Method and apparatus for generating high rate codes for recording information on a magnetic medium

<u>CROSS-REFERENCE TO RELATED APPLICATIONS</u>

0001This application is related to United States Patent Application Serial No. 08/515,445, Attorney Docket No. E. Soljanin 1, entitled "Method and Apparatus for Generating DC-Free Sequences," which was filed on August 15, 1995 and which is commonly assigned to the assignee of the present invention.

<u>FIELD OF THE INVENTION</u>

0002The present invention relates generally to the field of coding for digital systems, and, in particular, improved methods and apparatus for generating high rate codes that are dc-free and suitable for recording information on a magnetic medium.

<u>BACKGROUND OF THE INVENTION</u>

0003Information, such as signals representing voice, data, video or text, must typically be processed before the information can be transmitted over a communications channel or recorded on a medium. First, the information, if not already in digital form, is digitized, for example by an analog-to-digital converter. Next, the digital information may be "compressed" to represent the information by a fewer number of bits. Any savings due to compression are, however, partially offset by processing the compressed information using error correcting codes. Error correcting codes introduce additional bits to a signal to form an encoded signal. The additional bits improve the ability of a system to recover the signal when the encoded signal has been corrupted by noise introduced by a communications channel or by a recording medium.

0004A further type of coding used in transmission and recording systems is modulation coding. As with error correcting codes, modulation coding can improve a system's immunity to noise. Modulation codes also can advantageously be used to regulate timing and gain parameters in recording and communications systems.

0005For example, consider a system which reads information stored on a magnetic medium. In non-return-to-zero-inverse (NRZI) recording, for example, a binary "1" is recorded on a portion of the magnetic medium by causing a change in the magnetization or magnetic flux of that portion of the medium. A binary "0" is recorded by causing no change in magnetization. The bits are read by detecting a sequence of changes in a voltage signal caused by changes in the magnetization of portions of the medium. The voltage signal, however, may be corrupted by noise in the recording system. The voltage is typically a pulse each time a "1" is detected and just noise each time a "0" is detected. The position of the pulses is used to set timing parameters in the system, and the height of the pulses is used to set gain parameters in the system. If, however, a long string of zeros is read, there is no voltage output other than noise, and hence no timing or gain information, thereby leading to a loss of, or drift in, timing and gain parameters in the system.

0006Modulation coding thus may be used to ensure that the recording or transmission of a long string of binary zeros is avoided. Modulation coding may be implemented, for example, by dividing digital information that is to be recorded into sets of bits, called information words. Each information word is then used to select a codeword in a codebook. The codewords in the codebook are of length N bits where the codeword bits define a channel sequence, in other words, a sequence of symbols to be sent over a channel. For example, a binary "1" in a codeword may represent the symbol "-1" or negative magnetic flux, and a binary "0" in a codeword may represent the symbol "+1" or positive magnetic flux. If the codewords in the codebook do not contain a long string of zeros, then the selected codewords recorded on the medium will likewise not contain a long string of zeros, thereby obviating the timing and gain control problem.

0007Additionally, it is often desirable to use channel sequences that have a spectral null at zero (dc) frequency by which it is meant that the power spectral density function of the channel sequence at dc equals zero. Such sequences are said to be dc-free. One way to assure a dc-free sequence is to design a system in which the block digital sum, or the arithmetic sum, of symbols in a codeword transmitted over a channel is zero. However, efficient or high-rate modulation codes that can prevent long strings of zeros from occurring without adding an excessive number of redundant bits to the information to be recorded, and that are dc-free typically require both codewords and codebooks of larger sizes as discussed further below.

0008It is known that the power spectral density function of a channel sequence x, where x = ... x_-1,x₀,x₁ ... , vanishes at zero frequency if and only if its running-digital-sum (RDS), defined as<maths id="math0001" num=""><img file="EP0764950A2_D0001.tif" /></maths> is bounded. It is also known, for example, how to translate sequences of symbols from the symbol alphabet of the error-correcting code symbols into channel sequences with bounded RDS's by means of dc-free modulation codes which may be finite-state codes or block codes. Block codes, for example, take blocks of M symbols, called information words, and map them into blocks of N channel symbols or sequences called codewords. Several factors favor the use of block codes. One such factor is limited error propagation since the symbols used to encode one block are not used in encoding any other block and thus errors in encoding are typically confined to a particular block. Another factor is ease of implementation. One way to organize the mapping of information words to codewords is to form a codebook or look-up table of 2^M codewords and use an M-bit input word to specify or address an N-bit codeword in the codebook. The ratio M/N defines the rate R of the modulation code.

0009To ensure that an arbitrary sequence of codewords has a bounded RDS, each codeword w=w₀,w₁, ... w_N is required to have a block digital sum (BDS), defined as<maths id="math0002" num=""><img file="EP0764950A2_D0002.tif" /></maths> equal to zero. Codewords of bipolar symbols, for example, +1 and -1, and having a BDS equal to zero, are possible only if the codeword length N is even and if half the symbols are -1 and half the symbols are +1. The number of such codewords is then equal to<maths id="math0003" num=""><img file="EP0764950A2_D0003.tif" /></maths> where<maths id="math0004" num=""><img file="EP0764950A2_D0004.tif" /></maths> However, at most 2^M codewords having a BDS equal to zero can be used to form a codebook for an M/N rate code, where<maths id="math0005" num=""><img file="EP0764950A2_D0005.tif" /></maths> and where the function floor[x] returns the largest integer less than or equal to x. The code rate R = M/N indicates that for every M information bits, N channel bits are generated, with N ≥ M.

0010The above explanation is rendered more clear by use of a specific example. Consider a sequence having a block length N equal to 4. There are 16 possible sequences, 6 of which are dc-free. By using a block length of four bits, however, the value of M equals 2, and two of the dc-free codewords will not be used in the codebook. In some cases, the requirement that<maths id="math0006" num=""><img file="EP0764950A2_D0006.tif" /></maths> causes a substantial number of extra dc-free sequences not to be used. For example, if N=8, the number of dc-free sequences is 70, but the codebook is of size 64 and thus 6 dc-free sequences are not used. Similarly, if N=10, there are 252 possible dc-free sequences. The codebook, however, is of size 128. Thus, 124 sequences are discarded thereby lowering the code rate from approximately 0.8 to 0.7.

0011Furthermore, in magnetic recording applications, it is desirable that modulation codes have rates higher than 3/4 so that more information can be written on the recording medium. Codes having a relatively long block length are required for rates above 3/4. Also, large codebooks are required where the codewords in the codebooks are dc-free. For example, a code of rate 11/14 requires a block length of 14 and a codebook size of 2048, and a code of rate 13/16 requires a block length of 16 and a codebook of size 8192. Such large codebooks, however, typically require the implementation of more complex circuitry and often require large power consumption and large area on integrated circuits relative to other elements in the transmission or recording system. Also, the larger the codebook, the more time it takes to access codewords in the codebook. Although some techniques have been proposed to reduce the size of the codebooks, these techniques add additional complexity and do not substantially reduce the size of the codebooks. Thus, there is a need for a method and apparatus for generating high rate codes that are dc-free and suitable for recording information on a magnetic medium.

<u>SUMMARY OF THE INVENTION</u>

0012The present invention addresses the aforementioned problems by providing a novel method and apparatus for encoding digital information to be recorded on a magnetic medium. The method preferably includes the steps of receiving a sequence of (2^mn + d) user bits and mapping the sequence of user bits to 2^m dc-free codewords. In particular, the method preferably includes selecting 2^m dc-free codewords from among a plurality of non-intersecting subconstellations of dc-free codewords of a particular length, wherein the step of selecting comprises the step of identifying from which of the plurality of subconstellations each of the 2^m dc-free codewords will be selected by using a portion of the sequence of user bits in a predetermined order and the step of specifying the 2^m dc-free codewords based upon the remaining user bits in the sequence. The selected 2^m dc-free codewords may then be recorded on a magnetic medium.

0013A modulation coder, which preferably includes a codebook comprising the plurality of non-intersecting subconstellations of dc-free codewords, performs the mapping in a non-equiprobable manner such that a particular codeword from a larger subconstellation is more likely to be used than a particular codeword from a smaller constellation. In particular, codewords with more bit transitions or more frequent bit transitions preferably are assigned to subconstellations different from codewords with fewer or less frequent bit transitions. Specifically, codewords with a relatively high number of bit transitions are preferably assigned to the larger subconstellations, whereas codewords with a relatively low number of bit transitions are preferably assigned to the smaller subconstellations, thereby lessening the loss of timing and gain parameters in the magnetic recording system, as well as improving the transmission rate and efficient use of the set of possible dc-free sequences of a given length.

0014The present invention also provides a method and apparatus for decoding information stored on a magnetic medium. A more complete understanding of the present invention, as well as other features and advantages of the present invention, may be obtained with reference to the following detailed description and accompanying drawings.

<u>BRIEF DESCRIPTION OF THE DRAWINGS</u>

0015FIG. 1 is a block diagram of an exemplary system in which the invention may be practiced.

0016FIG. 2 is a flow chart showing the steps for designing a modulation coder with a codebook according to the principles of the present invention.

0017FIG. 2A ia a flow chart showing the steps for assigning codewords to subconstellations in the codebook in accordance with the present invention.

0018FIGS. 3A - 3C illustrate the creation of an exemplary constellation tree according to the principles of the present invention.

0019FIG. 4 is a table showing the relative sizes and the actual probabilities of utilization which result from using the encoding algorithm of the present invention with respect to an exemplary set of four subconstellations.

0020FIG. 5 is a flow chart showing the steps used in mapping user bits to codewords according to the present invention.

0021FIG. 5A is a flow chart showing the steps of the encoding algorithm in accordance with the present invention.

0022FIG. 6 and FIGS. 6A - 6D illustrate an exemplary sequence of user bits and the application of the encoding algorithm of the present invention using the exemplary sequence of user bits.

0023FIG. 7 is a flow chart showing the steps of recovering the digital information stored on a magnetic medium in accordance with the present invention.

<u>DETAILED DESCRIPTION OF THE INVENTION</u>

0024FIG. 1 is a block diagram of an exemplary system in which the invention may suitably be practiced. The system of FIG. 1 is particularly useful for recording digital information on and reading digital information from a magnetic medium such as those employed in digital audio tapes on disk drives.

0025The information to be recorded preferably is first "compressed" using a Lempel-Ziv compressor 110 so as to reduce the amount of information that must be recorded on the medium thereby saving time and money. Next, the compressed information preferably is sent as an input to an encoder 120 which encodes the compressed information using Reed-Solomon codes. The purpose of Reed-Solomon encoding is to adjoin extra symbols to the compressed information so that noise introduced in the reading process will not cause errors when the information is received. Lempel-Ziv and Reed-Solomon encoding are described in greater detail in Timothy C. Bell at al., Text Compression, Prentice-Hall, Englewood Cliffs, NJ, 1990 and S. Lin and D.J. Costello, Error Control Coding, Prentice-Hall, Englewood Cliffs, NJ, 1983, respectively, which are incorporated herein by reference. Other coding techniques which provide the same effect may suitably be employed.

0026The output of the encoder 120 is a series of symbols where each symbol is represented by a set of bits. The symbols are sent as an input to a modulation coder 130 which is designed to map each of the input symbols into a dc-free sequence or codeword having a predetermined number of bits. The modulation coder 130 may be implemented using a processor designed or programmed to perform the functions explained below. In particular, the modulation coder 130 may conveniently be implemented, for example, by a microprocessor programmed with appropriate firmware. For example, an 80C51 microprocessor manufactured by the Philips Semiconductors company, may suitably be used to implement the coder 130. Alternatively, the modulation coder 130 may be suitably implemented by a general purpose processor or in a specially designed semiconductor chip or other dedicated hardware. In any event, the dc-free sequences or codewords are then transmitted from the modulation coder 130 to a write head 135 and recorded on a magnetic medium 140, such as digital audio tapes on a disk drive.

0027Signals representing the information recorded on the magnetic medium 140 may then be read by a read head 145 and preferably sent to an equalizer 150 which controls intersymbol interference. Output signals from the equalizer are then sent to a modulation decoder 160, a Reed-Solomon decoder 170 and a Lempel-Ziv decompressor 180, respectively, so as to recover the information recorded on the magnetic medium 140.

0028FIG. 2 is a flow chart showing the steps for programming or designing the modulation coder 130 with a codebook for use in mapping a sequence of bits using a specified rate. First, as indicated by step 200, a constellation tree is created corresponding to the desired average rate, where the average rate β has the form β = (n + d/2^m) bits per codeword, where 2^m is the dimensionality of the code, n is a positive integer, and d is a positive integer less than 2^m. A generalized cross constellation is a set of non-intersecting subconstellations of channel sequences or codewords from which N-dimensional dc-free channel sequences are drawn. Each subconstellation is a subset of the entire constellation of channel sequences or codewords. The relative sizes of the subconstellations determine both the rate of the encoder and the dimensionality of the code. A specific code is thus uniquely defined.

0029The constellation tree comprises a root node and at least one additional level of nodes, including leaf nodes at the lowest level of the constellation tree. Each node, except the leaf nodes, has first and second branches such that each subsequent level of nodes has twice as many nodes as the previous level. This is illustrated, for example, in FIG. 3A. The root node represents the entire constellation Ω, whereas the subconstellations Ω₀₀, Ω₀₁, Ω₁₀, Ω₁₁ are represented in FIG. 3A by the leaf nodes at the lowest level of the constellation tree.

0030In order to aid in understanding the present invention, it will be helpful to use a particular example. Consider, for example, a rate of (7 + 3/4). Using this rate, n = 7, d = 3, and m = 2. Next, consider the binary expansion of<maths id="math0007" num=""><img file="EP0764950A2_D0007.tif" /></maths> The number of levels in the constellation tree, excluding the root node, is w, where w equals the number of indices i for which d_i is not equal to zero. Using the example indicated above, d = 3 = 2⁰ + 2¹. Thus, in the example, w equals 2, as shown in FIG. 3A.

0031Next, the branches of the constellation tree are labelled with a weight as explained below. Let p(0) &lt; p(1) &lt; . . . &lt; p(w-1) be the members of the set P which represents the ordering of indices i for which d_i is not equal to zero. Again, using the above example, P = (0,1), with p(0) = 0, and p(1) = 1. As indicated by step 204, the first branch of each node, or each of the left-going branches of the constellation tree, are labelled with a weight of zero. Next, as indicated by step 206, the second branch of each node, or each of the right-going branches, are labelled with the weight (m - p(j)), where j is the level of the tree and p(j) is defined as above. FIG. 3B illustrates the resulting constellation tree and relative weights of the branches for the above example.

0032Once the constellation tree is created, the relative weight r(i) of each subconstellation is determined, as indicated in step 210. To determine the relative weight of a particular subconstellation represented by one of the leaf nodes, the sum of the weights along the path commencing from the root node and terminating with the particular leaf node is calculated. The relative weights corresponding to the above example are indicated in FIG. 3C.

0033The size or number of entries of each subconstellation may be determined in step 220 from the relative weights according to the formula (2ⁿ x 2^-r(i)). Thus, as indicated by FIG. 3C, the subconstellation Ω₀₀ in the above example has a constellation size of 128, the subconstellation Ω₀₁ has a size of 64, the subconstellation Ω₁₀ has a size of 32, and the subconstellation Ω₁₁ has a size of 16. The total constellation Ω, in this example, therefore, contains 240 codewords. It will be noted that the sizes of the subconstellations are different from one another. In addition, the number of bits or length of the codewords is preferably chosen to be the minimum length N, where<maths id="math0008" num=""><img file="EP0764950A2_D0008.tif" /></maths> is greater than or equal to the total number of codewords in the constellation Ω. Thus, in the above example, each of the 240 dc-free codewords preferably has a length of ten bits.

0034As indicated by step 230, the codewords are then assigned to the subconstellations. Since, as explained above, there are 252 possible 10-bit dc-free channel sequences, a decision must be made as to which twelve of the 252 possible dc-free channel sequences will be discarded. Furthermore, a decision must be made as to how the remaining 240 codewords will be divided among and assigned to the subconstellations, Ω₀₀, Ω₀₁, Ω₁₀, Ω₁₁.

0035It will be evident from the encoding algorithm discussed below that the various subconstellations are not utilized in proportion to their relative sizes. For example, FIG. 4 is a table showing the relative sizes and the actual probabilities of utilization which result from using the mapping technique discussed below for the four subconstellations discussed above. While channel sequences or codewords from a particular subconstellation are selected with equal probability, the encoding algorithm discussed below favors channel sequences or codewords from the larger constellations at the expense of the smaller constellations. This non-equiprobable feature allows the less desirable dc-free sequences to be assigned to the smaller subconstellations, thereby decreasing the likelihood that they will be utilized during the coding process. Conversely, the more desirable dc-free sequences may be assigned to the larger subconstellations, thereby increasing the likelihood that they will be utilized during the coding process.

0036In certain communication systems using, for example, voiceband data transmission or fading channels, an important concern is the elimination of high energy symbols. In the magnetic recording context, however, eliminating high energy symbols is not an important issue because each bit in a channel sequence or codeword represents either a positive or negative magnetic flux of +1 or -1, respectively. Rather, as discussed above, a desirable feature for a magnetic recording system is the ability to generate high rate codes that are dc-free and that obviate the timing and gain problems which result from long strings of consecutive bits having the same value.

0037As indicated in step 232 of FIG. 2A, in a presently preferred embodiment, dc-free channel sequences that include a relatively long string of consecutive bits having the same value, and, in particular those channel sequences wherein the string of consecutive bits which have the same value appears at the beginning or at the end of the sequence, are discarded. Thus, for example, the channel sequences 1111100000 and 0000011111 would be discarded. Also, as indicated by step 234, other channel sequences containing relatively long strings of consecutive bits having the same value are preferably assigned to the smaller subconstellations. For example, the channel sequences 0001111100 and 1110000011 would be assigned to the subconstellation Ω₁₁. Conversely, as shown in step 236, channel sequences that do not contain a relatively long string of consecutive bits having the same value are preferably assigned to the larger subconstellations. The channel sequences 1010101010 and 0101010101 would, therefore, be assigned to the subconstellation Ω₀₀. In general, codewords with fewer bit transitions are preferably assigned to the smaller subconstellations, whereas codewords with more bit transitions are preferably assigned to the larger subconstellations.

0038Finally, as indicated by step 238, with respect to the remaining channel sequences, those codewords which are similar to one another are preferably assigned to different subconstellations, whenever possible. For example, channel sequences which are identical except for two bits which are interchanged would preferably be assigned to different subconstellations.

0039Assigning the possible dc-free channel sequences to the subconstellations in the above manner helps reduce the loss of timing and gain parameters in the system. It also reduces the likelihood of errors occurring in the transmission, recording and identification of the channel sequences. This is because, as a result of the assignment process for assigning codewords to the subconstellations according to the present invention, codewords with frequent bit transitions inherently have fewer nearest neighbors. Codewords with fewer nearest neighbors allow fewer errors to propagate into later stages of decoding.

0040As indicated in step 235, each codeword in a particular one of the subconstellations is also assigned a q-bit address, where q is the minimum number of bits required to address all the codewords in the particular subconstellation. Specifically, for a subconstellation of size Q, each address is of length q, where q = log₂Q. Once the codewords are assigned to the subconstellations, the codewords are preferably stored in a codebook 132 in the modulation coder 130, as indicated by step 250. The codebook 132 may be advantageously implemented in memory 131 such as read only memory or random access memory. The correspondence between the codewords and the subconstellations, as well as the addresses assigned to each codeword, also are stored in the memory 131. Although the embodiment shown in FIG. 1 includes a single codebook which incorporates all of the subconstellations, it should be understood that the various subconstellations may be stored in a plurality of codebooks. Thus, for example, each subconstellation may be stored in a separate codebook.

0041The determination of the number and size of the subconstellations, as well as the assignment of the dc-free codewords to the subconstellations, may be performed by a person designing the modulation coder 130. In that situation, the codewords, the correspondence between the codewords and subconstellations, and the address assigned to each codeword, are stored in the memory 131 when the modulation coder 130 is designed. In an alternative embodiment of the present invention, the modulation coder 130 is programmed to perform the steps 200 through 239 in FIG. 2, as well as the steps 232 through 238 in FIG. 2A, in response to receiving a signal indicating the desired rate β. For this purpose, the modulation coder 130 preferably has at least one input lead 133 for receiving the values of n, m and d. The values n, m and d, or their equivalents, may be entered into the coder 130 by a user of the system employing, for example, a keyboard. The modulation coder 130 would then perform the steps 200 through 239 in FIG. 2, as well as the steps 232 through 238 in FIG. 2A. The assigned codewords, the correspondence between the codewords and the subconstellations, and the address of each codeword, would then be stored in the memory 131.

0042The discussion that follows describes how the modulation coder 130 is further designed or programmed to map a sequence of 2^mβ = (2^mn + d) user bits received from the output of the Reed-Solomon coder 120 to 2^m dc-free channel sequences or codewords stored in the codebook 132. Returning to the example used earlier in which a rate of (7 + 3/4) bits per codeword was considered, m is equal to two, resulting in four channel sequences or codewords and a total of thirty-one user bits. Thus, the modulation coder 130 maps the thirty-one user bits into four 10-bit dc-free channel sequences or codewords. The four as-yet unspecified codewords conveniently may be labelled A₀, A₁, A₂ and A₃.

0043FIG. 5 is a flow chart showing the steps used in mapping the user bits to the codewords. In general, the modulation coder 130 utilizes a particular sequence of user bits to determine from which subconstellation the codewords will be selected and to identify the particular codewords which will be sent to the magnetic medium 140 and recorded thereon. More particularly, as explained in greater detail below, the modulation coder 130 preferably is designed to perform steps 300 through 340 in FIG. 5, as well as steps 311 through 318 in FIG. 5A.

0044As indicated by step 300, a partitioning technique is performed as follows. For each of the members of the set P defined above, the entire sequence of the as-yet unspecified 2^m codewords, A₀ . . . A_{2m - 1}, is partitioned into 2^P non-overlapping subdivisions T_p,i where 0 ≤ i &lt; 2^p. Again, referring to the example discussed above, P = {0,1}. Thus, for p = 0, there will be one subdivision T_0,0 which includes all four of the as-yet unspecified codewords, A₀, A₁, A₂ and A₃. Similarly, for p = 1, there will be two subdivisions, T_1,0 and T_1,1, each of which includes two of the four as-yet unspecified codewords. The partitioned subdivisions need not be contiguous sets of codewords and are disjoint for different p. Thus, for the purposes of illustration, the subdivision T_1,0 will include the as-yet unspecified codewords A₀ and A₃, and the subdivision T_1,1 will include the as-yet unspecified codewords A₁ and A₂.

0045Next, as indicated by step 310, the sequence of user bits that are received at the input to the modulation coder 130 are used to determine and identify which of the subconstellations each of the as-yet unspecified codewords comes from, according to an encoding algorithm, described further below with respect to FIG. 5A. The encoding algorithm may be viewed as "walking" each of the as-yet unspecified codewords down the constellation tree until each one is associated with one of the leaf nodes in the lowest level of the constellation tree.

0046As indicated by 311 in FIG. 5A, one assumes that the 2^m as-yet unspecified codewords are at the root of the constellation tree. As shown in step 312, each of the partitioned subdivisions T_p,i is addressed, in ascending order of p, by using a portion of the user bits in a predetermined order. Preferably, the user bits are used in sequential order. Next, as indicated by step 313, for each partitioned subdivision, the next user bit in the sequence determines whether all of the as-yet unspecified codewords in that subdivision proceed down the constellation tree over the first, or left, branch from each of their respective current positions, or whether exactly one as-yet unspecified codeword proceeds down the second, or right, branch, while the remaining as-yet unspecified codewords proceed down the first, or left, branch. For example, as shown by steps 314 and 315, respectively, a user bit of value "0" would indicate that all the as-yet unspecified codewords in the particular subdivision proceed down the first, or left, branch, whereas a user bit of value "1" would indicate that exactly one as-yet unspecified codeword in the particular subdivision proceeds down the second, or right, branch, while the remaining as-yet unspecified codewords in the particular subdivision proceed down the first, or left, branch. In the latter situation, as shown in step 316, the next (m - p) user bits in the sequence are used to identify and determine which particular as-yet unspecified codeword proceeds down the second, or right, branch. This process is continued, as indicated by step 317, until all the partitioned subdivisions have been addressed. Once all the partitioned subdivisions have been addressed, the encoding algorithm of step 310 ends, as shown by 318.

0047It should be noted that, although in the above discussion, with respect to steps 204, 206, 314, 315 and 316, the first branch of each node has been associated with the left-going branch and the second branch has been associated with the right-going branch, the association of the respective first and second branches may be reversed if done consistently for all relevant branches.

0048Referring again to the example discussed above, consider the exemplary sequence of thirty-one user bits 100011...b₃₁ shown in FIG. 6. The four as-yet unspecified codewords, designated A₀, A₁, A₂ and A₃, are assumed to be sitting at the root node Ω, as shown in FIG. 6A. The first user bit, whose value is "1", indicates that exactly one of the four codewords A₀, A₁, A₂ and A₃ in the subdivision T_0,0 proceeds down the right branch. The next two bits, which are both "0", are used to specify which one of the four codewords proceeds down the right branch. Again, for the purposes of illustration, A₀ is assumed to be the specified codeword that proceeds down the right branch. A₁, A₂ and A₃ thus proceed down the left branch. The result is shown in FIG. 6B.

0049The next user bit, which is a "0", indicates that both codewords in the next subdivision, T_1,0 for example, proceed down the left branches from their current positions. This situation is shown in FIG. 6C. The fifth user bit, which is a "1", indicates that exactly one of the unspecified codewords in the next subdivision, T_1,1, proceeds down the right branch from its current branch. The next user bit, which is also a "1", is used to determine which of the two unspecified codewords, A₁ or A₂, proceeds down the right branch. For the purposes of illustration, it is assumed that A₁ is specified as the codeword which proceeds down the right branch from its current position. Thus, A₂ proceeds down the left branch from its current position. FIG. 6D shows the resulting situation.

0050In the example discussed above, and as shown in FIG. 6D, two of the as-yet unspecified codewords, A₂ and A₃, will be selected from among the codewords in the subconstellation Ω₀₀. One of the as-yet unspecified codewords, A₁, will be selected from the codewords in the subconstellation Ω₀₁, and the remaining as-yet unspecified codeword, A₀, will be selected from among the codewords in the subconstellation Ω₁₀. In this particular example, none of the as-yet unspecified codewords will be selected from the codewords in the subconstellation Ω₁₁.

0051As should be clear form the example discussed above, the determination of which subconstellation each of the as-yet unspecified codewords will be selected from requires use of fewer than all the bits in the sequence of user bits. As indicated by step 320 and as shown in FIG. 6, the remaining user bits are utilized to determine which specific channel sequences or codewords are specified. In fact, the remaining user bits are exactly sufficient to address all of the codewords in the selected subconstellations for each of the as-yet unspecified codewords. The addressing may be achieved, for example, by the use of look-up tables. As noted above, for a subconstellation of size Q, q bits are required to specify the address of a particular one of the codewords in that subconstellation, where q = log₂Q.

0052In the example discussed above, six of the thirty-one user bits were required to determine from which of the four subconstellations, Ω₀₀, Ω₀₁, Ω₁₀ and Ω₁₁, each of the as-yet unspecified codewords, A₀, A₁, A₂ and A₃ will be selected. The remaining twenty-five user bits may be used to specify the addresses of the particular codewords within each of the selected subconstellations. Seven bits, for example bits b₇ through b₁₃, are required to determine which one of the 128 codewords in the subconstellation Ω₀₀ will be used for A₂. Likewise, seven bits, for example b₁₄ through b₂₀, are required to determine which one of the 128 codewords in the subconstellation Ω₀₀ will be used for A₃. Six user bits, for example b₂₁ through b₂₆, are required to determine which one of the 64 codewords in the subconstellation Ω₀₁ will be used for A₁, and five user bits, for example b₂₇ through b₃₁, are required to determine which one of the 32 codewords in the subconstellation Ω₁₀ will be used for A₀. In this manner, all thirty-one user bits received from the output of the Reed-Solomon coder 120 are used to map, and thereby encode, the thirty-one user bits to four 10-bit dc-free codewords obtained from the codebook 132.

0053Once the codewords that will be used for the encoding have been specified, the modulation coder 130 generates a sequence of bits corresponding to the selected 2^m dc-free codewords as indicated in step 330. A signal generating circuit or chip 134, for example, may be used to generate the bits corresponding to the selected codewords. Other suitable means may also be employed to generate the sequence of bits corresponding to the selected codewords. Then, as indicated by step 340, the coder 130 preferably transmits the specified codewords to the write head 135. Then, in step 350, the write head 135 records the specified codewords on the magnetic medium 140, preferably in a predetermined order, such as in the order corresponding to A₀, A₁, A₂ and A₃.

0054At some later time, as indicated by step 400 in FIG. 7, the codewords recorded on the magnetic medium 140 are read by the read head 145 and preferably sent to the equalizer 150. The (2^mn + d) user bits are recovered, as explained further below, by the modulation decoder 160. The modulation decoder 160 may be implemented, for example, using a processor appropriately designed or programmed to perform the inverse of the functions performed by the modulation coder 130. The modulation decoder 160 thus allows the original (2^mn + d) user bits to be recovered and generated from the codewords that are read from the magnetic medium 140.

0055In one embodiment, the modulation decoder 160 has a look-up table 162 which is used by the modulation decoder 160 to perform the inverse of step 320 by mapping each codeword to its corresponding portion of the sequence of user bits. The look-up table 162, which may be stored in a memory 163 such as a read-only memory or a random access memory, includes each codeword in the codebook 132 and its corresponding q-bit address assigned to it in step 239. The memory 163 also stores the correspondence between each codeword and the identity of the subconstellation to which it belongs.

0056As indicated by step 404, the modulation decoder 160 retrieves the q-bit address associated with each of the 2^m codewords read from the magnetic recording medium 140. Using the exemplary sequence of user bits discussed above with respect to FIG. 6, the modulation decoder 160 would retrieve the address bits b₇ through b₁₃ corresponding to the codeword that was designated by A2. Similarly, the modulation decoder 160 would retrieve the address bits b₁₄ through b₂₀, b₂₁ through b₂₆, and b₂₇ through b₃₁, corresponding, respectively, to the codewords that were designated by A₃, A₁ and A₀.

0057Next, as indicated by step 406, the modulation decoder 406 arranges the retrieved sets of address bits in the order in which the corresponding codewords were received. Again, with reference to the exemplary sequence of user bits discussed above, the modulation coder would arrange the bits b₇ through b₃₁ in sequential order, corresponding to the order of the received codewords A₂, A₃, A₁ and A₀.

0058The present invention also provides an added level of error detection. This added level of error detection is possible because the encoding algorithm discussed above with respect to FIG. 5A does not allow the 2^m codewords corresponding to a sequence of (2^mn + d) user bits to be selected from certain combinations of subconstellations. Thus, in the example discussed above, the four codewords may not all be selected from the subconstellation Ω₁₁. In fact, in that particular example, no more than one of the as-yet unspecified codewords may be selected from the subconstellation Ω₁₁. As indicated by step 408, the modulation decoder 160 retrieves from the memory 153 the identity of the subconstellations to which each of the 2^m received codewords belongs. Then, as indicated in 410, the modulation decoder 160 detects whether an impermissible combination of codewords has occurred. Thus, with reference to the example discussed above, the modulation decoder 160 preferably is programmed to detect the situation, for example, where two or more codewords are selected from the subconstellation Ω₁₁. The modulation decoder 160 is also programmed to detect other impermissible combinations of codewords. As indicated by step 415, if such an error is detected, the modulation decoder 160 preferably provides an error detection alarm or signal on an error detection electronic circuit 166, for example.

0059If no error is detected, then as indicated by step 420, the modulation decoder 160 generates the remaining bits in the original sequence of user bits based upon the combination and order of subconstellations to which the 2^m received codewords belong. It will be recalled that, in step 310, a portion of the sequence of user bits was used to determine from which subconstellation each of the as-yet unspecified codewords would be selected. The modulation decoder 160 now uses the identity and order of the subconstellations to recover that portion of the sequence of user bits. For this purpose, the modulation decoder 160 preferably includes a plurality of logic gates 164 or other dedicated hardware. Signals representing the identity of the subconstellations to which the 2^m received codewords belong are sent as inputs to the logic gates 164. The logic gates 164 preferably are hardwired so that the output signals correspond to the remaining portion of bits in the original sequence of user bits.

0060Referring again to the example discussed above, the identity of the subconstellations retrieved in step 408 is Ω₀₀, Ω₀₀, Ω₀₁, and Ω₁₀. The output of the logic gates 164 would then be "100011" which represents the first six bits in the original sequence of user bits as shown in FIG. 6. As indicated by step 422, these remaining bits are appended to the beginning of the arrangement of bits that resulted from step 406. In this manner, the entire sequence of user bits is recovered from the 2^m codewords recorded on the magnetic medium 140.

0061Finally, as indicated by step 425, the digital information originally entered into the system may be recovered by sending signals representing the recovered sequence of user bits to the Reed-Solomon decoder 170. The output of the Reed-Solomon decoder 170 is then preferably sent to the Lempel-Ziv decompressor 180, the output of which corresponds to the digital information originally entered into the system.

0062The present invention thus provides several advantageous features. First, because the cross-constellation technique maps over multiple dimensions, it allows, a fractional number of bits to be mapped into each dc-free codeword. This allows greater flexibility in the selection of a desired rate. It also provides a non-equiprobable coding mechanism for favoring certain codewords over others. Specifically, for example, codewords that do not have relatively long strings of consecutive bits having the same value may be favored over those that do by assigning the favored codewords to the larger constellations. In this manner, the present invention provides an improvement for obviating the timing and gain problems previously associated with reading information stored on magnetic media.

0063Furthermore, for codewords having a specified length, the size of the codebook is larger than the size of the codebook that is realized by using previously known techniques such as the one described above in the background section. Thus, in the example discussed above, using dc-free codewords which have a length of ten bits results in a codebook containing 240 codewords. This compares with the codebook size of 128 codewords for 10-bit codewords using the known technique described above in the background section. As previously noted, larger codebooks allow higher rates to be achieved. The present invention, therefore, improves the transmission rate and efficiently uses the set of possible dc-free sequences of a given length for encoding and recording information on magnetic media. The efficiency of the mapping may be measured by the coefficient expansion ratio which equals the number of codewords divided by 2^β. Thus, for the example discussed above, the coefficient expansion ratio equals 240/2^7.75 or 1.12.

0064Although the present invention has been described with reference to specific embodiments, it will be appreciated that other arrangements within the spirit and scope of the present invention will be readily apparent to persons of ordinary skill in the art. The present invention is, therefore, limited only by the appended claims.