Burst error compensation for a two-dimensional channel

In an optical channel for a digital holographic data storage system, burst errors, in the form of severe amplitude compression, are experienced in the vicinity of dust and other optical aberrations in the channel. The locations of these burst errors are approximately xed over a range of perturbations in magni cation and vertical-horizontal position. In a calibrated channel, lower error rates can be realized by allocating energy in the modulation code based on a measurement of the approximate location of these e ects. This paper discusses theoretical bounds on the energy inside the distortion region, and the modulation code which is used to approach this bound. 1. SYSTEM DESCRIPTION The digital holographic data storage system (DHDSS) is similar in many ways to a communication channel, with the data bits experiencing interference within a page similar to the inter-symbol interference (ISI) seen in a conventional communications channel. However, the DHDSS experiences ISI in two dimensions, with a spatial coordinate instead of a time coordinate. Additionally, the interference is not spatially causal. 1.1. Operation of the System A block diagram of the holographic data storage system is shown in gure 1. Polarization and other elements are omitted from the picture for simplicity. Storage of data in the medium is performed as follows: the beam from the laser source is split into two separate beams. One beam is used to illuminate the Experimental Data were obtained with the assistance of Optitek, Inc. All work was performed under NASA GSRP Fellowship NGT 5-22. Our special thanks to Dr. Hosae Lee of Murray State University whose computer programming expertise was invaluable to the timely completion of this paper. Spatial Light Modulator (SLM), a 1024 1024 array of pixels on which the image to be stored is formed. The SLM is controlled by a host computer, which sets the states of the pixels to \on" (1) or \o " (0). The beam re ected from the SLM is focused into the storage medium, where it interferes with the reference beam. The interference of the beams creates fringes in the material (here, LiNbO3). The angle of the reference beam, which is used for multiplexing the data, is controlled by a movable mirror. The data is later read from the medium as follows: only the reference beam is present (the object beam is blocked). If the reference beam is re ected at the same angle as a page of stored data, the fringes reconstruct the stored data, which appears at the charge-coupled device (CCD) camera for recording. Figure 1: Block Diagram of Digital Holographic Data Storage System 1.2. Channel Characteristics For the purposes of this study, the system is aligned so that the object beam is centered on the SLM and on the CCD, and the reference beam is approximately uniform across its region of illumination. The following types of channel perturbation and noise are addressed with the coding: Optical aberrations on the system components cause xed \burst" errors in the image plane, with the corresponding di raction rings causing compression in the gain of the pixel values. A typical example of these rings is shown in gure 2. The localized amplitude change can be approximated by the main lobe of the rst order, circularly symmetrical Bessel function J0(x; y). Fixed errors are caused by stuck pixels in the SLM and/or the CCD. These spots can appear as either bright or dark values. Stuck pixels are also seen in gure 2 as a small, very dark region. Distortion is caused by the nite aperture of the lens, resulting in a non-uniform focal surface in the image plane. The result, in the alignment used with this system, is a ring of distorted pixels near the in ection points of the Gaussian focal surface shape. This distortion complicates symbol detection. Background noise is caused by the non-zero reectivity of the \o " pixels in the SLM, by stray light from outside the system, and from scattering inside the system. The result is a noise oor at approximately 5% of the maximum pixel value. This noise is reasonably approximated by additive white Gaussian noise (AWGN). Figure 2: Typical Di raction Pattern Burst Error Unlike the conventional communications channel, the DHDSS channel cannot accurately be modeled as AWGN since the e ects of the burst errors and distortion far outweigh the e ects of the background noise. 1.3. Calibrated Channel The precise amplitude compression in the burst errors cannot be predicted deterministically in the channel; instead, the burst error locations have some probability distribution because of the small changes in the alignment that occur over time. The storage crystal and optics are considered to be a \ xed" system; i.e., the crystal will be used with the optics without removing the crystal (like a hard disk drive, where the platters are not removed from the heads). In this case, the system can be calibrated for the areas of highest distortion and other xed phenomena (e.g., aberrations in the optics). The information from calibration is used to allocate the storage capacity in the image plane, and all data are stored encoded for this capacity distribution. 1.4. Admissible Modulation Codes Since the most severe amplitude compression from the di raction rings of aberrations occurs for aberrations close to the SLM and CCD, and since the \main lobe" of the rings measures approximately 40 to 60 pixels in diameter, a di raction ring main lobe measuring 49 pixels in diameter is considered for analysis. Call this circular region R. For synchronization of the detector, the symbols are pixel run-length limited (RLL) to four o pixels. Near the center of R, low-energy codes that are RLL to 1 on pixel work well for reducing ISI. Near the edges (outside half the region radius), 2-3 on pixels are required to produce enough energy after the point-spread function (PSF) and the intensity attenuation so that the pixel values do not drop below the noise oor. These restrictions have been found by experimentation to facilitate acceptable performance in the detector. Call the class of all such modulation codes C. In R, the maximum energy after modulation coding could be bound by the signal which uses the \all-one" signal, Z. However, this bound is not reasonable since Z contains no information. On the other hand, it can be shown by exhaustive search that more than 99.8% of the members of C correspond to non-negative trigonometric polynomials, as described below. A similar result holds for main lobes of di erent diameters. Thus, a better energy bound for C is found by analyzing these non-negative trigonometric polynomials. 2. BOUNDING THE POWER The total image plane signals considered in this paper may be thought of as deriving from the uniform, 1024 1024, square lattice sampling of light intensity elds. They are therefore discrete, two-dimensional, spatial signals. Assume that the diameter of the circular region R is 2N+1. Let A be the set of two-dimensional indices over R, such that the (0; 0) point is at the center of R. Denote by I(n1; n2) the amplitude signal on R. Then I(n1; n2) can be represented in matrix form as