WO2004039089A2

WO2004039089A2 - Two-dimentional coding for high-density storage media applicatio ns

Info

Publication number: WO2004039089A2
Application number: PCT/US2003/033344
Authority: WO
Inventors: Chandra Mohan
Original assignee: Atlinks Usa, Inc.
Priority date: 2002-10-21
Filing date: 2003-10-21
Publication date: 2004-05-06
Also published as: AU2003282972A1; AU2003282972A8; WO2004039089A3

Abstract

There is provided a method for encoding a random bit stream in two-dimensions for storage on a storage medium. The random bit stream is encoded (3310) using Variable Aperture Coding (VAC) so as to generate a constant amplitude, varying pulse-width encoding that represents the random bit stream by a plurality of pulses separated using only transition widths included in a pre-specified set of transition widths. The encoding step includes the step of translating (step 3315) other pre-specified transition widths as amplitude combinations of the transition widths included in the pre-specified set of transition widths.

Description

TWO-DIMENSIONAL CODING FOR fflGH-DENSITY STORAGE MEDIA APPLICATIONS

BACKGROUND OF THE INVENTION

FIELD OF THE INVENTION

The present invention generally relates to signal coding and, more particularly, to two-dimensional coding for high-density storage media applications.

BACKGROUND OF THE INVENTION

Even though already a multi-billion dollar industry, the digital recording industry is expected to expand further in the future as an almost insatiable appetite for more storage continues to grow. This increase is partly fueled by the steady move towards digital systems, as has happened for example in the audio industry with the replacement of the analog Long Play (LP) disk by the digital Compact Disk (CD). Digital disk recording systems include magnetic and optical recording, the latter mainly for read only applications. Whether optical or magnetic, one of the main goals of ongoing research is to increase Areal density in bits per unit area.

Modulation codes, for most recording systems, focus on increasing linear density through the reduction of Inter-Symbol Interference (ISI). Further increases in storage density are potentially available by reducing track width and increasing track density. However, this results in undesirable Inter-Track Interference (ITI) and a reduction in signal-to-noise ratio (SNR). Consequently, typical magnetic recording systems have linear-to-track density ratios of only 25 to 1. Head misalignment or side reading (cross talk) that occurs between the read head and adjacent track data causes ITL This has been acknowledged as an important noise source that can be alleviated by employing sophisticated signal processing techniques whilst reading several adjacent tracks simultaneously with a multi-track head. An additional advantage of reading multiple tracks in parallel can be gained by employing two-dimensional run-length limited (d, k) modulation codes. These have attracted much attention in recent years as a means of increasing storage capacity by relaxing the timing constraint, k, along the tracks. Timing recovery is then achieved in a joint manner from information taken across a number of tracks. Currently, the storage capacity of media such as, for example, a Hard Disk Drive (HDD) or an optical drive is limited by the state of the art of head, media and write technology. Contiguous media provides the best opportunity to increase storage space using the best cost metrics. One of the vexing problems in media storage is the inability of magnetic media to handle transition flux changes in excess of 500-800 Kbpi due to overwrite issues.

Coding and other methods and apparatus have been proposed to reduce the number of such transitions. However, all of such existing approaches suffer from one or more deficiencies. For example, there is significant room for improvement in the current capacities of 1.3 bits per transition.

Multi head and multi track combinations along with the new perpendicular recording techniques have been used to increase the capacity of storage mediums. These methods are expensive and generally will suffer from reliability issues due to the increased number of heads as well as the characteristics of the new media used in the perpendicular recording system.

Accordingly, it would be desirable and highly advantageous to have a coding method and apparatus for high-density storage media applications that overcome the above-described deficiencies of the prior art.

SUMMARY OF THE INVENTION

The problems stated above, as well as other related problems of the prior art, are solved by the present invention, which is directed to a two-dimensional coding method and apparatus for high-density storage media applications.

According to an aspect of the present invention, there is provided a method for encoding a random bit stream in two-dimensions for storage on a storage medium. The random bit stream is encoded using Variable Aperture Coding (VAC) so as to generate a constant amplitude, varying pulse-width encoding that represents the random bit stream by a plurality of pulses separated using only transition widths included in a pre-specified set of transition widths. The encoding step includes the step of translating other pre-specified transition widths as amplitude combinations of the transition widths included in the pre- specified set of transition widths.

According to another aspect of the present invention, there is provided a method for storing a random bit-stream on a storage medium. The random bit stream is represented by a constant amplitude, varying pulse-width, VAC encoding having a plurality of pulses that are separated using only transition widths included in a pre-specified set of transition widths. The VAC encoding is transmitted along a data channel for storage on the storage medium. The representing step includes the step of translating other pre-specified transition widths as amplitude combinations of the transition widths included in the pre-specified set of transition widths.

These and other aspects, features and advantages of the present invention will become apparent from the following detailed description of preferred embodiments, which is to be read in connection with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a diagram illustrating a typical Hard Disk Drive (HDD) structure 100 to which the present invention may be applied, according to an illustrative embodiment of the present invention; FIGs. 2 A and 2B are diagrams respectively illustrating a longitudinal system 200 and a perpendicular system 250 for magnetic recording on a Hard Disk Drive (HDDD) to which the present invention may be applied, according to an illustrative embodiment of the present invention;

FIGs. 3 A and 3B are diagrams respectively illustrating a perpendicular magnetic media 300 having a magnetic under layer 310 and another perpendicular media 350 having a non-magnetic under layer 360, to which the present invention may be applied, according to an illustrative embodiment of the present invention;

FIG. 4 is a diagram illustrating an encoder 400 for encoding a Variable Aperture Coding (VAC) signal to which the present invention may be applied, according to an illustrative embodiment of the present invention;

FIG. 5 is a diagram illustrating an original signal and a Variable Aperture Coding (VAC) signal corresponding to the original signal, according to an illustrative embodiment of the present invention;

FIG. 6 is a diagram illustrating an original signal 600 and a Variable Aperture Coding (VAC) encoded signal 650 generated from equation (1), according to an illustrative embodiment of the present invention; FIG. 7 is a diagram illustrating an event generator 700 for generating signals, to which the present invention may be applied, according to an illustrative embodiment of the present invention;

FIG. 8 is a diagram illustrating a signal flow chart 800 for VAC signals with respect to four specified events, to which the present invention may be applied, according to an illustrative embodiment of the present invention;

FIG. 9 is a diagram illustrating a data string 900 and associated pulses 950 corresponding to an exemplary interpretation of the signal flow graph of FIG. 8, according to an illustrative embodiment of the present invention; FIGs. 10 A, 10B, and 10C are diagrams illustrating a signal flow chart simplification for computing T_11; according to an illustrative embodiment of the present invention;

FIGs. 11 A, 11B, and 11C are diagrams illustrating a signal flow chart simplification for computing T ₂, according to an illustrative embodiment of the present invention;

FIGS. 12 A, 12B, and 12C are diagrams illustrating a signal flow chart simplification for computing T₁₂, according to an illustrative embodiment of the present invention;

FIGs. 13 A, 13B, and 13C are diagrams illustrating a signal flow chart simplification for computing T₂₁, according to an illustrative embodiment of the present invention;

FIGs. 14A and 14B are diagrams illustrating a signal flow chart simplification for computing T₁₃, according to an illustrative embodiment of the present invention; FIGs. 15 A and 15B are diagrams illustrating a signal flow chart simplification for computing T₁ , according to an illustrative embodiment of the present invention;

FIGs. 16A and 16B are diagrams illustrating a signal flow chart simplification for computing T₂₃, according to an illustrative embodiment of the present invention;

FIG. 17 is a diagram illustrating a signal flow chart 1700 for Variable Aperture Coding (VAC) signals, according to an illustrative embodiment of the present invention;

FIG. 18 is a diagram illustrating a plot of the Power Spectral Density (PSD) 1800 of a Variable Aperture Coding (VAC) signal, according to an illustrative embodiment of the present invention;

FIG. 19 is a block diagram illustrating a Variable Aperture Coding (VAC) decoder 1900, according to an illustrative embodiment of the present invention;

FIG. 20 is a diagram illustrating an orthonormal basis 2000 for a Variable Aperture Coding (VAC) signal, according to an illustrative embodiment of the present invention; FIG. 21 is diagram illustrating a vector representation 2100 of Si, S_2j S₃, according to an illustrative embodiment of the present invention;

FIG. 22 is a diagram illustrating a Variable Aperture Coding (VAC) signal 2200 approximated as a Continuous Phase Modulation (CPM) signal with (π M) phase variation, according to an illustrative embodiment of the present invention;

FIG. 23 is a diagram illustrating a plot of a simulation result 2300 comparing the BER performance derived from Equations (6.15) and (6.20), respectively, according to an illustrative embodiment of the present invention;

FIG. 24 is a diagram illustrating a plot of simulation results 2400 corresponding to a Bit Error Rate (BER) performance comparison for M-VAC (M = 7, 9, 11), according to an illustrative embodiment of the present invention;

FIG. 25A is a diagram illustrating a data-packet 2500 with redundancy in space, according to an illustrative embodiment of the present invention;

FIG. 25B is a diagram illustrating the data packet 2500 of FIG. 25 A with redundancy added in time, according to an illustrative embodiment of the present invention;

FIGs. 26A, 26B, 26C, and 26D are diagrams illustrating the transitions on the 4-bus lines, according to an illustrative embodiment of the present invention;

FIG. 27 is a diagram illustrating Variable Aperture Signaling 2700, according to an illustrative embodiment of the present invention; FIG. 28 is a diagram illustrating a two-dimensional encoder 2800 for encoding in space and time, according to an illustrative embodiment of the present invention;

FIG. 29 is a diagram illustrating a plot of a variation 2900 of p_opt versus d, according to an illustrative embodiment of the present invention;

FIG. 30 is a diagram illustrating a two-dimensional encoding 3000 of a signal in both amplitude and time, according to an illustrative embodiment of the present invention;

FIG. 31 is a diagram illustrating a plot of a decoded waveform 3100, according to an illustrative embodiment of the present invention;

FIG. 32, which is a diagram illustrating various encoding combinations 3200 for a waveform 3210, according to an illustrative embodiment of the present invention; and FIG. 33 is a flow diagram illustrating a method for storing a random bit-stream on a storage medium, according to an illustrative embodiment of the present invention. DETAILED DESCRIPTION OF THE INVENTION

The present invention is directed to a two-dimensional coding method and apparatus for high-density storage media applications. The present invention provides a method and apparatus that utilizes the current Partial Response Maximum Likelihood (PRML) channel to reduce the number of transitions. However, this PRML channel is different from the existing PRML channels in that the PRML encoding is performed to represent only 3 distinct transition points 4T, 5T and 6T spacings, thereby avoiding Inter-Symbol Interference (ISI) related issues. Further, IT, 2T and 3T PRML encodings are translated into three distinct amplitudes at 4T, 5T and 6T spacings, so that the number of bits represented by a transition is maximized. The current capacities of 3 bits per transition can be progressively improved to 8 bits per transition, allowing tremendous improvement in storage density. Moreover, as the bandwidth occupied by the signal is very narrow, it allows for the introduction of sharp band pass filters that improve the SNR of the detection. It is to be appreciated that the present invention utilizes the existing head and media to achieve the capacity improvement.

It is to be understood that the present invention may be implemented in various forms of hardware, software, firmware, special purpose processors, or a combination thereof. Preferably, the present invention is implemented as a combination of hardware and software. Moreover, the software is preferably implemented as an application program tangibly embodied on a program storage device. The application program may be uploaded to, and executed by, a machine comprising any suitable architecture. Preferably, the machine is implemented on a computer platform having hardware such as one or more central processing units (CPU), a random access memory (RAM), and input/output (I/O) interface(s). The computer platform also includes an operating system and microinstruction code. The various processes and functions described herein may either be part of the microinstruction code or part of the application program (or a combination thereof) that is executed via the operating system. In addition, various other peripheral devices may be connected to the computer platform such as an additional data storage device and. a printing device.

It is to be further understood that, because some of the constituent system components and method steps depicted in the accompanying Figures are preferably implemented in software, the actual connections between the system components (or the process steps) may differ depending upon the manner in which the present invention is programmed. Given the teachings herein, one of ordinary skill in the related art will be able to contemplate these and similar implementations or configurations of the present invention.

In addition to reducing ISI, two-dimensional codes can also be exploited to reduce/control Inter-Track Interference (ITI). Further, the effects of controlled LTI can be incorporated into a Viterbi detector to aid detection in addition to the controlled ISI along the tracks. While we expect such a code to have slightly lesser linear density than a single dimension code, it would exhibit a much-improved performance through the reduction of the track width and guard band. If the increase in track density exceeds the loss in linear density, then the overall result will be the desired increase in Areal density.

The goal to increase the amount of information that can be stored per disk unit area in digital storage media continues to be important, together with the tendency to make recording systems physically smaller. An obvious way to increase information density in existing systems is to increase the number of bits per track and the number of tracks per unit area. Several detection schemes proposed for multi-track, multi-head, digital recording systems as well as theoretical studies about achievable radial information densities in these systems suggest that, if the construction of head-arrays simultaneously scanning multiple tracks becomes possible, Areal storage density can be greatly increased by means of track narrowing. The main advantage of using multiple-heads is their capability to partly recover the loss in performance due to inter-track-interference ( TI), which is a prominent noise source in narrow track systems. Since track narrowing also results in a decrease in signal-to- noise ratio (SNR), multi-dimensional modulation codes for improving noise immunity are needed to compensate for this. We show that codes designed for improving noise immunity in single-track, single-head systems provide the same coding gain when used in multi-track, multi-head systems when LTI satisfies certain conditions. These conditions are only sufficient but they do not impose significant requirements for most common LTI models where only adjacent tracks interfere. We show that multi-dimensional, coded systems potentially provide a substantial Areal density increase even in cases of high LTI.

One way Areal density can be increased along the radial direction is by narrowing the track width, thus allowing more tracks per unit radial distance. However, as a result of the track narrowing, there is a loss in signal-to-noise ratio (SNR) during read back, as well as increased inter-track interference (LTi). The loss in SNR can be compensated for by employing a code that improves noise immunity in each track which, in turn, decreases in- track density. A potential Areal density gain can be obtained provided high rate codes that can combat both inter-symbol interference (ISI) and ITI and make up for the SNR loss can be designed. It has been acknowledged that LTI is an important noise source in disk recording systems, whose effects can be significantly reduced by means of multiple recording heads. An additional benefit of writing and reading multiple tracks in parallel is that information for timing and gain control can be obtained from any track, which has been shown to reduce the required redundancy and enable substantial increases in Areal density. Thus, it is believed that simultaneous detection of read back signals from interfering tracks using array heads should be further investigated as a means of increasing Areal density in storage media. Although the use of multiple heads with simultaneous detection can reduce the effects of LTI on performance, there is nevertheless a residual performance loss incurred by track narrowing. This performance loss can be recovered by the use of coding, which as we will see later, can compensate for the loss, and also provide some extra gain. In general, the code must be designed to account for a two-dimensional interference pattern, ISI in the axial dimension and LTI in the radial direction. The present invention is concerned with the design of modulation codes capable of combating the performance loss incurred by both a decrease of SNR and the presence of LTI, in multi-track digital storage media. These codes must be of a high rate to provide an overall Areal density increase, and must satisfy certain run-length constraints to facilitate timing and gain recovery. Variable Aperture coding is a new class of Digital Bi-Phase coding that can drastically reduce the bandwidth efficiency (bits/sec/hertz) for any random digital bit stream. If "R" is the rate of the information signal, VAC coding does not reduce transition density as is the case with most of the higher order modulation schemes like Orthogonal Frequency Division Multiplexing (OFDM), Quadrature Amplitude Modulation (L-QAM) or Multiple Phase Shift Keying (MPSK) but rather compresses the power spectral density to be highly concentrated within a bandwidth of R 9 times the bit rate. A non-VAC encoding method would have required at least a bandwidth of "R" to successfully decode the signal. Due to the narrow occupied bandwidth of the VAC signal, a capacity increase in storage is possible through the introduction of di-bit encoded VAC that will increase capacity two-fold and/or having an orthogonal VAC bit stream introduced in the interval between the transitions. In order to increase the tracks per inch, alternate tracks can be run on orthogonal VAC streams. There is also the possibility of reducing the inter-track distances due to reduced LTI derived out of the signal orthogonality in adjacent tracks as well as due to the narrow PSD foot print that leads to lesser leakage into adjacent write domains. Since Longitudinal Recording/Optical recording is a 2-dimensional embedding process, and assuming spindle speeds to be constant, it is not possible to increase capacity without adding additional dimensions to the signal either in space or time. Perpendicular recording increases capacity by providing an additional dimension in space, where as VAC provides the added dimensions by altering the time domain. Consequently VAC symbols will vary in width, which will in turn represent the various signal transitions, the original data stream.

A description will now be given of Variable Aperture Coding (VAC) for high capacity Hard Disk Drives (HDDs), according to an illustrative embodiment of the present invention.

FIG. 1 is a diagram illustrating a typical Hard Disk Drive (HDD) structure 100 to which the present invention may be applied, according to an illustrative embodiment of the present invention. The HDD structure 100 includes one or more platters 110, an actuator 120, one or more arms 130, and one or more heads 140. Information (e.g., bits) is magnetically stored on the platters 110, usually on both surfaces of the platters 110. The bits are records in tracks that, in turn, are divided into sectors 150. The tracks include inner tracks 160, outer tracks 170, and other tracks (not labeled) there between. A sector 199 is the minimum unit for reading and writing. Typically, a sector 199 is 512 bytes up to a few Kbytes. The actuator 120 moves the head 140 at the end of the arm 140 over a track. The preceding process is called a "seek", during which the heads 140 are moved to the desired track and the head position is adjusted to be centered over the track. This analog adjustment is called "settling". The data between sectors identifies the current track and sector. The head 140 is moved to an adjacent track if the seek operation landed the head 140 on the wrong track. A set of tracks for each surface, at the same distance from the center, is called a cylinder.

For illustrative purposes, a description will now be given of a simple disk access model, according to an illustrative embodiment of the present invention. This description will be begin with a discussion on capacity.

Assume that a disk has "f ' surfaces and each surface has "t" tracks. Let each track have "s" sectors holding "b" bytes. The disk capacity, "C", is defined as follows:

C = f x b x t x b With respect to the simple disk access model, a description will now be given of access times, according to an illustrative embodiment of the present invention.

Let us assume that a Seek operation is performed at the rate of r tracks/second. The time taken to move from the i track to the J* tack cart be computed as |i - j| / r. If the rotation period Is P, then the average rotation latency is P/2.Therefσre the average time to access track j if currently at track ϊ is |i - j| / r + P/2.

In accessing data on a disk, latency is defined as the time it takes to position the proper sector under the read/write head.

Disk Latency = Seek Time + Rotation Time + Transfer Time + Controller Overhead

Seek Time depends on the number of tracks the arm has to move and also on the actuator tracking speed. Rotation time depends on the speed at which the disk rotates and how far the sector is from the head. Transfer time depends on the data rate (bandwidth) of the disk (bit density) and the size of the request.

Initially, to keep things simple, every track had the same number of sectors. Since outer tracks are longer than inner tracks, outer tracks can accommodate more sectors in comparison to inner tracks. HDD manufacturers have imposed a standard to maintain the same number of bits per inch (constant bit density) in both inner and outer tracks by having more sectors for the outer track versus the inner tracks, resulting in more capacity per disk. Since disks spin at a constant speed, outer tracks have a faster data rate in comparison to inner tracks. It can be computed that the outer track delivers 1.7 times more data than the inner track. In order to increase disc capacities and lower the cost of storing each bit of data, there has been on-going research specifically targeted at these areas.

With respect to lowering the cost per bit of storage, this means that the HDD vendor needs to store more bits per dollar. The current approach is to improve the Areal density by investing in new head technologies like Giant MagnetoResistive (GMR), using perpendicular recording methods to overcome the super-paramagnetic limits, and investing in new materials (such as, for example, FePt Cr) that have high anisotropy, allowing the balancing of the grain size to meet the best thermal and SNR requirements.

With respect to increasing capacity, increasing Areal density is the most important parameter to improve capacity. This has included research into new recording methods like perpendicular recording, new magnetic materials that have finer grain sizes and exhibit high level of anisotropy, and new recording head technology like GMR.

Magnetic disk drives are currently commercially available with Areal densities as high as 4.1Gb/ in². Laboratory tests have demonstrated the ability to achieve an Areal density of 10Gb/ in² using current technologies. According to some studies, the Areal densities of hard disk drives have been increasing at a rate of 60% per year since 1991. At this rate, hard drives will be able to store 100Gb/in² by 2006. Hence, we can expect to see 50Gb/in² drives commercially available somewhere in the range of 2004 - 2005.

There are many factors to consider in designing a disk drive with an Areal density of 50Gb/in². One of the most important factors is the inter-relation between Areal density with linear density and track density:

Area! Density D_a = D-x D*

, where D_a, Dl, and Dt are the Areal, linear, and track densities, respectively. The current state-of-the-art in commercially available hard drives is a linear density of 256.4kbpi (kilo bits per inch) and a track density of lόktpi (kilo tracks per inch). This yields a linear density to track density ratio of 16 to 1. Thus, a simple scaling of current properties to 50 Gb/in would give a track density of 56ktpi and a linear density of 895kbpi. However, recently established theoretical models indicate that it is advantageous to have a squarer bit cell at higher densities. The following factors limit the capability to achieve high Areal Densities in a recording media and must be overcome: magnetic relaxation (super-Para magnetic limit); head-to-medium separation; write head saturation; read head sensitivity; and servo tracking bandwidth. The fundamental problem blocking a simple scaling approach to achieving 50Gb/in² is the super-para magnetic limit. This limit is thought to be near 40Gb/ in² for currently used CoPtCr based media. Many engineering solutions have been proposed to extend magnetic recording beyond this "limit". These include alternate high-anisotropy media, perpendicular recording, patterned media, and keepered media. Patterned media, a scheme where the medium is lithographically patterned into an array of single-bit magnetic islands, holds great promise for ultra-high density recording in the distant future. High-density magnetic recording media has been developed to realize high-speed and large capacity data storage systems. Nowadays, the recording density of Hard Disk Drives (HDDs) has reached over 10Gb/in². However, this is not enough to deal with multi-media data. Much higher recording density is indispensable in the near future. FIGs. 2 A and 2B are diagrams respectively illustrating a longitudinal system 200 and a perpendicular system 250 for magnetic recording on a Hard Disk Drive (HDDD) to which the present invention may be applied, according to an illustrative embodiment of the present invention.

Currently, longitudinal recording systems for magnetic recording are widely used (see FIG. 2A). In longitudinal magnetic recording, the thickness of the magnetic layer has been decreased while coercivity has been increased to enable higher recording density. Due to this trend, several problems such as thermal instability of recorded bits and recording head writability will become serious issues in the near future.

Recently, perpendicular magnetic recording (see FIG. 2B) attracted attention as a magnetic recording system for the next generation due to its ability to overcome the super- para magnetism limit. Perpendicular magnetic recording can achieve high recording densities with larger thickness and lower coercivity of the magnetic layer than longitudinal recording, at least in theory. Therefore, perpendicular magnetic recording is an effective way to realize high recording density, by reducing the physical problems associated with longitudinal recording. However, perpendicular magnetic recording media have a low Signal to Noise Ratio (SNR) compared to longitudinal magnetic recording media due to lower number of grains in the recorded width and self-erasure of low frequency recording signals in the bit stream. It is important to fabricate a recording system with a high recording density and a high SNR. One approach to the solution for poor SNR problems is to apply various materials to the under layers of perpendicular magnetic recording media, including magnetic and nonmagnetic materials (see FIG. 3). FIGs. 3A and 3B are diagrams respectively illustrating a perpendicular magnetic media 300 having a magnetic under layer 310 and another perpendicular media 350 having a non-magnetic under layer 360, to which the present invention may be applied, according to an illustrative embodiment of the present invention. The magnetic materials commonly used for such applications range from a hard magnet with coercivity of around 1000 Oe to a semi-hard magnet with coercivity of around 100 Oe. The perpendicular magnetic recording media with these under layers are expectedly suitable for the read-write using a ring-type head (see FIG. 3A). As for non-magnetic under layers, researchers are focusing on controlling crystalline orientation of the magnetic layer. The media for these applications are realized by sputtering. Investigation into the influence of the under layers on magnetic properties, microstructure and read-write characteristics of the media is an on-going area of research.

Another alternative involves adding a film of soft magnetic material (the so-called "keeper layer") on top of the magnetic layer to stabilize the recorded data. Perpendicular recording, a long-championed yet never profitable commercialized alternative to longitudinal recording, still holds much promise as a high-density recording candidate. A much more conservative design approach to extending the super-Para magnetic limit and reaching 50 Gb/in 2 in the shortest possible time involves scaling the current technology and moving to alternate, high-anisotropy media. It is believed that this material strategy can push the super- Para magnetic limit by a factor of lOx to higher Areal densities.

Thermal energy causes small random fluctuations in the magnetization of a particle, just as it causes random Brownian motion of small particles. If the total anisotropy energy of a single-domain particle, KT_JV, becomes on the order of the thermal energy, kT, then the magnetization may be reversed as a statistical time-temperature effect. Here, KU is an anisotropy energy density constant, V is the particle volume, k is Boltzmann's constant and T is the absolute temperature. There is a critical volume VC given by:

, in which super-Para magnetism exists. Here, t is the time period of observation and fo is the Larmor frequency (about 10⁹ Hz). In the case of magnetic data storage, we want to be able to reliably store information for many years. Rearranging the above equation, we could solve for t in terms of the ratio KT V/kT. For a 100-year storage lifetime with respect to thermal stability, a ratio, Ku V/kT, of 43 is necessary. For a 5-year storage lifetime, a ratio of 40 is required.

As we move to higher Areal densities, the bit volumes become smaller. The signal to noise ratio, SNR, is dependent on the number of magnetic grains per bit. Thus, to preserve a reasonable SNR at higher recording densities, the size of each grain must shrink. Lower grain size results in less thermally stable media. The only ways to compensate for this effect is to either lower the temperature or raise the media anisotropy. Lowering the temperature is not economically feasible in the foreseeable future, though this is an alternative that may resurface in the distant future.

Therefore, we need to use a media that has as large an anisotropy constant as possible while still retaining favorable values for other magnetic properties. Grain size considerations are extremely important in designing high-density media. A lower limit on grain size is set by the requirements of thermal stability and an upper limit is set by the requirement of having a large number of grains per bit cell to get a good SNR. A thermal stability lifetime of 75 years, corresponding to a Ku V/kT value of 43, would greatly exceed the expect lifetime of the product and provide wide safety margins for use in a relatively high-temperature Environment. Assuming the grains are spherical, the minimum grain size (diameter) for thermal stability is 2.6nm. In actuality, the grains would probably be acicular cylinders, in which case minimum grain diameter depends on the cylinder aspect ratio. An upper limit is set on grain size by the requirement of having a large Signal-to-Noise Ratio (SNR). The SNR increases with an increasing number of grains per bit cell.

The SNR can be computed by the following equation:

, where b is the bit spacing, W the track width and D is the average domain size, respectively. If the average domain size D remains the same, then the SNR will be deteriorated with decreasing bit size and width. Thus, if we want a 25dB SNR, a minimum of 317 grains per bit cell is required, resulting in a maximum grain size of 9.4nm. Therefore, the media must be bounded by grain sizes of 2.6 nm and 9.4 nm. Using an average grain size midway between these two values (6nm) guarantees a SNR of greater than 25dB. As mentioned above, the coercivity of this media can be varied over a wide range. Typical coercivity of He = 5000 Oe, which is easily achievable.

The transition parameter is calculated to be 8.1 nm from the equation given below:

, where δ. is medium thickness (10 nm), Mr is remnant magnetization (760 emu /cc), d is sensor to medium separation (13 nm), and d_eff = sqrt [d (d+δ)] = 17nm. This transition parameter is a small fraction of the minimum distance between magnetic transitions (50.4 nm) for chosen linear density. A FePtCr film thickness of lOnm is presumed, which is a good compromise for a large read back signal and small sensor-to-medium separation. For 625kbpi in the FePtCr recording medium, a (1,7) run-length limited (RLL) code can be used in addition to an error correcting code (ECC). This corresponds to 504 kfci (50.4nm minimum separation between flux changes) in the recording medium. Using a sensor-to-medium separation of 13nm, the transition parameter for the medium is calculated to be 8.1nm (only a small fraction of the total flux reversal length). From the above discussions, it is clear that contemporary research focuses on the following areas to improve HDD capacity: overcoming super-paramagnetic limit by using perpendicular recording methods and development of High anisotropic materials for improved thermal stability; reducing the head-to-medium separation to optimize for a large read back signal; improving the read head sensitivity and linearity, using GMR heads; and improving the servo tracking bandwidth that allows for more tracks per inch. Using an efficient encoding and modulation method is yet another method for improving the Areal density, an area that has not been extensively researched by any of the leading research groups. Although EPRML and its derivatives have been adopted by the HDD industry as a standard, functionally, EPRML only enables the imprinting of a transition on a magnetic material with a smallest repeatable footprint, besides reducing the number of transitions in the magnetic media. However, using PRML, the closest transition could still occur at a distance of IT and this poses limitations with respect to ISI and LTI related issues. PRML combined with VAC reduces the transition density, while improving the number of bits per transition over a pure PRML encoded channel (in fact, a IT transition is never detected in a conventional PRML detector). Given the current state of the art media and head technologies, it is possible to increase capacity by assigning "multi bit weightage" for every transition. A description will now be given of motivations for the use of Variable Aperture Coding (VAC).

VAC was originally designed to minimize transmission bandwidth in digital communication applications. Further examination led us to look at how VAC can help increase storage capacity in the HDD industry.

The way Is and Os are stored on the disk to mark the duration of the original data from one transition to the next is to record a transition pulse on the disk. The distance (can be translated to time duration as we assume the disk is spinning at a constant spend) between transitions is that bit's duration of the incoming signal. This is really a simple and direct way to write on the disk. However, the duration differences of successive transitions vary so much (also in a random fashion) that they create inter-symbol interference such that the read process becomes unreliable. Further, as we increase the packing density, this inter symbol interference becomes a big problem. Consequently, several encoding schemes were used to correct the read error due to inter-symbol interference and other problems associated with data recovery during the read operation. These coding schemes have the basic concept that a certain combination (or pattern) of Is and 0s will help to minimize inter-symbol interference. The process is based on some mathematic formula such that, for so many bits, the maximum number of consecutive Is and 0s must meet a predetermined pattern. In most cases, additional 0s will be added to the input sequence to meet this coding requirement. As an example, a 1000 Is and 0s of an incoming signal become 1400 Is and 0s as they are being written onto the disk. This 40% increase in overhead is really not as bad as creating a redundant track to overcome the unreliable read operation.

PRML is another improvement for increasing storage capacity and read reliability. However, it is still dependent upon the channel coding scheme and, while the read reliability of PRML is much better than the peak detect method, the overhead is still there.

With VAC, we can encode the PRML encoded signal with a VAC encoder and achieve a data rate that is 1/5 the original data rate and put the maximum flux reversals in the magnetic media to be less than 500 to 800 Kilo flux changes per inch. Further, transitions are more predictable and the clock is reset at every transition so there is no timing error build up and therefore the clock accuracy is less critical. This make PRML detection even more effective than it is today. Also, we can offset adjacent track write operations by 90 degrees so that there is no track-to-track interference. Since VAC encoded signals occupy only a fraction of the bit duration, we can overlay another signal in the same space varying the amplitude. Then, the current PRML read channel can be used as is. This is further described herein below.

A description will now be given of a mechanism for increasing storage capacity using a VAC encoded PRML channel, according to an illustrative embodiment of the present invention.

The original data is encoded by PRML, for example, to allow combinations of 4T, 5T and 6T. This means that there will be 4 "0"s, 5 "0"s or 6 "0"s between adjacent "l"s. For example, we can have M = 5 representing a state that has 5 "0"s between a "1". The formation of the 8 bit word can be the concatenation of two 4 bit words that is formed after pre- processing the data into 2 bit, 3 bit, and 4bit sequences. Every 4T waveform conveys 6 bits of information, 5T conveys 7 bits of information and 6T conveys 8 bits of information. Hence, this VAC encoding and modulation process is now happening at l/5th of the original data rate and puts the max flux reversals in the magnetic media to be less than 500-800 Kilo flux changes per inch. In fact as "M" nears infinity, the signal will be almost like a perfect clock signal and detection will become very difficult.

Decoding clocks can be relatively easily synchronized for values of "M" up to 12-16. The resultant VAC encoded 4T, 5T and 6T waveform is a constant amplitude signal and can be limited (1 bit Analog-to-Digital Conversion), with the width variances embedded as instantaneous frequency/phase variations.

A description will now be given of some fundamentals of Variable Aperture Coding.

The description will begin with a discussion of the encoding process and encoding rules.

FIG. 4 is a diagram illustrating an encoder 400 for encoding a Variable Aperture Coding

(VAC) signal to which the present invention may be applied, according to an illustrative embodiment of the present invention.

The input signal to the encoder 400 is S(t), which is a random digital signal that assumes values {+1, -1 }. S_en(t) is the VAC encoded signal that also assumes values {+1, -1 } in amplitude. However, the encoded signal assumes three different widths to represent three different occasions in the original digital signal. The encoding rules are described as follows. Let T be the bit width of the original signal, T_c = T/M (where M is an integer) is the period of the sampling clock. A "0 to 1" transition in the original digital signal, a rectangular pulse with a width of "(M-l) T_c" is generated. A "1 to 0" transition in the original digital signal, a rectangular pulse with width of "(M+l) T_c" is generated. A "no change" (either a "0 to 0" or a "1 to 1" transition in the original digital signal, a rectangular pulse with width of "M T_c" is generated. The rectangular pulses generated in steps (2) to (4) are concatenated with alternative polarity to form VAC signal. FIG. 5 is a diagram illustrating an original signal and a Variable Aperture Coding

(VAC) signal corresponding to the original signal, according to an illustrative embodiment of the present invention. The VAC signal is illustrated with respect to the above encoding rules for a Variable Aperture Coding (VAC) signal. A one-bit delay is shown between the VAC signal 500 and the original signal 550. The first bit illustrates the width of the MTc clock periods.

A description will now be given of various characteristics of Variable Aperture Signals.

A qualitative investigation of the VAC encoding rules reveals some unique characteristics of the VAC signals. These characteristics may be summarized as follows. VAC "bits" assume alternative polarity. The VAC signal does not allow two consecutive "(M-l) T_c", or two "(M-l) T_c" separated by an integer multiple of "M T_c". The VAC signal does not allow two consecutive "(M+l) T_c", or two "(M+l) T_c" separated by an integer multiple number of "M T_c". If the original signal starts with a "-1", then all the VAC transitions happen only at nT, or [n+(M-l)/M] T, where n = 0, 1, 2, ... If the original signal starts with a "+1", then all the transitions of the encoded signal happen only at nT, or [n+(M+l)/M] T, where, n = 0, 1, 2, ...

A mathematical description will now be given of Variable Aperture Coding (VAC) signals.

Define: 1-bit duration of the original signal T = MT_C, where T_c is the period of the sampling clock, M is an integer. Define:

Assuming signal starts at t = 0, with a positive starting polarity, the VAC signal may be mathematically expressed as:

where T

P_kτ(t) is defined as above.

FIG. 6 is a diagram illustrating an original signal 600 and a Variable Aperture Coding (VAC) encoded signal 650 generated from equation (0), according to an illustrative embodiment of the present invention. The waveform is generated with setting M = 9. It is clear that the modulated VAC signal possesses memory. Therefore, a Markov chain can be derived to represent the VAC signals.

A description will now be given of Variable Aperture Coding (VAC) Spectral Analysis using a signal flow graph approach, according to an illustrative embodiment of the present invention. The description will begin with a discussion of some background corresponding thereto.

It has been shown that notation and techniques for finding the signal flow through linear systems of considerable complexity may be applied directly to the problem of finding the "response" of Markov processes of corresponding complexity. Moreover, the signal flow graph approach has been used to provide three examples of how to establish signal graphs for certain signals (telegraph signals, identical pulses with random spacing, and identical pulses of alternating polarity and random spacing), and how to compute the autocorrelation functions and power spectra.

To further illustrate how spectral density of signals can be computed in the signal flow sense, let us first look at the following example: FIG. 7 is a diagram illustrating an event generator 700 for generating signals, to which the present invention may be applied, according to an illustrative embodiment of the present invention. The two outputs of the event generator 700 are triggered by two events, E- and E_j. When E- happens, the generator outputs an impulse δ-, and when E- happens, the generator outputs an impulse δ_j. Wi, Wj are waveform generators (710 and 720, respectively) that are triggered by δ;, δ_j, respectively. The output of W- and Wj are two waveforms x(t) and y(t).

In order to find the cross spectral density of x(t) and y(t), one can first find, the cross- spectral density of δ-, δ_j, and use the well-known relation:

Ψχ_y(s) = W-(-s)W_j(s)Ψ_δiδj(s)

, where W-(s), W_j(s) denote the bilateral Laplace transform of Wi, W_j, respectively. Ψ_x (s) denotes the cross-spectral density of x(t), y(t), and Ψ_δi_δj(s) denotes cross-spectral density of the impulses δi and δ_j.

It is worth noting that if we connect x(t) and y(t) together in FIG. 7, the resulting signal can be viewed as one signal concatenated by x(t) and y(t). Therefore, the cross- correlation in this case becomes the autocorrelation function. Some procedures and results for finding spectral density functions are summarized as follows. Set up the signal flow chart to represent the Markov process. Find the transfer functions of each node and between each pair of nodes. Find the expectation density [1] or average message of the signal given a certain state happened at t=0 via the following relation:

, where X-(s) is the average message given state "i" happened at t=0, and U-_j(s) is the transfer function from State "i" to State "j". The autocorrelation function of the signal x(t) can be calculated as follows:

Φ_xx(s) = p-W_j(-s)X-(s)

, where pi is the steady state probability of x-(t), which can be evaluated by the residue of the transfer functions. The Power Spectral Density (PSD) can thus be obtained as follows:

Ψ_xxQω) = Φ⁺xx(s) + Φ⁺xx(-s) s=jω For convenience, some basic transformations in graph theory are listed below:

a ab • ►^*-

A description will now be given of signal flow graphs for Variable Aperture Coding (VAC) signals. It seems that some prior techniques for computing power spectra and correlation functions of signals with respect to network and control-systems can be applied to the spectral analysis of VAC signals. This is due to the following reasons. For any VAC signals, the transitions fall on two positions with respect to time: either {nT, T-T/M} or {nT, nT+T/M}, depending on the starting bit polarity of the original data. If the data starts with negative polarity ("-1"), the transitions of the corresponding VAC signal fall only at nT and (nT-T/M) positions on the time axis. However, if the data starts with positive polarity ("+1"), the transitions of the corresponding VAC signal fall only at nT and (nT+T/M) positions on the time axis. Hence, the VAC signal can be viewed as a clock signal whose transitions were perturbed by some random jitter in time. The jitter is random in time because the jitter in the VAC signal only occurs when there is a transition in the original data signal. Since the transitions in the original data are random in time, so are jitters in VAC. With respect to the case of identical pulse trains with random spacing and, particularly VAC, it can be viewed as passing a randomly (in the sense of time. i.e. WHEN a transition is laced at a certain position) placed pulse train with alternative polarity passing through a flip-flop whose transfer function can be characterized.

To set up the signal flow chart for VAC signals, it is necessary to divide the VAC into two cases as described immediately above with respect to (1) (a) and (l)(b).

The first case will now be described. Data starts with "-1", all VAC transitions will fall on nT and (nT-T/M) positions:

Let us define the following four events (states):

E₁ = a positive impulse is placed at nT positions (n = 0, 1, 2, ...)

E₂ = a positive impulse is placed at (nT-T/M) positions (n = 0, 1, 2, ...)

E₃ = a negative impulse is placed at nT positions (n = 0, 1, 2, ...)

E = a negative impulse is placed at (nT-T/M) positions (n = 0, 1, 2, ...)

FIG. 8 is a diagram illustrating a signal flow chart 800 for VAC signals with respect to four specified events, to which the present invention may be applied, according to an illustrative embodiment of the present invention. It is to be noted that the data start with a "- 1". In FIG. 8, the following designations are applicable: x = e^"sT where T is the mean bit duration of VAC signal. This indicates a delay of T in the time domain of the current bit from the previous bit

Tc = * M Clock period of VAC signal, Tc = T/M a ≡ x/2; b ≡ xe^sTc/2; c ≡ xe^'sTc/2

W(s) = 1/s, is the Laplace transform of the unit step signal

X(s) = the resulting VAC signal FIG. 9 is a diagram illustrating a data string 900 and associated pulses 950 corresponding to an exemplary interpretation of the signal flow graph of FIG. 8, according to an illustrative embodiment of the present invention. Assume the data string 900 being sent is as follows: {-1, -1, 1, -1, 1, 1, -1, -1, -1, 1, 1, 1 }. FIG. 9 shows the transmitted data string 900 and the associated pulses 950 placed along the time axis (marked with the corresponding events).

At t = 0, a positive impulse is placed, which is Ei as defined above. Since there is no change in the data string, a negative impulse is placed at time t = T, where T is the data bit duration. This corresponds to E₃. Since there is no change in the data string, a positive impulse is placed at time t = 2T. This corresponds to Ei. Since there is a leading edge in the data string, a negative pulse is place at (3T - T/M), which corresponds to E₄. Following VAC encoding rule, the subsequent impulses can be placed (and matched to the defined events) as shown. The generated pulse train passes through to the waveform generator, W(s) or -W(s), depending on the polarity of the pulse. Positive impulses (E- and E₂) trigger Unit Step Function u(t), while negative impulses trigger -u(t). This simulates "flip-flop" operation that translates the pulse train into an encoded VAC signal X(s).

The parameters "a", "b" and "c" in the signal flow graph can be interpreted as follows:

(a) a = x/2 = e^"sT/2: signal flows from current state to the next state with Vi probability, and a delay of T in time. For example, suppose current state is E_1; it can go to E₃ or E₄, with Vτ probability, respectively. When a state flows from Ej_. to E₃, there is a delay of T seconds. Therefore, a parameter "a = x/2" is placed on the path from E_t to E₃.

(b) b = xe^sT 2 = _e-^(M'1)T/M/2 (Tc = T/M) : signal flows from current state to the next state with Vz probability, and a delay of ((M-1)T/M) in time. For example, suppose current state is E_1; it can go to E₃ or E₄, with Vτ probability, respectively. When a state flows from Ei to E , there is a delay of ((M-1)T/M) seconds. Therefore, a parameter "b = xe^sTc/2" is placed on the path from Ei to E₄.

(c) c = xe^"sTo/2 = e ^(M+1)T/M/2 (Tc = T/M) : signal flows from current state to the next state with Vi probability, and a delay of ((M+1)T/M) in time. For example, suppose current state is E , it can go to Ei or E₂, with Vi probability, respectively. When a state flows from E₄ to Ei, there is a delay of ((M+1)T/M) seconds. Therefore, a parameter "c = xe^"sTc/2" is placed on the path from E to

Similarly, the transition parameters between states can be derived and are shown in FIG. 8. It is worth noting that the signal flow graph set forth is almost identical to the Markov chain setup in the previous derivation of power spectral density of VAC. A description will now be given of Variable Aperture Coding (VAC) functions with signal flow graphs, according to an illustrative embodiment of the present invention.

With the VAC signal flow graph shown in FIG. 8, one can derive the autocorrelation function of the generated VAC signal and, thus, derive the expression for the power spectral density of the VAC signal. Denote Ty (i = 1, 2, 3, 4, j = 1, 2, 3, 4) as the transfer function from node i to node j in the signal flow graph. Using signal flow chart simplification methods stated above, one can find Ty's as follows: Tn: The signal flow chart can be simplified as shown in FIGs. lOA-C. FIGs. 10A, 10B, and 10C are diagrams illustrating a signal flow chart simplification for computing Tn, according to an illustrative embodiment of the present invention. From FIG. 10(B), one can easily obtain the following equations:

From figure 4(c), one can easily obtain the following equations: (lb)

From (la) and (lb), we obtain :

1 i 2 2 π be j-ll = - ₂ [TΓ-„ +T i lJl]^J = bc+_n α + ₍₁ _ _fl ^₎₍₁ _ _6c) (1)

Similarly, Signal flow chart for computing T₂₂ can be simplified as shown in FIGs. 11 A- C. FIGs. 11 A, 11B, and 11C are diagrams illustrating a signal flow chart simplification for computing T₂₂, according to an illustrative embodiment of the present invention. Carrying out similar procedures of that computing Tn yields:

T₂₂ = 22+7I2] = bc_+a ²+ ^a^ (2)

2 (l-_a ^Λ)(l-bc)

Following the same procedures, one can find:

1 ²h

T33 = LT TIS] = bc+_a ²+ ^{a C (}3⁾

2 (l- )(l-bc)

T44 = 2 ή₄+T4²4l = ⁽4⁾

Notice that in the case of VAC, Tn = T₂₂ = T₃₃ = T₄₄. Signal flow chart for computing T₁₂ can be simplified as shown in FIGs. 12A-C.

FIGS. 12 A, 12B, and 12C are diagrams illustrating a signal flow chart simplification for computing T₁₂, according to an illustrative embodiment of the present invention. From FIGs. 12B and 12C, T₁₂ can be readily obtained as follows:

ab ab 2ab τ]ι = + (5a)

(l- a²)(l-bc) (l-a²)(l-bc) (l- _α ²)(l-bc)

Careful examination reveals that T₃ = T₁₂:

ab ab ₊ 1 (l-a²)(l-bc) ^

(l- α²)(l-bc) 2 ι_[ . ][ i ]

T₂₁: Signal flow chart for computing T₂₁ can be simplified as shown in FIG. 13s A-C.

FIGs. 13A, 13B, and 13C are diagrams illustrating a signal flow chart simplification for computing T₂₁, according to an illustrative embodiment of the present invention. From FIGs. 13B and 13C, T₁₂ can be readily obtained as follows:

ι ₌ ac ₊ ac _ 2ac ²¹ (l-α²)(l-bc) (l-_fl ²)(l-bc) (l-α²)(l-bc) ac ήι = {l-a²){l-bc)

(7b) ι-[- ac ab

(l-a²)(l-bc) ][^■ -a²)(l-bc) -]

Careful examination reveals that T ₃ = T₂₁:

T43 ^~ - T43⁺T43^

ac ₊ 1_ (l-a²)(l-bc)

(1 - _Ω ²)(l - be) 2 1 - [ M .][ s ^> ]

^L(l-α²Xl-fc_C)^JL(l-α²)(l-fcc)^J

T₁₃: Signal flow chart for computing T₁₃ can be simplified as shown in FIGs. 14A-B. FIGs. 14A and 14B are diagrams illustrating a signal flow chart simplification for computing T₁₃, according to an illustrative embodiment of the present invention. From FIGs. 14 A and 14B, the transfer function from node 1 to node 3 can be readily obtained as follows:

abc

Let g = —

(l-_α ²)(l-bc)² _r}_{3 =} ^{a +} s l — (a + g)a

l-(a + g)a

1 _{χ 2} 1 2Ω

7-13 - -jTis+Ti - 2 1^ ^{2a +} * (9)

+ g)a

Upon examination of the signal graph, one can find that T₁₃ = T₃₁ = T₂₄ = T ₂. Hence, we have:

r₂₄ = i 2[r -^] = i 2 - l--r(α + g)α oi)

T₁₄: Signal flow chart for computing T₁₄ can be simplified as shown in FIGs. 15A-B. FIGs. 15A and 15B are diagrams illustrating a signal flow chart simplification for computing Tι₄, according to an illustrative embodiment of the present invention.

T₁₄ can be easily obtained from elements (a) and (b) of FIGs. 15 A and 15B as follows:

a²b

Let h =

(l-bc)(l-a²)² b + h

T - (13a) l-(b + h)c b

T²4 = (13b) l-(c + h)b

1 _r 1 2 _ι _ l _r b + h b

^T1 ^~ _Ϊ, (13)

2U i4⁺ri4-l - 2 L l-(b + . « ι,)c ⁺ ι l-(c + , h_M)b

Investigation of the signal flow graph leads to T₁ = T₃₂:

Now the transfer functions left to compute are T₂₃and T₄₁.

T₂₃: Signal flow chart for computing T₂₃ can be simplified as shown in FIGs. 16A-B. FIGs. 16A and 16B are diagrams illustrating a signal flow chart simplification for computing T₂₃, according to an illustrative embodiment of the present invention. T₂₃ can be easily obtained from FIGs. 16A and 16B:

r-r-rJ- c + k

1 23 (15 ) l-(c + k)b

c + k 2 - τ;[T2 ⁺T2 - P t -+ ^■ ] (15)

2 l~(c + k)b l-(b + k)c

Investigation of the signal flow graph leads to T₄₁ = T₂ .

r« ⁼

The above analysis pertains to the case when data starts with "-1". The following analysis pertains to the case when the data starts with "+1".

The second case for setting up the signal flow chart for VAC signals will now be described. Data starts with "-1", all VAC transitions will fall on nT and (nT+T/9) positions:

Let us define the following four events (states):

E-i = a positive impulse is placed at nT positions (n = 0, 1 , 2, ...)

E₂ = a positive impulse is placed at (nT+T/9) positions (n = 0, 1 , 2, ...)

E₃ = a negative impulse is placed at nT positions (n = 0, 1 , 2, ...)

E₄ = a negative impulse is placed at (nT+T/9) positions (n = 0, 1 , 2, ...)

The signal flow chart can be described as shown in Figure 17. FIG. 17 is a diagram illustrating a signal flow chart 1700 for Variable Aperture Coding (VAC) signals, according to an illustrative embodiment of the present invention. The data starts with a "+1".

In FIG 17, the following designations are employed: x = e^"sT where T is the mean bit duration of VAC signal. This indicates a delay of T in the time domain of the current bit from the previous bit Tc = * 9 Clock period of VAC signal, Tc = T/9 a ≡ x/2; b ≡ xe^sTc/2; c = xe^"sTc/2

W(s) = 1/s, is the Laplace transform of the unit step signal

X(s) = the resulting VAC signal

It is easy to find that, because of symmetry, the case where the data starts with "+1" is identical to the case when he data starts with "-1" in the signal flow sense. The only difference is that the parameters "b" and "c" switched paths in the two cases. This does not affect the transfer functions because of the symmetry. Therefore, it will be sufficient to consider only one of the two cases for the calculation of the autocorrelation function and power spectrum density function.

A description will now be given of the Variable Aperture Coding (VAC) Autocorrelation function and power spectral density function, according to an illustrative embodiment of the present invention.

In the previous section, the transfer functions between nodes were obtained. With these transfer functions, one can derive the autocorrelation function of the VAC signals.

From the signal flow graph, one can easily observe the following: state E_1} E₂ pass through a positive transform W(s) to obtain positive components of the VAC signal; and state E_ls E₂ pass through a negative transform -W(s) to obtain negative components of the VAC signal.

The average message following the occurrence of a certain state at t=0 is defined as follows:

Xi(s): average message X(s) given Ei happened at t = 0;

X₂(s): average message X(s) given E₂ happened at t = 0;

X₃(s): average message X(s) given E₃ happened at t = 0; X₄(s): average message X(s) given E₄ happened at t = 0;

Then, the average messages after occurrence of E-, E , E₃, E , can be expressed as follows, respectively:

Xι(s) = T„(s)W(s) + T₁₂(s)W(s) + T₁₃(s)[-W(s)] + T₁₄(s)[-W(s)]

= T„(s)W(s) + T₁₂(s)W(s) - T₁₃(s)W(s) - T₁₄(s)W(s) (17)

X₂(s) = T₂₁(s)W(s) + T₂₂(s)W(s) - T₂₃(s) W(s) - T₂₄(s) W(s) (18)

X₃(s) = T₃₁(s)W(s) + T₃₂(s)W(s) - T₃₃(s) W(s) - T₃₄(s) W(s) (19)

X_t(s) = T₄₁(s)W(s) + T₄₂(s)W(s) - T₄₃(s) W(s) - T^Cs) W(s) (20)

Consequently, the bilateral Laplace Transform of autocorrelation function for the VAC signal can be obtained by:

Φ_xx(s) = _PlW(-s)X₁(s) + p₂W(-s)X₂(s) + p₃[-W(-s)]X₃(s) + p₄[-W(-s)]X₄(s)

= _PlW(-s)X₁(s) + p₂W(-s)X₂(s) - p₃W(-s)X₃(s) - p₄W(-s)X₄(s) (21)

Where p_l5 p₂, p₃, and p₄ are steady-state expectations of E_l5 E₂, E₃, E , respectively.

Since the occurrences of E_1; E₂, E₃, E₄ are random, with equal probability, hence,

Pi = p₂ = P₃ = P₄ = l/(4T) (22)

Therefore,

Φχx(s) = (l/4T)W(s)W(-s){ [Tn(s) + T₁₂(s) - T₁₃(s) - T₁₄(s)]

+[T₂₁(s) + T₂₂(s) - T₂₃(s) - T₂₄(s)] - [T₃₁(s) + T₃₂(s) - T₃₃(s) - T₃₄(s)] - [T₄₁(s) + T₄₂(s) - T₄₃(s) - T4 (s)] } (23) Recall :

T₃₂=T₁₄,

Φ_π(s) = (l/4T)W(s)W(-s)[4T„(s) + 2T₁₂(s) +2T₂₁(s) - 4T₁₃(s) - 2T₁₄(s) - 2T₂₃(s)]

= (l/2T)W(s)W(-s)[2T₁₁(s)+T₁₂(s)+T₂₁(s)-2T₁₃(s)-Ti₄(s)-T₂₃(s)] (24)

Hence, the power spectral density of the VAC signal is expressed by:

=^|w⁽ 7^"ω⁾|^2[2rιιω+ri2ω+r2iω-2ri3ω-rι₄ω-r23^(j)] _s=J-_fl, ⁽²⁵⁾

Note that W(s) is the Laplace transform of unit step function, therefore, W(s) = 1/s. Hence, IW(jω)l² = 1/ω² (26)

The terms in the bracket can be evaluated with Mathematica^®. The results are listed as follows:

(cos[2rcB]-ιsin[2rflj])(-6+17cos[2rffl]+15isin[2rffl])

4(28cos[rffl]-63cos[rffl] +13.(4sin[rffl]-5sin[3rffl])) ^{T ω ~} 272cos[2r«] -257cos[4rω]+5?^'(16t+ 48sin[2rω] - 51sin[4rω])

(30)

(32)

Insert (27) through (32) into (25), a closed form expression for the PSD of VAC signal can be obtained. 1

[ Tn(ω)+Ti2{ω) +T2^^ω) -2Ti3^ -Tu{ω) -T2 ^^{ω '} (³³)

2ω

The normalized Power Spectral Density (PSD) of VAC signal is shown in FIG. 18. FIG. 18 is a diagram illustrating a plot of the Power Spectral Density (PSD) 1800 of a Variable Aperture Coding (VAC) signal, according to an illustrative embodiment of the present invention.

A description will now be given of a Variable Aperture Coding (VAC) decoder, according to an illustrative embodiment of the present invention. The VAC decoder is one of the critical devices in the receiver. The resolution of the decoder directly determines the quality of the decoded signal. In other words, the ability to accurately measure time is a key parameter in determining the Bit Error Rate (BER) performance of the receiver.

FIG. 19 is a block diagram illustrating a Variable Aperture Coding (VAC) decoder 1900, according to an illustrative embodiment of the present invention. The example of FIG. 19 shows an example of the decoding process of a VAC signal, with M = 9, Data Rate R = 128 kbps, and sampling clock of F_c = 144*128 kHz = 18.432 MHz.

The VAC decoder 1900 includes a local clock generator 1910, a counter circuit 1920, a window decision circuit 1930, an error correction logic circuit 1940; and an output logic circuit 1950. The local clock generator 1910 generates a local clock signal whose frequency is an integer having a value that is a multiple of the transmitted data rate. The counter circuit 1920 is triggered by the edges of the demodulated VAC signal. The window decision circuit 1930 for VAC bit width that is based on the count registered by an output of the counter circuit 1920. The error correction logic circuit 1940 corrects detected VAC width errors based on the characteristics of the VAC signal (for example, two consecutive MT_c's are not allowed in the sequence). The output logic circuit converts the varying width VAC signal into the original data stream.

The decoding process shown in FIG. 19 is described as follows:

(1) The received VAC is an M = 9 (valid widths are "8T_C", "9T_C" and "10T_C", respectively) VAC signal. The data rate is at 128 kbps, and the sampling clock is 18.432 MHz, which an "8T_C" consists of 128 clock periods, a "9T_C" consists of 144 clock periods, and a "10T_C" consists of 160 clock periods. The counter is reset by a VAC edge. (2) When an edge arrives, the decoder 1900 first determines if the edge is an valid edge. This is accomplished by examining the output of the counter 1920. For example, between valid VAC edge, there shall be at least 128 clock periods. If an edge arrives before the couter 1920 counts to 128, the edge is a jitter rather than a valid VAC edge, and consequenctly, this edge is ignored and the counter 1920 continues counting. (3) When a valid VAC edge is detected, the counter 1920 gets reset and its output is sent to the window detection logic circuit 1930. For instance, if the counter 1920 output is less than 140, the detected bit width is considered as an "8", while a "10" is generated when the output of the counter 1920 is greater than 150. Any output of the counter 1920 in between 140 and 150 causes the generation of a "9".

(4) Then the VAC signal is sent to an error correction logic circuit 1940. If there is any invalid states, for example, two consecutive "8T_c's", or two "8T_c's" separated by a number of "9T_c's", the the error correction logic circuit 1940 corrects the error. (5) If the output state is valid, the decoder 1900 logic gives an output according to the encoding rules. For example, an"8Tc" casues a "-1" to "+1" transition.

It is obvious that the accuracy of the first VAC edge detection is critical. In order to ensure the correct detection of the first VAC edge, a training sequence (e.g. a "100001" sequence) may be used to train the receiver.

It can be seen that the higher the sampling clock, the more accurate the detection. However, a higher clock rate may not be practical for higher data rate transmission. The rule of thumb is that the sampling clock should be chosen to meet the BER requirement for the successful decoding of the signal. A theoretical analysis will now be given of a Variable Aperture Coding (VAC) signal, according to an illustrative embodiment of the present invention. The analysis will begin with a vector representation of VAC signals. As discussed herein, a VAC signal may be viewed as a concatenation of rectangular pulses with different widths. In order to calculate the error probability of VAC baseband signals in Additive White Gaussian Noise (AWGN) channel, one way is to represent the signal with an orthonormal basis, and calculate the Euclidean distances between the signal constellation points, and use the well-known relationship to calculate the theoretical Bit Error Rate (BER):

where <2(«)is the complementary error function (6.10) d is the Euclidean distance σ is the noise power

Since d is a function of bit energy E_b, BER vs. Eι N₀ relationship may be obtained from Equation (6.10).

FIG. 20 is a diagram illustrating an orthonormal basis 2000 for a Variable Aperture Coding (VAC) signal, according to an illustrative embodiment of the present invention.

ι(t), <p₂(t), and φ₃(t) can be mathematically described as:

rftμVϊAMT PtM-w t);

Vl7ϊ7Pτc[t-(M-l)Tc]; φ₃(t)= V 7P_Tc(t-MTc);

where PkTc(t) s defined as in Section 1.

It is obvious that in φ,-φ2-φ₃ coordinate, VAC signal (M-1)T_C (Si), MT_C (S₂), and (M+1)T_C (S₃) are positioned at

(A^(M -l)Tc, 0, 0), (A^(M -ϊ)Tc, A Tc^~, 0), and (A^](M -l)Tc, A-sJTc, AΛ[TC), respectively.

FIG. 21 is diagram illustrating a vector representation 2100 of Si, S₂, S₃, according to an illustrative embodiment of the present invention. A description will now be given of bit/symbol error probability, according to an illustrative embodiment of the present invention.

For convenience, we introduce the following notations and definitions:

8M-_I= bit energy of signal Si bit energy of signal S₂ 8M+I= bit energy of signal S₃

TM-I,M= correlation coefficient between signal S and S₂ correlation coefficient between signal Sj and S₃

orrelation coefficient between signal S₂ and S₃

Referring to FIG. 22, one can easily calculate the bit energy of signal set Si_, S₂, S₃ and the correlation coefficients between each signal pair. FIG. 22 is a diagram illustrating a Variable Aperture Coding (VAC) signal 2200 approximated as a Continuous Phase Modulation (CPM) signal with (π/M) phase variation, according to an illustrative embodiment of the present invention.

ε_M-ι = (M-l)A L ²TT_c .; r ε._M _= ΛM ΛA Λ 2^zrTp_c .; ε __M+ι _ =

(6.11)

Hence, the Euclidean distance d-_{, j} between each signal pair can be computed via the following formulae:

^d =

= £- +εj -2r -J 0 (6.15)

Using Equation 6.10, the BER of the VAC signal in the AWGN channel may be expressed as:

Another way to look at the VAC signal Bit Error Rate is to approximately represent the VAC signal as a Phase Modulation (PM) signal whose angle between signal vector pairs is confined to (π/M), and the vector "length" is varying. This can be depicted as shown in FIG.22.

The Euclidean distances can be calculated as follows:

d²M-l,M = (2M-T)A²T-2A²T^M(M -1) cos (6.17)

d² _M,M+_\ = 2M +1)A²T-2A²T^]M(M +1) cos (6.18)

d² _M-i,M+l=^:(^2M^A²T-2A²T-l(M-T)(M +l)cos(— ) (6.19) Considering the a priori probabilities of M-l, M, and M+l, we obtain:

"symbol t'b

1 ³ 2 where εj v ≡- ∑l jil

^J ι=l

Both Equations (6.15) and (6.20) give expressions for calculating BER versus Eb/Νo. Even though they look different, however, the basic concept of deriving the expressions of the BER performance is the same. That is, based on the minimum Euclidean distances between the signal pairs. Therefore, close results from (6.15) and (6.20) are expected. FIG. 23 is a diagram illustrating a plot of a simulation result 2300 comparing the BER performance derived from Equations (6.15) and (6.20), respectively, according to an illustrative embodiment of the present invention. The result shows the BER performance with M=9. From the simulation result, one can easily find that at BER=10^"6, the calculated results are very close to each other. A close-up look at the results show the difference between the two is around 0.2 dB. As mentioned above, this result is expected.

As alluded to earlier, at a fixed data rate, a different M will have an impact on the spectral occupancy of the VAC Power Spectral Density (PSD). It was expected the bandwidth occupancy decreases when M increases. However, increasing M implies the shrinkage of the aperture that, in turn, implies the degradation on the BER performance. FIG. 24 is a diagram illustrating a plot of simulation results 2400 corresponding to a

Bit Error Rate (BER) performance comparison for M-VAC (M = 7, 9, 11), according to an illustrative embodiment of the present invention. The BER vs. SΝR curves are generated by Equation (6.15). From FIG. 24, one can find the following two facts. First, when M increases, to maintain the same BER performance, power penalty has to be paid. In other words, for higher M, higher SNR is required to maintain the same BER as that uses lower M values. Second, the SNR requirement difference between M = 7 and M = 9 is around ldB. The SNR requirement different between M = 9 and M = 11 is also ldB. Even though it seems encouraging that, for a fixed data rate, bandwidth can be reduced through increasing M, the trade-offs need to be investigated.

The obvious trade-off is the power penalty that has to be paid. As indicated above, a higher SNR will be required to achieve the same BER for a higher M value.

A description will now be given of bit/symbol error probability, according to an illustrative embodiment of the present invention.

In general, the Euclidean distance dι_{ι j} between each signal pair can be computed via the following formulae:

d² _j =

ε.- + ε - -² £:£, (34)

Using general expression for calculating bit error probability,

is the noise power at receiver end, we can derive the equation for calculating bit error probability using our encoded data format as follows:

th ^symbol ⁼∑ Σ-P(^z ' J)P(J) where P(i\j) is the probability of deciding i bit j i≠j th th givenj bit sent. P(j) is the probability of j bit being sent by the transmitter.

In the case of encoded "(M-l) - M - (M+l)" waveform, we use i, j = M-l, M M+l to index the error probability. In this case, the symbol error probability is the same as the bit error probability since each symbol is composed of one bit. Hence, the symbol error probability of VAC signal can be expressed as:

P,_ymboi =∑∑P(i i J PU) = ∑ ∑P(i i J)PU)

J I ^«-'."'^{M +1} ,-M-l, ,ΛJ+l

= ∑{P(M-l\ j)P(j) + P(M \ j) + P(M +l\ j)}

= P(M -1 \M)P(M) + P(M -UM + T)P(M + Ϊ) + P(M \M -Ϊ)P(M -ϊ) +P(M \M +Ϊ)P(M +T)+P(M +1IM -1)P(M -1)+P(M +1IM)P(M)

= [P(M -T)+P(M)]P(M -1\M ) + [P(M -Ϊ)+P(M +1)]P(M -1IM +1) + [P(M) + P(M + T)]P(M I M + 1)] (35) - X

Note the fact that P(i I j) = P(j I /), and P(i I j^') = -* = Q Xlλ

2σ v

With the P(i\j) defined above, we immediately obtain the conditional probabilities:

P(M-l), P(M), and P(M+1) arc a priori probabilities of "M-l", "M" or "M+l" being sent.

It is obvious that a priori probabilities of "M-l", "M" or "M+l" being sent is:

P(M-l) = 4/16=1/4; P(M) = 8/16=1/2; P(M+1) = 4/16=1/4. (39)

Substituting equations (3) through (6) into equation (2), we obtain the symbol error probability of VAC: symbol b

A description will now be given of methods to increase Areal density by coding, according to an illustrative embodiment of the present invention. The description will begin with a discussion on coding in PRML Magnetic Recording Systems.

As mentioned earlier, modulation codes that are used in almost all contemporary storage products belong to the class of constrained codes, which translate random input sequences into sequences that satisfy certain constraints. Two types of constrained codes are of interest in PRML magnetic recording channels: codes for improving timing and gain control and simplifying the design of the Viterbi detector for the channel, and codes for improving noise immunity.

Codes of the first type impose run-length limitations (RLL) on sequences of recorded symbols. Various codes of this type are usually of rate 8=9.

Codes of the second type eliminate some of the possible recorded sequences thus increasing the minimum distance between those that remain. Matched-spectral-null codes are high rate single-track codes, which provide both run-length constraints required for timing and gain control and improved noise immunity. These codes have spectral nulls that match those of the channel. The rate 1/2 code with a DC-null, known as a bi-phase code, is a Matched Spectral Null (MSN) code for the (1 - D) channel that provides an increase in the SNR of approximately 4:8 dB. Higher rate MSN codes with a DC-null provide an increase in the SNR of approximately 3 dB for the (1 - D) channel.

A description will now be given corresponding to the increasing of the Areal Density by Track Narrowing, according to an illustrative embodiment of the present invention.

In disk recording systems, Areal density (density in bits per unit area) can be increased along the radial direction by narrowing the track width. A result of the track narrowing is a loss in the SNR. This loss can be compensated for by employing a code that improves noise immunity on each track which, in turn, decreases linear density. Ignoring several important technology issues, such as ITI, narrow-track width head design, and position-servo accuracy, allows a simple estimation of the overall Areal density increase achievable by using this approach. Suppose one begins with a channel using a rate r₀i_d code on a nominal track width of W₀i_d- If the track width is decreased to W_new, the amplitude of the read back signal, - B , as well as the power of the noise due to the medium, σ, would be reduced by approximately a factor of W₀i_d / W_ne - corresponding to an SNR loss of

SKBdu- SNK_11M = Ml-g^fV- W-JclB-

If a rate rnew code is used on each track to compensate for the incurred SNR loss (and provide the same features, such as RLL constraints, as the old code), the overall Areal density increases by a factor of:

^rnew v W ₀ld rold W_new

The potential gain indicated by the above equation can be obtained provided high rate codes can be designed. For example, let the nominal track width of a benchmark (1 - D) PRML system employing a rate 8/9 code be W_ol(j. Suppose that the original track is divided into three sub- tracks, each of width W_new = ^W°^ld ■ Suppose that the resulting SNR loss of 4.8 dB is

recovered by applying the rate 1/2 bi-phase code on each sub-track replacing the old code. The overall areal density of the original system this way increases by a factor of

without any loss in performance. A result of track narrowing is also the appearance of LTI, but existing codes for improving noise immunity in PRML systems, such as MSN codes, are not designed to account for it. The effects of ITI may, however, be alleviated through the use of multiple-head systems simultaneously writing and reading a number of adjacent tracks. Straightforward coding extensions in which transition signaling and redundancy in time are used for minimizing transition activity. The present invention employs two-dimensional codes with redundancy in both time and space for providing a capacity increase. These two- dimensional codes can be unrolled in either space or time in order to obtain new one- dimensional codes in the other dimension. Run-Length Limited (RLL) and phase- modulation techniques that use the extra freedom in the time domain are used for obtaining better codes for low power. Redundancy in amplitude can then be combined with time redundancy for obtaining other two-dimensional codes for low-power and high capacity.

A description will now be given of coding in space and time, according to an illustrative embodiment of the present invention. The description will begin by address coding in time.

Time encoding requires that the data be transmitted in packets but this is not a big constraint on global buses where there is now a clear trend of transmitting bursts of data for improved throughput. FIG. 25A is a diagram illustrating a data-packet 2500 with redundancy in space, according to an illustrative embodiment of the present invention. FIG. 25B is a diagram illustrating the data packet 2500 of FIG. 25A with redundancy added in time, according to an illustrative embodiment of the present invention. With the transmitted data arranged into K_t -word packets where each word is initially K_s -bit wide, the same coding techniques that are used for redundancy in space can now be applied in time. In particular, Bus-Invert and Limited-Weight codes with redundancy in time and transition signaling can use extra transfer cycles for encoding the K_t bits that are successively transmitted over each bus line in order to minimize the number of l's transmitted and hence the number of transitions. For example, if K_t = 4, then the equivalent Bus-Invert time encoding would look at the number of l's that are to be transmitted over each bus-line. If the number of 1 's for a bus-line is greater than K_t 12 = 2, then the K_t = 4 bits on that line will be inverted and this inversion will be signaled by a 1 in an extra (5th) transfer cycle. Otherwise, the K_t = 4 bits are transmitted as they are and the extra (5th) bit will be 0. The computation of the redundant bit needs to be done for each of the K_s bus-lines, in series or in parallel. FIGs. 26A, 26B, 26C, and 26D are diagrams illustrating the transitions on the 4-bus lines, according to an illustrative embodiment of the present invention.

FIG. 27 is a diagram illustrating Variable Aperture Signaling 2700, according to an illustrative embodiment of the present invention. By simple probabilistic reasoning it can be shown that with the same amount of redundancy, time encodings will have the same average bandwidth savings as the equivalent space encodings. As an example, transmitting the following 4-word packet takes 4 cycles and generates 8 transitions over a 4-line bus (with transition signaling, see FIG. 26A.). It is assumed that the 4-bit words are arranged in columns and are transferred from left to right:

i ϊ i o

0 t & ϋ

Ϊ α ti i

With redundancy in space or time (Bus-Invert with transition signaling), the number of l's (hence transitions) is reduced to 7 at the expense of an extra bus line (the first row on left) or an extra transfer cycle (the last column on right, see also figures 26b. and 26c):

& 1 ύ & O Q ϋ 1.

1 a t &

0 ft 0 0 3 0 0 1 i 1 0 1

& 0 D 1 β H i

Unfortunately, the peak power and simultaneous switching noise, which with space encoding were also reduced, are not affected by time encoding as it is still possible to have all bus lines switching simultaneously. Ultimately, choosing the use of space or time encoding lies with the designer since the techniques are similar but the trade-offs involve either extra dimensions or extra transfer cycles.

ΪABUES 'ϊ*fo-~ιi-&-flEk)π-l iϊ&Uϊi.u---* 4 lafora-iitlaπ kits i&<&Miύ isβ*. S-i£ώdsa_j-.i-

A description will now be given of coding in both time and space, according to an illustrative embodiment of the present invention.

If bandwidth needs to be further reduced, redundancy in both space and time can be used. Two-dimensional coding is a two-step process and there is a choice whether to apply redundancy first column- wise (in space) and then row- wise (in time), or vice- versa. The same average bandwidth reduction is obtained in both cases but lower peak simultaneous switching noise can be obtained by encoding first in time and then in space. For the previous example the number of l's can be reduced to 6 with two-dimensional coding (see also FIG. 26D.), column-wise encoding is done first on the left, row-wise encoding is done first on the right:

Table I shows the codewords of the smallest two-dimensional low-bandwidth codes, with column- wise encoding followed by row- wise encoding (or vice- versa, in parentheses). There are 16 such codewords, one for each of the 2 X 2 possible patterns of l's and O's. Two extra codebits are used in space and two extra codebits are used in time. The average bandwidth is reduced by 31% (compare with less than 25% for one-dimensional Bus-Invert. Table II shows two other two-dimensional codes. There is an extra 9^th bit that encodes in time, the space codebits (or vice-versa, in parentheses). The average bandwidth is reduced by 34%, slightly better compared to the previous codes. Again, there is a choice whether to encode column-wise followed by row-wise, or vice-versa (in parentheses) A useful application of such two-dimensional encoding is the generation of new one-dimensional codes by unrolling the two-dimensional code about one dimension. For example, by unrolling the two-dimensional code in Table I, the one-dimensional obtained code is a semi- perfect 2-Limited- Weight Code of length 8. Similarly by unrolling the code in Table LI the obtained code is a semi-perfect 2-LWC of length 9 (by-definition a semi-perfect M-LWC of length N includes the all-zeros pattern, all the N-bit patterns with 1, 2, ... M - l l's, some N-bit patterns with M l's and no other patterns). This is an important result for several reasons. On such reason is that the algorithmic generation of codes for low-bandwidth is intrinsically hard in the general case. Another such reason is that such unrolled two- dimensional codes provide a compromise between the two extremes of Bus-Invert (minimum redundancy, one extra dimension and one-hot encoding (minimum transition activity, one transition per cycle ), and offers another practical design alternative.

A description will now be given of an implementation of coding in space and time, according to an illustrative embodiment of the present invention. The techniques used for computing the code bits in space and time are similar, which means the circuits can also be similar. A key element is the efficient implementation of a majority voter, and this can be done in a digital or analog fashion.

FIG. 28 is a diagram illustrating a two-dimensional encoder 2800 for encoding in space and time, according to an illustrative embodiment of the present invention. The encoder 2800 is for the two-dimensional code in Table LI (time followed by space encoding).

For time redundancy, there are also issues related to accessing the entire data packet while encoding and decoding. For encoding, the entire packet must be stored and accessed, but decoding can be done on the fly if the extra code bits are transmitted before the data bits.

The majority voter in this case is an AND gate and it can be seen that although they are conceptually similar, time encoding is more expensive than space encoding because it needs to access the entire data packet at once.

A description will now be given of coding in amplitude and time, according to another illustrative embodiment of the present invention. When we discussed redundancy in time, we implicitly assumed that the time domain has exactly the same "integer" restrictions as the space domain but this needs not be the case. While the number of dimensions must always be an integer, time is continuous and we can use the extra freedom for building more bandwidth efficient codes. A description will now be given of modulating in time, according to an illustrative embodiment of the present invention.

To understand how this can be done, we must analyze the lower bounds on timing values on a bus line. A first bound has to do with the minimum possible width T_mi_n for a pulse that can be detected. This minimum width is determined by the decoder clocking speed and inter-symbol interference. Another bound is given by the maximum resolution Δ T with which the exact position in time of a transition can be determined.

This resolution depends on the amount of noise on the bus and is very much implementation dependent. Herein, we implicitly presumed that the two bounds are the same and equal to the "bus cycle". In most cases, the resolution ΔT can be much smaller than the minimum pulse width T_πύn (see FIG. 27), hence we can use the position in time of a transition for encoding several bits per transition. By considering a "virtual" cycle equal to Δ T, and T_mi_n as a multiple of Δ T, then in terms of transition signaling the lower bounds translate into the necessity of having a certain number of O's between any two consecutive l's (the minimum number of O's will determine T_mm ). Similar constraints appear naturally in magnetic recording devices and the coding community has developed the class of Run- Length Limited (RLL) codes for improving code efficiency when Δ T<T_min- For example, if T_mi_n = 3 XΔ T, then the very popular variable-length RLL (2,7) code can be used (a RLL(d; k) code has at least d and at most k O's between any two l's):

With the RLL(2,7) the average number of transitions is only slightly reduced over the un-encoded case, but for a given Tmi_n the transfer time is reduced by 50% (hence the energy- delay product is also reduced by 50%).

!'!;l!ώ' .

Typical values for T_mm = 80-100nsec and ΔT is 20-25 nsecs. This theoretically enables the use of RLL codes with large values for d and k (e.g., d = 16), but for such values RLL codes are impractical to implement. Phase modulation (see FIG. 27) is an inefficient (from the information theory point of view) RLL(d; k) code with d = (T_mm /Δ T ) - 1 and k = (d +2) X (p - 1) which encodes several bits of data in the position of a transition.

In such a scheme, each transition can have one of p different positions (e.g. p = 5) and the minimum pulse width is T_mi_n. With" p " positions we can transmit log₂ p bits per transition, and if "p" is large there is a potential for important capacity increase (in the un- encoded case the average rate is 2 bits per transition). From the low bandwidth coding point of view, phase modulation can be viewed as one-hot encoding with transition signaling with the extra constraint on T_mi_n which translates into a necessary string of d O's in-between any two one-hot code words. For example, if Tm = 10 X ΔT there will be a string of d = 9 O's between each pair of one-hot code words. Let's analyze the coding efficiency of the phase modulation scheme. For every log₂ p transmitted bits we have to transmit (d + p) Δ T periods. The code rate will then be:

log₂p rate = - —

(d + p)

Generally T_mi_n and Δ T are given and hence d is also given which means that the only variable is p. The rate is maximized at a value p_opt obtained by differentiating (A):

p₀pt X (ln2 log₂ op_t - 1) =d ^■ -(B)

FIG. 29 is a diagram illustrating a plot of a variation 2900 of p_opt versus d, according to an illustrative embodiment of the present invention.

Table III shows the values of p_opt (rounded to nearest integer) for different values of d, as well as the number of bits per transition and the average savings in the number of I/O transitions. FIG. 29 shows the growth of p_opt with d. Although very large values of d are not practical anyhow, it is interesting to note that the growth of p_opt with d is less than linear, hence the power savings are not impressive as d increases (see also table V). Extra bandwidth savings can be obtained by realizing that we may not want to use p_opt but a somewhat larger value. A description will now be given of modulation in both amplitude and time, according to an illustrative embodiment of the present invention.

As in the case of space and time, we can also define two-dimensional low-power codes that use modulation in both amplitude and time. By modulating the signal amplitude with 2X p_voit levels distanced Δ V apart, in each cycle we can transmit log ₂ p_voit bits in addition to the log ₂ p_ti_me bits transmitted in time for each transition. FIG. 30 shows such a two-dimensional encoding with p-i_me = 5 and p_voit = 5 that can transmit 10 bits per transition for savings of 80% in transition activity. FIG. 30 is a diagram illustrating a two-dimensional encoding 3000 of a signal in both amplitude and time, according to an illustrative embodiment of the present invention. A description will now be given of an implementation of amplitude and time encoding, according to an illustrative embodiment of the present invention.

By virtue of working on the PRML code, the rate at which the data is written has been reduced. The 4T, 5T and 6T waveforms must now be recovered. The read head signal is detected and passed through a limiting amplifier to suitable level and then decoded using a quadrature detector that discriminates between the three instantaneous Phase/frequency components. Alternate decoding techniques like IQ demodulation or a Phase Locked Loop (PLL) based detector will also work fine. The decoder consists of a high frequency clock running at 100 times the VAC encoded signal, lx clock is generated by dividing down the high frequency Master clock. Every edge of the incoming signal resets the divider counter. By adopting this architecture there are two benefits as follows: (a) the master clock need not be of exceptional stability as the drift in the master clock is divided down by a high value divider, whereby relaxing the stability requirement for the Master Oscillator; (b) the correction happens on a symbol by symbol basis. Transition jitters are a problem for detection closely spaced transitions and can be taken care of by providing aperture variation in the VAC -PRML coded signal that is magnitudes higher than jitter values. In the case of 4T, 5T, 6T encoding, the difference between adjacent the edges is 20% at a minimum and hence can compensate for the transition jitters of the order of magnitude of 6-10% without sacrificing the SNR. In a VAC-PRML system, detector at the receiver counts the number of cycles between transitions to decide if the data was a 4T, 5T or a 6T using a divided down high frequency clock. It is possible to set soft decision thresholds to fine-tune the system. hi order to reconstruct the data clock for the reconstructed data stream from the VAC decoder, a multiply by 5 PLL is used. There is also potential improvements to the SNR due to the possibility to introduce narrow Band pass filters in the read channel as the VAC- PRML signal occupies a very narrow band width.

Magnetic media is a 2-dimensional space and hence if we need to increase capacity, there are only 4 ways that this can be done, assuming constant spindle speed: (a) vary aperture; (b) vary amplitude; (c) add multiple heads; and (d) add multiple tracks.

Hence, to further increase capacity, it is possible to encode IT, 2T and 3T minimum distances as amplitude combinations of 4T, 5T and 6T. For example, IT would be 4T with amplitude "x", 2T would be 6T with amplitude "y" and 3T would be 5T with amplitude "z". In this case, the read channel cannot be subjected to limiting but the existing PRML read channel with non-limiting property would be sufficient. Thus, the payload per transition can be increased to 8 bits per transition. This will further allow the transition flux reversals to be lowered by a factor of 3. There are many possible implementations for a phase modulation scheme. For encoding and decoding it uses a PLL with a (p- d)-stage ring oscillator that can generate the p necessary phases and guarantees the minimum d zeros between two transitions.

It is clear from the BER characteristics (FIG. 30) as well as from a Spectral efficiency perspective that VAC is a good candidate for Magnetic recording. Applied to HDD recording, VAC helps to store more data in a given track by essentially representing the incoming random bit stream with (+/-) 3 distinctive widths. This process allows for more transitions to be applied in a given area. Further, as VAC will allow transitions to happen only at specific separation from a nominal bit width, the intra pulse interval does not carry any information and additional bit streams can be sent between the flux transitions representing adjacent encoded bits. In a traditional system, the bit patterns are random and it is not possible to utilize the area between the bit transitions, due to ISI issues. The critical detection issue is whether the width changes can be transferred to a medium and be read consistently. With a transition parameter of 8.1nm and a minimum of 50.4nm between flux changes, there is enough tolerance to clearly identify the three distinct widths about a nominal bit boundary. Also the power spectral density of the PRML signal with VAC will be spectrally compact occupying a very narrow band width and at half the rate of the data being stored. This greatly eases up the post processing computational horsepower required for realizing a high capacity HDD. Another important benefit of using the VAC encoding scheme is that clock recovery becomes very simple, leading to simpler implementation of multi stream data storage. A direct consequence of increasing the bits/inch (linear track density) using VAC, is in the ability to realize higher Areal density with the current state of Magnetic material, Servo and Coding technologies.

FIG. 31 is a diagram illustrating a plot of a decoded waveform 3100, according to an illustrative embodiment of the present invention. In the plot, the X-axis represents the elapsed time and Y-axis represents the amplitude of the detected waveform. In the simulation to which the plot corresponds, a 3-6-9 waveform was used. The section marked "D" is comprised of all "6" waveforms. Prior to introducing the 3-6-9 waveform, around 500bits of all"6" is sent in order to stabilize the DC offsets. The 3-6-9 waveform clearly shows an amplitude variation and this is due to the destructive influence of ISI. hi the detection of VAC demodulated signals, the detected signals are subjected to full wave rectification, whereby the negative peaks are folded over. The width WI between C^Λ and A (a 9 to 3 transition) signifies a width of WI. The ISI effects are not as pronounced for the peak A and hence it has an amplitude which is labeled Al. The next width W2 is the peak separation between A and B (a 3 to 6 transition). The ISI effects on B make its amplitude smaller than Al. The amplitude of B is designated as A2. Similarly, the transition between B and C produces a width W3. The amplitude of C is less than B and is designated as A3.

It can be easily surmised that by using a PRML type of detector, we have combinations of 3 amplitudes and 3 widths to make a correct decision. Error correction is easy, as we do not even have to detect C in the assigned slot, as the amplitude is so small. If no amplitude is sensed, it can automatically be concluded to be a "C".

The process uses a combination of amplitude and width to correctly decode the symbol. One could also look at amplitude alone and decode the expected width. Since in the demodulation process of VAC, the symbol widths are critical in determining the decoded data, the amplitude information can be directly mapped to give the demodulated VAC data stream.

Of particular significance is the ability to synthesize varying amplitudes at the receiver by manipulation of the encoded data stream at the transmitter. When applied to optical recording media, the widths WI, W2 and W3 can be made to have various pit depths by varying the intensity of the lasing device. Let the pit depths be designated as Dl, D2 and D3. We can now have 9 combinations: WI with Dl, D2, and D3; W2 with Dl, D2, and D3; W3 with Dl, D2, and D3. These combinations are shown in FIG. 32, which is a diagram illustrating various encoding combinations 3200 for a waveform 3210, according to an illustrative embodiment of the present invention. Thus, by choosing a combination of W and D, we can encode the incoming data stream at 3 bits/symbol. Extrapolating this result, we can use this method to store 15Gb information on a standard DVD disc than can normally hold only 4.7Gb, without the need to go to a blue laser. By including more widths and depths, a higher level of bandwidth efficiency is possible. To satisfy the storage capacity requirements of HDTV, some have proposed dual- layer storage on both sides of the optical disc. But such a tactic would increase manufacturing costs significantly, while doing nothing to bolster data transfer rates. The present invention provides a novel optical encoding technique, referred to herein as pit-depth modulation, that supports HDTV using single-layer recording on just one side of a disk even when using red-laser technology. Moreover, transfer rates are directly proportional to storage capacity, and quite significantly, the new encoding technique can be readily incorporated into existing CD/DVD production lines.

Many proposals for future DVD systems look to the blue laser as the enabling technology for high density. However, manufacturing tolerances for 120-mm DVD disks are already extremely tight. Blue lasers will surely compound matters and may necessitate new mastering and duplication equipment to meet acceptable quality standards. Even this new capital investment may not be sufficient to meet HDTV performance and capacity requirements

The capacity of the DVD drive increases by a factor proportional to the reduction in laser spot size. Thus, the long-sought-after blue laser, with its short 410nm wavelength, promises to increase the storage capacity by 2.4 times over similar DVD drives employing 635nm wavelength red lasers (635nm /410nm), which means that a DVD drive, even one with a blue laser inside, can not satisfy the storage requirements of HDTV (at least not with a single-sided, single layered disk). Assuming blue lasers were currently available, based on present DVD specs, it would take two layers to satisfy HDTV requirements; further, use of an ordinary red laser, a double-sided, four-layered disk would be required. Both of these solutions have drawbacks for end-users and manufacturers. Dual-sided disks have to be flipped over to accommodate a feature film. Dual-layer encoding requires an additional process step, adding significantly to the manufacturing cycle time.

The limitations of current read-only optical systems are a result of the pit-length encoding method in which streams of data are pressed into "pits" of variable lengths and fixed depths. Data is read using photoelectric cells, which detect the binary output of either a "pit" or "land" area. Pit-depth modulation, in contrast, is essentially a three-dimensional approach to data-stream recording: the depth of the data pit varies, while the length is fixed. The photocell detects the variable pit depths and produces a graded, rather than binary, output. An optical drive employing pit-depth modulation can support two-plus hours of high- resolution video, or at least 15 GB of data, while meeting the data transfer requirements of 19 Mbit/sec as defined in the Grand Alliance proposal for HDTV.

Moreover, this specification can be met using current red laser (635nm 650 nm) diodes. Of course, shorter wavelength lasers would provide additional benefits but they are not a prerequisite for production of an HDTV-ready drive employing this encoding technology. A significant point, especially for manufacturers who have invested in DVD production equipment, is that the new encoding technology requires only relatively minor changes to mastering benches. Also, existing replication gear can be used to produce the new media within accepted cycle times.

Based on laser and track-pitch specifications of current DVD drives, optical drives using fixed-length, variable-depth data pits can easily satisfy HDTV requirements. Data capacity and transfer speed can be enhanced further by modulating pit depths more finely and by shortening the lengths of the individual pits. The former can be achieved by improving the mastering process.

In conventional optical systems, streams of data are pressed onto disks in variable- length, fixed-depth pits (top). In the method according to the present invention, pit depth varies within a fixed pit length.

Current CD and DVD players cannot discriminate among pits deeper than a quarter- wavelength, because the reader response is sinusoidal in nature; as pit depth increases from zero to quarter-wavelength, the light reflected from the disk decreases from a maximum to minimum. Thus, an alternate readout method is needed, one that does not bottom out when pit depths exceed a quarter- wavelength.

Pits are usually shortened to improve performance. However, if the pits are too short, signals from neighboring pits will interfere with each other. This intersymbol interference can be pre-compensated for in the mastering process through slight adjustments to pit depth. Intersymbol interference stems from the pits being smaller than the laser reading spot: when the laser spot is centered over a pit, the diffracted light is affected not only by the pit of interest, but by the adjacent neighbors as well. By slightly adjusting each pit up or down from its nominal depth (the amount and direction is dependent on the depths of the neighboring pits), the reader can be made to produce the same electrical response for a particular depth pit that is independent of its neighbors. In practice, each reader head design varies slightly in the nature of its laser spot, so compensation for intersymbol interference will need to occur in both the mastering and readout stage (i.e., pre- and post-compensation). Post-compensation can be accomplished in the disk reader by using digital signal processing techniques to properly filter or equalize the data signal based on knowledge of intersymbol interference from varying pit depths. FIG. 33 is a flow diagram illustrating a method for storing a random bit-stream on a storage medium, according to an illustrative embodiment of the present invention.

Transition widths, for inclusion in a pre-specified set of transition widths, are selected based on a capability to reduce Inter-Symbol Interference (ISI) and/or Inter-Track Interference (LTI) during a read operation of a VAC encoding from the storage medium, to increase a number of bits per transition in a given storage area on the storage medium, and/or to reduce a Bit Error Rate (BER) of the VAC encoding during a peak detection operation performed on the VAC encoding (step 3305).

The random bit stream is represented by a constant amplitude, varying pulse-width, VAC encoding having a plurality of pulses that are separated using only the transition widths included in the pre-specified set of transition widths (step 3310). It is to be appreciated that step 3310 includes the step of translating other pre-specified transition widths as amplitude combinations of the transition widths included in the pre-specified set of transition widths (step 3315). The VAC encoding is transmitted along a data channel for storage on the storage medium (step 3320). It is to be appreciated that step 3320 may include the step of transmitting other VAC encodings along the data channel, within an intra-pulse interval of the VAC encoding, for storage on the storage medium (step 3325).

In conclusion, it may be stated that employing VAC encoding over existing PRML channels allows realization of higher areal density through its square like bit signature, coupled with the capability to utilize the unutilized intra- transition space to send additional data streams. Conservative estimate shows a 3:1 improvement in Capacity. VAC fundamentally alters the packing density of the original bit stream to allow 3 times more data to be packed in a given area, without having to use newer materials and newer head technologies.

Although the illustrative embodiments have been described herein with reference to the accompanying drawings, it is to be understood that the present invention is not limited to those precise embodiments, and that various other changes and modifications may be affected therein by one of ordinary skill in the related art without departing from the scope or spirit of the invention. All such changes and modifications are intended to be included within the scope of the invention as defined by the appended claims.

Claims

CLALMS:

1. A method for encoding a random bit stream in two-dimensions for storage on a storage medium, comprising the steps of: encoding (3310) the random bit stream using Variable Aperture Coding (VAC) so as to generate a constant amplitude, varying pulse-width encoding that represents the random bit stream by a plurality of pulses separated using only transition widths included in a pre- specified set of transition widths; wherein said encoding step comprises the step of translating (3315) other pre- specified transition widths as amplitude combinations of the transition widths included in the pre-specified set of transition widths.

2. The method of claim 1, wherein each of the amplitude combinations include an amplitude value selected from a pre-specified set of amplitude values and one of the transition widths included in the pre-specified set of transition widths.

3. The method of claim 1, wherein each of the transition widths in the predetermined set specify a different number of zeros between adjacent ones.

4. The method of claim 3, wherein the transition widths consist of four zeros, five zeros, and six zeros.

5. The method of claim 3, wherein the other pre-specified transition widths consist of one zero, two zeros, and three zeros.

4. The method of claim 1, further comprising the steps of: transmitting (3320) the VAC encoding along a data channel for storage on the storage medium; and transmitting (3325) other VAC encodings along the data channel, within an intra- pulse interval of the VAC encoding, for storage on the storage medium.

5. The method of claim 4, wherein the VAC encoding and the other VAC encodings are orthogonal with respect to each other.

6. The method of claim 1, further comprising the step of selecting (3305) the transition widths included in the pre-specified set of transition widths based on a capability to reduce at least one of Inter-Symbol Interference (ISI) and Inter-Track Interference (LTI) during a read operation of the VAC encoding from the storage medium.

7. The method of claim 1, further comprising the step of selecting (3305) the transition widths included in the pre-specified set of transition widths so as to increase a number of bits per transition in a given storage area on the storage medium.

8. The method of claim 1, further comprising the step of selecting (3305) the transition widths included in the pre-specified set of transitions so as to decrease a Bit Error Rate (BER) of the VAC encoding during a peak detection operation performed on the VAC encoding.

9. A method for storing a random bit-stream on a storage medium, comprising the steps of: representing (3310) the random bit stream by a constant amplitude, varying pulse- width, VAC encoding having a plurality of pulses that are separated using only transition widths included in a pre-specified set of transition widths; and transmitting (3320) the VAC encoding along a data channel for storage on the storage medium, wherein said representing step comprises the step of translating (3315) other pre- specified transition widths as amplitude combinations of the transition widths included in the pre-specified set of transition widths.

10. The method of claim 9, wherein each of the amplitude combinations include an amplitude value selected from a pre-specified set of amplitude values and one of the transition widths included in the pre-specified set of transition widths.

11. The method of claim 9, wherein each of the transition widths in the predetermined set specify a different number of zeros between adjacent ones.

12. The method of claim 11, wherein the transition widths consist of four zeros, five zeros, and six zeros.

13. The method of claim 11, wherein the other pre-specified transition widths consist of one zero, two zeros, and three zeros.

14. The method of claim 1, further comprising the step of transmitting (step 3325) other VAC encodings along the data channel, within an intra-pulse interval of the VAC encoding, for storage on the storage medium.

15. The method of claim 14, wherein the VAC encoding and the other VAC encodings are orthogonal with respect to each other.

16. The method of claim 9, further comprising the step of selecting (3305) the transition widths included in the pre-specified set of transition widths based on a capability to reduce at least one of Inter-Symbol Interference (ISI) and Inter-Track Interference (LTI) during a read operation of the VAC encoding from the storage medium.

17. The method of claim 9, further comprising the step of selecting (3305) the transition widths included in the pre-specified set of transition widths so as to increase a number of bits per transition in a given storage area on the storage medium.

18. The method of claim 9, further comprising the step of selecting the transition widths included in the pre-specified set of transitions so as to decrease a Bit Error Rate (BER) of the VAC encoding during a peak detection operation performed on the VAC encoding.