Finite-length Sequences Research Articles

On the autocorrelation of finite-length sequences

lt is well known that all autocorrelation sequences are positive (semi)definite. The correlation sequence is further constrained when, as is often the case, it is obtained from correlating a finite-length data sequence with itself. In this paper we study the effect of the data sequence length on the attainable correlations. We derive a formula for the sequence length needed to match an autocorrelation value at a given lag. The formula indicates that inordinately long sequences are needed to obtain a normalized autocorrelation coefficient near unity. We conclude that all-zero (moving-average) models may be ill suited for highly correlated signals. The formula for the single-lag case is then used to obtain a bound on the sequence length needed to match a set of autocorrelations simultaneously.

Arrival time determination using iterative signal reconstruction from the phase of the cross spectrum

A situation encountered in such applications as seismics, ocean acoustics, radar, sonar and others, is that of an unknown wavelet propagating nondispersively in a reverberatory or multipath environment. Two receivers placed in such an environment will each record the arrival of this wavelet numerous times and each time with a different attenuation factor. In this paper, we demonstrate that if 1) there is no noise, 2) the interreceiver delays are distinct, and 3) the arrival sequence at each receiver has no zero phase convolutional component, then the arrival times and attenuation factors at each receiver can be determined as well as the wavelet itself. We show this via the construction of a finite length sequence whose phase equals that of the cross spectrum of the two received signals. This reconstructed signal, under the conditions above, can then be inverted to recover the desired arrival time and attenuation information at each receiver. An example using synthetically generated data is provided.

Similarity Measures Between Strings Extended to Sets of Strings

Similarity measures between strings (finite-length sequences of symbols) are extended to apply to sets of strings in an intuitive way which also preserves some of the desired properties of the initial similarity measure. Two quite different measures are used in the examples: the first for applications where numerical computation of pointwise similarity is needed; the second for applications which depend more heavily on substring similarity. It is presumed throughout that alphabets may be nondenumerable; in particular, the unit interval [0, 1] is used as the alphabet in the examples.

Group characters: Sequences with good correlation properties

The structure of the group of integers relatively prime to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> under multiplication modulo <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> is reviewed, and the basic properties of characters defined on that group is developed. Appropriately chosen subcollections of the characters when viewed as periodic sequences are then shown to have relatively ideal autocorrelation and cross correlation properties. The results of a computer study indicate that the same subcollections when viewed as finite length sequences also have very good aperiodic autocorrelation and cross correlation properties.

Finite-Length Test Sequences in System Identification Techniques

A scheme is presented for the rapid measurement of weighting sequences of in-operation linear systems. Finite duration sequences are used as test signals to the system under test, whose response to the sequences is processed by an appropriate digital filter. The test sequences are here required to have high energy ratio and low sidelobes in their auto-correlation function. It is shown that sequences having desirable properties can be obtained by clipping Huffman sequences. Results of computer simulation tests performed in different systems and using different sequences are also given.

The Sequential Construction of Minimal Partial Realizations from Finite Input–Output Data

Any strictly proper transfer function matrix of a continuous or discrete, linear, constant, multivariable system can be written as the product of a numerator polynomial matrix with the inverse of another polynomial matrix, the denominator. Since a realization is easily constructed from the polynomial matrix representation, the minimal partial realization problem is translated to that of extracting -a minimal order partial denominator polynomial matrix from a finite length matrix sequence. It is shown that minimal partial denominator matrices evolve recursively that is, a minimal partial denominator matrix for any finite length sequence is a combination of the minimal partial denominator matrices of its proper subsequences. A computationally efficient algorithm that sequentially constructs a minimal partial denominator matrix for a given finite length sequence is presented. A theorem by Anderson and Brasch leads to a definition of uniqueness for the resulting denominator matrix based upon its invariant factors. Parameters used during execution of the algorithm are shown to be sufficient for enumerating all invariant factor sets in the equivalence class of minimal partial realizations. The results apply to continuous and discrete linear systems including finite state machines.

The Design of Optimal Convolutional Filters via Linear Programming

Computational algorithms are given for the design of optimal, finite-length, convolutional filters with finite-length input sequences. Design techniques are developed for minimum-weighted-mean-square-error filters (MWMSE), for minimum-weighted-absolute-error filters (MWAE), and for filters which minimize the maximum output error (minimax). It is shown that the coefficients of the MWAE and minimax filters can be obtained by using standard linear programming methods. Next, the problem of developing a filter whose function is to "sharpen" a particular input waveform is considered. The filter input sequence is assumed to be derived from a Ricker wavelet of the velocity type and the desired output is the Dirac delta function. Convolutional filters are developed for this problem using each of the three performance criteria described above. The output sequences of each of the three optimal filters are discussed. It is shown that the minimax filter gives significantly better discrimination than can be obtained from either the MWAE or MWMSE filters.

Analyses of subensembles ofM/G/1 queueing sequences

Although many queueing processes of various principles have extensively been investigated, little attention has been paid to the sampling aspect of the theory, by which the nature of sample sequences of finite or infinite length can be examined with respect to some given ensemble of queueing sequences. In this paper we wish to identify classes of sample sequences of an M/G/1 model and investigate several hitherto unknown properties of queueing phenomenon of a given particular service system over a finite or infinite length of time. The method to be used is an extension of both the method of imbedded Markow chains, cf. D. G. Kendall [4], and semi-Markovian processes, Smith [9], Levy [5], Pyke[7,8], Fabens [2], Neuts [6], etc.