Abstract:
An apparatus includes a memory and a controller. The memory may be configured to store data. The memory generally comprises a plurality of memory units each having a size less than a total size of the memory. The controller may be configured to generate a set of converted log likelihood ratios by scaling a set of original log likelihood ratios using a selected scalar value, wherein the controller determines the selected scalar value by generating a plurality of sets of scaled log likelihood ratios by scaling the set of original log likelihood ratios with a plurality of corresponding scalar values, calculating a plurality of respective correlation coefficients each measuring a similarity of a respective set of scaled log likelihood ratios to the set of original log likelihood ratios, and selecting the scalar value corresponding to the set of scaled log likelihood ratios whose respective correlation coefficient is highest as the selected scalar value.

Description:
[0001]    This application relates to U.S. Ser. No. 14/282,380, filed May 20, 2014, which is incorporated by reference in its entirety. 
       FIELD OF THE INVENTION 
       [0002]    The invention relates to data storage generally and, more particularly, to a method and/or apparatus for implementing a fixed point conversion of LLR values based on correlation. 
       BACKGROUND 
       [0003]    Due to aggressive process scaling, the raw bit error rate (BER) of NAND flash is becoming poorer and poorer. To maintain the same level of reliability, solid state drive (SSD) controllers are adopting error correction codes with soft decoding capability. For example, low density parity check (LDPC) codes have soft decoding capability. Error correction codes with soft decoding capability are more powerful in correcting errors but they use a soft input to the decoder. The soft input is in the form of a log likelihood ratio (LLR). Since conventional flash devices do not provide soft decision outputs, SSD controllers have to generate them using either hardware or software. LLR values are real values ranging from −Inf to +Inf. Once LLRs are generated, there is a practical problem of converting LLRs to fixed point numbers before decoding. 
         [0004]    It would be desirable to implement fixed point conversion of LLR values based on correlation in a SSD controller and/or drive. 
       SUMMARY 
       [0005]    The invention concerns an apparatus including a memory and a controller. The memory may be configured to store data. The memory generally comprises a plurality of memory units each having a size less than a total size of the memory. The controller may be configured to generate a set of converted log likelihood ratios by scaling a set of original log likelihood ratios using a selected scalar value, wherein the controller determines the selected scalar value by generating a plurality of sets of scaled log likelihood ratios by scaling the set of original log likelihood ratios with a plurality of corresponding scalar values, calculating a plurality of respective correlation coefficients each measuring a similarity of a respective set of scaled log likelihood ratios to the set of original log likelihood ratios, and selecting the scalar value corresponding to the set of scaled log likelihood ratios whose respective correlation coefficient is highest as the selected scalar value. 
     
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
         [0006]    Embodiments of the invention will be apparent from the following detailed description and the appended claims and drawings in which: 
           [0007]      FIG. 1  is a diagram illustrating an example embodiment; 
           [0008]      FIG. 2  is a diagram illustrating a more detailed embodiment; 
           [0009]      FIG. 3  is a diagram illustrating read reference voltages and decision regions when reading a single voltage; 
           [0010]      FIG. 4  is a diagram illustrating read reference voltages and decision regions when reading two voltages; 
           [0011]      FIG. 5  is a diagram illustrating read reference voltages and decision regions when reading three voltages; 
           [0012]      FIG. 6  is a diagram illustrating a graph of LLR values for reads of pages; 
           [0013]      FIG. 7  is a diagram illustrating a graph of correlation coefficients versus scalar values; 
           [0014]      FIG. 8  is a flow diagram illustrating converting original LLR values to fixed point LLR values; 
           [0015]      FIG. 9  is a flow diagram illustrating determining the best scalar value; and 
           [0016]      FIG. 10  is a flow diagram illustrating triggering the determination of a new scalar value. 
       
    
    
     DETAILED DESCRIPTION OF THE EMBODIMENTS 
       [0017]    Embodiments of the invention include providing a fixed point conversion of LLR values based on correlation that may (i) map continuous LLR values to fixed point integers, (ii) be implemented using a linear correlator, (iii) determine an optimal scalar value, (iv) update an optimal scalar value over the lifetime of the SSD, (v) correlate LLR values to the input precision of a decoder, and/or (vi) be implemented as one or more integrated circuits. 
         [0018]    Embodiments of the invention may handle the problem of converting LLRs to fixed point numbers before decoding. Generally, LLRs have real (e.g., continuous) values ranging from −Inf to +Inf. A linear correlator may select converted fixed point values. The original (e.g., real and/or continuous) LLRs may be mapped to fixed point integers (e.g., converted LLRs). The conversion of the original LLRs to converted LLRs may be implemented using correlation. The converted LLRs having fixed point values may be sent for error correction code decoding. 
         [0019]    Referring to  FIG. 1 , a block diagram of an example apparatus  50  is shown. The apparatus  50  generally comprises a block (or circuit)  60 , a block (or circuit)  70  and a block (or circuit)  80 . The circuit  70  may include a circuit  100 . The circuit  100  may be a memory/processor configured to store computer instructions (or firmware) or may be logic. The instructions, when executed, may perform a number of steps. The firmware  100  may include a redundancy control module  110 . The redundancy control module  110  may be implemented as part of the firmware  100  or as a separate module. While an example of redundancy implemented in the firmware  100  is shown, the redundancy may be implemented, in another example, in hardware (e.g., logic such as a state machine). 
         [0020]    A signal (e.g., REQ) may be generated by the circuit  60 . The signal REQ may be received by the circuit  70 . The signal REQ may be a request signal that may be used to access data from the circuit  80 . A signal (e.g., I/O) may be generated by the circuit  70  to be presented to/from the circuit  80 . The signal REQ may include one or more address bits. A signal (e.g., DATA) may be one or more data portions received by the circuit  60 . 
         [0021]    The circuit  60  is shown implemented as a host circuit. The circuit  70  reads and writes data to and from the circuit  80 . The circuit  80  is generally implemented as a nonvolatile memory circuit. The circuit  80  may include a number of modules  82   a - 82   n . The modules  82   a - 82   n  may be implemented as NAND flash chips. In some embodiments, the circuit  80  may be a NAND flash device. In other embodiments, the circuit  70  and/or the circuit  80  may be implemented as all or a portion of a solid state drive  90  having one or more nonvolatile devices. The circuit  80  is generally operational to store data in a nonvolatile condition. When data is read from the circuit  80 , the circuit  70  may access a set of data (e.g., multiple bits) identified in the signal REQ. The signal REQ may request data from the drive  90  or from one of a number of additional storage devices. 
         [0022]    Data within the circuit  80  is generally organized in a hierarchy of units, such as die, plane, block, and/or page units. The circuit  80  may contain multiple dies (e.g., in a single package or multiple packages). Generally, for enterprise applications the circuit  80  may be comprised of hundreds of flash memory dies. Flash memory may have multiple planes in the same die. The planes may be accessed in parallel to improve performance. 
         [0023]    A first type of redundancy may be implemented as a redundancy block. A redundancy block is a combination of blocks (e.g., a block from each nonvolatile memory die in the circuit  80 ) that can be combined to form a redundant array of silicon independent elements, similar to a redundant array of independent disks for magnetic media. The nonvolatile memory locations within the blocks may be written in a striped fashion. In some embodiments, organizing a plurality of blocks in redundancy blocks reduces an overhead of block management. A block is generally considered a smallest quantum of erasing. A page is generally considered a smallest quantum of writing. A read unit (or codeword or Epage or ECC-page) is a smallest correctable quantum of reading and/or error correction. Each block includes an integer number of pages. Each page includes an integer number of read units. 
         [0024]    In some embodiments, the circuit  80  may be implemented as a single-level cell (e.g., SLC) type circuit. A SLC type circuit generally stores a single bit per memory cell (e.g., a logical 0 or 1). In other embodiments, the circuit  80  may be implemented as a multi-level cell (e.g., MLC) type circuit. A MLC type circuit is generally capable of storing multiple (e.g., two) bits per memory cell (e.g., logical 00, 01, 10 or 11). In still other embodiments, the circuit  80  may implement a triple-level cell (e.g., TLC) type circuit. A TLC circuit may be able to store multiple (e.g., three) bits per memory cell (e.g., a logical 000, 001, 010, 011, 100, 101, 110 or 111). In yet another embodiment, the circuit  80  may implement a circuit with a cell level greater than a triple-level cell. Generally, the circuit  80  may be implemented as an n-level cell capable of storing n-bits per memory cell. 
         [0025]    In general, the controller  70  may include an erase/program unit that may implement redundancy across the modules  82   a - 82   n . For example, multiple blocks may be read from multiple dies  82   a - 82   n . The erase/program unit may be implemented as part of the firmware (or logic)  100 . 
         [0026]    The drive  90  may contain, in one example, multiple NAND Flash or memory modules  82   a - 82   n . Each of the memory modules  82   a - 82   n  may be fabricated as one or more dies (e.g., 1, 2, 4, 8, etc.). The dies (or modules)  82   a - 82   n  may operate to read or to write concurrently. The read and write bandwidth depends on how many of the dies  82   a - 82   n  are implemented, as well as the bandwidth of each of the dies  82   a - 82   n . Each of the dies  82   a - 82   n  may contain a plurality of planes. Each of the planes of the dies  82   a - 82   n  may contain a plurality of blocks  84   a - 84   n . The blocks  84   a - 84   n  of the planes of one of the dies  82   a - 82   n  may be accessed in parallel. If the SSD  90  receives the host command REQ, in order to achieve the best performance, and/or to address wear leveling issues, the drive  90  will walk through all of the dies  82   a - 82   n  (e.g., a first page of DIE 0 , DIE 1  . . . DIEn, then a next page of DIE 0 ). 
         [0027]    Referring to  FIG. 2 , a diagram illustrating a more detailed embodiment is shown. The controller  70  is shown connected to the memory circuit  80 . The controller  70  generally comprises the firmware  100 , a block (or circuit)  102 , a block (or circuit)  104 , a block (or circuit)  106 , and a block (or circuit)  108 . The circuit  102  may be a correlator. For example, the correlator  102  may be a linear correlator. The circuit  104  may be an ECC decoder. The circuit  106  may be a buffer. The circuit  108  may represent other modules. 
         [0028]    The ECC decoder  104  may be configured to perform error correction code decoding with soft decoding capability. Generally, the ECC decoder  104  is implemented using digital circuits. For example, the decoder  104  may be configured to generate low density parity check codes. Error correction codes with soft decoding capability may need soft input in the form of LLRs. The ECC decoder may have an input precision (e.g., 4-bits, 3-bits, etc.). Before LLRs are sent to the decoder  104 , the LLRs may be converted to fixed-point numbers by the correlator  102 . 
         [0029]    The correlator  102  may convert LLRs to fixed-point values. The conversion may determine which original LLR value to map to a specific fixed-point value. For example, the correlator  102  may convert the LLR to correspond with the input precision (e.g., 4-bits) of the ECC decoder  104 . The correlator  102  may scale down the magnitude of the original LLR values. The correlator  102  may determine a best scalar value. The correlator  102  may clip the scaled LLRs and round the scaled LLRs to integer values. Generally, when the input precision of the ECC decoder  104  is lower, heavier scaling is needed from the correlator  102 . Heavier scaling may result in more rounding noise. 
         [0030]    The controller  70  may generate LLRs. Generally, generating LLRs involves two steps. First, the desired read reference voltages are set based on the criterion of maximizing mutual information. Second, the LLRs are either generated by calculation if cell voltage distributions are known (or measured) when read back data is available. 
         [0031]    Referring to  FIG. 3 , a diagram illustrating read reference voltages and decision regions when reading a single voltage is shown. The read reference voltages (e.g., Vrefs), decision regions, and the corresponding decision patterns when the least-significant bit (LSB) pages of the MLC flash are read are shown. Four voltage distributions from left to right may be mapped to “11”, “01”, “00” and “10”, respectively. For an example of reading a single voltage (e.g., V 0 ), the voltage axis (e.g., the horizontal axis V) is divided into two decision regions (e.g., A 0 , and A 1 ). Corresponding decision patterns may be 1 for the region A 0 , and 0 for the region A 1 . Soft decisions corresponding to 0 or 1 may be obtained if the four distribution parameters are considered. The soft decision with a single read is a fairly rough approximation. A higher quality of soft decisions may be obtained with more reads. 
         [0032]    Referring to  FIG. 4 , a diagram illustrating read reference voltages and decision regions when reading two voltages is shown. The read reference voltages (e.g., Vrefs), decision regions, and the corresponding decision patterns when LSB pages of MLC flash are read are shown. Four voltage distributions from left to right may be mapped to “11”, “01”, “00” and “10”, respectively. For an example of reading two voltages (e.g., V 0 , and V 1 ), the voltage axis (e.g., the horizontal axis V) is divided into three decision regions (e.g., A 0 , A 1 , and A 2 ). Corresponding decision patterns may be 11 for the region A 0 , 01 for the region A 1 , and 00 for the region A 2 . 
         [0033]    Referring to  FIG. 5 , a diagram illustrating read reference voltages and decision regions when reading three voltages is shown. The read reference voltages (e.g., Vrefs), decision regions, and the corresponding decision patterns when LSB pages of MLC flash are read are shown. Four voltage distributions from left to right may be mapped to “11”, “01”, “00” and “10”, respectively. For an example of reading three voltages (e.g., V 0 , V 1 , and V 2 ), the voltage axis (e.g., the horizontal axis V) is divided into four decision regions (e.g., A 0 , A 1 , A 2 , and A 3 ). Corresponding decision patterns may be 111 for the region A 0 , 011 for the region A 1 , 001 for the region A 2 , and 000 for the region A 3 . 
         [0034]    A read operation of a NAND flash device (e.g., the memory  80 ) may include a process for applying threshold detection. Implementing multiple reads with varying read threshold voltages is equivalent to applying multiple threshold detection operations. The cell voltage is quantized into more than two regions leading to more than two decision patterns, as shown in  FIG. 4  and  FIG. 5 . The decision patterns are hard decisions. A soft decision may be calculated corresponding to each decision pattern. The soft decisions are usually in the form of LLR values 
         [0035]    Generally, the number of decision patterns for N reads of a lower page (e.g., a LSB page) is N+1. Generally, for a most-significant bit (e.g., MSB) page with conventional Gray coding, there may be a maximum of 2*N hard decision patterns. For example, a pair of read voltages is used for each read operation. 
         [0036]    Generally, each of the decision regions may have an associated LLR value. The associated LLR value may be an original LLR value. An original LLR value may be a real value number ranging from −Inf to +Inf. 
         [0037]    Referring to  FIG. 6 , a diagram illustrating a graph of LLR values for reads of pages is shown. The graph shows an example of original LLR values for seven reads of LSB pages. For example, the original LLR with LLR index position 0 may have a real value of approximately 14.5. In another example, the original LLR with LLR index position 1 may have a real value of approximately 10.2. Similarly, the original LLRs with index positions 2-7 have real values. 
         [0038]    The original LLRs having real values may be converted by the correlator  102  to fixed-point (e.g., integer) values before being sent to the decoder  104 . The original LLRs may need to be converted to fixed-point values since the decoder  104  may consist of digital circuits. 
         [0039]    In one example embodiment, the decoder  104  may be configured to receive 4-bit fixed-point numbers as inputs. A 4-bit number has 2̂4 possible integer values (e.g., ranging from −8 to 7). The conversion of an original LLR to a fixed-point number may be a mapping that determines which original LLR having a real value is mapped to which integer (e.g., fixed-point) value. For example, some of the original LLRs shown in  FIG. 6  are outside of the range of −8 to 7 (e.g., the original LLR with LLR index position 1 is shown having a real value of 10.2). 
         [0040]    The original LLR values may be converted to fixed-point integers (e.g., by the correlator  102 ). A scalar may be implemented to scale down the magnitude of the original LLR value. A version of the original LLR value having a scaled down magnitude may be clipped and/or rounded to an integer value. The LLR that has been scaled down, clipped, and/or rounded to an integer value may be a converted LLR. 
         [0041]    Generally, since min-sum decoding is linear, the scaling operation does not affect the decoded BER. An appropriate value for a scalar may be determined. The appropriate value for the scalar may be determined using the linear correlator  102 . The linear correlator  102  may calculate correlation coefficients between the original LLRs (e.g., as a sequence) and the converted LLRs (e.g., fixed-point integers). 
         [0042]    Generally, correlation coefficients may be a measure of “similarity” between two sequences. Selecting a scalar value that gives the largest correlation coefficient may keep the converted LLRs “similar” to the original LLRs. For example, the scalar giving the largest correlation coefficient may be the best scalar. The value of the best scalar may change based on the operating conditions of the SSD  90 . For example, the best scalar may minimize the distortion to the original LLRs introduced by the conversion to the converted LLRs. 
         [0043]    Referring to  FIG. 7 , a diagram illustrating a graph of correlation coefficients versus scalar values for a 4-bit input precision is shown. The y-axis may represent the correlation coefficient values. The x-axis may represent the scalar values. The correlation coefficient values versus scalar values of 0.2 to 1 for program/erase (e.g., P/E) cycles of 1000 and 5000 are plotted. Generally, the best scalar value is the scalar value resulting in a correlation coefficient closest to 1. For example, the best scalar value for P/E cycles of 1000 is 0.5. In another example, the best scalar value for P/E cycles of 5000 is 0.7. 
         [0044]    Generally, when the P/E cycle amount is smaller, the values of LLRs have a larger magnitude. A larger magnitude may need more scaling to reduce the distortion caused by clipping. For example, with 1000 P/E cycles a best scalar value of 0.5 results in more scaling than the best scalar value 0.7 with 5000 P/E cycles. In alternate embodiments, an alternate input precision for the decoder  104  may be implemented. For example, a 3-bit input precision may result in heavier (e.g., larger) scaling. With heavier scaling, the noise caused by rounding may be more significant. 
         [0045]    Referring to  FIG. 8 , a flow diagram illustrating a method (or process)  200  is shown. The method  200  may convert original LLR values to fixed point LLR values. The method  200  generally comprises a step (or state)  202 , a decision step (or state)  204 , a step (or state)  206 , a step (or state)  208 , a step (or state)  210 , a step (or state)  212 , a decision step (or state)  214 , a step (or state)  216 , and a step (or state)  218 . 
         [0046]    The state  202  may start the method  200 . Next, the decision state  204  may determine whether there are more original LLR values. If not, the method  200  moves to the state  206 , which ends the method  200 . If so, the method  200  moves to the state  208 . The state  208  may go to the next LLR index position. Next, the state  210  may retrieve the original LLR value. The state  212  may scale the magnitude of the original LLR value based on the range of the decoder input (e.g., the ECC decoder  104 ). Next, the method  200  moves to the decision state  214 . 
         [0047]    If the decision state  214  determines the scaled LLR is larger than the range of the decoder input (e.g., the ECC decoder  104 ), the method  200  moves to the state  216 . The state  216  may clip the scaled LLR value to the range of the decoder input (e.g., the ECC decoder  104 ). Next, the method  200  moves to the state  218 . If the decision state  214  determines the scaled LLR value is not larger than the range of the decoder input (e.g., the ECC decoder  104 ), the method  200  moves to the state  218 . The state  218  may round the LLR value to an integer (e.g., fixed point) value. Next, the method  200  returns to the decision state  204 . 
         [0048]    Generally, the magnitude of the original LLR values are larger than the range of the input (e.g., the input precision) of the ECC decoder  104 . The original LLR values may be scaled first. If the magnitude of the scaled LLR values are still larger than the range of the input of the ECC decoder  104 , then the LLR values may be clipped. 
         [0049]    For example, after scaling the original LLR values the scaled LLR values may be within the input range of the ECC decoder  104 , and the scaled LLR values may not need to be clipped. In another example, after scaling the original LLR values the scaled LLR values may not be within the input range of the ECC decoder  104 , and the scaled LLR values may be clipped. The range of the decoder input may be determined by the input precision of the ECC decoder  104 . Original LLR values may be scaled and/or clipped before being rounded to a fixed-point value so that the converted LLR values may be similar to the original LLR value (e.g., to maximize the correlation). 
         [0050]    Referring to  FIG. 9 , a flow diagram illustrating a method (or process)  240  is shown. The method  240  may determine the best scalar value. The method  240  generally comprises a step (or state)  242 , a decision step (or state)  244 , a step (or state)  246 , a step (or state)  248 , a step (or state)  250 , a step (or state)  252 , a step (or state)  254 , a step (or state)  256 , a decision step (or state)  258 , and a step (or state)  260 . 
         [0051]    The state  242  may start the method  240 . Next, the decision state  244  may determine whether there are more scalar values. If so, the method  240  moves to the state  250 . If not, the method  240  moves to the state  246 . The state  246  may use the scalar marked as the best scalar. Next, the state  248  may end the method  240 . 
         [0052]    The state  250  may use the next scalar value. Next, the state  252  may scale down the magnitude of LLRs (e.g., the original LLRs may be scaled down). The state  254  may clip scaled LLRs and round to integer values (e.g., converted LLRs). The state  256  may calculate a correlation coefficient between the original LLRs and the converted LLRs. Next, the method  240  moves to the decision state  258 . If the decision state  258  determines the scalar does not give the largest correlation coefficient, the method  240  returns to the decision state  244 . If the decision state  258  determines the scalar does give the largest correlation coefficient, the method  240  moves to the state  260 . The state  260  may mark the scalar as the best scalar. Next, the method  240  returns to the decision state  244 . 
         [0053]    Generally, optimal LLR values change with the use of the SSD  90 . During the life time of the SSD  90 , the LLRs may be updated by the SSD controller  70 . When the LLRs are updated by the SSD controller  70 , a new best scalar value may be determined. The new best scalar value may be determined by the correlator  102  by calculating the correlation coefficient for various scalar values (as shown in  FIG. 7 ). 
         [0054]    Referring to  FIG. 10 , a method (or process)  280  is shown. The method  280  may illustrate triggering the determination of a new scalar value. The method  280  generally comprises a step (or state)  282 , a decision step (or state)  284 , a step (or state)  286 , a step (or state)  288 , and a step (or state)  290 . 
         [0055]    The state  282  may start the method  280 . Next, the method  280  moves to the decision state  284 . If the decision state  284  determines the LLR lookup tables (LUT) has not been updated, the method  280  moves to the state  286 . The state  286  may continue using the scalar currently marked as the best scalar. Next, the method  280  moves to the state  290 , which ends the method  280 . If the decision state  284  determines the LLR lookup tables (LUT) has been updated, the method  280  moves to the state  288 . The state  288  may determine the best scalar (e.g., as described in  FIG. 9 ). Next, the method  280  moves to the state  290 , which ends the method  280 . 
         [0056]    Generally, the event of updating LLR LUTs may trigger determining a new best scalar value. The SSD  90  may be manufactured having LLR LUTs. The LLR LUTs may be updated during the lifetime of the SSD  90 . For example, an update to the firmware  100  may provide new/updated LLR LUTs. In another example, the LLR LUTs may be updated based on a number of program/erase cycles. 
         [0057]    When the best scalar is determined by the correlator  102 , the measured LLRs (e.g., the original LLR values having a high precision) may be converted to the size of the corresponding input precision of the decoder  104  (e.g., 4-bits) by clipping and/or rounding. For this purpose, the correlator  102  may be implemented by either firmware and/or hardware. Generally, the number of LLRs is not large. Since the number of LLRs is not large, the computation involved may be very light. The original LLR values may be calculated and/or measured. The original LLR values may be continuous values that may not be used directly by the ECC decoder  104 . To convert the original LLR values to converted LLR values with a maximum correlation, a best scalar value may be determined. 
         [0058]    The functions performed by the diagrams of  FIGS. 8-10  may be implemented using one or more of a conventional general purpose processor, digital computer, microprocessor, microcontroller, RISC (reduced instruction set computer) processor, CISC (complex instruction set computer) processor, SIMD (single instruction multiple data) processor, signal processor, central processing unit (CPU), arithmetic logic unit (ALU), video digital signal processor (VDSP) and/or similar computational machines, programmed according to the teachings of the specification, as will be apparent to those skilled in the relevant art(s). Appropriate software, firmware, coding, routines, instructions, opcodes, microcode, and/or program modules may readily be prepared by skilled programmers based on the teachings of the disclosure, as will also be apparent to those skilled in the relevant art(s). The software is generally executed from a medium or several media by one or more of the processors of the machine implementation. 
         [0059]    The invention may also be implemented by the preparation of ASICs (application specific integrated circuits), Platform ASICs, FPGAs (field programmable gate arrays), PLDs (programmable logic devices), CPLDs (complex programmable logic devices), sea-of-gates, RFICs (radio frequency integrated circuits), ASSPs (application specific standard products), one or more monolithic integrated circuits, one or more chips or die arranged as flip-chip modules and/or multi-chip modules or by interconnecting an appropriate network of conventional component circuits, as is described herein, modifications of which will be readily apparent to those skilled in the art(s). 
         [0060]    The invention thus may also include a computer product which may be a storage medium or media and/or a transmission medium or media including instructions which may be used to program a machine to perform one or more processes or methods in accordance with the invention. Execution of instructions contained in the computer product by the machine, along with operations of surrounding circuitry, may transform input data into one or more files on the storage medium and/or one or more output signals representative of a physical object or substance, such as an audio and/or visual depiction. The storage medium may include, but is not limited to, any type of disk including floppy disk, hard drive, magnetic disk, optical disk, CD-ROM, DVD and magneto-optical disks and circuits such as ROMs (read-only memories), RAMS (random access memories), EPROMs (erasable programmable ROMs), EEPROMs (electrically erasable programmable ROMs), UVPROM (ultra-violet erasable programmable ROMs), Flash memory, magnetic cards, optical cards, and/or any type of media suitable for storing electronic instructions. 
         [0061]    The elements of the invention may form part or all of one or more devices, units, components, systems, machines and/or apparatuses. The devices may include, but are not limited to, servers, workstations, storage array controllers, storage systems, personal computers, laptop computers, notebook computers, palm computers, personal digital assistants, portable electronic devices, battery powered devices, set-top boxes, encoders, decoders, transcoders, compressors, decompressors, pre-processors, post-processors, transmitters, receivers, transceivers, cipher circuits, cellular telephones, digital cameras, positioning and/or navigation systems, medical equipment, heads-up displays, wireless devices, audio recording, audio storage and/or audio playback devices, video recording, video storage and/or video playback devices, game platforms, peripherals and/or multi-chip modules. Those skilled in the relevant art(s) would understand that the elements of the invention may be implemented in other types of devices to meet the criteria of a particular application. 
         [0062]    The terms “may” and “generally” when used herein in conjunction with “is(are)” and verbs are meant to communicate the intention that the description is exemplary and believed to be broad enough to encompass both the specific examples presented in the disclosure as well as alternative examples that could be derived based on the disclosure. The terms “may” and “generally” as used herein should not be construed to necessarily imply the desirability or possibility of omitting a corresponding element. 
         [0063]    While the invention has been particularly shown and described with reference to embodiments thereof, it will be understood by those skilled in the art that various changes in form and details may be made without departing from the scope of the invention.