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TIMA Lab. Research Reports
TIMA Laboratory, 46 avenue Flix Viallet, 38000 Grenoble France
ON-CHIP TESTING OF LINEAR TIME INVARIANT SYSTEMS USING MAXIMUM-LENGTH SEQUENCES
Libor Rufer, Emmanuel Simeu and Salvador Mir
TIMA Laboratory 46 Av. Flix Viallet
38031 Grenoble FRANCE
Abstract: With the rapid development of system-on-chip (SoC) applications, containing digital and analogue parts, the need for fast and reliable on-chip testing methods has become obvious. We present a method for a fast and accurate broadband determination of the behaviour of analogue and mixed-signal circuits. This technique is based on impulse response (IR) evaluation using pseudo-random MaximumLength Sequences (MLS). This approach provides a large dynamic range and is thus an optimal solution for measurements in noisy environments and for low-power test signals. We will show the algorithms of the MLS generation and of the impulse response calculation which can be easily implemented on-chip. Copyright 2003 IFAC Keywords: Test, system-on chip, maximum-length sequence, response measurement.
The development of reliable System-on-Chip (SoC) containing mixed-signal systems requires seeking methods for facilitating their test, in particular Built-In Self-Test (BIST) techniques. In the research of cost effective testing of analogue parts, the techniques based on digital signals are often preferred since most of the circuitry in mixed signal systems is digital and most of the test equipment is devoted to them. A number of well-established approaches for the test of analogue functions using digital signals exist. These include impulse response and step response testing as an example.
Knowledge of the system impulse response provides enough information for system functional evaluation as well as for extraction of its parameters. A straightforward way to obtain the impulse response consists in the injection of a very short pulse of high amplitude (ideally close to a Dirac delta function) to a system and in measuring its response. This technique has two drawbacks: first, in the input pulse generation, the amplitude is limited by the range of linearity of the system, and second, the system behaviour that does not correspond to a steady-state regime. The measurement of the input-output cross-correlation function of the system stimulated by a white noise is another way to obtain the impulse response. This procedure fails sometimes due to the
limits in the signal to noise (S/N) ratio. Another source of uncertainty, the stochastic nature of the test signal, needs to be solved by using a certain number of averages. For these reasons efforts have been made during the last decade to reduce the influences of background noise by choosing special test signals which can replace the stochastic properties in a theoretically correct way. As a result of this effort, the technique of a system impulse response measurement based on Maximum-Length Sequences (MLS) has been developed (Rife and Vanderkooy, 1989). MLSs are now well-established test signals in various fields, as in electroacoustic transducer testing or in building acoustics.
Binary MLSs are periodic two-level deterministic sequences of the length N = 2m 1, where m is an integer denoting the order of the sequence. A simple pseudorandom sequence can be generated by an arrangement of N bits shift register clocked at fixed frequency using an exclusive-OR gate to generate the feedback signal from the nth bit (0
The method is based on the fact that input-output cross-correlation function of a Linear Time Invariant (LTI) system provides the system impulse response when the input signal has a flat frequency spectrum. An estimation of the system impulse response can be obtained by using MLS as an input signal, because its spectrum is flat in frequency band that is defined by the sequence length and by the clock frequency. The basic idea is to apply a MLS to a linear system, sample the resulting response, and then cross-correlate this response with the original sequence. Since the original sequence is a known pseudo-random sequence, there exists an efficient and very fast way to calculate the cross-correlation function, called Fast Hadamard Transform (FHT). A benefit of FHT is that it requires only N log2N operations. Since the MLS is represented by +1 and -1, the FHT consists of additions and subtractions only. For an LTI system, one period of the signal is sufficient for a cross-correlation computation and no averaging is required. The averaging can still be applied to reduce the system noise. Since the sequence is deterministic, it can be repeated precisely. It is therefore possible to increase signal to noise ratio by a synchronous averaging of the response sequence. This procedure reduces the effective background noise level by 3 dB per doubling of the number of averages because the exactly repeated periods of the test signal add up in phase while the background noise is not correlated between the different periods and only its energy is summed.
The use of the MLS-based method can be twofold. Firstly, we can perform a functional evaluation of the measured system based on the impulse response or on the transfer function. Secondly, we can define a system signature based on the impulse response and by checking the measured signature against the expected one we can verify the correctness of a device without measuring the original performance parameters (Pan and Cheng, 1997).
2. MAXIMUM-LENGTH SEQUENCE
Maximum-length sequences (also called pseudo-random sequences, pseudo-noise sequences or m-sequences) are certain binary sequences of the length N = 2m 1 with m denoting the order of the sequence. These sequences have been known for a long time in areas such as range-finding, scrambling, fault detection, modulation, synchronizing, acoustic measurements, etc. (Rife and Vanderkooy, 1989; Davies, 1966; MacWilliams and Sloane, 1976). Due to their specific properties, they are predestined for special measurement techniques as transfer function or impulse response measurements, where an important gain in speed and in S/N ratio can be obtained.
To construct a MLS of a given length N, we need a primitive polynomial p(x) of a degree m. An example of such a polynomial is given by the following expression
The value of the autocorrelation function is always equal to 1 for zero shift and drops to -1/N for any other shift, repeating after each period of the sequence. The power spectrum of the MLS that is schematically shown in Fig. 2b is a discrete spectrum whose upper 3 dB roll-off frequency is about 0.45 fc. By adjusting the clock frequency, the broadband signal over a wide frequency range can be generated.
3. IMPULSE RESPONSE MEASUREMENT
The LTI system can be completely specified either by its transfer function or by its impulse response. Both functions can be obtained by MLS-based measurements. The impulse response of the LTI system can be obtained as a result of the cross-correlation between its input and output signals. In the case of the discrete input and output sequences, we obtain the cross-correlation xy(k), that is related to auto-correlation of the input, xx(k) by a convolution with the periodic impulse response h(k):
)k(h*)k()k( xxxy = (2)
An important property of any MLS is that its auto-correlation function is essentially an impulse that can be represented by the Dirac delta function. We can see from the relation (2) that in the case of MLS-based measurements, cross-correlating of the system input and output sequences gives the impulse response. The basic idea of the measurement method is then shown in Fig.3.
Fig. 3. Block diagram of an MLS - based
measurement. A MLS signal excites a device under the test (DUT). The system impulse response is obtained as a result of the cross-correlation of the input and output sequences. The cross-correlation operation in the case of a discrete sequence is defined by:
The Equation (3) can also be described in terms of matrix multiplication:
xy = (4)
xy and Y are vectors whose elements are xy and y from the Equation (3), and the matrix X contains the circularly delayed version of the input sequence x. Since the elements of X are all 1, only additions and subtractions are required to perform the matrix multiplication. Finding each element of the correlation vector xy requires N-1 additions. The total number of additions necessary for the resulting vector is N (N-1). When N is a large number, the
number of operations can be a limiting factor for the measurements. Rapid calculations of the cross-correlation can be based upon the algorithm called Fast Hadamard Transform. The flow graph of FHT is similar to the flow diagram of the Fast Fourier Transform (FFT) and the total number of additions required to evaluate an N-point FHT is N log2N.
4. IMPLEMENTATION OF THE METHOD
The implementation of the MLS-based measuring method can be considered off-chip or on-chip. In the first case, the device under the test is connected with an auxiliary unit that provides the test sequence generation and necessary signal processing. This approach can serve for development purposes and for the extraction of system parameters for example. In the second case, the on-chip approach corresponds to a built-in self-test (BIST) technique for which the necessary operations for the MLS generation and processing are done on chip.
4.1 Off-chip simulation and measurements
In the case of an off-chip measurement, we have considered the scheme shown i