Brian Marcus

Holographic data storage

We present an overview of our research effort on volume holographic digital data storage. Innovations, developments, and new insights gained in the design and operation of working storage platforms, novel optical components and techniques, data coding and signal processing algorithms, systems tradeoffs, materials testing and tradeoffs, and photon-gated storage materials are summarized.

Finite-state modulation codes for data storage

The authors provide a self-contained exposition of modulation code design methods based upon the state splitting algorithm. They review the necessary background on finite state transition diagrams, constrained systems, and Shannon (1948) capacity. The state splitting algorithm for constructing finite state encoders is presented and summarized in a step-by-step fashion. These encoders automatically have state-dependent decoders. It is shown that for the class of finite-type constrained systems, the encoders constructed can be made to have sliding-block decoders. The authors consider practical techniques for reducing the number of encoder states as well as the size of the sliding-block decoder window. They discuss the class of almost-finite-type systems and state the general results which yield noncatastrophic encoders. The techniques are applied to the design of several codes of interest in digital data recording. >

Modulation coding for pixel-matched holographic data storage

We describe a digital holographic storage system for the study of noise sources and the evaluation of modulation and error-correction codes. A precision zoom lens and Fourier transform optics provide pixel-to-pixel matching between any input spatial light modulator and output CCD array over magnifications from 0.8 to 3. Holograms are angle multiplexed in LiNbO(3):Fe by use of the 90 degrees geometry, and reconstructions are detected with a 60-frame/s CCD camera. Modulation codes developed on this platform permit image transmission down to signal levels of ~2000 photons per ON camera pixel, at raw bit-error rates (BERs) of better than 10(-5). Using an 8-12-pixel modulation code, we have stored and retrieved 1200 holograms (each with 45,600 user bits) without error, for a raw BER of <2x10(-8).

Sofic systems and encoding data

Techniques of symbolic dynamics are applied to prove the existence of codes suitable for certain input-restricted channels. This generalizes the earlier work of Adler, Coppersmith, and Hassner on the same problem.

On codes with spectral nulls at rational submultiples of the symbol frequency

In digital data transmission (respectively, storage systems), line codes (respectively, recording codes) are used to tailor the spectrum of the encoded sequences to satisfy constraints imposed by the channel transfer characteristics or other system requirements. For instance, pilot tone insertion requires codes with zero mean and zero spectral density at tone frequencies. Embedded tracking/focus servo signals produce similar needs. Codes are studied with spectral nulls at frequencies f=kf_{s}/n , where f , is the symbol frequency and k, n are relatively prime integers with k \leq n; in other words, nulls at rational submultiples of the symbol frequency. A necessary and sufficient condition is given for a null at f in the form of a finite discrete Fourier transform (DFT) running sum condition. A corollary of the result is the algebraic characterization of spectral nulls which can be simultaneously realized. Specializing to binary sequences, we describe canonical Mealy-type state diagrams (directed graphs with edges labeled by binary symbols) for each set of realizable spectral nulls. Using the canonical diagrams, we obtain a frequency domain characterization of the spectral null systems obtained by the technique of time domain interleaving.

Factors and Extensions of Full Shifts.

Let ∑A be an irreducible shift of finite type with entropy logn. Then ∑A is a continuous extension of the fulln-shift. Also, if ∑A is a continuous factor of the fulln-shift, then it is shift equivalent to the fulln-shift.

Analyticity of Entropy Rate of Hidden Markov Chains

We prove that under mild positivity assumptions the entropy rate of a hidden Markov chain varies analytically as a function of the underlying Markov chain parameters. A general principle to determine the domain of analyticity is stated. An example is given to estimate the radius of convergence for the entropy rate. We then show that the positivity assumptions can be relaxed, and examples are given for the relaxed conditions. We study a special class of hidden Markov chains in more detail: binary hidden Markov chains with an unambiguous symbol, and we give necessary and sufficient conditions for analyticity of the entropy rate for this case. Finally, we show that under the positivity assumptions, the hidden Markov chain itself varies analytically, in a strong sense, as a function of the underlying Markov chain parameters

Maximum transition run codes for generalized partial response channels

A new twins constraint for maximum transition run (MTR) codes is introduced to eliminate quasi-catastrophic error propagation in sequence detectors for generalized partial response channels with spectral nulls both at dc and at the Nyquist frequency. Two variants of the twins constraint that depend on whether the generalized partial response detector trellis is unconstrained or j-constrained are studied. Deterministic finite-state transition diagrams that present the twins constraint are specified, and the capacity of the new class of MTR constraints is computed. The connection between (G,I) constraints and MTR(j) constraints is clarified. Code design methodologies that are based on look-ahead coding in combination with violation detection/substitution as well as on state splitting are used to obtain several specific constructions of high-rate MTR codes.

Two-dimensional low-pass filtering codes

We describe a framework for designing encoders that transform arbitrary data sequences into two-dimensional arrays satisfying certain constraints, in particular, constraints that guarantee arrays with limited high spatial frequency content. We also exhibit specific codes that produce such arrays. Such codes are useful for holographic recording systems.

Constrained systems with unconstrained positions

We develop methods for analyzing and constructing combined modulation/error-correcting codes (ECC codes), in particular codes that employ some form of reversed concatenation and whose ECC decoding scheme requires easy access to soft information (e.g., turbo codes, low-density parity-check (LDPC) codes or parity codes). We expand on earlier work of Wijngaarden and Immink (1998, 2001), Immink (1999) and Fan (1999), in which certain bit positions are reserved for ECC parity, in the sense that the bit values in these positions can be changed without violating the constraint. Earlier work has focused more on block codes for specific modulation constraints. While our treatment is completely general, we focus on finite-state codes for maximum transition run (MTR) constraints. We (1) obtain some improved constructions for MTR codes based on short block lengths, (2) specify an asymptotic lower bound for MTR constraints, which is tight in very special cases, for the maximal code rate achievable for an MTR code with a given density of unconstrained positions, and (3) show how to compute the capacity of the set of sequences that satisfy a completely arbitrary constraint with a specified set of bit positions unconstrained.