Input-output Cross-correlation Function Research Articles

Tuning of High-Speed Telecommunication Filters, via I/O Cross-Correlation Evaluation

Continuous-time (CT) filters are widely used in signal processing but, owing to the large spread in the process parameters which determine their time constants, they require a tuning system to align their frequency characteristics. Several techniques have already been proposed in the literature and are commonly employed, but usually they require either a doubling of the filter circuitry or some complex self-calibration scheme, resulting in sensible area overhead or high computational complexity. Here, we present a novel tuning approach which can be applied to both master-slave and self-calibration arrangements and is based on the evaluation of a few samples of the input-output cross-correlation function of the filter, when the input signal is a finite length sequence of pseudorandom pulses. The key advantages of the proposed technique are basically the ease of generation of the input bit-stream and the simplicity of the circuitry required to sample the filter output and to perform the cross-correlation operation. As a consequence, the area overhead associated to the tuning circuitry is kept very small. Some simulation results of the application to the tuning of a low-Q telecommunication filter are given and an extension to the tuning of high-Q active filters is also explored in order to assess the effectiveness and robustness of the proposed technique

Linear approximation of bilinear Processes

The linear approximation for a first-order bilinear process perturbed using pseudorandom binary and ternary signals is studied, in terms of the related linear dynamics consisting of the underlying linear dynamics and bias caused by nonlinear distortions. The related time constant is derived from the input-output cross-correlation function. The analysis is extended by simulation to the case of multilevel input signals. Simulation results are also given for a second-order bilinear process, as theoretical results are not available.

Application of multi-level signals to the identification of direction-dependent processes

Theoretical results for the output and the input–output crosscorrelation function are derived for first order direction-dependent processes perturbed using pseudo-random maximum length ternary (MLT) signals with unsymmetrical signal levels. The analytical results are valid if the process output is either always increasing or always decreasing when the input is at its median level of zero. It is shown that, although this is not normally the case for the whole of the signal period, good approximations to the analytical results are obtained when it is the case for most of the signal period, as can be achieved with a suitable choice of signal levels. MLT signals with unequal spacing between signal levels can also be used to minimise the nonlinear distortion. This is equivalent to compensating for the direction-dependent behaviour of the system by preceding it with a static nonlinearity. Based on the theoretical and simulation results obtained, a novel technique is proposed to allow the best linear approximation of the process to be estimated from the ratio between the amount of time when the output increases to that when it decreases. The proposed method is shown to be applicable even when the number of signal levels is greater than three, and is less susceptible to the effects of noise than the method of correlation analysis and least squares.

Identification of direction-dependent processes using maximum length ternary signals

Analytical results have been obtained for first-order direction-dependent processes perturbed using maximum length ternary (MLT) signals. In particular, the process output and the input-output crosscorrelation function are derived, leading to a theoretical expression for the combined time constant of the process. The importance of doing this is that the direction-dependent behaviour can be detected, and the combined linear dynamics of the process can be estimated, from the input-output crosscorrelation function. In the above derivations, it is assumed that the process dynamics are faster in the upward direction, and that the clock-pulse interval is chosen such that the output always decreases when the input is at signal level zero. The first of these assumptions does not affect the terms in the crosscorrelation function because the input signal is inverse-repeat. The second assumption is shown to be valid during most of the signal period. The errors caused by the deviation from the ideal case are assessed, and are found to be small. Hence, the ideal case can be used as an effective benchmark against which results from an identification test as specified above can be measured. Advantages and disadvantages of using the MLT signal, instead of the maximum length binary or its inverse-repeat signal, are discussed. A simulation example is also shown to verify the obtained theoretical results.

Identification of processes with direction-dependent dynamics

The identification of systems which have different dynamics when the output is increasing compared with those when the output is decreasing is considered. The dynamics in each direction will be assumed to be linear. It is shown that when such systems are perturbed by pseudorandom binary signals based on maximum length sequences, there are coherent patterns in the input–output crosscorrelation function, but there is no coherent pattern in either the gain response or the phase response in the frequency domain. The crosscorrelation terms are developed in detail for a process with first-order dynamics in the two directions, perturbed by a maximum length binary (MLB) signal and the results are confirmed by simulation. Similar theoretical expressions and simulation results are given for such a process perturbed by an inverse-repeat signal based on an MLB signal. The crosscorrelation function patterns obtained using an MLB signal are not present when other classes of pseudorandom binary signals are used. The linear dynamics for the process perturbed by an MLB signal and its corresponding inverse-repeat MLB signal are estimated, and found to agree more closely with the theoretical value when the latter type of signal is used. The theory cannot be readily extended to processes with direction-dependent dynamics of higher order, but simulation results presented for such second-order processes show that the departure from linearity can still be detected from the crosscorrelation function when an MLB perturbation signal is used.

Identification of physiological systems: estimation of linear time-varying dynamics with non-white inputs and noisy outputs.

A new technique to identify linear time-varying systems from ensembles of input-output realisations is presented. First, a correlation-based least-squares method is derived. This method consists of solving, for each sampling time, a matrix equation involving estimates of the input autocorrelation and input-output cross-correlation functions computed from data across the ensemble. Then, the matrix inverse needed to solve this matrix equation is replaced with a pseudo-inverse. The model is thus constrained to describe only those components of the dynamics that can be reliably identified. Ignoring 'unidentifiable' components has virtually no adverse effect on the predicted outputs. Simulation results demonstrate that the pseudoinverse technique yields more reliable estimates of the dynamics than a previously proposed least-squares technique when the inputs are coloured and the output signal-to-noise ratio (SNR) is low. With the input spectrum flat up to approximately 10% of the sampling rate and an output SNR of 5dB, the mean variance accounted for (VAF) between the true instantaneous impulse response functions (IRFs) and the instantaneous IRFs estimated with the least-squares technique was 0.2%. In contrast, the mean VAF between the true instantaneous IRFs and the instantaneous IRFs estimated with the pseudoinverse technique was 89.0%.

The mechanical waveform of the basilar membrane. III. Intensity effects.

Mechanical responses in the basal turn of the guinea-pig cochlea were measured with broad-band noise stimuli and expressed as input-output cross-correlation functions. The experiments were performed over the full range of stimulus intensities in order to try to understand the influence of cochlear nonlinearity on frequency selectivity, tuning, signal compression and the impulse response. The results are interpreted within the framework of a nonlinear, locally active, three-dimensional model of the cochlea. The data have been subjected to inverse analysis in order to recover the basilar-membrane (BM) impedance, a parameter function that, when inserted into the (linearized version of that) model, produces a model response that is similar to the measured response. This paper reports details about intensity effects for noise stimulation, in particular, the way the BM impedance varies with stimulus intensity. In terms of the underlying cochlear model, the decrease of the "activity component" in the BM impedance with increasing stimulus level is attributed to saturation of transduction in the outer hair cells. In the present paper this property is brought into a quantitative form. According to the theory [the EQ-NL theorem, de Boer, Audit. Neurosci. 3, 377-388 (1997)], the BM impedance is composed of two components, both intrinsically independent of stimulus level. One is the passive impedance Zpass and the other one is the "extra" impedance Zextra. The latter impedance is to be multiplied by a real factor gamma (0 < or = gamma < or = 1) that depends on stimulus level. This concept about the composition of the BM impedance is termed the "two-component theory of the BM impedance." In this work both impedances are entirely derived from experimental data. The dependence of the factor gamma on stimulus level can be derived by using a unified form of the outer-hair-cell transducer function. From an individual experiment, the two functions Zpass and Zextra are determined, and an approximation (Zpass + gamma Zextra) to the BM impedance constructed. Next, the model response (the "resynthesized" response) corresponding to this "artificial" impedance is computed. The same procedure is executed for several stimulus-level values. For all levels, the results show a close correspondence with the original experimental data; this includes correct prediction of the compression of response amplitudes, the reduction of frequency selectivity, the shift in peak frequency and, most importantly, the preservation of timing in the impulse response. All these findings illustrate the predictive power of the underlying model.

The mechanical waveform of the basilar membrane. I. Frequency modulations ("glides") in impulse responses and cross-correlation functions.

The purpose of this investigation is to present evidence from experimental as well as model results on temporal variations of the frequency of oscillation in the basilar membrane's impulse response. Stimuli were either clicks leading to a direct estimate of the impulse response, or bands of pseudo-random noise (one or two octaves wide) which lead to an indirect estimate of the impulse response via a cross-correlation procedure. The noise bands were centered at the best frequency of the BM location under observation. Responses were obtained from the basal turn of the guinea-pig cochlea, from a location with a best frequency (for the weakest stimuli) between 17.0 and 18.5 kHz. Data acquisition was done with a sample frequency of 208 kHz. Input-output cross-correlation functions were found to share with impulse responses the property that the initial oscillations have a noticeably lower frequency than the later ones. During the impulse response the frequency of oscillation increases gradually. This increase occurs and continues to beyond the time that the oscillations reach the largest amplitude. This frequency variation is called a "glide." Using the "analytic signal" method the frequency of oscillation is found to increase continually throughout the duration of the main lobe of oscillation, even at the lowest tested stimulus intensities (about 20 dB SPL). At high stimulus intensity both the direct and indirect impulse response change their appearance drastically but the glide retains its basic form. In the case of the direct impulse response estimate the glide can be attributed to temporal variation of the degree of nonlinearity. For the indirect impulse response this is not true, because with a constant level noise stimulus there is no regular temporal variation of nonlinearity. In this case the glide should be interpreted as an intrinsic property of the cochlear system. From our and others' data the glide was found to exist over a topographic frequency range of best frequencies of at least from 1.76 to 18 kHz. Two examples of present-day models of the cochlea are discussed of which one is found to demonstrate the glide phenomenon in its response, and the other one does not.

Nonlinear system identification by m-pulse sequences: application to brainstem auditory evoked responses

The purpose of this paper is to introduce a method for characterizing the nonlinear behavior of the auditory system. The method uses an m-pulse sequence as the stimulus and employs a general nonlinear framework for the auditory system. Like Sutter's binary m-sequence approach, the m-pulse sequence approach is computationally efficient since calculation of the first-order input-output cross-correlation function is all that is necessary for obtaining the nonlinear characteristics of the system. The nonlinear system characteristics are reflected in pulse kernels in contrast to binary kernels associated with the binary m-sequence approach. By assuming the system under study is a third-order nonlinear system, binary and pulse kernels are shown to be related to Volterra kernels. The results suggest that the m-pulse sequence can be used to study the system nonlinear effects of varying the stimulus repetition rate more effectively than conventional methods. Preliminary physiological data obtained by applying m-pulse sequences to the brainstem auditory evoked response (BAER) clearly illustrates the feasibility of obtaining replicable evoked responses using this method.

Volterra Systems with Random Inputs: A Formalized Approach

The object of this work is to investigate the problem of Volterra systems driven by random inputs. General expressions for the second-order output statistics of a Volterra system are obtained and formalized. The input-output crosscorrelation and output autocorrelation functions are expressed in terms of higher order input autocorrelation functions. In the frequency domain, the cross-power and output power spectral densities are expressed in terms of higher order input power spectral densities. These expressions are used to obtain closed-form equations for the cross-power and output power spectral densities of a Volterra system driven by a real, stationary, zero-mean Gaussian input.

Cross-correlation function of a class η1 system

This paper investigates the cross-correlation function of a class η1 system with pseudorandom input signal. It is shown that the input-output cross-correlation function is proportional to the impulse function of the linear filter. The constant of proportionality is dependent upon the non-linearity and the input statistics. Extension to multivariate situation is then briefly discussed.

Fault diagnosis using time domain measurements

A voting technique is used to diagnose fault conditions down to component level in a feedback control system using only the input-output cross-correlation function measured at suitable time delays. These time delays are chosen using a new formula based on Bayes's theorem, which ranks the time delays in order of usefulness in fault diagnosis; it can be readily applied at the design stage, thus assisting the integration of system design and test functions. The fault conditions necessary to set up the scheme may be obtained by direct fault generation in an actual system, or by simulation of the system mathematical model. A learning approach to the design of a fault diagnosis scheme is described which makes full use of any available failure data, together with the ranking formula for time delay selection, in the creation of an optimum scheme.Results obtained on a complex electro-hydraulic servo are presented which show that the scheme works satisfactorily in the presence of measurement noise and parameter drift, two factors which often cause a breakdown in conventional pattern recognition techniques. The computational requirements are extremely modest, and are well within the capacity of present day mini-computers proposed for use in automatic test equipment. In many instances, the scheme is suitable for manual and partially automated test sets, and can be used for fault diagnosis of a wide range of circuits and systems.

Stability of systems with power-law non-linearities

It is shown that a sufficient condition for the asymptotic stability-in-the-large of an autonomous system containing a linear part with transfer function G( jω) and a non-linearity belonging to a class of power-law non-linearities with slope restriction [0, K] in cascade in a negative feedback loop is ReZ(jω)[G(jω) + 1 K ] ≥ 0 for all ω where the multiplier is given by, Z(jω) = 1 + αjω + Y(jω) − Y(−jω) with a real, y( t) = 0 for t < 0 and ∫ 0 ∞ |y(t)| dt < 1 2 c 2 , c 2 being a constant associated with the class of non-linearity. Any allowable multiplier can be converted to the above form and this form leads to lesser restrictions on the parameters in many cases. Criteria for the case of odd monotonic non-linearities and of linear gains are obtained as limiting cases of the criterion developed. A striking feature of the present result is that in the linear case it reduces to the necessary and sufficient conditions corresponding to the Nyquist criterion. An inequality of the type |R(T) − R(− T)| ≤ 2c 2R(0) where R( T) is the input-output cross-correlation function of the non-linearity, is used in deriving the results.

Input-Output cross-correlation functions for some memory-type nonlinear systems with Gaussian inputs

IN a previous paper1 the author described the application of the method of quasi-linearization (that of Booton2) to the determination of input-output cross-correlation functions for relay servomechanisms. Experimental verification of the validity of this analysis was presented in the form of analog-computer simulation results for several specific configurations. The present work is an extension of the earlier effort, presenting the results for simulated systems with memory-type nonlinearities. The theoretical analysis of the memory-type system is accomplished by combining quasi-linearization with a well-known analog-computer technique for generating hysteresis-type nonlinearities by placing nonmemory nonlinearities within closed subloops.