Fig. 15 shows the uncorrected output spectrum in dark gray, the corrected output spectrum in light gray, and the stationary output spectrum in black. We clearly see that the corrected output spectrum almost lies on top of the stationary output spectrum and hence represents significant amelioration compared with the uncorrected output spectrum.
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where the last equality holds due to the fact that u0 (t = 0) for t < 0. We can simplify the last expression further to establish N −1 ∞ −jωk nTs u0 (nTs − t)e dt. Y (ωk ) = g(t) n=0
0
Let u ˜0 be the periodically repeated version of u0 such that u ˜0 (t + T ) = u0 (t). Hence, we obtain N −1 ∞ −jωk nTs dt u ˜0 (nTs − t)I(t≤nTs ) e Y (ωk ) = g(t) n=0
0
Fig. 16. Signal fading with overlap: (black) stationary output spectrum; (light gray) corrected output spectrum; and (dark gray) uncorrected output spectrum.
2) Signal Fading: The automatic correction algorithm has been used on the data in Fig. 13 with an overlap percentage of 80%. Fig. 16 represents the uncorrected output spectrum in dark gray, the corrected output spectrum in light gray, and the stationary output spectrum in black. After correction, the output spectrum is 2 dB closer to the stationary output spectrum compared with the uncorrected spectrum. This difference is small. However, note that the considered case of fading is quite difficult due to the fact that all periods are contaminated by nonstationarities. In the previous case of fallout, at least one period contained no nonstationarities.
where I{A} = 1 if condition A is satisfied and I{A} = 0 otherwise. Let m be the index such that (m − 1)Ts < t ≤ mTs , we find that N −1 ∞ −jωk nTs g(t) u ˜0 (nTs − t)e dt Y (ωk ) = ∞ =
A PPENDIX A P ROOF OF THE L EAKAGE E XPRESSION D UE TO S YSTEM T RANSIENTS
u ˜0 (nTs )e
e−jωk t dt
n=m
where the periodicity of u ˜0 was used to establish the last equality. Finally, we compute N −1
u ˜0 (nTs )e−jωk nTs = U0 (ωk ) ∗ Dt (ωk )
where N −1 1 −jωk nTs D(t)(ωk ) = e N n=m
=−
1 1 − e−jωk mTs N 1 − e−jωk kTs
1 −j ωk (m−1)Ts sin ω2k mTs 2 . =− e N sin ω2k Ts
A PPENDIX B P ROOF OF THE L EAKAGE E XPRESSION D UE TO S ENSOR FAILURE We compute the DFT of the measured signal y(nTs )
We compute N −1 n=0 N −1
N −1 n=0
Y (ωk ) =
y(nTs )e−jωk nTs
0
∞ 0
N −1
y(nTs )e−jωk nTs
n=0
nT s g(t)u0 (nTs − t)dt e−jωk nTs
n=0
=
−jωk nTs
n=m
The presented automatic correction algorithm allows compensating all kinds of nonstationary effects present in measured data. Compared to existing techniques, the presented algorithm is fully automated without need for user interaction and no parametric model of system and applied signals are needed. The performance of the algorithm has been proven on measurements performed on a wireless link communication system and on simulations. Even under severe nonstationary conditions, the algorithm is able to reduce systematic errors in the measured spectra.
=
g(t)
N −1
0
VIII. C ONCLUSION
Y (ωk ) =
n=m
0
g(t)u0 (nTs − t)dt e−jωk nTs
=
N −1
y0 (nTs )h(nTs )e−jωk nTs
n=0
where h(t) = 0 for t ∈ [nTs , nTs ] and 1 elsewhere. As a result, the last equality further simplifies to ˜ k) Y (ωk ) = Y0 (ωk ) ∗ D(ω
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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 61, NO. 8, AUGUST 2012
with n−1 N −1 1 −jωk nTs −jωk nTs ˜ D(ωk ) = e + e N n=0 n=n+1
e−jωk (n+1)Ts − e−jωk nTs 1 − e−jωk Ts
=
1 N
=
e−jωk (n+1)Ts N
=
e−j
ωk 2
(n+n)Ts
N
1 − e−jωk (n−n−1)Ts 1 − e−jωk Ts
sin (ωk (n − n − 1)Ts ) . sin(ωk Ts )
R EFERENCES [1] L. Ljung, System Identification: Theory for the User. Engelwood Cliffs, NJ: Prentice-Hall, 1987. [2] J. Douce and L. Balmer, “Transient effects in spectrum estimation,” Proc. Inst. Elect. Eng. D, Control Theory Appl., vol. 132, no. 1, pp. 25–29, Jan. 1985. [3] B. Porat, Digital Signal Processing of Random Signals: Theory and Methods. Engelwood Cliffs, NJ: Prentice-Hall, 1994. [4] P. M. T. Broersen, Automatic Autocorrelation and Spectral Analysis. London, U.K.: Springer-Verlag, 2006. [5] T. Söderström, “Errors-in-variables methods in system identification,” Automatica, vol. 43, no. 6, pp. 939–958, Jun. 2007. [6] J. F. Callejon, A. R. Bretones, and R. G. Martin, “On the application of parametric models to the transient analysis of resonant and multiband antennas,” IEEE Trans. Antennas Propag., vol. 46, no. 3, pp. 312–317, Mar. 1998. [7] R. Pintelon, J. Schoukens, G. Vandersteen, and K. Barbé, “Estimation of nonparametric noise and FRF models for multivariable systems. Part I: Theory,” Mech. Syst. Signal Process., vol. 24, no. 3, pp. 573–595, Apr. 2010. [8] R. Pintelon and J. Schoukens, “Identification of continuous-time systems with missing data,” IEEE Trans. Instrum. Meas., vol. 48, no. 3, pp. 736–740, Jun. 1999. [9] D. Brillinger, Time Series: Data Analysis and Theory. New York: McGraw-Hill, 1981. [10] K. Barbé, R. Pintelon, and G. Vandersteen, “Finite record effects of the errors-in-variables estimator for linear dynamic systems,” IEEE Trans. Instrum. Meas., vol. 60, no. 2, pp. 642–654, Feb. 2011. [11] P. D. Welch, “The use of fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms,” IEEE Trans. Audio Electroacoust., vol. AU-15, no. 2, pp. 70–73, Jun. 1967. [12] K. Barbé, R. Pintelon, and J. Schoukens, “Welch method revisited: A nonparametric power spectrum estimator via circular overlap,” IEEE Trans. Signal Process., vol. 58, no. 2, pp. 553–565, Feb. 2010. [13] W. Rudin, Functional Analysis. New York: McGraw-Hill, 1991. [14] W. Kaplan, Advanced Calculus. Reading, MA: Addison-Wesley, 2002. [15] R. Pintelon, J. Schoukens, and G. Vandersteen, “Frequency domain system, identification using arbitrary signals,” IEEE Trans. Autom. Control, vol. 42, no. 12, pp. 1717–1720, Dec. 1997. [16] K. Barbé, J. Schoukens, and R. Pintelon, “Frequency-domain, errorsin-variables estimation of linear dynamic systems using data from overlapping subrecords,” IEEE Trans. Instrum. Meas., vol. 57, no. 8, pp. 1529–1536, Jan. 2008. [17] N. L. Johnson and S. Kotz, Continuous Univariate Distributions. Boston, MA: Houghton Mifflin, 1970. [18] R. F. Engle, “Wald, likelihood ratio, and Lagrange multiplier tests in econometrics,” in Handbook of Econometrics. II, M. D. Intriligator and Z. Griliches, Eds. New York: Elsevier, 1983, pp. 796–801.
Kurt Barbé received the M.S. degree in mathematics (option statistics) and the Ph.D. degree in electrical engineering from Vrije Universiteit Brussel (VUB), Brussels, Belgium, in 2005 and 2009, respectively. Currently, he is an Assistant Professor in the Department of Fundamental Electricity and Instrumentation (ELEC), VUB. Furthermore, he is a Postdoctoral Research Fellow with the Flemish Research Foundation (FWO). His main interests are in the field of system identification, time series analysis, and signal processing for biomedical applications. Dr. Barbé has served as an Associate Editor of the IEEE T RANSACTIONS ON I NSTRUMENTATION AND M EASUREMENT since 2010. He was the recipient of the 2011 Outstanding Young Engineer Award from the IEEE Instrumentation and Measurement Society.
Wendy Van Moer received the Engineer and Ph.D. degrees in applied sciences from Vrije Universiteit Brussel (VUB), Brussels, Belgium, in 1997 and 2001, respectively. She is currently an Associate Professor in the Department of Fundamental Electricity and Instrumentation (ELEC), VUB. She is a Visiting Professor of the RF measurement technology of the university of Gavle, Gavle, Sweden. Her main research interests are nonlinear measurement and modeling techniques for medical and high-frequency applications. Since 2007, Dr. Van Moer has been an Associate Editor of the IEEE T RANSACTIONS ON I NSTRUMENTATION AND M EASUREMENT, and in 2010, she became an Associate Editor of the IEEE T RANSACTIONS ON M ICROWAVE T HEORY AND T ECHNIQUES. She was the recipient of the 2006 Outstanding Young Engineer Award from the IEEE Instrumentation and Measurement.
Lieve Lauwers received the M.S. and Ph.D. degrees in electrical engineering from Vrije Universiteit Brussel (VUB), Brussels, Belgium, in 2005 and 2011, respectively. Currently, she is a Postdoctoral Researcher in the Department of Fundamental Electricity and Instrumentation (ELEC), VUB. In October 2011, she became a Postdoctoral Fellow with the Flemish Research Foundation (FWO) to investigate novel fMRI signal detection methods. Her research interests are in the field of nonlinear system identification and signal analysis for biomedical applications.
Niclas Björsell was born in Falun, Sweden, in 1964. He received the B.Sc. degree in electrical engineering and the Lic.Ph. degree in automatic control from Uppsala University, Uppsala, Sweden, in 1994 and 1998, respectively, and the Ph.D. degree in telecommunication from the Royal Institute of Technology, Stockholm, Sweden, in 2007. He has several years of experience on research and development (R&D) projects that fostered collaborations between the industry and the academy. For more than 15 years, he has held positions in the academy and in the industry, and he has worked as a Project Manager for some of the R&D projects. Currently, he is the Head of the Center for RF Measurement Technology, University of Gävle, Gävle, Sweden. He has published more than 40 papers in peer-reviewed journals and conferences, and his research interests include radio-frequency measurement technology and analog-to-digital conversion. He is involved in the research projects QUASAR and MEMON. Dr. Björsell is a Voting Member of the IEEE Instrumentation and Measurement TC-10.
