Abstract:
Low-voltage CMOS (Complementary Metal Oxide Semiconductor) circuits, suitable for analog decoders, for example, are provided. The circuits include multiplier modules that receive first input signals and respective ones of a plurality of second input signals. Each multiplier module generates as output signals products of the first input signals and its respective second input signals. Dummy multiplier modules that respectively correspond to the multiplier modules receive the second input signals, and each dummy multiplier module forms products of the second input signal of its corresponding multiplier module and the other second input signals. The dummy multiplier modules reduce the overall voltage requirements of the circuit, thereby providing for low-voltage operation. In some embodiments, a connectivity module receives output signals from the multiplier modules and generates as output signals sums of predetermined ones of the output signals, and a renormalization module receives and normalizes the output signals from the connectivity module to generate output signals that sum to a predetermined unit value.

Description:
FIELD OF THE INVENTION  
       [0001]     The present invention relates to signal decoding, and in particular to analog decoders.  
       BACKGROUND OF THE INVENTION  
       [0002]     In the mid-1990&#39;s, a new class of error correcting codes called “Turbo codes” was introduced [ 3 ]. Turbo codes achieve performance, typically measured in BER (bit error rate), very close to a long sought-after limit known as the Shannon bound. The decoding algorithms used in Turbo codes have since been generalized into a class of decoding algorithms based on factor graphs [ 17 ,  6 ]. These algorithms are commonly referred to as message-passing algorithms. Turbo-like codes which employ message-passing algorithms are generally referred to as iterative decoders.  
         [0003]     An important general form of message passing algorithms is the sum-and-product algorithm [ 9 ], which implements probability propagation on a graph. The sum-and-product algorithm is a very general algorithm which describes decoders for many codes, including trellis codes, LDPC (low-density parity check) codes, block product codes, BCJR (Bahl, Cocke, Jelinek, Raviv) codes [ 2 ], and Turbo codes. The sum-and-product algorithm also describes algorithms used in artificial intelligence.  
         [0004]     There has been great interest in implementing iterative decoders, but conventional digital implementations are often complex and demand expensive resources. Analog circuits for iterative decoding have been proposed and demonstrated by various researchers in recent years [ 10 ,  7 ,  12 ,  13 ,  14 ,  8 ,  15 ,  18 ]. CMOS (Complementary Metal Oxide Semiconductor) circuits are of particular interest for some applications because they can be implemented in ordinary so-called “plain vanilla” CMOS processes, which are low-cost compared to high-end alternatives such as BiCMOS and SiGe. CMOS analog decoders can also be more easily integrated with other CMOS components for single-chip receiver solutions.  
         [0005]     In many cases, analog decoders offer significant advantages over digital designs. For example, the operations required for implementing the sum-and-product or BCJR algorithms can be implemented in analog decoders with fewer transistors. Analog circuits also require significantly fewer wire connections between components. Such efficient use of space allows parallel circuit implementations to enable decoding operations to occur completely in parallel, which thereby provides for high data throughput.  
         [0006]     Analog decoders are also intrinsically low-power, and eliminate the need for high speed analog-to-digital (A/D) conversion in a receiver front-end. A typical A/D converter by itself consumes a significant amount of power and silicon real-estate. An analog decoder may thus be thought of as an information A/D converter, specially designed to convert coded analog channel information into decoded bits.  
       SUMMARY OF THE INVENTION  
       [0007]     In one aspect, the invention provides an electronic circuit comprising a plurality of multiplier modules for receiving a plurality of first input signals and respective ones of a plurality of second input signals, each multiplier module configured to generate as output signals products of the plurality of first input signals and its respective one of the plurality of second input signals, and a plurality of dummy multiplier modules for receiving the plurality of second input signals, each dummy multiplier module corresponding to a respective one of the plurality of multiplier modules and configured to form products of the respective one of the plurality of second input signals of its corresponding multiplier module and the second input signals other than the respective one of the plurality of second input signals.  
         [0008]     Each of the multiplier modules preferably comprises a plurality of transistors for receiving respective ones of the plurality of first input signals and the respective one of the plurality of second input signals. Each of the plurality of transistors preferably comprises a control terminal for receiving the respective one of the plurality of first input signals, a first switched terminal for receiving the respective one of the plurality of second input signals, and a second switched terminal on which the output signals are generated.  
         [0009]     The dummy multiplier modules preferably have a similar structure, comprising a plurality of transistors for receiving the respective one of the plurality of second input signals and respective ones of the other second input signals, with each transistors comprising a first switched terminal for receiving the respective one of the plurality of second input signals, a control terminal for receiving the respective one of the other second input signals, and a second switched terminal on which the products are formed.  
         [0010]     In one embodiment, a connectivity module receives the output signals from the plurality of multiplier modules and generates as output signals sums of predetermined ones of the output signals from the multiplier modules. The connectivity module preferably comprises inputs for receiving the output signals from the plurality of multiplier modules and outputs for outputting the output signals of the connectivity module. Connections between the inputs and the outputs determine the predetermined ones of the output signals from the multiplier modules.  
         [0011]     A renormalization module is also provided in some embodiments for receiving the output signals from the connectivity module and normalizing the output signals from the connectivity module to thereby generate output signals that sum to a predetermined unit value.  
         [0012]     The invention also provides, in another aspect, a sum-product circuit comprising a plurality of first inputs for receiving respective first input signals, a plurality of second inputs for receiving second input signals, a plurality of multiplier modules connected to the first inputs and a respective one of the second inputs, each multiplier module configured to generate as intermediate output signals products of the first input signals and the second input signal on its respective one of the second inputs; a plurality of dummy multiplier modules connected to the second inputs, each dummy multiplier module corresponding to a respective one of the plurality of multiplier modules and configured to form products of the second input signal of its corresponding multiplier module and the second input signals other than the second input signal of its corresponding multiplier module; a connectivity module for receiving the intermediate output signals and configured to generate as output signals sums of predetermined ones of the intermediate output signals; and a plurality of output terminals for outputting the output  
         [0013]     The sum-product circuit may also include a plurality of reset switches connected across the first inputs, the second inputs, and the outputs and configured to short the first inputs, to short the second inputs, and to short the outputs in response to a reset signal.  
         [0014]     Other aspects and features of the present invention will become apparent, to those ordinarily skilled in the art, upon review of the following description of the specific embodiments of the invention. 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0015]     The invention will now be described in greater detail with reference to the accompanying diagrams, in which:  
         [0016]      FIG. 1  is a factor graph of a constraint function of three variables;  
         [0017]      FIG. 2  is a block diagram of a Gilbert-based sum-product circuit;  
         [0018]      FIG. 3   a  is a schematic diagram of an array of source-connected transistors as an example implementation of the blocks  16 ,  18 , and  20  of  FIG. 2 ;  
         [0019]      FIG. 3   b  is a schematic diagram of diode-connected transistors for converting voltages to currents;  
         [0020]      FIG. 4  is a schematic diagram of a first translinear loop;  
         [0021]      FIG. 5  is a schematic diagram of a folded Gilbert-based circuit topology;  
         [0022]      FIG. 6  is a schematic diagram representing a non-saturation MOS model;  
         [0023]      FIG. 7  is a schematic diagram of a second translinear loop;  
         [0024]      FIG. 8  is a schematic diagram of a third translinear loop;  
         [0025]      FIG. 9  is a block diagram of a low-voltage CMOS sum-product circuit according to an embodiment of the invention;  
         [0026]      FIG. 10  is a schematic diagram of an example renormalization circuit;  
         [0027]      FIG. 11  is a block diagram representing iterated sum-product functions with renormalization.  
         [0028]      FIG. 12  is a block diagram of an example reset circuit;  
         [0029]      FIG. 13   a  shows an example trellis stage;  
         [0030]      FIG. 13   b  is a block diagram of an example low-voltage sum-product architecture according to an embodiment of the invention for computing on trellis graphs;  
         [0031]      FIG. 14   a  shows an example Tanner Graph;  
         [0032]      FIG. 14   b  shows an example normalized Tanner Graph;  
         [0033]      FIG. 15  is a block diagram representation of the construction of a 4-edge node from two 3-edge nodes;  
         [0034]      FIG. 16  is a schematic diagram of a 3-edge equality node; and  
         [0035]      FIG. 17  is a schematic diagram of a 3-edge check node. 
     
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS  
       [0036]     The sum-product algorithm [ 9 ,  17 ] is a general framework for expressing probability propagation on factor graphs. The purpose of the algorithm is to compute global conditional probabilities using only local constraints. Constraints are often expressed by factor graphs [ 5 ,  9 ,  17 ]. Special cases of factor graphs include trellis graphs and Tanner graphs, which are also referred to as constraint graphs. According to one embodiment of the invention, the sum-product algorithm is implemented on graphs which express boolean functions on discrete-type variables.  
         [0037]     For example, consider the implementation of graphs which can be simplified to local boolean constraints on three variables.  FIG. 1  is a factor graph of such a constraint function of three variables. The function ƒ expresses a relationship between variables x, y, z, which can take values from discrete alphabets A x , A y , A z , respectively. The function ƒ(x,y,z) is a boolean constraint on (x,y,z) if ƒ(x,y,z)ε{0,1} for all x, y, and z, and the constraint ƒ is said to be satisfied if and only if ƒ(x,y,z)=0.  
         [0038]     The constraint ƒ and the variables (x,y,z) have, up to this point, been defined as deterministic. When (x,y,z) are random variables, denoted herein with boldface type, they are characterized by probability, masses instead of exact values. The goal of probability propagation is to determine the probability mass of z, written ρ z , based on known masses ρ x  and ρ y . Normal typeface for these variables indicates particular samples or values of the random variables.  
         [0039]     The local operations of the sum-product algorithm are described as follows. The constraint ƒ is mapped to a processing node which receives probability masses for. variables x and y. These variables are assumed to be independent of each other. The processing node then computes the probability mass of z based on the constraint ƒ and the masses of x and y. Let S ƒ  be the set of combinations of (x,y,z) for which ƒ(x,y,z) is satisfied. Let Sƒ(j) be the subset of S ƒ  for which z=j, where jεA z . We then compute, for each j, the function  
                   P   ⁡     (     z   =   j     )       =     η   ·       ∑       (     k   ,   l     )     ∈       S   f     ⁡     (   j   )           ⁢       P   ⁡     (     x   =   k     )       ·     P   ⁡     (     y   =   l     )               ,     
     ⁢     or   ⁢           ⁢   equivalently       ⁢     
     ⁢           ρ   z     ⁡     (   j   )       =     η   ·       ∑       (     k   ,   l     )     ∈       S   f     ⁡     (   j   )           ⁢         ρ   x     ⁡     (   k   )       ·       ρ   y     ⁡     (   l   )               ,             (   1   )             
 
 where kεA x  and lεA y , η is any non-zero constant real number, and x, y, and z are random variables. The constant η is typically chosen so that ΣjP(z=j)=1. In principle, though, the accuracy of the algorithm is indifferent to η. 
 
         [0040]     The local computation ( 1 ), is the heart of the sum-product algorithm. A complete sum-product decoder consists of many interconnected instances of ( 1 ).  
         [0041]     One approach for CMOS analog decoder designs, elaborated in [ 10 ], has emerged as a popular topology for analog decoder designs. This topology is based on a generalized Gilbert multiplier.  FIG. 2  is a block diagram of a Gilbert-based sum-product circuit, comprising a renormalization module  12 , a connectivity module  14 , a of column input transistors, three of which are shown at  15 ,  17 ,  19 , which behave as analog current sources. Because of its popularity, the architecture of  FIG. 2  is referred to herein as the canonical topology.  
         [0042]     In typical implementations of a Gilbert circuit, each multiplier module  16 ,  18 ,  20  includes an array of source-connected transistors, such as the array  17  shown in  FIG. 3   a.    
         [0043]     In  FIG. 2 , X denotes an ordered sequence of row inputs (voltages) X=&lt;x 1 , . . . , x N &gt;, and Y denotes an ordered sequence of column inputs (voltages) Y=&lt;y 1 , . . . , y M &gt;. Many circuit implementations convert current inputs Ix i  and Iy k  into voltage inputs x i  and y k  using diode-connected transistors, for example, such as shown in  FIG. 3   b , corresponding to transistors 21, 22 and 23, 24. All of the transistor devices implementing the canonical topology typically have the same dimensions.  
         [0044]     For the purposes of simplifying circuit analysis, assume that Ix i ∝ρ x (i) and that Iy j ∝ρ x (j), so that the current-mode column and row inputs represent probability masses.  
         [0045]     Intermediate outputs emerge from the top of the multiplier modules  16 ,  18 , and  20  in  FIG. 2 . There are N×M such outputs, which will be referred to as Iz ij , corresponding to one row position i and one column position j. These intermediate outputs represent multiplication of row and column inputs, Iz ij ∝Ix i ·Iy j , thus performing the “product” portion of ( 1 ). To complete the computation of ( 1 ), summation of desired products is accomplished by shorting wires in the connectivity module  14 . As those skilled, in the art. will appreciate, unused products, namely those for which the constraint ƒ is not satisfied, are shorted to Vdd.  
         [0046]     The Gilbert multiplier consists of MOS transistors biased in the sub threshold region, meaning v qs &lt;V rh  for each transistor, where v qs refers to the voltage between the gate and source of an MOS device, and V th  refers to the threshold voltage. In digital design, sub threshold transistors are usually regarded as “off.” A more precise model of their behaviour is given by  
                 I   D     =       I   0     ⁢     W   L     ⁢       exp   ⁡     (       κ   ·     v   gs         U   T       )       ⁡     [     1   -     exp   ⁡     (     -       κ   ·     v   ds         U   T         )         ]           ,           (   2   )             
 
 where I 0  is a technology constant with units of amperes, W and L are the width and length of the transistor device, respectively, K≈0.7 is a unit less technology constant, and U T ≈25 mV is the well-known thermal voltage. In the subthreshold region, I D  is usually below 1 μA, resulting in very low power consumption. The subthreshold region is also commonly known as weak inversion, because the mobile charge density in the transistor&#39;s channel is very small. Circuits based on this subthreshold model were popularized by Vittoz, et. al. [ 16 ]. 
 
         [0047]     If v ds  is sufficiently large (around 150 mV), then it has little effect on I D  in ( 2 ). When v ds  is large enough to be neglected, the device is said to be in saturation. In the canonical approach, all transistors are assumed to be in saturation. To ensure this, the reference voltage V ref ≈0.3 V is used at the source of transistors  21 ,  22  and other row input current to voltage conversion devices. This maintains a sufficiently high voltage at the drain of each column input transistor  15 ,  17 ,  19  for the column input transistors to remain in saturation.  
         [0048]     The Gilbert multiplier is based on the translinear, principle. According to this principle, in a closed (Kirchoff) loop of devices in which the current (I D ) is an exponential function of the voltage (V qs ), a sum of voltages is equivalent to a product of currents. Because the sum of forward voltage drops equals the sum of backward voltage drops around the loop, the product of forward currents equals the product of backward currents.  
         [0049]     By taking a closed loop beginning and ending at V ref  and traversing the V qs  of four devices  25 ,  26 ,  27 ,  28  as shown in  FIG. 4 , we arrive at  
                       Iz   ij     ·     Ix   k       =       Iz   kj     ·     Ix   i                     ⇒     Iz   kj       =       Ix   k     ⁢         Iz   ij       Ix   i       .                     (   3   )             
 
         [0050]     If the column input transistors  15 ,  17 ,  19  are saturated, their drains simply replicate the current inputs at corresponding current to voltage conversion devices  23 ,  24 . Because the sources of their constituent transistors are all connected together, the sum of intermediate outputs from the j th  multiplier module is equal to Iy j . Thus  
                 ∑   k     ⁢     Iz   kj       =     Iy   j             (   4   )                 ⇒         Iz   ij       Ix   i       ⁢       ∑   k     ⁢     Ix   k           =     Iy   j             (   5   )                 ⇒     Iz   ij       =           Ix   i     ·     Iy   j           ∑   k     ⁢     Ix   k         .             (   6   )             
 
         [0051]     Because the algorithm specifies probability masses as the input and output of processing nodes, it may be assumed that the denominator of ( 6 ) is equal to one, in probability terms, and thus may be neglected.  
         [0052]     The above requirement for saturation imposes a minimum allowed supply voltage on Gilbert-based circuits. It is common to use one Gilbert-based circuit made of NMOS devices, of which the outputs are “folded” into a second Gilbert-based circuit made of PMOS devices. A “slice” of this topology is illustrated in  FIG. 5 . The PMOS Gilbert-based circuit comprises a plurality of transistors  29 ,  30 ,  31 ,  32 , and can perform either a second sum-product operation or provide renormalization of currents to boost the output currents so that their sum equals a desired unit current. The renormalization module  12  in  FIG. 2  represents this renormalization function. The PMOS transistors  29 ,  30 ,  31 ,  32  must also be in saturation, and must have their own separate V ref . In  FIG. 5 , the separate V ref &#39;s are indicated by V ref (N) and V ref (P). The operation of the folded topology shown in  FIG. 5  will be apparent to those skilled in the art.  
         [0053]     It is clear in  FIG. 5  that the terminals of four devices are traversed between Vdd and ground. The transistor devices  30 ,  33 , and  36  must remain in saturation for the circuit to function properly. As a rough analysis, a voltage of at least 300 mV between the drain and source of each device may be required. The drain of the device  33  is typically ˜350 mV below V ref (P). The drain of the device  30  is commonly in the range V ref (P)±50 mV. The source of the device  33  is typically V ref (N)±50 mV. Assuming the extreme cases, saturation requires a supply voltage no less than 1.35 V. If the saturation requirement is relaxed to a v ds  of 150 mV, then the supply voltage can be no less than 0.9 V.  
         [0054]     This result is based on the saturation assumption. In practice, saturation is only an approximate condition. The output of the folded circuit does always depend slightly on the other inputs. As V ref (P) is increased and V ref (N) is decreased, the device becomes less saturated, and the output begins to change dramatically. In the extreme case where V ref (P)=V dd  and V ref (N)=0, we must use the full model expressed in (2), without neglecting V ds .  
         [0055]     When a transistor is not in saturation, the translinear principle still applies. The complete non-saturation MOS device model is illustrated in  FIG. 6 . The total device current is divided into two oppositely directed components. The forward component, I ƒ , is controlled by v gs . The backward component, I r , is controlled by v gd . The total current is I D =I ƒ −I r . To solve circuits with unsaturated transistors, translinear loops must traversed v gd  as well as V gs .  
         [0056]     It is clear from the foregoing that the voltage needs of the Gilbert multiplier can be reduced if V ref =0. This results in the column input transistors  15 ,  17 ,  19  becoming unsaturated. This situation can be analyzed using the translinear principle. The circuit consists of the translinear loops shown in  FIGS. 4, 7 , and  8 . The first loop, shown in  FIG. 4 , was analyzed above, and yields the equation 
 
Ix i ·Iz kj =Iz y ·Ix k   (7) 
 
 such that in  FIG. 7  
 
Iƒ=Iy j .   (8) 
 
         [0057]      FIG. 7  shows a column input current to voltage conversion transistor  38  connected to an unsaturated column input transistor  37 , with its additional component I r . The current I dj  is defined as 
 Id j ≡Iƒ−Ir.   (9)  
         [0058]     The role of the source-connected transistors, from (4), may be expressed as:  
               Id   j     =       ∑   k     ⁢       Iz   kj     .               (   10   )             
 
         [0059]     To solve for all currents in the circuit, we need one more equation, which is provided by the third loop, through the devices  43 ,  39 ,  44  shown in  FIG. 8  also shown:  
                 Iz   ij     ·   If     =     Ir   ·     Ix   i               (   11   )                 ⇒   Ir     =       Iy   j     ·       Iz   ij       Ix   i                 (   12   )             
 
 Although the devices  40 ,  41 ,  42  are shown in  FIG. 8  for completeness, the above particular example third loop does not include these devices. 
 
         [0060]     Combining (7) through (12), we arrive at  
               Id   j     =       Iy   j     -       Iy   i     ·       Iz   ij       Ix   i                   (   13   )                 ⇒       ∑   k     ⁢     Iz   kj         =       Iy   j     -       Iy   j     ·       Iz   ij       Ix   i                   (   14   )                 ⇒           Iz   ij       Ix   i       ⁢       ∑   k     ⁢     Ix   k         +       Iy   j     ·       Iz   ij       Ix   i             =     Iy   j             (   15   )                 ⇒     Iz   ij       =           Iy   j     ·     Ix   i             ∑   k     ⁢     Ix   k       +     Iy   j         .             (   16   )             
 
         [0061]     The result (16) is almost the same as the normal Gilbert multiplier output (6), except there is an additional term in the denominator. In probability terms, the denominator of (16) can no longer be neglected. To solve this problem, in accordance with an embodiment of the invention, additional transistors are provided with their sources connected to Id j . If these transistors represent a current I 8 ≡Σ l≠j Iy l , then the output becomes  
               Iz   ij     =         Iy   j     ·     Ix   i             ∑   k     ⁢     Ix   k       +     I   δ     +     Iy   j                 (   17   )                 ⇒     Iz   ij       =         Iy   j     ·     Ix   i             ∑   k     ⁢     Ix   k       +       ∑     l   ≠   j       ⁢     Iy   l       +     Iy   j                 (   18   )                 ⇒     Iz   ij       =           Iy   j     ·     Ix   l             ∑   k     ⁢     Ix   k       +       ∑   l     ⁢     Iy   l           .             (   19   )             
 
         [0062]     In probability terms, the denominator of (19) is a constant and can be neglected. The addition of redundant transistors therefore corrects the probability calculation of the Gilbert multiplier when V ref =0. Because these new transistors do not provide useful outputs, they are referred to herein as dummy transistors or dummy inputs. The drains of these transistors are preferably connected to Vdd.  
         [0063]      FIG. 9  is a block diagram of a low-voltage CMOS sum-product circuit according to an embodiment of the invention, comprising a renormalization module  52 , a connectivity module  54 , a plurality of multiplier modules  56 ,  58 , and  60 , a plurality of dummy multiplier modules  66 ,  68 , and  70 , and a plurality of column input transistors generally designated by  71 . In  FIG. 9 , X denotes an ordered sequence of row inputs (voltages) &lt;X1, . . . , X N &gt;, and Y k  denotes a set of column inputs (voltages) {y l } M   l=1  excluding the input Y k . The members of Y k  need have no particular order. If necessary, current inputs Ix i  and Iy l  are converted into voltage inputs x i  and y l  by diode-connected transistors as shown in  FIG. 3   b  and described above.  
         [0064]     All transistor devices used to implement the sum-product circuit in  FIG. 9  preferably. have the same dimensions, with the possible exception of the transistors, illustratively PMOS transistors, used in the renormalization module  52 . Renormalization is described in further detail below. As described above, desired products from the multiplier modules  56 ,  58 , and  60  are summed together by shorting wires in the connectivity module  54 , and unused products are shorted to Vdd.  
         [0065]     In a practical setting, the sum-product algorithm is carried out repeatedly in a chain of computations. The output of one computation provides input for the next. In an analog implementation, each computation is performed by a separate sum-product module. A complete analog error control decoder consists of cascades of sum-product modules.  
         [0066]     For a low-voltage decoder, renormalization of currents between modules is preferred. In a canonical Gilbert-based sum-product circuit with outputs expressed by (6), the denominator is equivalent to ‘1’ and can be truly ignored. In the low voltage output expressed by (19), however, the denominator is equivalent to ‘2’, which results in substantial attenuation at the output of each circuit. While, in principle, linear attenuation will not change the. result of decoding, consistent attenuation in a large network will cause the outputs to approach zero, making it impossible to extract any results, and causing the circuit to slow to a halt.  
         [0067]     By inserting the renormalization module  52  between sum-product circuits, the current outputs are prevented from approaching zero. A renormalization circuit that may be implemented in the sum-product circuit of  FIG. 9 .is shown in  FIG. 10 . The parameters m and n indicate that the transistor&#39;s width is increased by a factor of m or n relative to other transistors used in a sum-product circuit. Canonical Gilbert-based sum-product circuits typically use m=1 and n=1.  
         [0068]     The circuit of  FIG. 10 .is basically a low-voltage sum-product circuit with only a single column input I u , which represents a global reference current, and a plurality of PMOS transistors  72 ,  73 ,  74 ,  75 ,  76 ,  77 . In probability terms, I u ∝1. The behaviour of  FIG. 10 . is described by the equation  
                 Out   i     =       n   ·   m   ·     I   u     ·     In   i           n   ·     I   u       +     m   ·       ∑   k     ⁢     In   k               ,           (   20   )             
 
 which is a generalization of (16). As demonstrated by (20), the renormalization circuit of  FIG. 10  boosts each input by a constant factor, and is thus a linear transformation. The probability information is therefore not subject to distortion due to renormalization. 
 
         [0069]     Because sum-product circuits will normally be situated in a network, the. iterated behaviour of (20) should also be considered. This can be simplified to a one-dimensional problem by using a summary variable k for each set of probabilities, defined as  
               k   x     ≡           ∑   i     ⁢     Ix   i         I   u       .             (   21   )             
 
         [0070]     This allows the treatment of (20) as a simple one-dimensional transfer function,  
               k   out     =         n   ·   m   ·     k   In         n   +     m   ·     k   In           .             (   22   )             
 
         [0071]     Iteration of (22) is illustrated in  FIG. 11 . The blocks  104  and  108  labeled ƒ n  represent the transfer function (22), and are respectively connected to the outputs of the sum-product circuits  102  and  106 . To determine the dynamic behaviour of this system, the fixed points (where k Out =k In ) are identified, and a determination is then made as to whether they are stable. As those skilled in the art will appreciate, it can be shown that the fixed points occur at  
               k   0     =     0   ⁢           ⁢   and             (   23   )                 k   1     =     n   -       n   m     .               (   24   )             
 
         [0072]     It is well known that a fixed point is stable and non-oscillating if and only if the slope of the transfer function, ƒ′ n (k), satisfies 0≦ƒ′ n (k j )&lt;1 at the fixed point k ƒ . Also, a fixed point k ƒ  is unstable (i.e. it is a repeller) if and only if ƒ′ n (k ƒ )&gt;1. It can also be shown that  
                 f   n   ′     ⁡     (   0   )       =     m   ⁢           ⁢   and             (   25   )                   f   n   ′     ⁡     (     k   1     )       =       1   m     .             (   26   )             
 
         [0073]     Equations (25) and (26) show that there is always a stable fixed point above zero when m&gt;1. Known renormalizers use m=1, and therefore cause all currents to approach zero in a low-voltage network. By using m&gt;1, this can be avoided.  
         [0074]     In a low-voltage design, use of a reset circuit is also preferred. An example of such a circuit is shown in  FIG. 12 , comprising transistors  112 ,  114 , and  116  connected across inputs and outputs of a sum-product circuit  110 . The transistors  112 ,  114 , and  116  provide for restoration of a sum-product network to an unbiased state when decoding of one block of information is complete. When the digital ‘rst’ signal in  FIG. 12  is high, all probabilities are shorted together at the inputs and outputs. of the sum-product circuit  110 , which may also be considered a local processing node. The transistors  112 ,  114 , and  116  act as switches and are shown in  FIG. 12  as NMOS transistors. After a short settling time, the ‘rst’ signal falls low again, the transistors  112 ,  114 , and  116  become open switches, and the sum-product circuit  110 , as well as other such circuits connected in a network, is ready to process new information.  
         [0075]     Reset circuits are well known in Gilbert-based designs, having been studied, for example, in the PhD theses of Felix Lichtenberger [11] and Jie Dai (4). Their value seems debatable in canonical Gilbert circuits. For low-voltage circuits as in  FIG. 9 .and other types of circuits that may have a stronger memory effect, the use of reset circuits is generally preferred.  
         [0076]     One very common class of decoders employ the BCJR algorithm. This algorithm is used with concatenated convolutional codes, such as Turbo codes and Turbo. equalizers. The BCJR algorithm [2] is a special case of the sum-product algorithm (1), as shown in [9]. In the BCJR algorithm, variables are often not binary.  
         [0077]     A trellis stage is a portion of a trellis graph which describes a boolean constraint function. The graph of a trellis stage consists of two columns of states, connected by branches. An example trellis stage is shown in  FIG. 13   a . The states on the left represent the possible values of a random state variable a. The states on the right represent branches represent the possible values of a branch variable, y. If a branch exists for a particular combination (α i , y j , Out k ), then the constraint is satisfied for that combination.  
         [0078]     An example of the low-voltage sum-product architecture for computing on trellis graphs is shown in  FIG. 13   b . As shown, the sum-product circuit comprises a renormalization module  122  having a plurality of transistor pairs  117 ,  118 ,  119 ,  120  and a single column input transistor  121 , a connectivity module  124 , a plurality of multiplier modules  126 ,  128 ,  130 ,  132 , a plurality of dummy multiplier modules  136 ,  138 ,  140 ,  142 , and a plurality of column input transistors generally designated by  143 . The branch variable takes values from the set {a, b, c, d}. The sum-product equation for this particular trellis section can be written in matrix form as  
               [         Out0           Out1           Out2           Out3         ]     =       [         a                   d                       b                   c                                   c                   b                       d                   a         ]     ×       [         a0           a1           a2           a3         ]     .               (   27   )             
 
         [0079]     The particular trellis function is determined by the connectivity module  124 .  
         [0080]     The sum-product circuit is shown with the state probabilities as row inputs and the branch probabilities as column inputs, but these roles can be reversed without affecting the results. Every stage of the BCJR algorithm consists of a matrix multiplication of the form (27). Low-voltage sum-product circuits can therefore be easily produced to implement a complete BCJR decoder.  
         [0081]     Decoders for binary LDPC codes are mapped from the code&#39;s normalized Tanner Graph, which is a direct visualization of the code&#39;s binary parity check matrix. The Tanner Graph contains two types of nodes, check nodes and variable nodes. A variable node denotes a particular bit in a codeword. A check node represents a parity check equation, which is a single row in the parity check matrix. All variables in the graph are binary. For implementations, this means that all probability masses will have only two components.  FIG. 14   a  shows an example Tanner Graph that contains a variable node  150  and a plurality of check nodes  152 .  
         [0082]     In a normalized Tanner Graph,, also known as a Forney Graph, equality nodes are inserted between variable nodes and check nodes [5]. This is illustrated in  FIG. 14   b , in which an equality node  156  is inserted between a variable node  154  and a plurality of cheek nodes  158 .  
         [0083]     The purpose of equality node insertion is to provide explicit representation for the constraint which occurs at variable nodes: all connected edges convey the same variable. Let U i  be the set of check nodes connected to variable i, and let u j  be the binary value from check node j. The equality constraint is satisfied if and only if 
 
u j =u k , ∀j, kεU i .   (28) 
 
         [0084]     The check node is only slightly more complicated. Let V j  be the set of variables which are connected to check node j, and let v i  be the value of variable i. Parity check j is then satisfied if and only if  
                   ⊕     i   ∈     V   j         ⁢     v   i       =   0     ,           (   29   )             
 
 where ‘⊕’ denotes modulo-2 summation. 
 
         [0085]     The constraints in (28) and (29) can be conveniently broken down into recursive operations, allowing, the construction of nodes with many edges by connecting 3-edge nodes in a chain. This is illustrated in  FIG. 15 , for example, in which a 4-edge node 160 is shown with an equivalent implementation as two 3-edge nodes  162  and  164 . Therefore, the construction of larger nodes using 3-edge nodes will be apparent from the description of the implementation of 3-edge nodes.  
         [0086]     For a 3-edge check node, the constraint function is simply a logical XOR operation. Mapping this to a sum-product implementation, and labeling the three edges X, Y, and Z, we obtain. 
 
 P ( Z =0)= P ( X= 0)· P ( Y =0)+ P ( X =1)· P ( Y =1)   (30) 
 
 P ( Z =1)= P ( X= 1)· P ( Y =0)+ P ( X =0)· P ( Y =1)   (31) 
 
         [0087]     For a 3-edge equality node, we obtain 
 
 P ( Z =0)∝ P ( X =0)· P ( Y =0)   (32) 
 
 P ( Z =1)∝ P ( X =1)· P ( Y =1)   (32) 
 
         [0088]     The proportionality symbol is used to indicate that the algorithm is invariant to multiplication by any non-zero normalizing constant.  
         [0089]     Applying the general circuit of  FIG. 9 , and varying the connectivity to produce the appropriate functions, we arrive at the circuits of  FIGS. 16 and 17  for a 3-edge equality node and a 3-edge check node, respectively. As above, the ‘y’ inputs are the column. inputs, and the ‘x’ inputs are the. row inputs. In  FIGS. 16 and 17 , the transistors  172 ,  174 ,  176 ,  178  and  192 ,  194 ,  196 ,  198  form the multiplier modules, the transistors  184 ,  186  and  204 ,  206  are column input transistors, and the transistors  180 ,  182  and  200 ,  202  form the dummy multiplier modules. The renormalization module is not shown in  FIGS. 16 and 17 , but is preferably present. Complete decoders for linear binary block codes, including LDPC codes, can be constructed from these 3-edge circuits. Of course, as those skilled in the art will appreciate, conversion modules, voltage to current conversion modules in the example circuits shown in  FIGS. 16 and 17 , may be provided between circuits where circuit outputs are to be converted for input to a next circuit.  
         [0090]     What has been described is merely illustrative of the application of the principles of the invention. Other arrangements and methods can be implemented by those skilled in the art without departing from the spirit and scope of the present invention.  
         [0091]     For example, the invention is in no way limited to the particular implementations shown as illustrative examples in the drawings. Alternate implementations, using different elements and/or different types of components, will be apparent to those skilled in the art.  
         [0092]     The invention may also be applied to other types of decoding. than the trellis and LDPC decoding described above.  
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