BACKGROUND

[0001]
Due to the presence of noise on a communications channel, the signal received is not always identical to the signal transmitted. Channel coding, or equivalently, error correction coding, relates to techniques for increasing the probability that a receiver in a communications systems will be able to correctly detect the composition of the transmitted data stream. Typically, this is accomplished by encoding the signal to add redundancy before it is transmitted. This redundancy increases the likelihood that the receiver will be able to correctly decode the encoded signal and recover the original data.

[0002]
Turbo coding, a relatively new but widely used channel coding technique, has made signaling at power efficiencies close to the theoretical limits possible. The features of a turbo code include parallel code concatenation, nonuniform interleaving, and iterative decoding. Because turbo codes may substantially improve energy efficiency, they are attractive for use over channels that are power and/or interference limited.

[0003]
A turbo decoder may be used to decode the turbo code. The turbo decoder may include two softinput/softoutput (SISO) decoding modules that work together in an iterative fashion. The SISO decoding module is the basic building block for established iterative detection techniques for a system having a network of finite state machines, or more generally, subsystems.

[0004]
[0004]FIGS. 7A and 7B respectively show block diagrams of typical turbo encoder 10 and turbo decoder 11 arrangements. In this example, the turbo encoder 10 uses two separate encoders, RSC1 and RSC2, each a Recursive Systematic Convolutional encoder. Each of the encoders RSC1 and RSC2 can be modeled as a finite state machine (FSM) having a certain number of states (typically, either four or eight for turbo encoders) and transitions therebetween. To encode a bit stream for transmission over a channel, uncoded data bits b_{k }are input both to the first encoder RSC1 and an interleaver I. The interleaver I shuffles the input sequence b_{k }to increase randomness and introduces the shuffled sequence a_{k }to the second decoder RSC2. The outputs of encoders RSC1 and RSC2, c_{k }and d_{k }respectively, are punctured and modulated by block 12 to produced encoded outputs x_{k}(0) and x_{k}(1), which are transmitted over a channel to the decoder 11.

[0005]
At the decoder 11, the encoded outputs are received as noisy inputs Z_{k}(0) and Z_{k}(1) and are demodulated and depunctured by block 13, which is the softinverse of puncture and modulation block 12. The output of block 13, M[c_{k}(0)], M[c_{k}(1)] and M[d_{k}(1)], is “soft” information—that is, guesses or beliefs about the most likely sequence of information bits to have been transmitted in the coded sequences c_{k}(0), c_{k}(1) and d_{k}(1), respectively. The decoding process continues by passing the received soft information M[c_{k}(0)], M[c_{k}(1)] to the first decoder, SISO1, which makes an estimate of the transmitted information to produce soft output M[b_{k}] and passes it to interleaver I (which uses the same interleaving mapping as the interleaver I in the encoder 10) to generate M[a_{k}]. The second decoder, SISO2, uses M[a_{k}] and received soft information M[d_{k}(1)] to reestimate the information. This second estimation is looped back, via the soft inverse of the interleaver, I^{−1}, to SISO1 where the estimation process starts again. The iterative process continues until certain conditions are met, such as a certain number of iterations are performed, at which point the final soft estimates become “hard” outputs representing the transmitted information.

[0006]
Each of the SISO decoder modules in the decoder 11 is the softinverse of its counterpart RSC encoder in the encoder 10. The conventional algorithm for computing the softinverse is known as the “forwardbackward” algorithm such as described in G. David Forney, Jr., “The ForwardBackward Algorithm,” Proceedings of the 34^{th }Allerton Conference on Communications, Control and Computing, pp. 432446 (Oct. 1996). In the forwardbackward algorithm, an estimate as to a value of a data bit is made by recursively computing the least cost path (using addcompareselect, or ACS, operations) forwards and backwards through the SISO's “trellis”—essentially an unfolded state diagram showing all possible transitions between states in a FSM. Each path through the SISO's trellis has a corresponding cost, based on the received noisy inputs, representing a likelihood that the RSC took a particular path through its trellis when encoding the data. Typically, the lower a path's total cost, the higher the probability that the RSC made the corresponding transitions in encoding the data. In general, the forward and backward ACS recursions performed by a SISO can be computed either serially or in parallel. Performing the recursions in parallel, the faster of the two methods, yields an architecture with latency O(N), where N is the block size of the encoded data. As used herein, “latency” is the endtoend delay for decoding a block of N bits.

[0007]
The present inventors recognized that, depending on the latency of other components in a decoder or other detection system, reducing the latency of a SISO could result in a significant improvement in the system's throughput (a measurement of the number of bits per second that a system architecture can process). Consequently, the present inventors developed a treestructured SISO that can provide reduced latency.
SUMMARY

[0008]
Implementations of the treestructured SISO may include various combinations of the following features.

[0009]
In one aspect, decoding an encoded signal (for example, a turbo encoded signal, a block encoded signal or the like) can be performed, e.g., in a wireless communications system, by demodulating the received encoded signal to produce soft information, and iteratively processing the soft information with one or more softin/softoutput (SISO) modules. At least one of the SISO modules uses a tree structure to compute forward and backward state metrics, for example, by performing recursive marginalizationcombining operations, which may in various embodiments include minsum operations, min*sum operations (where min*=min(x,y)−ln(1+e^{−xy})) sumproduct operations, and/or maxproduct operations.

[0010]
The encoded signal may comprise at least one of a turbo encoded signal, a block turbo encoded signal, a low density parity check coded signal, a product coded signal, a convolutional coded signal, a parallel concatenated convolutional coded signal, and/or a serial concatenated convolutional coded signal.

[0011]
The iterative processing may be terminated upon occurrence of a predetermined condition, for example, the completion of a predetermined number of iterations. The iterative processing may include performing parallel prefix operation or parallel suffix operations, or both, on the soft information. Moreover, the iterative processing may include using soft output of a first SISO as soft input to another SISO, and/or may include performing marginalizationcombining operations which form a semiring over the softinformation.

[0012]
The tree structure used by at least one SISO may be a tree structure that results in the SISO having a latency of O(log_{2 }N), where N is a block size, a BrentKung tree, or a forwardbackward tree, e.g., having a tree structure recursion that is bidirectional.

[0013]
Processing performed by at least one SISO may include tiling an observation interval into subintervals, and applying a minimum halfwindow SISO operation on each subinterval.

[0014]
In another aspect, a SISO module may include a plurality of fusion modules arranged into a tree structure and adapted to compute forward and backward state metrics. Each fusion module may be defined by the equation:
$C\ue8a0\left({k}_{0},{k}_{1}\right)\ue89e\stackrel{\Delta}{=}\ue89eC\ue8a0\left({k}_{0},m\right)\ue89e{\otimes}_{C}\ue89eC\ue8a0\left(m,{k}_{1}\right)\iff C\ue8a0\left({s}_{{k}_{0}};{s}_{{k}_{1}}\right)=\underset{{s}_{\mathrm{ni}}}{\mathrm{min}}\ue89e\left[C\ue8a0\left({s}_{{k}_{0}},{s}_{m}\right)+C\ue8a0\left({s}_{m},{s}_{{k}_{1}}\right)\right]\ue89e\forall \text{\hspace{1em}}\ue89e{s}_{{k}_{0}},{s}_{{k}_{1}}$

[0015]
where C(k, m) is a matrix of minimum sequence metrics (MSM) of state pairs s_{k }and s_{m }based on softinputs between s_{k }and s_{m}. At least one of the fusion modules may compute forward and backward state metrics by performing recursive marginalizationcombining operations.

[0016]
In another aspect, a SISO module may include one or more complete fusion modules (CFMs) for performing marginalizationcombining operations in both a forward direction and a backward direction, one or more forward fusion modules (fFMs) for performing marginalizationcombining operations only in the forward direction, and one or more backward fusion modules (bFMs) for performing marginalizationcombining operations only in the backward direction. The one or more CFMs, fFMs, and bFMs are arranged into a tree structure (e.g., BrentKung tree, forwardbackward tree). An amount of the CFMs may be set to a minimum number needed to compute a softinverse. In general, fFMs and bFMs may be used in the tree structure in place of CFMs wherever possible.

[0017]
In another aspect, iterative detection may include receiving an input signal (e.g., a turbo encoded signal or a convolutional coded signal) corresponding to one or more outputs of a finite state machine (FSM), and determining the soft inverse of the FSM by computing forward and backward state metrics of the received input signal using a tree structure. The forward and backward state metrics may be computed by one or more SISO modules, for example, using a treestructured set of marginalizationcombining operations.

[0018]
Determining the soft inverse of the FSM may include iteratively processing soft information, for example, performing parallel prefix operation or parallel suffix operations, or both, on the soft information. Moreover, the iterative processing may include using soft output of a first SISO as soft input to another SISO. At least one SISO further may the an observation interval into subintervals, and apply a minimum halfwindow SISO operation on each subinterval.

[0019]
In another aspect, a turbo decoder includes a demodulator adapted to receive as input a signal encoded by a FSM and to produce soft information relating to the received signal, and at least one SISO module in communication with the demodulator and adapted to compute a softinverse of the FSM using a tree structure. The tree structure may implement a combination of parallel prefix and parallel suffix operations.

[0020]
The turbo decoder may include at least two SISO modules in communication with each other. In that case, the SISO modules may iteratively exchange soft information estimates of the decoded signal. In any event, at least one SISO may compute the softinverse of the FSM by computing forward and backward state metrics of the received signal.

[0021]
In another aspect, iterative detection may be performed by receiving an input signal corresponding to output from one or more block encoding modules, and determining the soft inverse of the one or more block encoding modules by computing forward and backward state metrics of the received input signal using a tree structure. The input signal may include a block turbo encoded signal, a low density parity check coded signal, and/or a product coded signal.

[0022]
Determining the soft inverse of the block encoding module may include iteratively processing soft information, for example, performing parallel prefix operation or parallel suffix operations, or both, on the soft information.

[0023]
In another aspect a block decoder may include a demodulator adapted to receive as input a signal encoded by a block encoding module and to produce soft information relating to the received signal, and at least one SISO module in communication with the demodulator and adapted to compute a softinverse of the block encoding module using a tree structure. The tree structure used may implement a combination of parallel prefix and parallel suffix operations. The block decoder may further include at least two SISO modules in communication with each other, wherein the SISO modules iteratively exchange soft information estimates of the decoded signal. In any event, at least one SISO may compute the softinverse of the block encoding module by computing forward and backward state metrics of the received signal.

[0024]
In another aspect, iterative detection may include receiving an input signal (e.g., a block error correction encoded signal, a block turbo encoded signal, a low density parity check coded signal, and/or a product coded signal) corresponding to one or more outputs of a module whose softinverse can be computed by running the forwardbackward algorithm on a trellis representation of the module (e.g., a FSM or a block encoding module), and determining the soft inverse of the module by computing forward and backward state metrics of the received input signal using a tree structure.

[0025]
One or more of the following advantages may be provided. The techniques and methods described here result in a treestructure SISO module that can have a reduced latency as low as O(lg N) [where N is the block size and “lg” denotes log_{2}] in contrast to conventional SISOs performing forwardback recursions in parallel, which have latency O(N). The decrease in latency comes primarily at a cost of chip area, with, in some cases, only a marginal increase in computational complexity. This treestructure SISO can be used to design a very high throughput turbo decoder, or more generally an iterative detector. Various subwindowing and tiling schemes also can be used to further improve latency.

[0026]
These reducedlatency SISOs can be used in virtually any environment where it is desirable or necessary to run the forwardbackward algorithm on a trellis. For example, the treeSISO finds application in wireless communications and in many types of decoding (turbo decoding, block turbo decoding, convolutional coding including both parallel concatenated convolutional codes and serial concatenated convolutional codes, parity check coding including low density parity check (LDPC) codes, product codes and more generally in iterative detection). The reducedlatency SISOs are particularly advantageous when applied to decode FSMs having a relatively small number of states, such as turbo code FSMs which typically have either 4 or 8 states.

[0027]
The details of one or more embodiments are set forth in the accompanying drawings and the description below. Other features, objects, and advantages of the invention will be apparent from the description and drawings, and from the claims.
DRAWING DESCRIPTIONS

[0028]
[0028]FIG. 1 shows the C fusion process.

[0029]
[0029]FIG. 2 shows a fusion module array for combining the complete set of C matrices on [k_{0}, k_{0}+K] and [k_{0}+K, k_{0}+2K] to obtain the complete set on [k_{0}, k_{0}+2K].

[0030]
[0030]FIG. 3 shows a treeSISO architecture for N=16.

[0031]
[0031]FIG. 4 shows a tiled subwindow scheme based on the forwardbackward algorithm.

[0032]
[0032]FIG. 5 shows a tiled subwindow approach with 4 treeSISOs of window size 4 for N=16 to implement a d=4 MHW SISO.

[0033]
[0033]FIG. 6 is a graph showing a simulation of a standard turbo code for SISOs with various halfwindow sizes, N=1024, and ten iterations.

[0034]
[0034]FIGS. 7A and 7B show examples of a turbo encoder and a turbo decoder, respectively.

[0035]
[0035]FIGS. 8A and 8B show a forwardbackward treeSISO (FBTSISO) implementation.

[0036]
[0036]FIG. 9 shows an embodiment of a BrentKung TreeSISO.

[0037]
[0037]FIG. 10 is a graph showing transistor count estimations for the BrentKung TreeSISO approach of FIG. 9.

[0038]
[0038]FIG. 11 shows the trellis structure for the parity check for a standard Hamming (7,4) code.

[0039]
[0039]FIG. 12 shows a tree structured representation of the Hamming code parity check, in which the forwardbackwardtreeSISO algorithm may be run on this tree with an inward (up) set of message passes to the node labeled V6, followed by an outward (down) set of message passes to produce the desired softout metrics on the coded bits.

[0040]
[0040]FIG. 13 shows an example of an LDPC decoder (overall code rate of 0.5) in which beliefs are passed from the broadcaster nodes to the parity check nodes through a fixed permutation, and in which the parity check nodes can be implemented as a treestructured SISO.
DETAILED DESCRIPTION

[0041]
The present inventors have developed a reformulation of the standard SISO computation using a combination of prefix and suffix operations. This architecture—referred to as a TreeSISO—is based on treestructures for fast parallel prefix computations used in Very Large Scale Integration (VLSI) applications such as fastadders. Details of the TreeSISO and various alternative embodiments follow.

[0042]
Calculating the “softinverse” of a finitestate machine (FSM) is a key operation in many data detection/decoding algorithms. One of the more prevalent applications is iterative decoding of concatenated codes, such as turbo codes as described in Berrou, et al., “Near shannon limit errorcorrecting coding and decoding: turbocodes,” International Conference on Communications, (Geneva, Switzerland), pp. 10641070, May 1993; and Berrou, et al., “Near optimum error correcting coding and decoding: turbocodes,” IEEE Trans. Commun., vol. 44, pp. 12611271, October 1996. However the SISO (softin/softout) module is widely applicable in iterative and no n iterative receivers and signal processing devices. The softoutputs generated by a SISO may also be thresholded to obtain optimal hard decisions (e.g., producing the same decisions as the Viterbi algorithm (G. D. Forney, “The Viterbi algorithm,” Proc. IEEE, vol. 61, pp. 268278, March 1973) or the Bahl algorithm (L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inform. Theory, vol. IT20, pp. 284287, March 1974)). The general trend in many applications is towards higher data rates and therefore fast algorithms and architectures are desired.

[0043]
There are two basic performance (speed) aspects of a data detection circuit architecture. The first is throughput which is a measurement of the number of bits per second the architecture can decode. The second is latency which is the endtoend delay for decoding a block of N bits. Nonpipelined architectures are those that decode only one block at a time and for which the throughput is simply N divided by the latency. Pipelined architectures, on the other hand, may decode multiple blocks simultaneously, shifted in time, thereby achieving much higher throughput than their nonpipelined counterparts.

[0044]
Depending on the application, the throughput and/or latency of the data detection hardware may be important. For example, the latency associated with interleaving in a turbocoded system with relatively low data rate (less than 100 Kb/s) will likely dominate the latency of the iterative decoding hardware. For future highrate systems, however, the latency due to the interleaver may become relatively small, making the latency of the decoder significant. While pipelined decoders can often achieve the throughput requirements, such techniques generally do not substantially reduce latency. In addition, sometimes latency has a dramatic impact on overall system performance. For example, in a data storage system (e.g., magnetic hard drives), latency in the retrieval process has a dramatic impact on the performance of the microprocessor and the overall computer. Such magnetic storage channels may use highspeed Viterbi processing with turbocoded approaches.

[0045]
The standard SISO algorithm is the forwardbackward algorithm. The associated forward and backward recursion steps can be computed in parallel for all of the FSM states at a given time, yielding an architecture with O(N) computational complexity and latency, where N is the block size. The key result of this paper is the reformulation of the standard SISO computation using a combination of prefix and suffix operations, which leads to an architecture with O(lg N) latency. This architecture, referred to as a “treeSISO”, is based on a treestructure for fast parallel prefix computations in the Very Large Scale Integration (VLSI) literature (e.g., fast adders).

[0046]
This exponential decrease in latency for the treeSISO comes at the expense of increased computational complexity and area. The exact value of these costs depends on the FSM structure (e.g., the number of states) and the details of the implementation. However, for a fourstate convolutional code, such as those often used as constituent codes in turbo codes, the treeSISO architecture achieves O(lg N) latency with computational complexity of O(N lg N). Note that, for this fourstate example, the computation complexity of treeSISO architecture increases sublinearly with respect to the associated speedup. This is better than linearscale solutions to the Viterbi algorithm (e.g., such as described in Fettweis, et al., “Parallel Viterbi algorithm implementation: Breaking the ACSbottleneck” IEEE Trans. Commun., vol. 37, pp. 785790, August 1989); the generalization of which to the SISO problem is not always clear. For this 4state code example, the area associated with the O(lgN)latency treeSISO is O(N).

[0047]
SoftIn SoftOut Modules

[0048]
Consider a specific class of finite state machines with no parallel state transitions and a generic Sstate trellis. Such a trellis has up to S transitions departing and entering each state. The FSM is defined by the labeling of the state transitions by the corresponding FSM input and FSM output. Let t_{k}=(s_{k},a_{k},s_{k}+1)=(s_{k}, a_{k})=(s_{k},s_{kp30 1}) be a trellis transition from state s_{k }at time k to state s_{k+1 }in response to input a_{k}. Since there are no parallel state transitions, t_{k }is uniquely defined by any of these representations. Given that the transition t_{k }occurs, the FSM output is x_{k}(t_{k}). Note that for generality the mapping from transitions to outputs is allowed to be dependent on k.

[0049]
Consider the FSM as a system that maps a digital input sequence a
_{k }to a digital output sequence x
_{k}. A marginal softinverse, or SISO, of this FSM can be defined as a mapping of softin (SI) information on the inputs SI(a
_{k}) and outputs SI(x
_{k}), to softoutput (SO) information for a
_{k }and/or x
_{k}. The mapping is defined by the combining and marginalization operators used. It is now wellunderstood that one need only consider one specific reasonable choice for marginalization and combining operators and the results easily translate to other operators of interest. Thus, the focus is on the minsum marginalizationcombining operation with the results translated to maxproduct, sumproduct, min*sum, and max*sum in a standard fashion. In all cases, let the indices K
_{1 }and K
_{2 }define the time boundaries of the combining window or span used in the computation of the softoutput for a particular quantity u
_{k }(e.g., u
_{k}=s
_{k}, u
_{k}=a
_{k}, u
_{k}=t
_{k}, u
_{k}=x
_{k}, =(s
_{k},s
_{k+d}), etc.). In general, K
_{1 }and K
_{2 }are functions of k. For notational compactness, this dependency is not explicitly denoted. For minsum marginalizationcombining, the minimum sequence metric (MSM) of a quantity u
_{k }is the metric (or length) of the shortest path or sequence in a combining window or span that is consistent with the conditional value of u
_{k}. Specifically, the MSM is defined as follows. As is the standard convention, the metric of a transition that cannot occur under the FSM structure is interpreted to be infinity.
$\begin{array}{cc}{\mathrm{MSM}}_{{K}_{1}}^{{K}_{2}}\ue8a0\left({u}_{k}\right)\ue89e\stackrel{\Delta}{=}\ue89e\underset{{t}_{{K}_{1}}^{{K}_{2}}\ue89e{u}_{k}}{\mathrm{min}}\ue89e{M}_{{K}_{1}}^{{K}_{2}}\ue8a0\left({t}_{{K}_{1}}^{{K}_{2}}\right),& \left(1\right)\\ {M}_{{K}_{1}}^{{K}_{2}}\ue8a0\left({t}_{{K}_{1}}^{{K}_{2}}\right)\ue89e\stackrel{\Delta}{=}\ue89e\sum _{m={K}_{1}}^{{K}_{2}}\ue89e\text{\hspace{1em}}\ue89e\mathrm{Mm}\ue8a0\left({t}_{m}\right)& \left(2\right)\\ \mathrm{Mm}\ue8a0\left({t}_{m}\right)\ue89e\stackrel{\Delta}{=}\ue89e\mathrm{SI}\ue8a0\left({a}_{m}\right)+\mathrm{SI}\ue8a0\left({x}_{m}\ue8a0\left({t}_{m}\right)\right),& \left(3\right)\end{array}$

[0050]
where the set of transitions starting at time K
_{1 }and ending at time K
_{2 }that are consistent with u
_{k }is denoted
${t}_{\mathrm{K1}}^{\mathrm{K2}}:{u}_{k}\ue89e\text{\hspace{1em}}\ue89e\mathrm{and}\ue89e\text{\hspace{1em}}\ue89e{t}_{\mathrm{K2}}^{\mathrm{K1}}$

[0051]
implicitly defines a sequence of transitions t_{k1}, t_{k1+1}, . . . , t_{K2}.

[0052]
Depending on the specific application, one or both of the following “extrinsic” quantities will be computed
$\begin{array}{cc}{\mathrm{SO}}_{{K}_{1}}^{{K}_{2}}\ue8a0\left({x}_{k}\right)\ue89e\stackrel{\Delta}{=}\ue89e{\mathrm{MSM}}_{{K}_{1}}^{{K}_{2}}\ue8a0\left({x}_{k}\right)\mathrm{SI}\ue8a0\left({x}_{k}\right)& \left(4\right)\\ {\mathrm{SO}}_{{K}_{1}}^{{K}_{2}}\ue8a0\left({a}_{k}\right)\ue89e\stackrel{\Delta}{=}\ue89e{\mathrm{MSM}}_{{K}_{1}}^{{K}_{2}}\ue8a0\left({a}_{k}\right)\mathrm{SI}\ue8a0\left({a}_{k}\right)& \left(5\right)\end{array}$

[0053]
Because the system on which the SISO is defined is an FSM, the combining and marginalization operations in (2)(3) can be computed efficiently. The traditional approach is the forwardbackward algorithm which computes the MSM of the states recursively forward and backward in time. Specifically, for the standard fixedinterval algorithm based on softin for transitions t
_{k}, k=0,1, . . . , N−1, the following recursion based on addcompareselect (ACS) operations results:
$\begin{array}{cc}{f}_{k}\ue8a0\left({s}_{k}+1\right)\ue89e\stackrel{\Delta}{=}\ue89e{\mathrm{MSM}}_{0}^{k}\ue8a0\left({s}_{k}+1\right)=\underset{{t}_{k}:{s}_{k+1}}{\mathrm{min}}\ue89e\left[{f}_{k}1\ue89e\left({s}_{k}\right)+{M}_{k}\ue8a0\left({t}_{k}\right)\right]& \left(6\right)\\ {b}_{k}\ue8a0\left({s}_{k}\right)\ue89e\stackrel{\Delta}{=}\ue89e{\mathrm{MSM}}_{k}^{N1}\ue8a0\left({s}_{k}\right)=\underset{{t}_{k}:{s}_{k}}{\mathrm{min}}\ue89e\left[{b}_{k}+1\ue89e\left({s}_{k}+1\right)+{M}_{k}\ue8a0\left({t}_{k}\right)\right]& \left(7\right)\end{array}$

[0054]
where f
_{−1 }(s
_{0}) and bN(S
_{N}) are initialized according to available edge information. Note that, since there are S possible values for the state, these state metrics can be viewed as (S×1) vectors f
_{k }and b
_{k}. The final softoutputs in (4)(5) are obtained by marginalizing over the MSM of the transitions t
_{k}
$\begin{array}{cc}{\mathrm{SO}}_{0}^{N1}\ue8a0\left({u}_{k}\right)=\underset{{t}_{k}:{u}_{k}}{\mathrm{min}}\ue89e\left[{f}_{k}1\ue89e\left({s}_{k}\right)+{M}_{k}\ue8a0\left({t}_{k}\right)+{b}_{k+1}\ue8a0\left({s}_{k+1}\right)\right]\mathrm{SI}\ue8a0\left({u}_{k}\right)& \left(8\right)\end{array}$

[0055]
where u_{k }is either x_{k }or a_{k}. The operation in equation (8) is referred to as a completion operation.

[0056]
While the forwardbackward algorithm is computationally efficient, straightforward implementations of it have large latency (i.e., O(N)) due to ACS bottleneck in computing the causal and anticausal state MSMs.

[0057]
Prefix and Suffix Operations

[0058]
A prefix operation is defined as a generic form of computation that takes in n inputs y_{0}, y_{1}, . . . , y_{n−1 }and produces n outputs z_{0}, z_{1}, . . . , z_{n−1 }according to the following:

z _{0} =y _{0} (9)

z _{i} =y _{0} ⊕ . . . ⊕y _{i}, (10)

[0059]
where ⊕ is any associative binary operator.

[0060]
Similarly, a suffix operation can be defined as a generic form of computation that takes in n inputs y_{0}, y_{1}, . . . , y_{n−1 }and produces n outputs z_{0}, z_{1}, . . . , z_{n−1 }according to

z _{n−1} =y _{n−1} (11)

z _{i} =y _{i} ⊕ . . . ⊕y _{n−1}, (12)

[0061]
where ⊕ is any associative binary operator. Notice that a suffix operation is simply a (backward) prefix operation anchored at the other edge.

[0062]
Prefix and suffix operations are important since they enable a class of algorithms that can be implemented with low latency using treestructured architectures. The most notable realizations of this concept are VLSI Nbit tree adders with latency O(lgN).

[0063]
Reformulation of the SISO Operation

[0064]
The proposed lowlatency architecture is derived by formulating the SISO computations in terms of a combination of prefix and suffix operations. To obtain this formulation, define C(s
_{k},s
_{m}), for m>k, as the MSM of state pairs s
_{k }and s
_{m }based on the softinputs between them, i.e., C(s
_{k},s
_{m})=MSM
_{k} ^{m−1 }(s
_{k},s
_{m}). The set of MSMs C(s
_{k},s
_{m}) can be considered an (S×S) matrix C(k,m). The causal state MSMs f
_{k−1 }can be obtained from C(0,k) by marginalizing (e.g., minimizing) out the condition on s
_{0}. The backward state metrics can be obtained in a similar fashion. Specifically, for each conditional value of s
_{k}
$\begin{array}{cc}{f}_{k1}\ue8a0\left({s}_{k}\right)=\underset{{s}_{0}}{\mathrm{min}}\ue89eC\ue8a0\left({s}_{o},{s}_{k}\right)& \left(13\right)\\ {b}_{k}\ue8a0\left({s}_{k}\right)=\underset{{s}_{N}}{\mathrm{min}}\ue89eC\ue8a0\left({s}_{k},{s}_{N}\right)& \left(14\right)\end{array}$

[0065]
With this observation, one key step of the algorithm is to compute C(O,k) and C(k, N) for k=0, 1, . . . , N−1. Note that the inputs of the algorithm are the onestep transition metrics which can be written as C(k, k+1) for k=0, 1, . . . , N−1. To show how this algorithm can be implemented with a prefix and suffix computation, a minsum fusion operator on C matrices is defined that inputs two such matrices, one with a leftedge coinciding with the rightedge of the other, and marginalizes out the midpoint to obtain a pairwise stateMSM with larger span. Specifically, given C(k
_{0},m) and C(m,k
_{1}) , we define a C Fusion Operator, or ⊕c operator is defined by
$\begin{array}{cc}C\ue8a0\left({k}_{0},{k}_{1}\right)\ue89e\stackrel{\Delta}{=}\ue89eC\ue8a0\left({k}_{0},m\right)\ue89e{\otimes}_{C}\ue89eC\ue8a0\left(m,{k}_{1}\right)\iff C\ue8a0\left({s}_{{k}_{0}},{s}_{{k}_{1}}\right)=\underset{{s}_{m}}{\mathrm{min}}\ue89e\left[C\ue8a0\left({s}_{{k}_{0}},{s}_{m}\right)+C\ue8a0\left({s}_{m},{s}_{{k}_{1}}\right)\right]\ue89e\forall {s}_{{k}_{0}},{s}_{{k}_{1}}& \left(15\right)\end{array}$

[0066]
Note that the ⊕c operator is an associative binary operator that accepts two matrices and returns one matrix. This is illustrated in FIG. 1. With this definition C(0,k) and C(k, N) for k=0, 1, . . . , N−1 can be computed using the prefix and suffix operations as follows:

C(0,k)=C(0,1)⊕_{C} C(1,2)⊕_{C }. . . ⊕_{C} C(k−1,k)

C(k,N)=C(k,k+1)⊕_{C } . . . C(N−2,N−1)⊕_{C} C(N−1,N)

[0067]
In general, a SISO algorithm can be based on the decoupling property of stateconditioning. Specifically, conditioning on all possible FSM state values at time k, the shortest path problems (e.g., MSM computation) on either side of this state condition may be solved independently and then fused together (e.g., as performed by the cfusion operator). More generally, the SISO operation can be decoupled based on a partition of the observation interval with each subinterval processed independently and then fused together. For example, the forwardbackward algorithm is based on a partition to the singletransition level with the fusing taking place sequentially in the forward and backward directions. In contrast, other SISO algorithms may be defined by specifying the partition and a schedule for fusing together the solutions to the subproblems. This may be viewed as specifying an association scheme to the above prefixsuffix operations (i.e., grouping with parentheses).

[0068]
The Cfusion operations may be simplified in some cases depending on the association scheme. For example, the forwardbackward algorithm replaces all Cfusion operations by the much simpler forward and backward ACSs. However, latency is also a function of the association scheme. An architecture based on a pairwise treestructured grouping is presented below. This structure allows only a small subset of the Cfusion operations to be simplified, but facilitates a significant reduction in latency compared to the forwardbackward algorithm, by fusing solutions to the subproblems in a parallel, instead of sequential, manner.

[0069]
LowLatency TreeSISO Architectures

[0070]
Many lowlatency parallel architectures based on binary treestructured groupings of prefix operations can be adopted to SISOs such as described in Brent, et al., “A regular layout for parallel adders,” IEEE Transactions on Computers, vol. C31, pp. 260264, March 1982; T. H. Cormen, et al., Introduction to Algorithms. Cambridge, Mass.: The MIT Press, 1990; and A. E. Despain, “Chapter 2: Notes on computer architecture for high performance,” in New Computer Architectures, Academic Press, 1984. All of these have targeted nbit adder design where the binary associative operator is a simple 1bit addition. The present inventors were the first to recognize that parallel prefixsuffix architectures could be applied to an algorithm based on binary associative operators that are substantially more complex than 1bit addition. Conventional parallel prefix architectures trade reduced area for higher latency and account for a secondary restriction of limited fanout of each computational module. This latter restriction is important when the computational modules are small and have delay comparable to the delay of wires and buffers (e.g., in adder design). The fusion operators, however, are relatively large. Consequently, given current VLSI trends, it is believed that they will dominate the overall delay for the near future. Thus, an architecture described herein minimizes latency with the minimal number of computational modules without regard to fanout.

[0071]
Specifically, the forward and backward metrics, f_{k−1 }and b_{n−k}, for k=1, 2, . . . , N can be obtained using a hierarchal treestructure based on the fusionmodule (FM) array shown in FIG. 2. A complete set of C matrices on the interval {k_{0}, k_{0}+K} as the 2K−1 matrices C(k_{0},k_{0}+m) and C(k_{0}+m,k_{0}+K) for m 1,2, . . . , K−1 along with C(k_{0},k_{0}+K). This is the MSM information for all state pairs on the span of K steps in the trellis with one state being either on the left or right edge of the interval. The module in FIG. 2 fuses the complete sets of C matrices for two adjacent spanK intervals to produce a complete set of C matrices on the combined span of size 2K. Of the 4K−1 output C matrices, 2K are obtained from the 2(2K−1) inputs without any processing. The other 2K−1 output C matrices are obtained by 2K−1 C Fusion Modules, or CFMs, which implement the ⊕c operator.

[0072]
The basic spanK to span2K FM array shown in FIG. 2 can be utilized to compute the C matrices on the entire interval in lgN stages. This is illustrated in FIG. 3 for the special case of N=16. Note that, indexing the stages from left to right (i.e., increasing span) as i=1, 2, . . . , n=lgN it is clear that there are 2^{n−i }FM arrays in stage i.

[0073]
Because the final objective is to compute the causal and anticausal state metrics, however, not all FM's need be CFMs for all FM arrays. Specifically, the forward state metrics f
_{k−1 }can be obtained from f
_{m−1 }and C(m,k) via
$\begin{array}{cc}{f}_{k1}\ue8a0\left({s}_{k}\right)=\underset{{s}_{m}}{\mathrm{min}}\ue89e\left[{f}_{m1}\ue8a0\left({s}_{m}\right)+C\ue8a0\left({s}_{m},{s}_{k}\right)\right]& \left(16\right)\end{array}$

[0074]
Similarly, the backward state metrics can be updated via
$\begin{array}{cc}{b}_{k}\ue8a0\left({s}_{k}\right)=\underset{{s}_{m}}{\mathrm{min}}\ue89e\left[{b}_{m}\ue8a0\left({s}_{m}\right)+C\ue8a0\left({s}_{k},{s}_{m}\right)\right]& \left(17\right)\end{array}$

[0075]
A processing module that produces an f vector from another f vector and a C matrix, as described in equation (16), is referred to as an f Fusion Module (fFM). A b Fusion Module (bFM) is defined analogously according to the operation in equation (17). FIG. 3 indicates which FM's may be implemented as fFMs or bFMs.

[0076]
The importance of this development is that the calculation of the state metrics has O(lgN) latency. This is because the only data dependencies are from one stage to the next and thus all FM arrays within a stage and all FMs within an FM array can be executed in parallel, each taking O(1) latency. The cost of this low latency is the need for relatively large amounts of area. One mitigating factor is that, because the stages of the tree operate in sequence, hardware can be shared between stages. Thus, the stage that requires the most hardware dictates the total hardware needed. A rough estimate of this is N CFMs, each of which involves S^{2 }Sway ACS units with the associated registers. A more detailed analysis that accounts for the use of bFMs and fFMs whenever possible is set out below in the section heading “Hardware Resource Requirements”. For the example in FIG. 3 and a 4state FSM (i.e., S=4), stage 2 has the most CFMs (8), but stage 3 has the most processing complexity. In particular, the complexity of stages i=1,2,3,4 measured in terms of 4 4way ACS units is 26, 36, 32, and 16, respectively. Thus, if hardware is shared between stages, a total of 36 sets of 4 4way ACS units is required to execute all FMs in a given stage in parallel. For applications when this number of ACS units is prohibitive, one can reduce the hardware requirements by as much as a factor of S with a corresponding linear increase in latency.

[0077]
The implementation of the completion operation defined in equation (8) should also be considered. The basic operation required is a Qway ACS unit where Q is the number of transitions consistent with u_{k}. Assuming that at most half of the transitions will be consistent with u_{k}, Q is upper bounded by S^{2}/2. Consequently, when S is large, lowlatency, areaefficient implementations of the completion step may become an important issue. Fortunately, numerous lowlatency implementations are wellknown (e.g., such as described in P. J. Black, Algorithms and Architectures for High Speed Viterbi Decoding. PhD thesis, Stanford University, California, March 1993.). One straightforward implementation may be one that uses a binary tree of comparators and has latency of O(lgS^{2}) . For small S, this additional latency generally is not significant.

[0078]
The computational complexity of the state metric calculations can be computed using simple expressions based on FIGS. 2 and 3. As described below in the section heading “Computation Complexity Analysis”, the total number of computations, measured in units of S Sway ACS computations is

N _{S,S} =N((lgN−3)S+2)+4S−2 (18)

[0079]
For the example in FIG. 3 and a 4state FSM, an equivalent of 110 sets of 4 4way ACS operations are performed. This is to be compared with the corresponding forwardbackward algorithm which would perform 2N=32 such operations and have baseline architectures with four times the latency. In general, note that for a reduction in latency from N to lgN, the computation is increased by a factor of roughly (½)(lgN−3)S+1. Thus, while the associated complexity is high, the complexity scaling is sublinear in N.

[0080]
Optimizations for Sparse Trellises

[0081]
In general, the above architecture is most efficient for fullyconnected trellises. For sparser trellis structures, however, the initial processing modules must process Cmatrices containing elements set to ∝, accounting for MSMs of pairs of states between which there is no sequence of transitions, thereby wasting processing power and latency. Alternative embodiments may incorporate optimizations that address this inefficiency.

[0082]
Consider as a baseline a standard 1step trellis with S=M^{L }states and exactly M transitions into and out of each state, in which, there exists exactly one sequence of transitions to go from a given state at time s_{k }to a given state s_{k+L}. One optimization is to precollapse the onestep trellis into an Rstep trellis, 1≦R≦L, and apply the treeSISO architecture to the collapsed trellis. A second optimization is to, wherever possible, simplify the C fusion modules. In particular, for a SISO on an Rstep trellis, the first lg(L/R) stages can be simplified to banks of additions that simply add incoming pairs of multistep transition metrics.

[0083]
More precisely, precollapsing involves adding the R metrics of the 1step transitions that constitute the transition metrics of each supertransition t
_{kR} ^{(k+1)R}, for k=0,1, . . . , (N−1)/R. The SISO accepts these inputs and produces forward and backward MSMs, f
_{kR−1}(s
_{k}) and b
_{(k+1)R}(s
_{(k+1)R}), for k=0, 1, . . . , N/R. One key benefit of precollapsing is that the number of SISO inputs is reduced by a factor of R, thereby reducing the number of stages required in the state metric computation by lgR. One potential disadvantage of precollapsing is that the desired softoutputs must be computed using a more complex, generalized completion operation. Namely,
$\begin{array}{cc}\begin{array}{c}{\mathrm{SO}}_{O}^{\mathrm{NR}1}\ue8a0\left({u}_{\mathrm{kR}+m}\right)=\text{\hspace{1em}}\ue89e\underset{{t}_{\mathrm{kR}}^{\left(k+1\right)\ue89eR}:{u}_{\mathrm{kR}+m}}{\mathrm{min}}\ue89e[{f}_{\mathrm{kR}1}\ue8a0\left({s}_{k}\right)+{M}_{\mathrm{kR}}^{\left(k+1\right)\ue89eR}\ue89e\left({t}_{\mathrm{kR}}^{\left(k+1\right)\ue89eR}\right)+\\ \text{\hspace{1em}}\ue89e{b}_{\left(k+1\right)\ue89eR+1}\ue8a0\left({S}_{\left(k+1\right)\ue89eR}\right)]\mathrm{SI}\ue8a0\left({u}_{\mathrm{kR}+m}\right)\\ m=\text{\hspace{1em}}\ue89e0,1,\dots \ue89e\text{\hspace{1em}},R1\end{array}& \left(19\right)\end{array}$

[0084]
One principal issue is that for each u_{kR+m }this completion step involves an (M^{L+R}/2)way ACS rather than the (M^{L+1}/2)way ACS required for the 1step trellis.

[0085]
In order to identify the optimal R (i.e., for minimum latency) assuming both these optimizations are performed, the relative latencies of the constituent operations are needed. While exact latencies are dependent on implementation details, rough estimates may still yield insightful results. In particular, assuming that both the precollapsing additions and ACS operations for the state metric and completion operations are implemented using binary trees of adders/comparators, and their estimated delay is logarithmic in the number of their inputs. One important observation is that the precollapsing along with lgR simplified stages together add L 1step transition metrics (producing the transition metrics for a fullyconnected Lstep trellis) and thus can jointly be implemented in an estimated lgL time units. In addition, the state metric (M^{L})way ACS units take lgM^{L }time units and the completion units (M^{L+R}/2)way ACSs take lg(M^{L+R}/2) time units. Assuming maximal parallelism, this yields a total latency of

lgL+lg(N/R)lg(M ^{L})+lg(M ^{L+R}/2) (20)

[0086]
It follows that the minimum latency occurs when R−L lgR is minimum (subject to 1≦R≦L), which occurs when R=L. This suggests that the minimumlatency architecture is one in which the trellis is precollapsed into a fullyconnected trellis and more complex completion units are used to extract the soft outputs from the periodic state metrics calculated.

[0087]
The cost of this reduced latency is the additional area required to implement the trees of adders that produce the Lstep transition metrics and the larger trees of comparators required to implement the more complex completion operations. Note, however, that this area overhead can be mitigated by sharing adders and comparators among stages of each tree and, in some cases, between trees with only marginal impact on latency.

[0088]
Use in Tiled SubWindow Schemes

[0089]
The present inventors recognized that using smaller combining windows reduces latency and improves throughput of computing the softinverse. Minimum halfwindow (MHW) algorithms are a class of SISOs in which the combing window edges K_{1 }and K_{2 }satisfy K_{1}≦max(0, k−d) and K_{2}≧min(N, k+d), for k=0, . . . , N−1—i.e., for every point k away from the edge of the observation window, the softoutput is based on a subwindow with left and right edges at least d points from k.

[0090]
The traditional forwardbackward algorithm can be used on subwindows to obtain a MHWSISO. One particular scheme is the tiled subwindow technique in which combining windows of length 2d+h are used to derive all state metrics. In this scheme, as illustrated in FIG. 4, the windows are tiled with overlap of length 2d and there are
$\frac{N2\ue89ed}{h}$

[0091]
such windows. The forwardbackward recursion on each interior subwindow yields h soft outputs, so there is an overlap penalty which increases as h decreases.

[0092]
For the i
^{th }such window, the forward and backward state metrics are computed using the recursions, modified from that of equations (6) and (7):
$\begin{array}{cc}{f}_{k}^{\left(t\right)}\ue8a0\left({s}_{k+1}\right)\ue89e\stackrel{\Delta}{=}\ue89e{\mathrm{MSM}}_{\mathrm{ih}}^{k}\ue8a0\left({s}_{k+1}\right)& \left(21\right)\\ {b}_{k}^{\left(i\right)}\ue8a0\left({s}_{k}\right)\ue89e\stackrel{\Delta}{=}\ue89e{\mathrm{MSM}}_{k}^{\mathrm{th}+2\ue89ed+h1}\ue8a0\left({s}_{k}\right)& \left(22\right)\end{array}$

[0093]
If all windows are processed in parallel, this architecture yields a latency of O(d+h).

[0094]
The treeSISO algorithm can be used in a MHW scheme without any overlap penalty and with O(lgd) latency. Consider N/d combining windows of size d and let the treeSISO compute C(id, id+j) and C((i+1)d, (i+1)d−j) for j=0, . . . , d−1 and i=0, . . . , N/d−1. Then, use one additional stage of logic to compute the forward and backward state metrics for all k time indices that fall within the i
^{th }window, i=0, . . . , N/d−1, as follows (this should be interpreted with C(s
_{d}, s
_{0}) replaced by initial leftedge information and similarly for C(s
_{N−1}, S
_{N+d−1}):
$\begin{array}{c}{f}_{k}^{\left(i\right)}\ue8a0\left({s}_{k+1}\right)\ue89e\stackrel{\Delta}{=}\ue89e\text{\hspace{1em}}\ue89e{\mathrm{MSM}}_{\left(i1\right)\ue89ed}^{k}\ue8a0\left({s}_{k+1}\right)\\ =\text{\hspace{1em}}\ue89e\underset{{s}_{\mathrm{id}}}{\mathrm{min}}\ue89e\left\{\left[\underset{{s}_{\left(i1\right)\ue89ed}}{\mathrm{min}}\ue89eC\ue8a0\left({s}_{\left(i1\right)\ue89ed},{s}_{\mathrm{id}}\right)\right]+C\ue8a0\left({s}_{\mathrm{id}},{s}_{k+1}\right)\right\}\end{array}$ $\begin{array}{c}{b}_{k}^{\left(i\right)}\ue8a0\left({s}_{k}\right)\ue89e\stackrel{\Delta}{=}\ue89e\text{\hspace{1em}}\ue89e{\mathrm{MSM}}_{k}^{\left(i1\right)\ue89ed}\ue8a0\left({s}_{k}\right)\\ =\text{\hspace{1em}}\ue89e\underset{{s}_{\mathrm{id}}}{\mathrm{min}}\ue89e\left\{C\ue8a0\left({s}_{k},{s}_{\mathrm{id}}\right)\ue8a0\left[\underset{{s}_{\left(i+1\right)\ue89ed}}{\mathrm{min}}\ue89eC\ue8a0\left({s}_{\mathrm{id}},{s}_{\left(i+1\right)\ue89ed}\right)\right]\right\}\end{array}$

[0095]
The inner minimization corresponds to a conversion from C information to f (b) information as in equations (13) and (14). The outer minimization corresponds to an fFM or bFM. The order of this minimization was chosen to minimize complexity. This is reflected in the example of this approach, shown in FIG. 5, where the last stage of each of the four treeSISOs is modified to execute the above minimizations in the proposed order. The module that does this is referred to as a 2Cfb module. This module may be viewed as a specialization of the stage 2 center CFMs in FIG. 3. The above combining of subwindow treeSISO outputs adds one additional processing stage so that the required number of stages of FMs is lg(d)+1.

[0096]
Computational Complexity Comparison

[0097]
The computational complexity of computing the state metrics using the forwardbackward tiled scheme is the number of windows times the complexity of computing the forwardbackward algorithm on each window. In terms of S Sway ACSs, this can be approximated for large N via
$\begin{array}{cc}\frac{N2\ue89ed}{h}\ue89e2\ue89e\left(d+h\right)\approx \frac{2\ue89eN}{h}\ue89e\left(d+h\right)& \left(23\right)\end{array}$

[0098]
The computational complexity of computing the state metrics using the treeSISO tiled scheme in terms of S Sway ACSs can be developed similarly and is
$\begin{array}{cc}\frac{N}{d}\ue89ed\ue89e\text{\hspace{1em}}\ue89e\mathrm{lg}\ue8a0\left(d\right)\ue89eS+2\ue89eN=N(S\ue89e\text{\hspace{1em}}\ue89e\mathrm{lg}\ue8a0\left(d\right)+2& \left(24\right)\end{array}$

[0099]
Determining which scheme has higher computational complexity depends on the relative sizes of h and d. If h is reduced, the standard forwardbackward scheme reduces in latency but increases in computational complexity because the number of overlapped windows increase. Since the tiled treeSISO architecture has no overlap penalty, as h is decreased in a tiled forwardbackward scheme, the relative computational complexity tradeoff becomes more favorable for the treeSISO approach. In fact, for
$h<\frac{2\ue89ed}{S\ue89e\text{\hspace{1em}}\ue89e\mathrm{lg}\ue89ed},$

[0100]
the computational complexities of the treeSISO are lower than for the tiled forwardbackward scheme.

[0101]
A Design Example: 4state PCCC

[0102]
The highly parallel architectures considered require large implementation area. In this example, an embodiment is described in which the area requirements are most feasible for implementation in the near future. Specifically, an iterative decoder based on 4state sparse (onestep) trellises is considered. Considering larger S will yield more impressive latency reductions for the treeSISO. This is because the latencyreduction obtained by the treeSISO architecture relative to the parallel tiled forwardbackward architecture depends on the minimum halfwindow size. One expects that good performance requires a value of d that grows with the number of states (i.e., similar to the ruleofthumb for traceback depth in the Viterbi algorithm for sparse trellises). In contrast, considering precollapsing will yield less impressive latency reductions. For example, if d=16 is required for a singlestep trellis, then an effective value of d=8 would suffice for a twostep trellis. The latency reduction factor associated with the treeSISO for the former would be approximately 4, but only {fraction (8/3)} for the latter. However, larger S and/or precollapsing yields larger implementation area and is not in keeping with our desire to realistically assess the nearterm feasibility of these algorithms.

[0103]
In particular, a standard parallel concatenated convolutional code (PCCC) with two 4state constituent codes [1], [2] is considered. Each of the recursive systematic constituent codes generates parity using the generator polynomial G(D)=(1+D^{2})/(1+D+D^{2}) with parity bits punctured to achieve an overall systematic code with rate ½.

[0104]
In order to determine the appropriate value for d to be used in the MHWSISOs, simulations were performed where each SISO used a combining window {k−d, . . . k+d} to compute the softoutput at time k. This is exactly equivalent to the SISO operation obtained by a tiled forwardbackward approach with h=1. Note that, since d is the size of all (interior) halfwindows for the simulations, any architecture based on a MHWSISO with d will perform at least as well (e.g., h=2 tiled forwardbackward, dtiled treeSISO, etc.). Simulation results are shown in FIG. 6 for an interleaver size of N=1024 with minsum marginalization and combining and ten iterations. The performance is shown for various d along with the performance of the fixedinterval (N=1024) SISO. No significant iteration gain is achieved beyond ten iterations for any of the configurations. The results indicate that d=16 yields performance near the fixedinterval case. This is consistent with the ruleofthumb of five to seven times the memory for the traceback depth in a Viterbi decoder (i.e., roughly d=7×2=14 is expected to be sufficient).

[0105]
Since the required window size is d=16, the latency improvement of a treeSISO relative to a tiled forwardbackward scheme is close to 4=16/lg(16). The computational complexity of these two approaches is similar and depends on the details of the implementation and the choice of h for the tiled forwardbackward approach. A complete fair comparison generally would require a detailed implementation of the two approaches. Below a design for the treeSISO based subwindow architecture is described.

[0106]
A factor that impacts the area of the architecture is the bitwidth of the data units. Simulation results suggest that an 8bit datapath is sufficient. Roughly speaking, a treebased architecture for this example would require 1024 sets of sixteen 4way ACS units along with associated output registers to store intermediate state metric results. Each 4way ACS unit can be implemented with an 8bit 4to1 multiplexor, four 8bit adders, six 8bit comparators, and one 8bit register. Initial VLSI designs indicate that these units require approximately 2250 transistors. Thus, this yields an estimate of 16×2250×1024≈40 Million transistors. This number or logic transistors pushes the limit of current VLSI technology but should soon be feasible. An architecture is considered in which one clock cycle is used per stage of the tree at a 200 MHz clock frequency. For d=16, each SISO operation can be performed in 6 such clock cycles (using one clock for the completion step). Moreover, assume a hardwired interleaver comprising two rows of 1024 registers with an interconnection network. Such an interleaver would be larger than existing memorybased solutions, but could have a latency of 1 clock cycle. Consequently, one iteration of the turbo decoder, consisting of two applications of the SISO, one interleaving, and one deinterleaving, requires 14 clock cycles. Assuming ten iterations, the decoding of 1024 bits would take 140 clock cycles, or a latency of just 700 ns.

[0107]
This latency also implies a very high throughput which can further be improved with standard pipelining techniques. In particular, a nonpipelined implementation has an estimated throughput of 1024 bits per 700 ns=1.5 Gb/second. Using the treeSISO architecture one could also pipeline across interleaver blocks as described by Masera et al., “VLSI architectures for turbo codes,” IEEE Transactions on VLSI, vol. 7, September 1999. In particular, 20 such tiled treeSISOs and associated interleavers can be used to achieve a factor of 20 in increased throughput, yielding a throughput of 30 Gb/second.

[0108]
Moreover, unlike architectures based on the forwardbackward algorithm, the treeSISO can easily be internally pipelined, yielding even higher throughputs with linear hardware scaling. In particular, if dedicated hardware is used for each stage of the treeSISO, pipelining the treeSISO internally may yield another factor of lg(d) in throughput, with no increase in latency. For window sizes of d=16, the treebased architecture could support over 120 Gb/second. That said, it is important to realize that with current technology such hardware costs may be beyond practical limits. Given the continued increasing densities of VLSI technology, however, even such aggressive architectures may become costeffective in the future.

[0109]
Computation Complexity Analysis

[0110]
The number of required stages is n=lgN, with 2
^{n−i }FM arrays in stage i. Each of these FM arrays in stage i span 2
^{i }steps in the trellis and contains of 2
^{i}−1 FMs. Thus, the total number of FMs in stage i is n
_{FM}(i)=(2
^{i}−1(2
^{n−i}. The total number of fusion operations is therefore
$\begin{array}{cc}\begin{array}{c}{N}_{F\ue89e\text{\hspace{1em}}\ue89eM}=\sum _{i=1}^{n}\ue89e\text{\hspace{1em}}\ue89e{n}_{F\ue89e\text{\hspace{1em}}\ue89eM}\ue8a0\left(i\right)\\ =\sum _{i=1}^{n}\ue89e\text{\hspace{1em}}\ue89e{2}^{n}{2}^{n}\ue89e{2}^{i}\\ =\mathrm{Nn}N\ue89e\sum _{i=1}^{n}\ue89e\text{\hspace{1em}}\ue89e{2}^{i}\\ =N\ue8a0\left(\mathrm{lg}\ue89eN1\right)+1\end{array}& \left(25\right)\end{array}$

[0111]
For the example in FIG. 3, this reduces to N_{FM}=49.

[0112]
Using FIG. 3 as an example, it can be seen that the number of FMs that can be implemented as fFMs in stage i is n_{f}(i)=2^{i−1}. In the special case of i=n, this is interpreted as replacing the 2K−1 CFMs by K fFMs and K bFMs. For example, in the fourth stage in FIG. 3, the 15 CFMs implied by FIG. 2 may be replaced by 8 fFMs and 8 bFMs, as shown. The number of FMs that can implemented as bFMs is the same—i.e., n_{b}(i)=n_{f}(i)=2^{i−1}. It follows that the number of CFMs required at stage i is

n _{C}(i)=n _{FM}(i)−n _{b}(i)−n _{f}(i)+δ(n−i) (26)

=2^{n}−2^{n−1}−2^{1}+δ(n−i) (27)

[0113]
where δ(j) is the Kronecker delta. The total number of fusion modules is therefore
$\begin{array}{cc}{N}_{f}={N}_{b}=\sum _{i=1}^{n}\ue89e\text{\hspace{1em}}\ue89e{n}_{f}\ue8a0\left(i\right)=\sum _{i=1}^{n}\ue89e\text{\hspace{1em}}\ue89e{2}^{i1}=N1& \left(28\right)\\ \begin{array}{c}{N}_{C}=\text{\hspace{1em}}\ue89e\sum _{i=1}^{n}\ue89e\text{\hspace{1em}}\ue89e{n}_{C}\ue8a0\left(i\right)\\ =\text{\hspace{1em}}\ue89eN\ue8a0\left(n1\right)+1\left(\sum _{i=1}^{n}\ue89e\text{\hspace{1em}}\ue89e{2}^{i}\right)+1\\ =\text{\hspace{1em}}\ue89eN\ue8a0\left(\mathrm{lg}\ue89eN3\right)+4\end{array}& \left(29\right)\end{array}$

[0114]
Comparing equations (25) and (29), it is seen that, for relatively large N, the fraction of FMs that must be CFMs is (lg N−3)/(lg N−1). For smaller N, the fraction is slightly larger. For example, in FIG. 3, N_{f}=N_{b}=15 and there are 20 CFMs.

[0115]
The CFM is approximately S (i.e., the number of states) times more complex than the fFM and bFM operations. This can be seen by comparing equation (15) with equations (16) and (17). Specifically, the operations in equations (15), (16), and (17) involve Sway ACSs. For the CFM, an Sway ACS must be carried out for every possible state pair (s_{k} _{ 0 },s_{k} _{ 1 }) in (15)—i.e., S^{2 }state pairs. The Sway ACS operations in equations (16), and (17) need only be computed for each of the S states s_{k}. Thus, taking the basic unit of computation to be S Sway ACS on an Sstate trellis, the total number of these computations required for stage i is

n _{S,S}(i)=Sn _{C}(i)+n _{f}(i)+n _{b}(i) (30)

[0116]
Summing over stages, the total number of computations is obtained, measured in units of S Sway ACS computations

N _{S,S} =SN _{C}+2N _{f} =N((lg N−3)S+2)+4S−2 (31)

[0117]
For the example in FIG. 3 and a 4state FSM, an equivalent of 110 sets of S Sway ACS operations are performed. This is to be compared with the corresponding forwardbackward algorithm which would perform 2N=32 such operations and have baseline architectures with four times the latency. In general, note that the for a reduction in latency from N to lg N, the computation is increased by a factor of roughly (½)(lg N−3) S+1. Thus, while the associated complexity is high, the complexity scaling is sublinear in N. For small S, this is better than linearscale solutions to lowlatency Viterbi algorithm implementations (e.g.,such as described in P. J. Black, Algorithms and Architectures for High Speed Viterbi Decoding. PhD thesis, Stanford University, California, March 1993; and Fettweis, et al., “Parallel Viterbi algorithm implementation: Breaking the ACSbottleneck” IEEE Trans. Commun., vol. 37, pp. 785790, August 1989).

[0118]
Hardware Resource Requirements

[0119]
The maximum of n_{S,S }(i) over i is of interest because it determines the minimum hardware resource requirements to achieve the desired minimum latency. This is because the fusion modules can be shared between stages with negligible impact on latency.

[0120]
The maximum of n_{S,S }(i) can be found by considering the condition on i for which n_{S,S }(i)≧n_{S,S }(i−1). Specifically, if i<n,

n _{S,S}(i)≧n _{S,S}(i−1) (32)

<=>2^{n}≧2^{2i−1}(1−S ^{−1}) (33)

[0121]
[0121]
$\begin{array}{cc}\iff i\le \frac{n+1\mathrm{lg}\ue8a0\left(1{S}^{1}\right)}{2}& \left(34\right)\end{array}$

[0122]
It follows that n
_{S,S }(i) has no local maxima and
$\begin{array}{cc}{i}^{*}=\lfloor \frac{n+1\mathrm{lg}\ue8a0\left(1{S}^{1}\right)}{2}\rfloor & \left(35\right)\end{array}$

[0123]
can be used to find the maximizer of n
_{S,S}(i). Specifically, if equation (35) yields i*<n−1, then the maximum occurs at i*, otherwise (i*=n−1), the i=n−1 and i=n cases should be compared to determine the maximum complexity stage. For S≧4, equation (35) can be reduced to
$\begin{array}{cc}{i}^{*}=\lfloor \frac{n+1}{2}\rfloor & \left(36\right)\end{array}$
$0.5\le \frac{11\ue89eg\ue8a0\left(1{S}^{1}\right)}{2}\le 0.71$

[0124]
Other Embodiments of a TreeStructured SISO

[0125]
Generally, in implementing a treestructured SISO, any of various tree structures can be used that represent a tradeoff between latency, computational complexity and IC area according to the system designer's preferences. FIGS. 8A and 8B show an embodiment referred to as a ForwardBackward TreeSISO (FBTSISO). In the FBTSISO, the tree structure recursion is bidirectional—inward and outward. As a result, the FBTSISO has twice the latency of the TreeSISO, but it also achieves a significant decrease in computational complexity and uses less area on an integrated circuit (IC) chip.

[0126]
Details of message passing using the generalized forwardbackward schedule are shown in FIGS. 8A and 8B in minsum processing. The inward messages are shown in FIG. 8A. Specifically, initially MI[tk] is set to uniform and the algorithm begins by activating the first level of subsystems to compute Mk[tk] from MI[x(tk)] and MI[a(tk)]. The messages passed inward to the next level of the tree are MSMk[sk, sk+1] which is simply Mk[tk] if there are no parallel transitions. This inward message passing continues with the messages shown. When the two messages on s4 reach V4s, the outward propagation begins and proceeds downward as shown in FIG. 8B. Again, all nodes at a given level of the tree are activated before activating any of the nodes at the next level. At the bottom level, the input metric of (sk, sk+1) is MSM{k}c[sk, sk+1]—i.e., the sum of the forward and backward state metrics in the standard forwardbackward algorithm. Thus, the final activation of the nodes on the bottom level produces the desired extrinsic output metrics.

[0127]
This FBTSISO has twice the latency of the TreeSISO because the messages must propagate both inward and outward. This modest increase in latency is accompanied by a significant reduction in computational complexity. Specifically, the FBTTree SISO has O(K) computational complexity and O(log2K) latency. This is to be compared to O(K log2K) computational complexity and O(log2K) latency for the TreeSISO and O(K) computational complexity and O(K) latency for message passing on the standard trellis.

[0128]
Other embodiments of a treestructured SISO may be advantageous depending on the designer's goals and constraints. In general, a treestructured SISO can be constructed that uses virtually any type of tree structure. For example, advantageous results may be obtained with a treestructured SISO that uses one or more of the tree structures described in R. P. Brent and H. T. Kung, “A regular layout for parallel adders,” IEEE Transactions on Computers, C31:260264 (March 1982) (the “BrentKung tree”); A. E. Despain, “New Computer Architectures,” Chapter 2: Notes on computer architecture for high performance, Academic Press (1984); T. H. Cormen, C. Leiserson, and R. L. Rivest, “Introduction to Algorithms,” The MIT Press (1990); and/or S. M. Aji and R. J. McEliece, “The generalized distributive law,” IEEE Trans. Inform. Theory, 46(2):325343 (March 2000).

[0129]
[0129]FIG. 9 shows a representation of a BrentKung TreeSISO in which calculation of the branch metrics in a d=16 subwindow uses a modified version of the BrentKung model. There are six pipeline stages in the BrentKung TreeSISO which are indicated by the dotted lines. Nodes shaded gray represent a 2way ACS while the black nodes represent a 4way ACS.

[0130]
The BrentKung TreeSISO is based on the BrentKung parallel adder. It was created to solve the fanout problem of the prefix adders. Additionally it reduces the amount of hardware needed for the calculation of all the metrics. The fusion operation is also used here, but the amount of fusion operations needed is significantly reduced. In the BrentKung adder, instead of additions, fusion operations are performed at each node. In this form only the forward information can be calculated. The model was modified so that the calculation of both the forward and backward metrics could be performed at the same time.

[0131]
This model firstly reduces the number of units that have to be driven by the output of an ACS. In the worst case here, 4 units feed on the output of a single stage (i.e. node C0,8). This is almost half of what the previous approach needed. It is also observed that there is a significant number of operations that only need 2way ACS units. Considering that the 2way ACS unit uses less than half of the hardware needed in a 4way ACS unit, the total size of the circuitry is expected to drop considerably. An analysis similar to the one performed before, verifies this intuitive assumption. The results are summarized in FIG. 10.

[0132]
Advantages presented by this approach may be counterbalanced with a reduction in the system's performance. A close look at the block diagram in FIG. 10 indicates that the longest path for the calculations is now 6 cycles long. Adding the three extra cycles needed for the forward and backward metric calculation, the completion step and the interleaving process brings the total delay of the SISO module to 9 cycles. This means that for the whole decoding process takes 180 cycles. Assuming that the same cells are used for implementation, this yields a total time of 540 nsec for processing one block of 1024 bits of data. The throughput of the system in this case will be 1.896 Gbps. If the unfolded design pipelined after each ACS step is used, that would allow one block to be processed after each step, and a throughput of 16.82 Gbps could be realized. If 20 SISOs were used to unfold the loop, the throughput would be 341.32 Gbps. For the first case, which is of interest due to the size of the design, the performance degradation is only about 24.1%. At the same time the number of transistors needed to implement this design is reduced by almost 43.8%.

[0133]
Decoding Block Codes and Concatenated Blocks with the TreeStructure SISO

[0134]
There are other relevant decoding problems in which the conventional solution is to run the standard forwardbackward algorithm on a trellis. The methods and techniques described herein also can be applied in place of the forwardbackward algorithm in order to reduce the processing latency in these applications. One example of such an application is computing the soft inversion of a block error correction code using the trellis representation of the code. A method for decoding a block code using a trellis representation was described in Robert J. McEliece, “On the BCJR Trellis for Linear Block Codes,” IEEE Transactions on Information Theory,” vol. 42, No. 4, July 1996. In that paper, a method for constructing a trellis representation based on the paritycheck structure of the code is described. Decoding, or more generally softinversion of the block code, can be accomplished by running the forwardbackward algorithm on this trellis. The methods and techniques described herein can be applied to determine the softinversion of a block code in a manner similar to the treestructured methods for softinversion of an FSM. Specifically, the forward and backward state metrics can be computed to find the softinversion of a block code by using a parallel prefix computational architecture with treestructure.

[0135]
The application of the treeSISO architecture to implementing the softinverse of a block code via the parity check structure is significant as a practical matter because several attractive turbolike codes are based on the concatenation of simple parity check codes. These codes are often referred to as “turbolike codes” since they have similar performance and are decoded using iterative algorithms in which softinverse nodes accept, update and exchange softinformation on the coded bits. Specific examples of turbolike codes include: LowDensity Parity Check Codes (LDPCs) also known as Gallager codes and product codes (also known as turboblock codes), which are described in J. Hagenauer, E. Offer, and L. Papke, “Iterative decoding of binary block and convolutional codes,” IEEE Transactions on Information Theory, 42(2):429445, March 1996, and R. G. Gallager, “LowDensity ParityCheck Codes,” MIT Press, Cambridge, Mass., 1963.

[0136]
As a specific example, the parity check trellis for a standard Hamming(7,4) code is shown in FIG. 11. The forwardbackward algorithm may be run on this trellis with latency on the order of 7. Alternatively, a treeSISO architecture can be used resulting in the computational structure shown in FIG. 12. As in the case of FSM softinversion, the treestructured computation yields the same forward and backward state metrics as the conventional solution method (i.e., running the standard forwardbackward algorithm on a trellis), but has logarithmic latency (roughly ½ the latency in this simple example). In general, for a block code with length N, the conventional approach based on the forwardbackward algorithm will have latency O(N), whereas the same computation can be computed using the treeSISO with latency O(lg N).

[0137]
As a specific example of the use for a treestructured parity check node SIS0 such as the one illustrated in FIG. 12, consider the iterative decoder for an LDPC, as illustrated in FIG. 13. This decoder has two basic types of softinverse processing nodes: the soft broadcaster nodes and the paritycheck SISO nodes. In one embodiment, the decoder executes by activating the softbroadcaster nodes in parallel, followed by permuting the generated softinformation which becomes the input to the parity check SISO nodes. These nodes are activated producing updated beliefs on the coded bit values that are passed back through the permutation and become the input to the softbroadcaster. This may be viewed as a single iteration. Several iterations are required before the decoding procedure is complete. As in turbo decoding, the stopping criterion is part of the design, with a common choice being a fixed number of iterations. The computation shown in FIG. 11 is performed in each parity check SISO node in a preferred embodiment. Alternatively, the treestructured architecture can be used for the computation at each parity check SISO node. In the most common form, the parity check nodes in LDPCs are singleparity checks, and therefore have a twostate trellisrepresentation.

[0138]
Conclusion

[0139]
Based on the interpretation of the SISO operation in terms of parallel prefix/suffix operations, a family of treestructured architectures are described above. Compared to the baseline forwardbackward algorithm architecture, the treeSISO architecture reduces latency from O(N) to O(lgN). Alternative treestructured SISOS, such as the FBTSISO, trade a linear increase in latency for substantially lower complexity and area.

[0140]
An efficient SISO design generally is not built using a single treeSISO, but rather using treeSISOs as important components. For example, as described above, many treeSISOs were used to comprise a SISO using tiled subwindows. Latency in that example reduced from linear in the minimum halfwindow size (d) for fullyparallel tiled architectures based on the forwardbackward algorithm, to logarithmic in d for tiled treeSISOs.

[0141]
In general, potential latency advantages of the treeSISO are clearly most significant for applications requiring large combining windows. For most practical designs, this is expected when the number of states increases. In the one detailed 4state tiledwindow example considered, the latency was reduced by a factor of approximately 4. For systems with binary inputs and S states, one would expect that d≅8 lg(S) would be sufficient. Thus, there is a potential reduction in latency of approximately 8 lg(S)/lg(8 lg S) which becomes quite significant as S increases. However, the major challenge in achieving this potential latency improvement is the area required for the implementation. In particular, building a highspeed Sway ACS unit for large S is the key challenge. Techniques to reduce this area requirement without incurring performance degradations (e.g., bitserial architectures) are promising areas of research. In fact, facilitating larger S may allow the use of smaller interleavers, which alleviates the area requirements and reduces latency.

[0142]
Various implementations of the systems and techniques described here may be realized in digital electronic circuitry, integrated circuitry, specially designed ASICs (application specific integrated circuits) or in computer hardware, firmware, software, or combinations thereof.

[0143]
A number of embodiments of the present invention have been described. Nevertheless, it will be understood that various modifications may be made without departing from the spirit and scope of the invention. Accordingly, other embodiments are within the scope of the following claims.