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Entropy and Power Spectrum of Asymmetrically Dc-Constrained Binary Sequences
Abstract
The eigenstructure of bidiagonal Hessenberg-Toeplitz matrices is determined. These matrices occur as skeleton matrices of finite-state machines generating certain asymmetrically DC-constrained binary sequences that can be used for simulating pilot tracking tones in digital magnetic recording. The eigenstructure is used to calculate the Shannon upper bound to the entropy of the finite state machine as well as the power spectrum of the maxentropic process generated by it.
... The classes of runlengthlimited constraints and spectral-null constraints have already been introduced. In addition, there are constraints that generate spectral lines at specified frequencies, called pilot tracking tones, which can be used for servo tracking systems in videotape recorders [118], [115]. Certain channels require a combination of time and frequency constraints [128], [157], [160]; specifically DC-balanced RLL sequences have found widespread usage in recording practice. ...
... The classes of runlengthlimited constraints and spectral-null constraints have already been introduced. In addition, there are constraints that generate spectral lines at specified frequencies, called pilot tracking tones, which can be used for servo tracking systems in videotape recorders [118], [115]. Certain channels require a combination of time and frequency constraints [128], [157], [160]; specifically DC-balanced RLL sequences have found widespread usage in recording practice. ...
Constrained codes are a key component in digital recording devices
that have become ubiquitous in computer data storage and electronic
entertainment applications. This paper surveys the theory and practice
of constrained coding, tracing the evolution of the subject from its
origins in Shannon's classic 1948 paper to present-day applications in
high-density digital recorders. Open problems and future research
directions are also addressed
The encoding of independent data symbols as a sequence of discrete amplitude, real variables with given power spectrum is considered. The maximum rate of such an encoding is determined by the achievable entropy of the discrete sequence with the given constraints. An upper bound to this entropy is expressed in terms of the rate distortion function for a memoryless finite alphabet source and mean-square error distortion measure. A class of simple dc-free power spectra is considered in detail, and a method for constructing Markov sources with such spectra is derived. It is found that these sequences have greater entropies than most codes with similar spectra that have been suggested earlier, and that they often come close to the upper bound. When the constraint on the power spectrum is replaced by a constraint On the variance of the sum of the encoded symbols, a stronger upper bound to the rate of dc-free codes is obtained. Finally, the optimality of the binary biphase code and of the ternary bipolar code is decided.
In digital transmission systems, the transmission channel often does not pass d-c. This causes the well- known problem of baseline wander. One way to overcome this difficulty is to restrict the d-c content in the signal stream using suitably devised codes. It is shown that, for a d-c constrained code, the limiting efficiency is related to the number of allowable running digital sum states in a very simple way.
We derive the limiting efficiencies of dc-constrained codes. Given bounds on the running digital sum (RDS), the best possible coding efficiency η, for a K-ary transmission alphabet, is η = log2 λmax/log2 K, where λmax is the largest eigenvalue of a matrix which represents the transitions of the allowable states of RDS. Numerical results are presented for the three special cases of binary, ternary and quaternary alphabets.
The signal-to-noise ratio of pilot tracking tones embedded in binary coded formats has been assessed. It has been found that the signal-to-noise ratio can be improved when the redundancy of the code is increased. Since the signal-to-noise ratio of the pilot tracking tone limits the maximum attainable bandwidth of the servo system, a compromise has to be found between the redundancy of the code and the performance of the servo system. The signal-to-noise ratio of pilot tones based on the polarity switch technique, which is attractive as no lookup tables are required for encoding and decoding, is substantially smaller than that of codes based on the fixed disparity format.
This paper deals with the spectral analysis of digital processes obtained through memoryless functions of stationary Markov chains. Under the assumption that the Markov chain is ergodic, not necessarily acyclic, closed form formulas are derived for both the discrete part (spectral lines) and the continuous part of the spectral density. The results are applied at the output of a finite state sequential machine driven by a stationary Markov chain, which represents a general model of several situations of engineering interest. Finally, the theory is used to evaluate the spectrum of encoded and modulated digital signals.
A limit on the absolute value of the running digital sum of a
sequence is known as the charge constraint. Such a limit imposes a
spectral null at DC. The maximum-entropy distribution of a
charge-constrained sequence is presented. A closed-form expression for
the power spectral density of maximum-entropy charge-constrained
sequences is given and plotted
Asymptotic results for the Perron-Frobenius eigenvalue of a
nonnegative Hessenberg-Toeplitz matrix as the dimension of the matrix
tends to ∞ are given. The results are used and interpreted in
terms of source entropies in the case where the Hessenberg-Toeplitz
matrix arises as the transition matrix of a finite-state machine
generating certain constrained sequences of ±1's
Spectral analysis of low frequencies, where for an encoded sequence a = aOal * * . aL, functions of Markov chains with applications
- G Bilardi
- R Padovani
- G Pierobon
G. Bilardi, R. Padovani, and G. Pierobon, "Spectral analysis of low frequencies, where for an encoded sequence a = aOal * *. aL,
functions of Markov chains with applications," IEEE Trans. Com-RDS, of a is defined by
mun., vol. COM-31, pp. 853-861, July 1983.