Patent Publication Number: US-8972831-B2

Title: Belief propagation processor

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is the National Stage of International Application No. PCT/US2011/020794, filed on Jan. 11, 2011, and is a Continuation-in-Part of PCT Application No. PCT/US10/25956, titled “Belief Propagation Processor,” filed Mar. 2, 2010, and published as WO2010/101944 on Sep. 10, 2010, and claims the benefit of the following applications:
         U.S. Provisional Application No. 61/293,999, titled “Belief Propagation Processor,” filed Jan. 11, 2010, and   U.S. Provisional Application No. 61/380,971, titled “Current Mode Analog Logic,” filed on Sep. 8, 2010.
 
Each of the above-referenced applications is incorporated herein by reference.
       

     This application is related to, but does not claim the benefit of the filing dates of, the following applications:
         U.S. patent application Ser. No. 12/716,148, titled “Circuits for Soft Logical Functions,” filed Mar. 2, 2010,   U.S. patent application Ser. No. 12/716,113, titled “Signal Mapping”, filed Mar. 2, 2010, and   U.S. application Ser. No. 12/537,060, titled “Storage Devices with Soft Processing,” filed Aug. 6, 2009,
 
the contents of which are incorporated herein by reference.
       

    
    
     BACKGROUND 
     This document relates to an analog belief propagation processor. 
     “Belief Propagation” (BP) is an efficient approach to solving statistical inference problems. The approach exploits underlying structure of a network of stochastic elements and its constraints and Bayesian laws of probabilities to find the most optimal set of valid outputs that satisfy constrains and network structure requirements. 
     Belief Propagation includes a class of techniques for performing statistical inference using a system model that is in the form of a graph. The term “graph” here refers to the mathematical definition of a graph, which represents the connectedness of a set of abstract objects. The objects are often referred to as “nodes” and the connections between objects are often referred to as “edges.” One common type of graph used in such models is referred to as a “factor graph.” In a factor graph (a particular style of factor graph called a “Forney factor graph”) the nodes represents statistical relationships between values, which are represented as edges. Other types of graphs, such as Bayesian networks, and Markov random fields are also commonly used for statistical inference. 
     Examples of Belief Propagation approaches operate by passing messages between nodes in the graph, where each message represents a summary of the information known by that node through its connections to other nodes. Such approaches are known by various names, including belief propagation, probability propagation, message passing, and summary-product algorithms, among others. Particular forms of these approaches include sum-product, max-product, and min-sum. 
     A large variety of approaches to coding, signal processing, and artificial intelligence may be viewed as instances of the summary-product approach (or belief/probability propagation approach), which operates by message passing in a graphical model. Specific instances of such approaches include Kalman filtering and smoothing, the forward backward algorithm for hidden Markov models, probability propagation in Bayesian networks, and decoding algorithms for error correcting codes such as the Viterbi algorithm, the BCJR algorithm, and the iterative decoding of turbo codes, low-density parity check codes, and similar codes. 
     Graphs on which belief propagation may operate include two types: graphs with loops (cyclic graphs) and graphs with no loops (acyclic graphs). Graphs with no loops are also known as “trees.” Belief propagation procedures differ fundamentally between these two types of graphs. For a tree, belief propagation approach can proceed in a well-defined order with a well-defined number of steps to compute the result. And assuming ideal computation, this result is always known to be correct. For a graph with loops, on the other hand, belief propagation approaches are generally iterative, meaning the same set of computations must be repeated successively until a result is reached. In this case, the computation typically converges to a useful result, but does not always do so. In some cases, the computation may not converge to a single result, or if it does, the result in some cases is inaccurate. For a cyclic graph, the performance of belief propagation can depend on the order in which the computations are performed, which is known as the message passing ‘schedule.’ 
     In one particular application mentioned above, Belief Propagation has been adopted as an efficient method of implementing decoders for various forward error correcting codes. In this case BP uses structure of the code and constraints to infer the correct valid codeword from the input codeword that contains noise, for instance, with each element (e.g., bit) of the input codeword being represented as a distribution rather than a discrete value. In some implementations of Belief Propagation for forward error correction a Digital Signal Processor is used to perform various arithmetic computations required by the algorithm with all the statistical data being processed in digital format. 
     Observing the fact that “soft”—probabilistic data is continuous in nature, i.e., represented by real values in a finite interval, it is possible to implement belief propagation algorithm using analog electrical circuits. Since only one signal is associated with the unit of statistical data rather than multiple signals for different digits (e.g., binary digits, bits) of the digital signal representing the same data, the savings in hardware and power dissipation can be very significant. 
     Several architectures had been proposed that utilize analog circuits to perform efficient decoding of various codes, including convolutional codes, Low Density Parity Check Codes (LDPC) or linear block codes. These include analog implementations that use a so-called full flat architecture, where each input data symbol is associated with dedicated computing element. 
     SUMMARY 
     In one aspect, in general, an analog processor includes an intermediate value memory, a plurality of analog computation elements, and a controller. The analog processor may be used for error correction of a parity check code having a plurality of parity constraints on input values of an input frame. The intermediate value memory comprising a plurality of memory elements, each memory element being associate with a corresponding constraint and a corresponding input value. Each analog computation modules is used for performing an analog computation associated with a parity constraint of the code. Each input of the computation modules is connectable (e.g, via passive and/or active signal paths) to a selected plurality of the memory elements and each output of the computation modules is connectable to a selected memory element. The controller is used for sequencing operation of the processor in a series of iterations, and in each iteration, in a series of cycles. In each cycle the controller configures connections of inputs and outputs of each analog computation module to perform a computation associated with one of the parity constraints of the code, including connecting each output of the module to a memory element associated with the parity constraint and connecting each input of the module to a memory element associated with an input value used in the parity constraint. In each cycle, no memory element connected to both an input and an output of any of the analog computation modules. 
     The arrangement of intermediate value memory can permit the controller to sequence connections of the memory elements to the analog computation modules without requiring that any memory element is both read from and written to in one cycle, while also allowing computations for multiple constraints to be performed in each cycle. The arrangement of the memory can be applied to efficiently process LDPC codes. 
     In another aspect, in general, an analog processor has a first memory module and a second memory module. The first memory module is for storing a first set of storage values in respective storage elements each representing a respective input to the processor. The second memory module is for storing a second set of storage values in analog form in respective storage elements. The second set of storage values includes intermediate values determined during operation of the processor. The analog processor also includes an analog computation module coupled to the first and the second memory modules. This processor is configurable such that in each of a set of operation cycles the analog module determines values for at least some of the second set of storage values based on at least some of the first and the second sets of storage values. An output module is use for generating a set of outputs from at least some of the second set of storage values. 
     Aspects may include or more of the following features. 
     The first storage module is configured to store the first set of storage values in analog form. 
     The analog computation module is linked to the first and the second memory modules via analog signal paths. For example, the analog signal paths are each configured to carry a value on a conductor represented as at least one of a voltage and a current proportional to the value. 
     The analog module is configurable to determine values for a different subset of the second set of storage values in each of a plurality of operation cycles. 
     The processor includes input selection circuitry configurable to couple the analog computation module to outputs of selected memory elements of the first and the second memory modules. 
     The processor further includes, for each analog computation module, a plurality of signal busses, each bus providing an input value to the analog computation module and being switchably coupled to a plurality of the storage elements of the second memory module. 
     The storage elements are coupled to switchably provide a current representation of a storage value stored in the storage element such that the input value provided to the analog computation module is represented as a current that is substantially proportional to a sum of the currents representations provided by the storage elements. 
     The processor further include output section circuitry configurable to accept outputs of selected memory elements of the first and the second memory modules, and to determine outputs of the analog processor. 
     The processor includes multiple analog computation modules being concurrently operable to determines values for different subsets of the second set of storage values in each operation cycle. 
     The second memory module includes a plurality of section, each associated with a corresponding different one of the analog computation modules for storing values determined by the associated computation module. 
     The second memory module is configured such that in a single operation cycle, each storage element can provide a storage value to one or more of the analog computation modules and can accept a determined value to storage in the storage element for providing in a subsequent operation cycle. 
     Each storage element is associated with two storage locations such that in any one cycle, one storage location is used for accepting a determined value and one storage location is used for providing a value. 
     The second memory module includes multiple memory sections. Groups of the sections form banks, wherein for each of the analog computation modules each of a set of inputs to the module is associated with a different bank of the memory sections. 
     The processor is configurable such that in each cycle, each memory section includes memory elements that either provide values to one ore more analog computation modules or memory elements that are updated with values from one or more analog computation modules. 
     The processor is configured to implement a belief propagation computation. 
     The processor of claim is configured to implement a factor graph computation. 
     The analog computation module is configured to accept and provide analog signals that are substantially logarithmic representations of at least one of probabilities, likelihoods, and likelihood ratios. 
     The processor is configured to implement a decoder for a low density parity check (LDPC) code. 
     The processor further includes a controller configured to control operation of the processor to perform a set of iterations of computation, each iteration comprising a set of computation cycles. 
     The set of computation cycles is substantially the same in each iteration, each cycle being associated with a configuration of the first and the second storage modules to provide inputs and output to one or more analog computation modules. 
     The processor is configured and/or configurable to implement a decoder parity check code, and each cycle is associated with one or more parity check constraints, and wherein the cycles of each iteration are together associated with all the parity check constraints of the code. 
     The analog computation module implements a network of analog processing elements. 
     The analog processing elements include elements that represent soft logical operations. For example, the soft logical operations include soft XOR operations. 
     The network of elements is acyclic. 
     The network of elements includes at least one cycle of elements, the analog computation module being configured to implement an relaxation computation. 
     The analog computation module includes inputs for configuring one or more gain characteristics in the network of processing elements. 
     In another aspect, in general, a decoder includes a first memory for storing code data having a length in bits, and a second memory for storing intermediate data in analog form. The decoder includes an analog decoder core coupled to the first memory and to the second memory. The decoder core has an input length less than the length of the code data and an output length less than a number of constraints represented in the code data. The decoder further includes a controller for, in each of a set of cycles, coupling the inputs of the decoder code to selected values from the first and the second memories, and coupling outputs of the decoder core for storage in the second memory. An output section of the decoder is coupled to the second memory for providing decoded data based on values stored in the second memory. 
     In another aspect, in general, a method is used for forming a data representation of an analog processor. The method includes forming: a data representation of a first memory module for storing a first set of storage values in respective storage elements each representing a respective input to the processor; a data representation of a second memory module for storing a second set of storage values in analog form in respective storage elements, the second set of storage values including intermediate values determined during operation of the processor; a data representation of an analog computation module coupled to the first and the second memory modules, the processor being configurable such that in each of a set of operation cycles the analog module determines values for at least some of the second set of storage values based on at least some of the first and the second sets of storage values; and a data representation of an output module for generating a set of outputs from at least some of the second set of storage values. 
     In some examples, forming the data representations includes forming Verilog representations of the processor. 
     The method can further include fabricating a integrated circuit implementation of the analog processor according to the formed data representation. 
     In some examples, the method further includes accepting a specification of a parity check code and forming the data representations to represent an implementation of a decoder for the code. 
     In another aspect, in general, software stored on a computer readable medium includes instructions for and/or data imparting functionality when employed in a computer component of an apparatus for forming an integrated circuit implementation of any of the analog processor described above. 
     In another aspect, in general, decoding method includes, in each of a series of cycles of a decoding operation, applying a portion of code data and a portion of an intermediate value data to an analog decoder core, and storing an output of the decoder coder in an analog storage for the intermediate data. Data, including intermediate value data from the analog storage, are combined to form decoded data representing an error correction of the code data. 
     In some examples, each of the series of cycles is associated with a corresponding subset of less that all of a plurality of parity-check constraints of the code. The intermediate value data may include values each associated with a different one of the parity check constraints of the code. 
     In another aspect, in general, a processor includes a first memory module for storing a first set of storage values each representing a respective input, and a second memory module for storing a second set of storage values in analog form. An analog module is coupled to the first and the second memory modules. The analog module is configured to, in each operation cycle of at least one iteration, update at least some of the second set of storage values based on the first and the second sets of storage values. An output module is for generating a set of outputs from at least some of the second set of storage values. 
     The analog module may be configured for updating a different subset of the second set of storage values in each of at least two operations cycles of an iteration. 
     The analog module may include a set of distributed components each configured to update a different subset of the second set of storage values using a different subset of the first set of storage values and the second set of storage values. 
     In another aspect, in general, a decoder includes a first memory for storing code data having a length in bits, and a second memory for storing intermediate data in analog form. An analog decoder core is coupled to the first memory and to the second memory, the decoder core having an input length less than the length of the code data and an output length less than a number of constraints represented in the code data. A controller in the decoder is for, in each of a plurality of cycles, coupling the inputs of the decoder code to selected values from the first and the second memories, and coupling outputs of the decoder core for storage in the second memory. An output section is coupled to the second memory for providing decoded data based on values stored in the second memory. 
     In another aspect, in general, a decoding method includes, in each of a number of cycles of a decoding operation, applying a portion of code data and a portion of an intermediate value data to an analog decoder core, and storing an output of the decoder coder in an analog storage for the intermediate data. Data, including intermediate value data from the analog storage, is then combined to form decoded data representing an error correction of the code data. 
     Advantages of one or more aspects may include the following: 
     Use of analog computations and/or analog storage of intermediate values provides lower power and/or smaller circuit area implementations as compared to a digital implementations, for instance in applications of iterative decoding or error correcting codes. 
     Iterative use of one or more analog computation cores provides lower power and/or smaller circuit area as compared to fully parallel relaxation implementations of similar decoding algorithms. In some examples, a partially relaxation implementation in which parts of a computation are implemented in relaxation from in each of a succession of cycles may also provide similar advantages over a fully parallel relation implementation. 
     Approaches are applicable to decoding of block codes without requiring that the size and/or power requirements of an implementation grow substantially with the length of the code. 
     Other features and advantages of the invention are apparent from the following description, and from the claims. 
    
    
     
       DESCRIPTION OF DRAWINGS 
         FIG. 1  is an example factor graph for a length 8 LDPC code; 
         FIG. 2A  is a diagram that illustrates transformation of a variable node with bidirectional links to a set of variable nodes with directed links, and  FIG. 2B  is a diagram that illustrates a similar transformation for a constraint node; 
         FIG. 3  is a portion of the graph shown in  FIG. 1 ; 
         FIG. 4  is a portion of a directed graph corresponding to the portion of the bidirectional graph shown in  FIG. 3 ; 
         FIG. 5  is a diagram illustrating a module implementation corresponding to the portion of the graph shown in  FIG. 4 ; 
         FIG. 6  is a diagram illustrating output calculation; 
         FIG. 7A  is a diagram that shows a relationship between input and outputs or a module, and  FIG. 7B  illustrates the corresponding code matrix; 
         FIG. 8  is a diagram of an implementation of a decoder for a length 8 LDPC code using a shared module; 
         FIG. 9  is a table that specifies inputs and outputs for the shared module shown in  FIG. 8 ; 
         FIG. 10  is a block diagram of a decoder with two shared modules; 
         FIG. 11  is a tabular representation of a parity matrix for a (1056, 352) LDPC code; 
         FIG. 12  is a diagram of a shared module for use with the code shown in  FIG. 11   
         FIG. 13  is a block diagram of a decoder for a (1056, 352) LDPC code with eight shared modules (of which two are illustrated); 
         FIG. 14  is a circuit implementation of a variable node; 
         FIG. 15A  is a circuit implementation of a constraint node; 
         FIG. 15B  is an alternative implementation of a constraint node; 
         FIG. 16  is a diagram that illustrates a distributed bus implementation of a variable node; 
         FIG. 17  is a diagram of an alternative shared module; 
         FIG. 18  is a block diagram of a decoder that uses distributed bus implementations of variable nodes; 
         FIG. 19  is a block diagram of an alternative architecture for a decoder for a (1056, 352) LDPC code; 
         FIG. 20  is a block diagram that shows multiple update modules in the architecture shown in  FIG. 17 ; and 
         FIGS. 21A-B  are a block diagram of an implementation of a decoder. 
         FIG. 22  is a schematic including differential current storage cell and a current input Soft Equals gate. 
     
    
    
     DESCRIPTION 
     An approach to using an analog processor for belief propagation is described in PCT Publication WO2010/101944 and is included below. The description below first provides in Section 1 a description of a general approach to analog belief propagation, and provides in Section 2 a description of one or more additional embodiments, which include improvements to the general approach, for instance, based on arrangements and/or scheduling of memory access, and use of current-mode analog storage elements. Section 3 provides further details of further embodiments, implementations and/or applications of the approach. 
     1 Overview of Analog Belief Propagation 
     Referring to  FIG. 1 , in one example of an analog-based implementation of a belief propagation processor, a decoder for a Low Density Parity Check (LDPC) code is based on a factor graph  100  in which one variable node  110  is associated with each different input bit (b j ), and one check (constraint) node  120  is associated with each constraint. In  FIG. 1 , an example with eight input bits in an input frame with four checks (constraints) on the input bits is shown. The code can be represented in matrix form in which each column is associated with a different input bit, and each row is associated with a different check or constraint. An (i, j) entry is 1 if the j th  input is used in the i th  constraint and 0 otherwise. In the LDPC example, the constraint is that the XOR of the inputs for a constraint is 0. This example length 8 LPDC code can be represented according to the following check matrix (note that the rows are dependent modulo 2 in this illustrative example, which is not necessarily true in general): 
     
       
         
           
               
             
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                     1 
                   
                 
                 
                   
                     1 
                   
                   
                     1 
                   
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                   
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                     1 
                   
                   
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                     1 
                   
                   
                     1 
                   
                   
                     1 
                   
                 
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                   
                     1 
                   
                   
                     0 
                   
                   
                     1 
                   
                   
                     0 
                   
                 
               
               ] 
             
           
         
       
     
     In  FIG. 1 , each edge is bidirectional. Referring to  FIGS. 2A-B , an equivalent directed (unidirectional) graph can be formed by replacing each n-edge node with n separate nodes, each of the n nodes having n−1 inputs and one output, and forming unidirectional edges between the nodes to achieve the connectivity of the original graph. Referring to  FIG. 2A , for instance, each 3-edge variable node  110  can be replaced with three 2-input/1-output variable nodes  210 ,  212 . Referring to  FIG. 2B , each 4-edge check node  120  can be replaced with four 3-input/1-output check nodes  220 . 
     One approach to analog implementation of a decoder corresponding to the factor graph shown in  FIG. 1  is to implement a circuit element for each node of the equivalent unidirectional graph. Referring to  FIG. 3 , a portion of the graph shown in  FIG. 1  is illustrated showing check node 0 ( 120 ), the bidirectional edges and variable nodes 1, 3, 4 and 7 ( 110 ) linked to that check node, as well as the other check nodes 1, 2 and 3 ( 120 ) linked to those variable nodes. Referring to  FIG. 4 , a portion of the corresponding directed graph is shown in which check node 0 ( 120 ) is expanded as four 3-input/1-output check nodes  220 , for instance, labeled “0/1” to indicate that this is part of the expansion of check node 0 with the output link coupled to variable node 1. Similarly variable node 1 ( 110 ) is shown in its expansion into three 2-input/1-output nodes  210 ,  212 , for instance, labeled “1/0” to indicate that this is part of the expansion of variable node 1 with the output link coupled to check node 0, or labeled “1/out” to indicate that the output link provides an output of the factor graph. 
     In the example, which is partially illustrated in  FIG. 4 , a full implementation has four circuit elements for each check node (i.e., 16 total expanded unidirectional check nodes  220 ), and three circuit elements for each variable node (i.e., 24 total expanded unidirectional variable nodes  210 ,  212 ). Out of the three circuit elements for a variable node, two (i.e., 16 total expanded variable nodes  210  for all variable nodes) are used for message passing in an iterative stage of decoding operation, and one (i.e., 8 total expanded variable nodes  212  for all variable nodes) is used for generating the decoder output (i.e., the “belief”) in an output stage of decoding operation, as will be described further below. 
     In operation, input signals y i  are used to determine corresponding analog representations of input messages, which may be determined in a signal mapping circuit. In some examples, the inputs messages form representations of the probabilities corresponding to bits b i , but the reader should recognize that the discussion below with respect to computations involving representations of bit probabilities is illustrative of a particular form of input and internal messages that are stored or passed during computation. These messages are provided to the inputs of the variable nodes  210 , for example, as outputs of analog input registers  260 . As discussed further below, in some embodiments the representations of the bit probabilities are provided as analog signals from the input registers  260  encoding a (prior) log likelihood ratio (LLR) which is typically of the form 
               log   ⁡     (       Pr   ⁡     (       b   i     =     0   |     y   i         )         Pr   ⁡     (       b   i     =     1   |     y   i         )         )       ,         
In the case of equal prior bit probabilities P(b i =0)=P(b i =1) is equal to
 
               log   ⁡     (       P   ⁡     (         y   i     |     b   i       =   0     )         P   ⁡     (         y   i     |     b   i       =   1     )         )       .         
In some examples, these bit probabilities are encoded as voltage or current in single-ended or differential form (e.g., using a pair of conducting paths for each unidirectional signal).
 
     The approach partially illustrated in  FIG. 4  is one of a number of approaches to implementation of a decoder corresponding to the graph shown in  FIG. 1  that involve introducing an analog memory element  230  to break some or all cycles in the directed graph. In the approach shown in  FIG. 4 , the memory elements are introduced at the outputs of the check nodes. Other versions have such memory elements introduced at the output of the variable nodes instead of or in addition to the memory elements at the outputs of the check nodes. Note that in yet other embodiments, some or all cycles remain without memory elements, and operation is at least partially based on a “relaxation” form of computation as signals propagate through the cycles. In some embodiments, as combination of relaxation and memory based computation is used. 
     As illustrated in the example partially illustrated in  FIG. 4 , memory elements  230  in this embodiment store values in analog form, and are introduced at each output of the check nodes  220 ; that is, 16 memory elements are introduced. For notational simplicity, these locations are indexed as (i, j) and labeled “Ci,j”, for the output from check node i that is linked to variable node j. Note that each location corresponds to one of the non-zero entries in the check matrix of the code. The (i, j) memory location corresponds to the row i, column j, non-zero entry of the check matrix of the code. 
     In a number of approaches that make use of analog memory elements, the memory is introduced in the circuit implementation of the graph such there remain no cycles in the directed graph by breaking all cycles in the directed graph. The circuit implementation is then operated in a series of clocked cycles, such that at each cycle analog values read from some or all of the analog memory elements are propagated through analog circuit elements to inputs of some or all of the memory elements where they are stored at the end of the clock cycle. As discussed in detail below, such clocked (“discrete time”) implementation can be used to decode with a result that is similar to that which would result from a relaxation (“continuous time”) implementation. 
     Referring to  FIG. 5 , another partial illustration of the example shown in  FIG. 4  includes outputs of the four expanded check nodes  220  associated with the original check node 0 ( 120 ). A circuit block  390  forms an analog computation module that includes implementations of the expanded variable nodes 1/0, 3/0, 4/0, and 7/0 ( 210 ) which have outputs to the four expanded check nodes  220 . Note that check node 0 corresponds to row 0 of the matrix representation of the code, which is reproduced in  FIG. 7B . Note that the outputs of the circuit block  390  correspond to the memory locations row 0 of the matrix representation, as illustrated in  FIG. 7A . The inputs of the circuit block  390  correspond to the non-zero entries in each column of the matrix representation that has a non-zero entry in row 0, omitting those entries in row 0. In this illustration, the inputs correspond to the non-zero entries in columns 1, 3, 4 and 7 in rows 1, 2 and 3. This results in four memory cell inputs, C1,1, C3,3, C3,4 and C2,7, in addition to the inputs from the input bit probabilities, B1, B3, B4, and B7. 
     An example of a full clocked circuit implementation of a decoder for the length 8 LDPC has a memory element  230  at the output of each unidirectional check node  220 , and four copies of the circuit block  390 , one corresponding to each row of the code matrix. In the first stage of decoding operation, each unidirectional variable node  210  (i.e., a total of 16 circuit elements) takes its input from an output of a memory element  230 , and one of the input bit probabilities  260 . (Note that in general for other size codes, the variable nodes are associated with more than two check nodes, and therefore variable nodes would take as input values from multiple memory elements). The memory cells  230  as a whole form a memory that is configured so that effectively all the values are updated at once at the end of each clock cycle. One implementation of such a memory uses a “double buffering” approach in which two banks of memory are used, and in each clock period, one bank is read from and the other bank is written to, with the banks switching role between each clock period. 
     In some examples, the decoder may perform memory updates in successive clock cycles, each clock cycle corresponding to a full update of all memory cells of the memory  250 . The number of clock cycles to be performed in the first stage of decoding operation may be pre-determined, for example, based on design preference, or depend upon the satisfaction of certain convergence conditions, for example, satisfaction of the code constraints (i.e., full error correction) or a condition based on a rate of change of output values between iterations. 
     Referring to  FIG. 6 , in some examples, once the iterations of memory updates are completed, the decoder proceeds to the output stage of decoding operation to generate decoder outputs representing bit estimates. Here, the decoder outputs are denoted as {circumflex over (b)} j , each being an estimate of a corresponding input bit (b j ) based on the entire input signal. In some examples, as illustrated in  FIG. 6 , the variable node  212  outputs a message that includes a representation of the bit probability after decoding, for example, as an LLR, which can be considered to approximate 
             log   ⁡     (       P   ⁡     (       b   i     =     0   |     y     \   ⁢   i           )         P   ⁡     (       b   i     =     1   |     y     \   ⁢   i           )         )           
where y \i  denotes the observations not including y i . The output of variable node  212  is combined in a combination element  312  with the input bit probability representation from input register  260  to form the representation of the bit probability based on all the inputs and the constraints between the decoded bits. Recall that the output of input register  260  can be considered to represent
 
             log   ⁡     (       Pr   ⁡     (       b   i     =     0   |     y   i         )         Pr   ⁡     (       b   i     =     1   |     y   i         )         )           
and therefore the combined probability output from combination element  312  is computed as a sum approximates
 
             log   ⁡     (       Pr   ⁡     (       b   i     =     0   |   y       )         Pr   ⁡     (       b   i     =     1   |   y       )         )           
where y represents all the input values. Optionally the combined bit probability is passed through a hard decision, which in the case of binary outputs and logarithmic representations determines {circumflex over (b)} j  to take on the value of either 0 or 1 based on a thresholding of the combined log likelihood ratio as either greater or less than zero. For example, the output element that uses memory elements C0,1 and C1,1 and the input B1 to generate bit estimate {circumflex over (b)} j . In some implementations, the set of eight output elements may be configured to operate in a parallel fashion to generate the full set of bit estimates {circumflex over (b)} j  in a single clock cycle. Note that as illustrated in  FIG. 6 , elements  212  and  312  are drawn as separate. However, each effectively computes a sum of its inputs, and the two summations may be combined into a single circuit element  315 .
 
     Referring to  FIG. 8 , in another example of a clocked circuit implementation each of the nodes of the directed graph is not required to correspond to a different circuit element. That is, certain circuit elements form analog computation modules (“cores”) that are reused multiple times with different input and output connections (i.e., shared) within each iteration. The functions performed by multiple modules  390  in one clock cycle in the previous example are carried out in a series of clock cycles such that at in each of the series of clock cycles, only some of the memory elements  230  are updated, with all the memory elements being updated at the end of the series of clock cycles. Similarly, in the output stage of decoding operation, one or more shared circuit elements (e.g., element  315 ) may be reused in an output section  395  for generating one or more bit estimates in each of a series of clock cycles. In the discussion below, the entire series of clock cycles that updates all the memory elements in  FIG. 3  is referred to as an “iteration.” 
     Continuing to refer to  FIG. 8 , a shared module  390  is coupled to input selection circuitry  370  and output circuitry  380 , which together provide interfaces to the memory elements  230  in the memory  250 . For example, the input circuitry  370  couples each input of a variable node  210  to the output of an appropriate memory cell  230  and to an appropriate input register  260 , which collectively form an input memory module  265 , and the output circuitry  380  passes the outputs of the check nodes  220  to the inputs of appropriate memory cells  230 , which collectively form an intermediate memory module  250 . In this example, the shared module  390  includes all the variable nodes  210  and check nodes  220  needed to compute all the outputs corresponding to one of the bidirectional check nodes  120  in the factor graph illustrated in  FIG. 1 . During each successive clock cycle of an iteration, the input circuitry  370  and the output circuitry  380  is effectively reconfigured to change the connection of the variable nodes  210  and check nodes  220  to the memory  250  and the input bits. 
     As an example of a multiple cycle iteration using the shared module  390  illustrated in  FIG. 8 , the table shown in  FIG. 9  illustrates the configurations during the four clock cycles of an iteration. Note that the configuration indicated for cycle 0 corresponds to the configuration illustrated in  FIG. 5 . 
     In some examples, multiple shared modules  390  are implemented in a single integrated circuit. For example, the example shown in  FIG. 8  may be modified to have two shared modules, thereby providing eight new values for memory cells  230  in each clock cycle, with the iteration to update all the memory cells taking a total of two cycles (i.e., four constraints per iteration divided by two constraints per cycle yielding two cycles per iteration). Similarly, in some examples, a shared module may update fewer cells, for example, updating only a single cell in each clock cycle (i.e., using a single check node  220  and three variable nodes  210 ). 
     In the example illustrated above in  FIG. 8 , the updated values to the memory  250  are not passed through to the outputs of the memory until after the entire iteration is completed. In some examples, the updated values determined in one clock cycle may be presented at the output of the memory during subsequence clock cycles within the same iteration. In such examples, the order in which the outputs of the check nodes are computed (the “schedule”) may be significant. Examples of schedules include a sequential updating of the outputs associated with each of the check nodes  120  (see the factor graph in  FIG. 1 ), and random updating in which different nodes are updated at each clock cycle. 
     Referring to  FIG. 10 , in some examples, multiple modules  390  are used (but not a sufficient number so that an iteration may be completed in a single cycle), and the input selection circuitry  370 , output selection circuitry  380 , and memory  250 , are distributed among a set of local processing elements  490 , and each local processing element  490  has one shared module  390 . Each local processing element has a local output circuitry  480  and a local input selection circuitry  470 . The memory is distributed such that the memory cells  230  in the memory  450  of a local processing element are those cells that are updated by the shared module  390  in the various clock cycles of an iteration. As illustrated, each row of memory cells is updated in one clock cycle. A control input controls the configuration of the input and output circuitry according to the cycle in the iteration being performed. Note that in general, a shared module  390  at one local processing element  490  requires outputs of memory cells  230  in a local memory  470  of its own local processing element and/or another (or more generally, one or more other) local processing element. The local input selection circuitry  470  selects the memory cells required by each of the local processing elements and passes those values onto a global selection unit  440 , which then determines the proper subsets of the memory values to be passed onto each one of those local processing elements. In the output stage, the memory cells are coupled through the selection circuitry  470  to the output section  495  to determine the outputs. The configuration shown in  FIG. 10  can also be understood as the function of input selection logic  370  shown in  FIG. 8  is distributed among blocks  470  and  442 , and the output logic  380  is distributed among the blocks  480 . 
     In some examples, the global selection unit  440  may include a set of selection units  442 , each coupled to inputs of a respective local processing element to provide the corresponding subset of memory values to the shared module  390 . For example, one selection unit  442  may receive 8 signals representing memory values provided by the two local input selection circuitries  470  to generate four output signals representing the memory values to be provided to the local processing element shown on the left of  FIG. 10 . 
     Referring again to  FIG. 8 , in some examples, the memory  250  as a whole is configured such that effectively all the values are updated exactly once in an iteration. In one implementation of such a memory using a “double buffering” approach two banks of memory are used. In iteration k, the write circuitry always writes into memory bank #1, and the read circuitry always reads from memory bank #2. By the end of iteration k, memory bank #1 has achieved a full update. In the next iteration k+1, the write circuitry switches to write into memory bank #2, and the read circuitry reads from memory bank #1 which was just updated in the last iteration. In this case, the memory  250  would need a capacity twice the amount of the outputs from the local check nodes to keep two different copies for read and write operations respectively. 
     The approaches described above in the context of a length 8 code is applicable to a larger example of an (1056,352) LDPC code, such as is used in IEEE 802.16 based communication. The check matrix of the code can be represented in tabular form breaking the 0,1 matrix into 8 rows by 24 columns of 44 by 44 blocks, with each block being either all zero, or being a shifted (circularly offset) diagonal with one non-zero entry in each row and in each column. This tabular representation of the code is shown in  FIG. 11 . The upper-left (0,0) block (showing the number of “0”) in the tabular representation is a diagonal matrix. The (0,2) block shown as a “8” is a off-diagonal block M=[m i,j ] such that m i,j =1 if j=i+8 (mod 44) and 0 otherwise. The full factor graph is not illustrated, but can be derived from the matrix representation in the same manner as the example illustrated in  FIG. 1 . 
     Referring to  FIG. 12 , a module  590  is configured to include variable nodes  510  and constraint nodes  520  for the code shown in  FIG. 11 . Note that the nodes illustrated in  FIG. 12  are unidirectional nodes in which links are either input or output links. Module  590  is analogous to module  390  for the length 8 code discussed above. Note that each row in the code matrix shown in  FIG. 11  has ten non-zero entries in all row blocks, except row block 6, which has eleven non-zero entries per block. In order to implement constraints outside row block 6, the module  590  has ten (unidirectional) variable nodes  510  and ten (unidirectional) constraint nodes  520 , and for rows in row block 6, eleven (unidirectional) variable nodes  510  and eleven (unidirectional) constrain nodes  520 . Each variable nodes accepts inputs for memory cells corresponding to non-zero entries in a particular column of the code matrix. Therefore, variable nodes corresponding to columns in the range 0 through 15 have four inputs (three inputs for memory cells corresponding to entries in the code matrix and one input for the bit probability) and one output. Variable nodes for columns 16 through 23 have two or three inputs depending on the column and the block row. In some examples, the module  590  has the maximum number of variable nodes and inputs necessary, and is configurable during different cycles to accommodate the specific number of variable nodes and inputs needed, for instance, by ignoring certain inputs. 
       FIG. 13  illustrates one type of implementation of a decoder operable to perform the iterative stage of decoding operation for use with the (1056, 352) LDPC code shown in  FIG. 11 . In such an implementation, the decoder includes an analog input memory  660  that stores representations of the input bit probabilities (e.g., as voltages encoding log likelihood ratios) corresponding to the 1056 bits (i.e., 24 blocks of 44) of the LDPC code illustrated in  FIG. 11 . These input bits are then distributed by memory selection circuitry  672  to be processed in a set of local processing elements  690 . Each local processing element  690  has a shared module  590  that includes the variable nodes and check nodes needed to compute all the outputs corresponding to one of the check nodes of the full factor graph. The structure of each processing element  690  is similar to each processing element  490  shown in  FIG. 10  to implement the decoder for a length 8 code. 
     Each local processing element  690  also includes a local output circuitry  680 , which directs the output of the local check nodes into appropriate cells  630  of a memory  650 . In this example, the memory is distributed among the set of local processing elements  690  as a set of local memories  650 , each of which includes memory cells  630  updated by the shared module  590  of its local processing element  690  (not other local processing elements) in the various clock cycles of an iteration. As described before, in general, each shared module  590  at one local processing element  690  requires outputs of memory cells in a local memory  650  of its own local processing element and/or one or more of other local processing elements. These outputs are obtained by a set of local read circuitry  670  that retrieve values from the local memory  650  and send them to a global selection unit  640 , which then determines the appropriate combinations of output values to be sent to the individual local processing elements  690  at various clock cycles. The global selection unit  640  includes a separate input selection unit  642  associated with each of the local processing elements, and provides as outputs the values stored in the memories  650  required for input that that unit on each iteration. 
     Implementations of the type illustrated in  FIG. 13  can have different numbers of processing elements, and use different schedules of applying each of the 352 constraints in different cycles of a decoding iteration. Referring back to  FIG. 11  in the matrix representation of the (1056, 352) LDPC, out of the total 8 blocks of rows, rows in 7 of the blocks of rows (i.e., row blocks, 0-5, and 7) contains 10 non-zero entries, and row in one block (row block 6) each contains 11 non-zero entries. Therefore, of the 192 (i.e., 24*8) blocks of entries, only 81 are non-zero, and each of the non-zero blocks has exactly 44 non-zero entries, for a total of 3564 (81 times 44) entries. 
     The exemplary arrangement shown in  FIG. 13  uses a set of 8 local processing elements  690  each configurable to perform the computation associated with one check node (i.e., one row), of a corresponding block of rows of the check matrix. In other words, each local processing element  690  is used repeatedly in 44 cycles to compute the outputs for the 44 constraints represented by the 44 rows of the code matrix in the corresponding block. For example, the 8 elements apply constraints 0, 44, 88, . . . , 308, respectively, on the first cycle, constraints 1, 45, 89, . . . , 309, respectively, and finally 43, 87, 131, . . . , 351, respectively, on the last cycle of an iteration. 
     Note that because of differences in each of the row blocks in the code matrix, the shared module  590  in each local processing element  690  may be have differences. Consider a shared module  590  for performing the computation associated with a row in the first (row block 0) block. The corresponding check node in the bidirectional graph has 10 edges linked to variable nodes. Each of the first 8 variable node has five edges, four to check nodes and one to a bit input; the 9 th  variable node has four edges, three to check nodes and one to a bit input, and the 10 th  variable node has three edges, two to check nodes and one to a bit input. The shared module  590  therefore has circuits for 10 (directional) check nodes, each with 9 inputs and one output. The 10 outputs of the check nodes update 10 locations in the local memory. The local processor has circuits for 10 (directional) variable nodes  510 , each with four, three, or two inputs and one output. Each node  510  provided an input to 9 of the 10 (directional) check nodes  520 . Of the inputs for each variable, one input is for an input bit probability and the remainder are for values from the local memories. 
     Shared modules  590  in the other local processing elements have the same structure as that associated with row block 0, with differences including the shared module  590  for row block 6 having 11 check nodes, and 11 variable nodes, and the share module  590  for blocks 1 through 5 each having two variable nodes with two inputs and the other variable nodes having four inputs. 
     In operation, at each clock cycle, the variable nodes of shared module  590  for row block 0 reads 10 sets of inputs from the input memory  660 , one set for each variable node and updates 10 locations of the local memory  650 . The values from memory  650  are passed through the blocks  670  of multiple of the local processing elements  690  and through the control unit  642  associated with the destination processing element. Over 44 clock cycles of an iteration, the shared module  590  provides updated values for all 440 (44 times 10) locations in the local memory. 
     As outlined above, in some embodiments, each one of the shared modules  590  may be implemented as a combination of 10 variable nodes and 10 check nodes (also referred to as a 10×10 shared module), except for the shared module  590  for row block 6 which is implemented as a 11×11 module. 
     A number of different circuit arrangements and signal encodings can be used within the approaches described above. For certain soft decoding applications, each variable node circuit can be formed using a soft Equals gate, and each check node circuit can be formed by a soft XOR gate. In the example of  FIG. 13 , each variable node takes the form of a 4-input (or 3-input or 2-input) soft Equals gate and each check node takes the form of a 9-input (or 10-input) soft XOR gate. Therefore, for each shared module  590 , besides reading the 10 (or 11) of input bits (one each soft Equals gate) from the input memory  660 , it also requires 10 (or 11) sets of values from the local memories  650  (one set for each soft Equals gate). Note that in this example these values come from the memory cells in the other local processing element(s), and not from the memory  650  in the same processing element. 
     One approach for providing the proper combinations of memory values needed as input to the shared modules  590  includes forming, in the global control unit  640 , a set of 8 individual selection units  642 , each of which selects or combines the outputs of the local processing elements  690  as needed for the input values for a corresponding shared module  590 . In some examples, each one of the read circuitry  670  is selectively coupled to the set of 8 selection units, for example, using a set of 8 buses with each bus containing 10 (or 11) wires for sending a total of 10 (or 11) output values to an individual selection unit in one clock cycle. The selection unit  642  then chooses a set of 10×3 (or 11×3) output values for input to the shared module  690 . 
     By arranging the decoder into local processing elements, in some embodiments, all of the XOR signals become local to the local processing elements in which they are formed. The inputs to the Equals gates become globally routed signals that come from multiple local processing elements. In some examples, the local processing elements  690  can be configured in a way such that each shared module  690  requires only output values from a pre-defined set of three other local processing elements. As a result, the coupling between each local processing element to the global control unit  640  can be reduced, for example, with read circuitry  670  now being coupled to only 3 (instead 8) selection units. In some examples, the local processing elements  690  can be further arranged such that all of the even-numbered (i.e., 0, 2, 4, and 6) local processing elements communicate with each other but not with the odd-numbered (i.e., 1, 3, 5, and 7) local processing elements (except for the last eight block columns of the check matrix). 
     Note that, in some applications relating to soft decoding, the decoder described above is used for converting input “soft” bits based on individual measurements of each bit to soft bits each based on the entire block of soft bits, taking into account the constraints that the original bits of the block satisfied. These output soft bits can then be further processed, or converted by hard decision into output “hard” bits taking values 0 or 1. The input soft bits may be provided in the probability domain, for example, as the probability of a bit being value of 1 or 0. Alternatively, the input soft bits may be provided in the log domain, for example, as the log likelihood ratio of a bit (e.g., as defined by 
               log   ⁡     (       p   ⁡     (       b   i     =     0   |     y   i         )         p   ⁡     (       b   i     =     1   |     y   i         )         )       .         
In either case, the shared module  590  can be implemented using a set of analog circuit components that perform analog computation functions appropriate for the particular application. Implementations of some of these analog circuit components (such as soft Equals and soft XOR) are illustrated in detail in U.S. patent application Ser. No. 12/716,148, titled “Circuits for Soft Logical Functions,” filed Mar. 2, 2010.
 
     Referring to  FIG. 14 , in some implementations, the soft equals makes use of differential voltage inputs, each representing a log likelihood ratio to produce a voltage that is proportional to the sum of the inputs. Each differential voltage input is passed through a voltage to current converter  712 , and the resulting currents are summed on a bus  714 . The current on the bus is passed through a current to voltage converter  716 . The output voltage then branches to the soft XOR circuits that require the output of this equal node. Exemplary circuit implementations are shown in the figure. A variety of alternative circuits can be used, including alternative soft Equals circuits described in U.S. patent application Ser. No. 12/716,148. 
     Referring to  FIG. 15A , in some implementations, the soft XOR circuits make use of log domain differential voltages as produced by the circuit shown in  FIG. 14 . In the exemplary implementation of the soft XOR circuit shown in  FIG. 15A , which approximates an ideal soft XOR function for log domain processing, one differential voltage input is passed to a circuit  812 . The second and further inputs to the soft XOR circuit are passed to circuits  814 , each of which performs an analog computation that approximates multiplication of the current provided by the previous element according to that input. The resulting current approximates the ideal soft XOR function each of the inputs and is passed through a current to voltage converter  816 , to provide the differential voltage output of the soft XOR. Note that unlike the soft Equals circuit shown in  FIG. 14 , the output of the overall soft XOR circuit does not fan out on any particular cycle, because the output of the soft XOR circuit provides the input to only a single memory cell. The circuit parameters, for instance, resistance values, transistor dimensions, and voltage scaling, are chosen to best approximate the ideal function of a soft XOR and/or to optimize higher level (e.g., overall decoding) system performance. 
     Referring to  FIG. 15B , an alternative arrangement  820  of circuit elements to the soft XOR circuit  520  shown in  FIG. 15A  implements the directional soft XOR circuit using a branching tree structure, optionally sharing signals between different trees associated with a same bidirectional check node. Specifically, groups of circuit elements  818  effectively form two-input, one-output, voltage based soft XOR circuits using the circuit elements  812 ,  814 , and  816  introduced with reference to  FIG. 15A . These groups of circuit elements  818  are then arranged in a tree structure, preferably a binary tree structure that is as balanced as possible to form the circuit arrangement  820  shown in the figure. In some implementations, the branching structure shown in  FIG. 15B  may have preferable characteristics, for instance, providing a better approximation of the ideal soft XOR function with LLR representations. Furthermore, when multiple modules  820  are implemented for a set of unidirectional XOR circuits, certain computations can be shared, for example, by passing a signal  825  from one module to another where a portion of the tree in that other module can be eliminated. 
     Referring to  FIG. 16 , in some implementations, the bus  714  associated with each equal gate is distributed. Each memory cell  230 , which stores its value as a differential voltage, has at its output a corresponding voltage to current converter  912 . These voltage to current converters are similar to the converters  712  shown in  FIG. 14 . The current output is passed to a set of switches  913 , at most one of which is enabled if the corresponding cell&#39;s value is to be injected as a current on a current bus  714  corresponding to that switch  913 . Each bus  714  similarly includes a portion onto which current associated with an appropriate input bit is injected at each cycle to account for the input to the equal gate corresponding to the bit input. Note that the bus  714  for each equal node may have a complex structure, for example, having numerous branches. Nevertheless, all the current injected onto the bus passes to the current to voltage converter  716  for the equal node, from where it branches to the unidirectional XOR circuits that require that output. 
     Referring to  FIG. 17 , in some implementations that make use of a distributed bus  714  to perform a current summation function, the module  590  illustrated in  FIG. 12  is replaced by a module  592  in conjunction with distributed busses  714  and voltage-to-current converters  712 . In module  592 , each variable node corresponds to a current-to-voltage converter  716 , which outputs a voltage proportional to the total injected current on the corresponding bus  714 , and then that voltage branches to the appropriate check node circuits  520 . Note that in yet other implementations, the current-to-voltage converters  716  are themselves distributed, and a module  593  (i.e., a portion of module  592 ) receives voltage inputs, which are internally distributed to the appropriate check node circuits. 
     Referring to  FIG. 18 , a second example of an implementation of a decoder operable to perform the iterative stage of decoding operation for use with the (1056, 352) LDPC code shown in  FIG. 11  provides the same or similar functionality to the implementation shown in  FIG. 13 . In this example, modules  592  are used, as illustrated in  FIG. 17 . The circuitry that implements each soft Equals circuit associated with a variable node includes a current-to-voltage converter in the module  592 , with the distributed busses corresponding to the 10 unidirectional variable nodes of the module  592  being distributed. Each memory  652  includes circuitry to inject current to the appropriate busses corresponding to the soft Equal circuits for different variable nodes via read switching circuits  671 . The bus section  644  effectively includes 81 busses, each associated with a different current-to-voltage converter  716  at the input of a module  592 . Therefore, the soft Equal circuit is distributed in a manner effectively forms interconnection paths between the memories  652  and the analog computation modules  592 . 
     2 Additional Embodiments 
     2.1 Memory Access Scheduling 
     In some examples, system is configured using an incremental “scheduled” approach such a subset of values is updated and available for use at the end of each cycle rather than at the end of an entire iteration composed of multiple cycles. In some such examples, for example using a single block  390  as shown in  FIG. 8 , there is no conflict between memory cells that are read in a cycle and locations that are being written in a cycle. The reason that there is no conflict can be understood with reference to  FIGS. 7A-B . For example, applying a constrain corresponding to row 0, the memory cells corresponding to non-zero entries in that row are written, while memory cells in other rows, specifically in columns that have non-zero entries in row 0, are read. 
     Referring again to  FIG. 10 , when two or more update modules  490  are used, it is possible that a memory cell being updated in one module is to be read in the same cycle in another module. This can be understood again with reference to  FIGS. 7A-B , for instance, in a case where one module is applying the constraint associated with row 0 and another is applying a constrain associated with row 1. The application of the row 0 constraint writes C0,1 and reads C1,1, while application of the row 1 constraint writes C1,1 and reads C0,1. 
     In another implementation a memory  250  for an incremental approach does not required twice the capacity of the entire memory. Here, in the first half of clock cycle k, the read circuitry retrieves some values from the memory, some of which may have been updated in a preceding clock cycle k−1. The outputs of the XOR gates are written back into the same memory during the 2 nd  half of clock cycle k. In some examples, by using edge-sensitive (instead of level-sensitive) memory, the read and write operation in the same clock cycle would not interfere with each other. 
     2.2 Memory Arrangement 
     In another implementation of a decoder for the (1056, 352) LDPC code uses a different arrangement of memory cells, and takes advantage of the distributed current summing approach to implementing the log likelihood ratio based equals gates. The implementation makes use of a schedule in which multiple constraints are applied at each cycle of an iteration, and the updated values are used in the next cycle of the iteration. Referring to  FIG. 11 , a previously described approach applies eight constraints in each cycle, with each constraint being associated with a different row block. As discussed above, there are situations in which the same cell  230  is both written and read in the same cycle, and therefore approaches such as double buffering and multiple phases (update, write, etc.) per cycle are used. In the present implementation, multiple constraints are applied in each cycle, and these constraints are selected such that there is no read and write conflict on the memory cells, permitting a scheduled approach without the need for approaches such as double buffering. One way to avoid the conflict is for each of the constraints being applied in a cycle to come from the same row block. The reason this avoids conflicts may be understood with reference to  FIG. 7A , which relates to a simpler length 8 code. As illustrated, a constraint corresponds to a row in the code matrix, and the cells that are updated are associated with the non-zero entries in that row. The cells that are read are in the columns in which that the constraint&#39;s row has non-zero entries. If another constraint is applied in the same cycle, but the row associated with that constraint as no non-zero entries in the same columns as the row for the first constraint, then the cells updated for the second constraint are not read for applying the first constraint, and the cells read to apply the second constraint are disjoint from those read for the first constraint. Turning back to the code matrix illustrated in  FIG. 11 , it is evident due to the offset diagonal structure of each non-zero block that multiple rows from the same row block have the property that no two rows have ones in any of the same columns. Therefore, multiple constraints corresponding to multiple rows in one row block can be applied in one cycle without read/write conflicts. As another observation, two rows one in an even block 2n and one in an odd block 2n+3 mod 8 (or odd block 2n+5 mod 8) similarly cannot have any columns with non-zero entries in common due to the overall block structure of the code matrix. 
     Referring to  FIG. 19 , a partial illustration of an instance of this approach arranges the memory cells  230  of the memory  950  into groups  955  of 44 memory cells. Each of these groups  955  corresponds to a different non-zero block of the code matrix shown in  FIG. 11 . Further, these groups are arranged into banks  958  each corresponding to a column in the same way that they are arranged in the code matrix. For the first 24 columns, there are exactly four memory groups  955  corresponding to the four non-zero blocks in each column block. Each of the last eight columns have two or three memory groups each. 
     In  FIG. 19 , a single module  590  is shown, representing the update logic to apply a single constraint in a cycle. As discussed previously, each equal node in the module receives up to four inputs, up to three from the memory cells within the memory  950  and one from the input registers in memory  960 , which is also arranged into groups  965 , each associated with a different column block. The equal node uses a distributed current bus approach, as described above, onto which the current encoding log likelihood ratios are injected.  FIG. 19  illustrates application of a constraint in the first row block. In the first bank of memory groups  955 , one memory cell in the first memory group  955  is updated, and one memory cell from each of the other memory groups  955  is read as well as one input from an input register in a register group  965  associated with that column block. Each of the 10 (or 11 depending on the row block) inputs to the module  590  comes from one bank of memory groups  955 . 
     As is discussed further below, the off-diagonal structure of the blocks constrains which combinations of memory cells are coupled to the current bus in any one cycle. Specifically, the combinations of memory cells must be found in a same column of the check matrix. Therefore, if the memories  955  are indexed by column within the block column, then in any particular cycle a same index is applied to each memory  955  in a bank  958  and the input memory group  965 . Note also that with an arrangement indexed by column, contiguous columns are naturally indexed contiguously, treating wrapping from the highest index (43) to zero as contiguous. 
     In some examples, multiple constraints in a row block are applied in each cycle. For example, four constraints corresponding to four contiguous rows may be applied in each cycle. Referring to  FIG. 20 , as compared to  FIG. 19 , such an example has four copies of the module  592 , and each distributed equal node has as a separate input bus. For updates of four contiguous constraint rows within a row block, for each block column, four contiguous columns are read or written in each bank  958  of the memory groups  955 . As in the example shown in  FIG. 19 , the memory locations in a same column are coupled to a same bus, and therefore outputs of four contiguous columns for a memory  955  are each output to a different of the four busses. Similarly, the updates to the updates of the memory group  955  in contiguous groups of four memory cells in one of the groups  955 . Each memory group  955  is arranged in four groups  966  of eleven memory cells each, with each group  966  being coupled to a corresponding one of the four busses. For instance, one memory group  966  has memory cells for the within-block column indices 0, 3, 7, . . . 40 (i.e., columns indices 0 mod 4), a second for column indices 1, 4, . . . 41 (i.e., 1 mod 4), and so forth. In any cycle, one cell from a group  966  is coupled to the bus or one cell from the group is written to. In each block column, the column that is applied associated with the first module  590  depends on the offset of the block diagonal in the block being updated. Therefore, if the first constraint row has a non-zero entry in that block column at an index i, a shifter  970  “rotates” current bus i mod 4 to the 0 th  module  590 , bus (i+1) mod 4 to the 1 st  module, (i+2) mod 4 to the 2 nd , and (i+3) mod 4 to the 3 rd . 
     In some examples, the function of a soft Equal circuit is distributed such that the output of a memory group  966  is a current, which is summed on a bus as illustrated in  FIG. 20 . Then the shifter  970  includes a current to voltage conversion circuit for each bus at its input, and then shifts the resulting voltages to modules when each voltage encoded input is applied to the inputs of appropriately selected soft XOR circuits. The module  592  is then replaced with a module  593 , which accepts voltage rather than current inputs. 
     Note that a number of further optimizations can be used to take advantage of structure of the code. For example, four constraints from row block 2n can be applied at the same time as four constrains from row block (2n+3 mod 8), thereby allowing eight constraints to be applied in each cycle. Note that of the first 16 banks of memory groups  955 , memories in an even block row are updated based on an update based only on even block rows in the first 16 block columns, and odd block rows in the last 8 block columns. Similarly, memories in an odd block row are updated based only on odd block rows in the first 16 block columns and even blocks rows in the last 8 block columns (with the exception of block (6,16) which is used to update block (0,16) and vice versa. 
     In various examples, different schedules for associating sets of constraints with cycles can be used. For example, with four modules  590 , the constraints applied in each cycle can proceed as {0, 1, 2, 3}, {4, 5, 6, 7}, . . . {40, 41, 42, 43}, {44, 45, 46, 47}, {348, 349, 350, 351}, in 88 cycles for each iteration. As another example using eleven modules, the cycles can proceed as {0, 1, . . . 10}, {11, 12, . . . , 21}, {341, 342, . . . , 351} in 32 cycles in each iteration. As another example, with two sets of four modules  590  (i.e., eight total, four for block 2n and four for block 2n+3), the constraints applied can proceed as {0, 1, 2, 3, 132, 133, 134, 135}, {4, 5, 6, 7, 136, 137, 138, 139}, . . . , {40, 41, 42, 43, 172, 172, 173, 174}, {88, 89, 90, 91, 220, 221, 222, 223}, . . . in 44 cycles. 
     Referring to  FIGS. 21A-B , an overall diagram for an implementation of a decoder for the code with check matrix shown in  FIG. 11  is shown. An input memory  1660  holds the 1056=24×44 input values, represented as voltages proportional to the LLR inputs. In some examples, these values are loaded, for example, four at a time, in a serial manner before the iterative stage begins. In some examples, the LLR values are obtained from a signal mapping circuit that accepts one or more of the signal values y i  and outputs one or more of the LLR values. In some examples, one signal input generates a set of LLR inputs, for example, when multi-level encoding is used, such as when four bit values are encoded in a sixteen-level analog signal. Generally, the architecture shown in  FIGS. 21A-B  has separate parts for the “even” and the “odd” block rows, which as introduced above allows concurrent processing of constraints in different block rows without read-write contention for memory cells. A set of memory banks  1650  include memories  1652 , each holding 44 entries corresponding to a corresponding block of the check matrix. Each bank includes only either odd or even block rows in a particular block column of the check matrix. Together, these banks correspond to the memories  955  shown in  FIG. 19 . Two parity check processors  1590  are illustrated. Each implements four analog computation modules generally of the form of module  593  that is illustrated in  FIG. 17 . The memory banks  1650  are coupled to the parity check processors  1590  via rotator modules  1970 . Generally, each rotator module  1970  includes a set of four rotators each associated with one of the four analog computation modules that are in the corresponding parity check processor  1590 . Note that these rotator modules include current to voltage conversion circuits such that the links between the memory banks  1650  and the rotator modules  1970  are differential current encoded, while the links between the rotator modules  1970  and the parity check processors  1590  are differential voltage encoded. The outputs of the parity check processors  1590  are passed to the appropriate memory banks  1590  where the determined values are stored. An output section  1400  is coupled to the memory banks  1650  and to the input memory  1660 , and includes the circuitry for computing the output values, which may be computed multiple outputs at a time an multiplexed. Not shown is the control logic that coordinates the configurations of the rotators and read and write control logic within the memory banks. 
     2.3 Signal Representations 
     It should also be understood that various modifications of the approaches described are possible. For example, memory elements are not necessarily analog and/or continuous valued. For example, digital (e.g., storage for binary (base 2) digits, possibly in “soft” forms) and/or quantized storage may be used. Other representations of the values passing between nodes can be used. In some examples, log likelihood ratios are used. Other possibilities include passing of linear probabilities. With different representations, different circuit implementations of the equal and constraint nodes would be used. Various encodings of the representations may be use. In some examples described above, differential voltage and differential current encodings are used. Alternatives include stochastic (e.g., dithered) representation, digital or quantized representations. 
     2.4 Current-Mode Storage Cell and Processing Elements 
     Referring to  FIG. 22 , a differential current memory cell  110  works as follows. In a writing mode, an input value is represented by the differential current on inputs  2214  (labeled IP and IN). In the writing mode, switches S 1 , S 2 , S 3 , and S 4  are closed and switches S 5  and S 6  are open. The input current causes a state to be stored in the capacitors  2216  at gates of transistors M 1  and M 2 . The write speed depends on the transconductance of M 1  and M 2 . The cascode transistors M 3  and M 4  are used to reduce the message error due to channel length modulation of transistors M 1  and M 2 . 
     In a later reading mode, switches S 5  and S 6  are closed and the remaining switches are open. Because the gate voltages, which are maintained in the capacitors, match the voltages needed to reproduce the original input currents, the output currents through the output leads  2212  match the input currents, and are not sensitive to any transistor mismatches. 
     Referring to  FIG. 22 , a differential current input Soft Equals accepts multiple differential current encoded signals on input lines  2212  (i.e., from multiple memory cells), and produces a differential voltage encoded output on output  2220  such that the voltage is proportional to the sum of the input currents. 
     In the circuit shown in  FIG. 22 , transistors M 7  and M 8  are biased in the triode region. Source followers M 9 -M 12  are added to eliminate the soft-equal resistance dependency on its output signal swing and widen the soft-equal linear range. Two NMOS transistors M 5  and M 6  provide the virtual ground at the current mode memory cell output. The soft-equal output signal amplitude is determined by the linearized resistance of PMOS transistors M 7  and M 8  which is regulated by the control voltage applied at the gate of the NMOS current source transistors M 11  and M 12 . 
     The description in prior sections above includes implementations of an LDPC decoder in which each equal gate includes a current summation and an explicit current-to-voltage conversion stage using resistors coupled to a reference voltage (e.g., V DD ). In another implementation, these Soft Equals gates are replaced with the circuit implementations shown in  FIG. 22 . 
     In one or more of the above implementations described in prior sections, the outputs of the Soft XOR gates also include explicit current-to-voltage conversion elements (e.g., resistors), to produce voltage signals to produce voltage inputs to the memory cells at the outputs of the Soft XOR gates. In the present implementation, this last current-to-voltage stage is not required because the memory cells as shown in  FIG. 22  directly accept current encoded inputs. 
     In one or more of the earlier implementations, the output of each memory cell includes a voltage-to-current conversion before passing the current encoded signals for summation in the Soft Equals gates. In the present implementation, such conversion is not required. Note that in some implementations, the previous implementation allowed the voltage output of the memory cell to drive multiple voltage-to-current converters permitting a “fanout.” In some implementations using current mode memory cells, the fanout is introduced before a set of memory cells at a voltage stage in the Soft XOR. 
     2.5 Gain Control 
     The Gilbert multiplier based Soft XOR gate shown in  FIG. 23  takes the soft-equal (check node) output and depending on applications, the voltage output Soft XOR might be required to perform message passing/belief propagation before the Soft XOR with current output sends the belief to the current-mode memory. The voltage input and voltage output Soft XOR has linearized PMOS triode transistors which can be adjusted to meet the proper gain requirement. 
     A fraction of gain variation for the signal chain could affect the bit-error-rate (BER) by one order of magnitude. Essentially, there are two gain stages for the current-mode memory based signal chain. One is from the input of voltage-input current-out soft-xor to the output of soft-equal. The other one is simply the voltage-in voltage-out soft-xor. 
     The gain calibration from the input of voltage-in voltage-out soft-xor to the output of soft-equal involves two major periods: offset sampling and gain calibration. Initially, the common mode reference voltage is connected to the XOR input differential pair M 1  and M 2 . In the same time, the peak differential inputs VPKP and VPKN will be applied to the other input differential pair M 3 -M 6 . The switches S 1 , S 2 , S 5  and S 6  in the current mode memories will be constantly closed; and S 3  and S 4  open. The auto-zeroing networks samples and holds the offset voltage during this period. Next, the input reference voltage VRPI and VRNI will be applied to M 1  and M 2 . The differential output voltage VOP-VON of soft-equal will be regulated to be equal to VRPO-VRNO by the fully differential difference amplifier, the other high gain amplifier, and the source follower branches M 9 -M 12  in the loop with the offset voltages being eliminated by the auto-zero networks. 
     In some implementations, a soft equal circuit features an active MOS resistor that senses the voltage across itself and adjusts it&#39;s own bias (Gate Voltage) as to keep on resistance constant independent of the voltage swing across it, for instance, instead of using just a MOS (in this case PMOS) transistor as the resistive load to the summer of the currents. We achieve this feature by adding a source follower to the MOS resistor that takes as it&#39;s input a drain voltage of the controlled MOS transistor and produces MOS transistor gate control voltage as it&#39;s output. This way the ON resistance of the controlled MOS resistor can be set by the geometries of the controlled MOS resistor and the source follower device and can be finely controlled by the source followers bias current. 
     In some examples, the device may have somewhat unpredictable characteristics, for instance, that depend on particular fabrication time (process) or run-time conditions. For example, gains of transistors may depend on process characteristics or on operating temperature of the device. Some designs include selectable and/or configurable gain elements, for example, to adjust the gains of soft XOR circuits, gains of current to voltage conversion circuits, gains of current outputs of memory cells, etc. In some examples, gain is adjusted by controlling resistance elements, such as but not limited to, resistive elements in current to voltage converters  716  (see  FIG. 14 ) or circuits  812  and  814  (see  FIG. 15A ) or circuits  816  (see  FIG. 15B ), for example, at the last stage of the tree of two-input XOR stages. The gain control can be provide in digital form, and passed through a digital-to-analog converter to an analog-controllable resistive element (e.g., an appropriately biased transistor). In some examples, these gains are adjusted to optimize overall system performance, for instance, in a system calibration stage before operational use or in an ongoing (e.g., feedback) adaptation of the elements to maintain optimal or near optimal performance. 
     In some examples, an autozeroed process-voltage-temperature compensated automatic gain control for analog logic gates is used. The circuits has analog logic gates with fixed gain from input to output that is set to be independent of manufacturing process-supply voltage-operating temperature variations. Such feedback control loops have not been previously used to implement gates that perform statistical computations. 
     3 Alternatives, Applications and/or Implementations 
     It should be understood that the decoder applications described above are only one example of an application of an analog belief propagation processor. The techniques employed in these examples are applicable to other uses of belief propagation. 
     Implementations of the belief propagation processors may have different degrees of customization to particular applications. Generally, a controller (not shown) sequences the application of different constraints in different cycles of an iteration, and sequences the series of iterations to complete a computation. The controller can be a special-purpose controller or sequencer that is configured for a particular code, for a particular class of codes, or to some other class of computations. The controller may also be a general purpose controller that may be used to implement a wide variety of computations. In some implementations, the modules that include the variable and constraint nodes may have fixed structures, or may be configurable. Circuit configuration may occur through the operation of the controller and/or the configuration may be implemented through a field programmable approach in which certain connections between circuit elements are enabled using personality data that is applied to the processor. In some cases the controller is implemented on the same device that implements the analog circuit portions of the processor, while in other cases, the controller is fully or partially implemented in a separate device that is in communication with the device implementing the analog processing. 
     The belief propagation processor may be used in many applications. One application relates to communication systems, where the belief propagation processor is used as a soft decoder subsequent to a soft demapper that converts signals transmitted over a noisy communication channel into soft bits. The soft bits may be represented in the probability domain (e.g., as probabilities or differential probabilities), or alternatively, in the log domain (e.g., as log likelihood ratios or log-odds). Examples of soft demappers are described in detail in U.S. patent application Ser. No. 12/716,113, titled “Signal Mapping,” which is incorporated herein by reference. 
     One application of a decoder implemented in using an analog belief propagation processor is in a communication system in which blocks of transmitted information are received, and the decoder performs an error correction of the received information. Another application of such a decoder is as a component of a data storage system, for instance a semiconductor memory (e.g., flash memory, which may include multilevel cell storage) or a disk storage memory, in which the decoder performs an error correction of the information retrieved in the storage system, for example, retrieved as storage levels in memory cells or as degrees of magnetization in a disk storage system. In some examples, the decoder in integrated on the same device as a semiconductor memory, while in other examples, the decoder is implemented in a separate device (e.g., on a separate integrated circuit). Some examples of memory systems that can make use of decoder implementations as described in the present application are found in co-pending U.S. application Ser. No. 12/537,060, titled “STORAGE DEVICES WITH SOFT PROCESSING,” filed Aug. 6, 2009, which is incorporated herein by reference. 
     Examples described above are implemented in various ways. In one example, a particular code, for example, represented as a check matrix, is transformed using a computer implemented (optionally human assisted) technique that produces a data representation of switching and interconnect circuitry, and optionally of the circuit implementations of soft Equals and soft XOR circuits. In some examples, this data representation (e.g., data structures or instructions) is stored on a machine readable medium and is later used to impart functionality when employed in a computer-based device layout and fabrication system. In other words, when employed in such a system, the data representation is read and determines the physical circuit implementation. An example of a data representation includes a representation that follows Verilog-A or Verilog-AMS specifications. In other examples, a specific code is not specified and the resulting device is configurable to accommodate a variety of different codes. Furthermore, the approach is not limited to devices used to decode codes, as many other analog graph-based computations may be implemented using the techniques described above. 
     It is to be understood that the foregoing description is intended to illustrate and not to limit the scope of the invention, which is defined by the scope of the appended claims. Other embodiments are within the scope of the following claims.