Impulse Response Of Linear Systems Research Articles

The criterion for the existence of the weight of the orthogonality of equidistant elementary signals

To reduce the level of intersymbol interference and interchannel interference, it is proposed to form the transmitted signals using the basis functions obtained by shifting the impulse responses of linear systems by multiple time intervals. An algorithm for calculating the weight of the orthogonality of basis functions is proposed. A criterion for the existence of the indicated weight of orthogonality is formulated.

Zero Crossings, Overshoot and Initial Undershoot in the Step and Impulse Responses of Linear Systems

We consider functions in the time and the frequency domain, and show how the occurrence of real zeros in the frequency domain is related to the number of zero crossings, overshoot and initial undershoot in the time domain. This leads to simple proofs of known facts for linear time-invariant systems and their generalizations in various directions. In particular, we consider time-delay systems and differential systems of fractional order.

Linking regularization and low-rank approximation for impulse response modeling

In the last years, nonparametric linear dynamical systems modeling has regained attention in the system identification world. In particular, the application of regularization techniques that were already widely used in statistics and machine learning, has proven beneficial for the estimation of the impulse response of linear systems. The low-rank approximation of the impulse response obtained by the truncated singular value decomposition (SVD) also leads to reduced complexity estimates. In this paper, the link between regularization and SVD truncation for finite impulse response (FIR) model estimation is made explicit. The SVD truncation is reformulated as a regularization problem with a specific choice of the regularization matrix. Both approaches (regularization and SVD truncation) are applied on a FIR modeling example and compared with the classic prediction error method/maximum likelihood approach. The results show the advantage of these techniques for impulse response estimation.

Unified approach to the impulse response and green function in the circuit and field theory part II : multi–dimensional case

In the circuit theory the concept of the impulse response of a linear system due to its excitation by the Dirac delta function δ(t) together with the convolution principle is widely used and accepted. The rigorous theory of symbolic functions, sometimes called distributions, where also the delta function belongs, is rather abstract and requires subtle mathematical tools [1-4]. Nevertheless, the most people intuitively well understand the delta function as a derivative of the (Heaviside) unit step function 1(t) without too much mathematical rigor. In the previous part [5] the concept of the impulse response of linear systems was approached in a unified manner and generalized to the time-space phenomena in one dimension (transmission lines). Here the phenomena in more dimensions (static and dynamic electromagnetic fields) are treated. It is shown that many formulas in the field theory, which are often postulated in an inductive way as results of the experiments, and therefore appear as “deux ex machina” effects, can be mathematically deduced from a few starting equations.

Unified approach to the impulse response and green function in the circuit and field theory, part I: one–dimensional case

In the circuit theory the concept of the impulse response of a linear system due to its excitation by the Dirac delta function ƍ(t) together with the convolution principle is widely used and accepted. The rigorous theory of symbolic functions, sometimes called distributions, where also the delta function belongs, is rather abstract and requires subtle mathematical tools [1], [2], [3], [4]. Nevertheless, the most people intuitively well understand the delta function as a derivative of the (Heaviside) unit step function 1(t) without too much mathematical rigor. The concept of the impulse response of linear systems is here approached in a unified manner and generalized to the time-space phenomena in one dimension (transmission lines), as well as in a subsequent paper [5] to the phenomena in more dimensions (static and dynamic electromagnetic fields).

New approach to pulse propagation in nonlinear dispersive optical media

We develop an intuitive approach for studying propagation of optical pulses through nonlinear dispersive media. Our new approach is based on the impulse response of linear systems, but we extend the impulse response function using a self-consistent time-transformation approach so that it can be applied to nonlinear media as well. Numerical calculations based on our new approach show excellent agreement with the generalized nonlinear Schrodinger equation in the specific case of the Kerr nonlinearity in both the normal and anomalous dispersion regimes. An important feature of our approach is that it works directly with the electric field associated with an optical pulse and can be applied to pulses of arbitrary width. Numerical calculations performed using single-cycle optical pulses show that our results agree with those obtained with the finite-difference time-domain technique using considerably more computing resources.

An interactive impulse response extraction system

A cross-correlation method of measuring the impulse response of linear systems is presented. In theory, by using this method, the impulse response of any linear system can be obtained while the system is being used under normal operation with no significant distortion of the normal system output. A cross-correlation system was modeled and constructed using discrete components and simulation and experimental results are presented. Unfortunately, these results show that the extracted impulse response is severely distorted by the normal system input and the system noise. However, by averaging and interleaving methods, the impulse response of any linear system can be extracted with a high degree of accuracy.

The application of linearly swept frequency measurements

The theory of linearly swept frequency measurements has been studied [M. A. Poletti, J. Audio. Eng. Soc. 36(6) (1988)]. This theory shows that the Fourier transform of a linear system impulse response may be produced as a function of scaled time by convolving, multiplying, and reconvolving the impulse response with infinite duration, complex linear frequency-modulated (FM) signals. The first convolution is implemented by injecting a linear FM signal into the system. The multiplication may be viewed as a synchronous detector that removes the carrier from the output of the system. The third operation may be viewed as a quadratic phase filter that removes the skewing distortion to produce the true transfer function as a function of time. The application of the theory using finite real signals is discussed. It is shown that for a finite duration impulse response a suitably time-limited signal may be injected into the system which produces the response that would be produced with an infinite signal, and that the response to a complex input signal may be produced by sequentially exciting the system with the real and imaginary parts of the complex signal. The second and third complex signals may also be successfully applied to produce the true frequency and phase response of the system without any errors due to the chosen sweep rate.

Two nonparametric methods for identifying the impulse response of linear systems

This paper addresses the problem of impulse response identification using nonparametric methods. Although the techniques developed herein apply to the truncated, untruncated, and circulant models, we focus on the truncated model which is useful in certain applications. Two methods of impulse response identification will be presented. The first is based on the minimization of the C L statistic, which is an estimate of the mean-square prediction error; the second is a Bayesian. approach, For both of these methods, we consider the effects of using both the identity matrix and the Laplacian matrix as weights on the energy in the impulse response. In addition, we present a method for estimating the effective length of the impulse response. Estimating the length is particularly important in the truncated case. Finally, we develop a method for estimating the noise variance at the output which is needed in the approach involving the C L statistic. Often, prior information on the noise variance is not available, and a good estimate is crucial to the success of estimating the impulse response with a nonparametric technique.

Statistical Technique for Radar Amplitude Calibration

A statistical technique for amplitude calibration of radar systems is presented. As distinct from the input-output crosscorrelation measurement of a linear system impulse response [1], this technique measures the amplitude transfer function of a memoryless, nonlinear system. A generalized theory is developed and calibration accuracy bounds are derived. Used in the calibration of a modern pulsed radar, the technique is compared with conventional reference pulse calibration. The relative merits of the deterministic and statistical approaches are compared.

Simultaneous analysis of two or more samples on the same chromatographic column

Pseudo random binary and ternary sequences (PRBS and PRTS) have already been used to identify impulse responses of linear systems—in particular, we have shown [1,2], how decorrelation techniques allow the chromatogram of a given sample to be obtained by injecting the product following a PRBS or PRTS law. In this paper an additional difficulty is overcome by injecting in parallel two different samples onto one column, then obtaining simultaneously but separately the two chromatograms by use of numerical decorrelation. We present the theoretical justification and the result of numerical simulation in the case of a pseudo random ternary sequence injection law. A successful experiment on an actual chromatograph using a pseudo random binary sequence injection law is presented.

Power spectral representation of nonstationary random processes defined over semi-infinite intervals

A new sequence of locally time-averaged power spectra is defined that describes the time evolution of the frequency content of nonstationary random processes and linear system impulse response functions. The exact input-response relations for these spectral sequences are shown to be finite discrete convolutions of the input and system spectral sequences. Expressions for the coefficients of a Laguerre function expansion of the time-varying mean square response are derived from the response spectral sequence, and the time resolution and convergence of the expansion are discussed. Simple formulas are derived for generation of the spectra from recorded sample functions using Fourier transform computational algorithms. The relationship of the spectra to the Laplace transform is also developed. Physical interpretation of the spectra is discussed in detail and related to the uncertainty principle for Fourier transforms. Subject Classification: [43]45.40; [43]60.20; [43]20.50; [43]30.30; [43]40.35.

Analysis of afferent responses from isolated semicircular canal of the guitarfish using rotational acceleration white-noise inputs. I. Correlation of response dynamics with receptor innervation.

The small-signal linear characteristics of afferent responses from the isolated semicircular canal were described by the use of white-noise rotational acceleration inputs. The results, based on cross-correlation analysis, showed a striking and systematic variation in linear system impulse response characteristics from afferents which innervated different regions of the receptor. Afferents from centrally located nerve bundles innervating the crest region of the crista exhibited an initial maximum response amplitude followed by a rapid decay. In contrast, afferents from extreme rostral and caudal nerve bundles innervating the crista slopes exhibited an initial rise up to a low-amplitude maximum followed by a slower decay. These results imply that the afferents innervating a single canal do not merely carry redundant information concerning current head acceleration, but could be considered an ensemble of specific classes of filters that are tuned individually to specific classes of head movements. On the basis of these considerations, a new hypothesis of matched filter detection was proposed as relevant to information processing and dynamic control in central vestibular pathways.

Practical Determination of the Linear System Impulse Response of a Room

A review is made of the possible methods to obtain the impulse response of a room. Included in this review are shock stimulus, swept frequency analysis, and correlation technique using a white noise input. The latter technique is readily adaptable to computer processing (either hard wired or soft wired). The difficulties associated with each method are put forward as are the advantages. Emphasis is placed on the correlation technique because of the advantages that can be gained from further processing of the gathered data. It is pointed out that, with the correlation technique used, although the effects of the system up to the source input (say a speaker) are totally removed, the speaker modification to the actual room impulse response is a difficult quantity to deal with. Because of the inherent rise time in any impulse response, the speaker impulse response is not minimum delay (or minimum phase) and hence an exact convolution inverse operator is not stable. Methods to deal with this are discussed and practical results from a Wiener filter approach are presented. [Supported in part by two grants from NASA.]

Evaluation of a linear system impulse response and its Fourier transform†

Impulse response of a linear time invariant system is partitioned by dividing the time axis into equal intervals of time. Then the impulse response is expressed as a sum of these partitioned portions. Each individual portion is approximated by a finite sum of orthogonalized sinusoids satisfying integral squared error criteria. Four different sets are given for this purpose. If the time reversed functions from these sets are applied to the system then the sampled values of the system response at the partitioning instants directly yield the system coefficients as required for the least integral squared error. Knowing these coefficients the best approximation to the impulse response can be constructed as illustrated by the examples considered. Sampled values of the Fourier transform of system impulse response are obtained as a by-product.

Identification of Impulse Response of Linear Systems and Synthesis of System Model

It is a well-known fact that the impulse response completely identifies a linear system. In this investigation the impulse response h(t) is approximated by a set of ortho-normalized damped exponentials satisfying the integral squared error criterion. An analogue computer instrumentation is suggested to evaluate the generalized Fourier coefficients of impulse response. To evaluate these coefficients the time reversal of the orthonormal set is essential. This is achieved on an analogue computer by inserting inverters in series with all integrators. The method Suggested in this investigation directly leads to the synthesis of a system model approximating the system.

Evaluation of Impulse Response of Linear Systems by using Legendre Polynomials

Impulse response of linear time invariant systems is partitioned by dividing time axis into equal intervals of time. Then impulse response is expressed as a sum of partitioned portions of impulse response. Each individual portion of impulse response is approximated by a finite sum of Legendre polynomials, satisfying integral squared error criterion. In this investigation an experimental procedure is suggested to evaluate the coefficients of these polynomials to cahieve the best approximation. Knowing these coefficients the best approximation to impulse response is obtained us illustrated by the examples considered. If the coefficients of the input signal are also known then it is shown that the values of the output response at the partitioning instants can be computed.

A New Method for Recording the Impulse Response of Linear Systems by Correlation Measurement

A new method similar to the stroboscopic observation for measuring the correlation functions and the impulse responses of linear systems is described. The method essentially makes use of a peculiar periodic property of the autocorrelation function of a signal derived from a maximal-length binary shift register sequence which is applied to the input. A quick impulse response can thus be recorded directly on a rather slow pen writing oscillograph with the scale of time magnified by a factor of several thousands. An example of oscillogram showing the response of a sonic system is presented.