Stochastic Monitoring and Testing of Digital LTI Filters

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1 Stochastic Monitoring and Testing of Digital LTI Filters CHRISTOFOROS N. HADJICOSTIS Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign 148 C&SRL, 1308 West Main Str., Urbana, IL U.S.A Abstract: - The increasing use of digital signal processing (DSP) architectures in a variety of digital devices has led to a pressing need for methodologies that can test or monitor the correctness of their functionality in a cost-effective manner. In this paper, we introduce an approach for testing or monitoring a digital linear time-invariant (LTI) filter based on perturbations of the statistical properties of its output. More specifically, we assume that the input to the filter can be modeled as a wide-sense stationary (WSS) random process, and we apply stochastic techniques to analyze how faults will influence the mean or the average power of the output. Our analysis describes how to estimate the quantities of interest using a small amount of additional hardware and how to chose the different parameters in order to achieve a desirable probability of fault detection or a desirable probability of false alarm. We also discuss how this approach relates to previous methods for concurrent testing of digital LTI filters. Key-Words: - Testing, monitoring, stochastic, digital signal processing systems 1 Introduction Concurrent testing and monitoring techniques traditionally rely on an external mechanism that is designed to check whether a given system operates as expected. Concurrent testing schemes assume that the external mechanism has access to the system input, whereas concurrent monitoring schemes do not require access to the input. Both monitoring and testing schemes are performed on-line and impose an additional hardware or computational overhead on the system. Depending on the objective, monitoring/testing strategies may focus simply on detecting the existence of faults (such as quality testing of digital circuits [1], or build-in self-test mechanisms, [2]), or they may also perform fault identification. The latter is essential in fault-tolerant systems where an external error-correcting mechanism needs to first perform error detection and identification, and then initiate appropriate corrective action, [3, 4]. The traditional way of coping with faults is to use modular redundancy schemes. For example, by duplicating the original system (double modular redundancy), we can perform the desired computation in two separate modules. Any inconsistencies between the outputs of the two system replicas will then indicate the presence of a fault. Full duplication is usually sufficient to capture most of the faults and the challenge in concurrent testing/monitoring is to achieve high fault coverage in a cost-effective manner, [5]. Parity check and algorithm-based fault tolerance schemes perform a coarser ( parity ) operation using separate hardware alongside the original computation. By comparing the result of the parity hardware against the result of the original system, one is able to concurrently detect faults, [2, 6, 7]. The goal in such schemes is to come up with an invariant property of the system that is amenable to fault detection and that can be tested using a small amount of additional hardware/computation. The re-

2 duction of hardware in most parity schemes comes at the cost of reduced fault coverage; more specifically, the faults in the part of the original system that is orthogonal to the parity operation are undetectable (see, for example, the discussions in [2, 7]). As discussed in [5], invariant-based concurrent testing of a digital LTI filter faces two basic challenges: 1) identifying an appropriate invariant property and, (ii) ensuring that several practical issues (such as the inaccuracies of finite precision arithmetic) are taken into account to reduce false alarms while providing an acceptable detection rate. This has motivated the study of tolerance-based invariant methods for concurrent test of digital LTI filters, [5, 8]. These methods introduce latency in fault detection but avoid extensive additional hardware. The authors in [5] used an invariant based on the DC gain of a digital LTI filter and discussed digital design approaches that ensure the monotonicity of every possible fault (so that faults result in accumulative errors that eventually exceed the tolerance chosen for the invariant under consideration). In this paper, we discuss invariants that are based on the statistical properties of the output and input of the filter. Our approach is applicable in cases where the input can be modeled as a wide-sense stationary (WSS) random process. If the mean and autocovariance of the input random process are known, then the invariants can be tested without knowing exactly what the input is, resulting in a concurrent monitoring scheme. This paper is organized as follows: in Section 2 we provide some background on the theory of WSS processes through LTI filters; in Section 3 we analyze how the mean and autocovariance of the output are perturbed under a fault in the implementation of a digital LTI filter; in Section 4 we discuss how to estimate the quantities of interest in order to check the invariant. We conclude in Section 5 with a summary and some discussion on future work. 2 Background on WSS Random Processes Through LTI Filters Throughout this paper we assume that the input x[n] to a given digital LTI filter with impulse response h[n] can be modeled as a WSS random process X[n], i.e., a random process that satisfies the following: E(X[n]) = µ x, E(X[n + k]x[n]) = R xx [k]. The mean E(X[n]) of the input process is constant (independent of n) and its autocorrelation E(X[n + k]x[n]) is only a function of k. Equivalently, one can require the mean of the random process to be constant and its autocovariance, defined as E((X[n + k] µ x )(X[n] µ x )) = C xx [k], to only be a function of k. Note that C xx [k] = R xx [k] µ 2 x. Given such an input, the output y[n] of the filter is a random process that is also WSS so that µ y E(Y [n]) = µ x H(0), (1) R yy [k] E(Y [n + k]y [n]) = R hh [k] R xx [k], (2) where H(0) = + h[j] is the DC gain of the filter, R hh [k] = h[k] h[ k] is the deterministic autocorrelation of the impulse response (symbol denotes linear convolution), [9, 10]. The only requirement is that h[n] corresponds to a stable system. Example 2.1 Suppose that the input to a digital LTI filter is a white WSS random process, i.e., a WSS random process X[n] with mean µ x and autocovariance C xx [k] = p 2 δ[k]. Then, the output process Y [n] will satisfy µ y = µ x H(0), (3) R yy [k] = µ 2 y + p 2 R hh [k]. (4) In particular, the average power E((Y [n]) 2 ) of the output process will be E((Y [n]) 2 ) = R yy [0] = µ 2 y + p 2 R hh [0] = µ 2 y + p 2 + h 2 [j].

3 3 Perturbation Analysis Consider a situation where a single fault (that affects addition or multiplication in the hardware implementation of a digital LTI filter) causes the impulse response of the filter to be different in one coefficient. Under a permanent failure, this corrupted impulse response can be written as h f [n] = h[n] + cδ[n n 0 ] where h[n] is the fault-free impulse response, c is a constant and n 0 is an integer. Given a finite impulse response (FIR) filter implemented using fixed point arithmetic, this faulty h f [n] could be used to indicate a single stuck-at fault in the adder or multiplier of the n 0 th coefficient; the constant c would be of the form c = ±2 l, where l is an integer indicating the position of the bit effected by the fault. Given the WSS random process X[n] as an input, the output of this faulty filter will be a random process Y [n] whose mean and autocorrelation will be µ y = µ x + h f [j] = µ x (c + H(0)) = µ y + cµ x, (5) R y y [k] = h f [k] h f [ k] R xx [k] = (h[k] + cδ[k n 0 ]) (h[ k] + cδ[k + n 0 ]) R xx [k] = ( R hh [k] + c 2 δ[k] + ch[k + n 0 ]+ +ch[ k n 0 ]) R xx [k] = R yy [k] + ( c 2 δ[k] + ch[k + n 0 ]+ +ch[ k n 0 ]) R xx [k]. (6) The above expressions relate the mean and autocorrelation of the output Y [n] of a faulty digital LTI filter to the mean and autocorrelation of the output Y [n] of a fault-free digital filter. Essentially, they describe how the statistical invariants of Eqs. (1) and (2) will be affected when a fault is present in the implementation of a digital LTI filter. Therefore, they can be used to perform stochastic testing or monitoring. Example 3.1 Suppose that the digital LTI filter in Example 2.1 is a causal FIR filter with M coefficients and impulse response h[n] = M 1 h j δ[n j], where h j are the filter coefficients. Eqs. (3) and (4) then simplify to M 1 µ y = µ x h j, M 1 E((Y [n]) 2 ) = µ 2 y + p 2 h 2 j. If the filter implementation uses fixed point arithmetic with coefficients in the form 1.7 (one bit for the sign and seven bits for the fraction, [5]), then a faulty filter h f [n] could be given by h f [n] = h[n] ± 2 l δ[n n 0 ], where l { 7, 6,..., 1, 0} and n 0 {0, 1, 2,..., (M 1)}. In such case, the output process Y [n] will have mean and average power given by µ y = µ y ± 2 l µ x, E((Y [n]) 2 ) = E((Y [n]) 2 ) + (see Eqs. (5) and (6)). +(2 2l ± 2 l (h n0 + h n0 ))p 2 Note that the discussion in this example applies to any white WSS random process, including processes for which samples are independent, identically distributed (i.i.d) random variables (with mean µ x and variance p 2 ). Due to space limitations, we only analyze a stochastic monitoring scheme that is based on perturbations in the mean of the output random process. Another possibility, that requires more complicated analysis (and also the fourth order moments of the input process X[n] to be stationary) is to analyze perturbations in the average power.

4 4 Statistical Analysis 4.1 Estimating the Mean of the Input and Output In order to study perturbations using the mean of the input random process X[n] and the output random process Y [n], we have to estimate their means from available data. If the input random process is statistically known, then µ x is known and does not have to be estimated (so that the resulting scheme is actually a concurrent monitoring scheme). If not, we would have to form estimates for both µ x and µ y. To estimate the means we can use the following estimators: ˆµ x = 1 N x[j], ˆµ y = 1 N y[j], i.e., we time-average samples of the input and output over a window of length N. These estimators are unbiased, i.e., E(ˆµ x ) = µ x and E(ˆµ y ) = µ y. An indication of how good these estimators are can be obtained by looking at their variances. These variances depend on the autocovariance C xx [k] of the input random process and the impulse response h[n] of the digital LTI filter; more specifically, Var(ˆµ x ) = E(ˆµ 2 x) µ 2 x = µ 2 x + 1 = µ 2 x + 1 = 1 i= () i=0 i= () R xx [j i] (N i )R xx [i] (N i )C xx [i]. (7) This variance goes to zero for large N if the autocovariance C xx [k] decays to zero for large k. Similarly, the estimator ˆµ y has variance Var(ˆµ y ) = 1 i= () (N i )C yy [i]. (8) Example 4.1 In this example we consider again the setup of Example 3.1 with the additional assumption the input random process is Gaussian, [9, 10]. Samples of a Gaussian random process are jointly Gaussian random variables. In this particular case, since X[n] is white, samples of the process will be i.i.d. Gaussian random variables (with mean µ x and variance p 2 ). Given the above setup, the variance (σˆµx ) 2 of the estimator ˆµ x can be calculated from Eq. (7) as (σˆµx ) 2 = 1 N p2 = p2 N. Furthermore, ˆµ x is a linear combination of independent Gaussian random variables, so it will be a Gaussian random variable (with mean µ x and variance p2 N ). Similarly, the estimator ˆµ y will be a Gaussian random variable with mean µ y and variance ( ) (σˆµy ) 2 = p2 R hh [0] + 2 (N i)r hh [i]. i=0 (9) Note that the estimators discussed in this example can be implemented on-line using very simple hardware: all that needs to be done is keep track of an accumulated sum of x[n] (for ˆµ x ) and the accumulated sum of y[n] (for ˆµ y ). 4.2 Choosing the Threshold We know that under fault-free conditions, the mean µ y has to be equal to H(0)µ x, where H(0) = + h[j]. One approach for detecting failures would be to choose a certain tolerance T and declare a fault whenever our estimates ˆµ y and ˆµ x do not satisfy H(0)ˆµ x T < ˆµ y < H(0)ˆµ x + T. The value of T allows us to adjust the probability of false alarm or (equivalently) the probability of a miss, [9, 10]. The probability of a false alarm is the conditional probability that no faults are present but the rule indicates that a fault is present; the probability of a miss is the conditional probability that a fault is present but the rule indicates that no fault is present.

5 Example 4.2 Continuing the previous example (and assuming that µ x is known) we calculate the probability of a false alarm and the probability of a miss in terms of the tolerance parameter T. The probability of a false alarm is the probability that, given that no faults are present, ˆµ y evaluates outside the interval [H(0)µ x T, H(0)µ x + T ]. Since ˆµ y is a Gaussian random variable (with mean H(0)µ x and variance (σˆµy ) 2 as in Eq. (9)), the probability of a false alarm is Pr[False Alarm] = Q( T σˆµy ) + (1 Q( T σˆµy )) = 2Q( T σˆµy ), where the Q function is defined as Q(x) = + x 1 2π e x2 2 dx. Clearly, the larger T, the smaller the probability of false alarm. On the flip side, however, the probability of a miss, also increases with T. More specifically, if a fault results in a faulty impulse response h f [n] = h[n] + cδ[n n 0 ] (where c = ±2 l for an appropriate integer l), then the probability of a miss will be given by Pr[Miss] = Q( T cµ x σ ˆµ ) Q( T cµ x y σ ˆµ ), y where (σ ˆµ y ) 2 is again given by Eq. (9) but with h f [n] used instead of h[n]. The reason is that ˆµ y is a linear combination of jointly Gaussian random variables; therefore, ˆµ y is also a Gaussian random variable, with mean (cµ x + H(0)µ x ) and variance (σ ˆµ y ) 2. 5 Conclusions and Future Work We have introduced a stochastic approach for monitoring and testing permanent faults in digital LTI filters. Our approach is based on analyzing faultinduced perturbations in the mean of the input and output random processes. The method allows us to choose parameters of the testing/monitoring scheme in order to trade-off the probability of detection with the probability of false alarm. Our approach is essentially a stochastic version of the scheme described in [5] (because our estimates ˆµ x and ˆµ y are exactly what is used in the invariant analysis there) and only requires half of the hardware if the mean of the input random process is known. The examples presented here analyzed explicitly the case of inputs that are i.i.d. Gaussian random variables; future analysis should obtain a characterization of the probabilities of false alarm and miss for other input random process. Also, we intend to pursue a similar analysis using perturbations in the average power of the output of the digital LTI filter. Techniques that use the cross-correlation between the input and output should also be investigated. 6 Acknowledgments This research has been supported in part by NSF award CAREER References: [1] M. Abramovici, M. Breuer, and D. Friedman, Digital Systems Testing and Testable Design, Computer Science Press, [2] J. Wakerly, Error Detecting Codes, Self- Checking Circuits and Applications, Elsevier Science, Amsterdam/New York, [3] D.P. Siewiorek and R.S. Swarz, Reliable Computer Systems: Design and Evaluation, A.K. Peters, [4] B.W. Johnson, Design and Analysis of Fault- Tolerant Digital Systems, Addison-Wesley, Reading, Massachusetts, [5] I. Bayraktaroglu and A. Orailoglu, Cost effective digital filter design for concurrent test, in Proceedings of ICASSP 2000, the IEEE Int. Conf. on Acoustics, Speech and Signal Processing, [6] T. R. N. Rao, Error Coding for Arithmetic Processors, Academic Press, New York, [7] C. N. Hadjicostis, Coding Approaches to Fault Tolerance in Dynamic Systems, Ph.D. thesis,

6 EECS Department, Massachusetts Institute of Technology, Cambridge, Massachusetts, [8] A. L. Narasimha R. and P. Banerjee, Algorithm-based fault detection for signal processing applications, IEEE Transactions on Computers, vol. 39, pp , October [9] C. W. Helstrom, Probability and Stochastic Processes for Engineers, Macmillan Publishing Company, New York, [10] K. S. Shanmugan and A. M. Breipohl, Random Signals: Detection, Estimation and Data Analysis, Wiley, New York, 1988.

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