LDPC code and encoder/decoder regarding same

[0001] This application claims the benefit of U.S. Provisional Application No. 60/413,373, filed Sep. 25, 2002, which is incorporated herein by reference in its entirety.

[0002] This invention was made with Government support under Grant No. DA/DAAG55-98-1-0315 and DA/DAAD19-01-1-0705, Disclosure Z02232, awarded by the Army Research Office. The Government has certain rights in this invention.

[0003] The present invention relates to data transmission. More particularly, it relates to error-correcting codes for data transmission.

[0004] Codes on graphs have become a topic of great current interest in the coding the-ory community. The prime examples of codes on graphs are low-density parity-check (LDPC) codes that have been widely considered as next-generation error-correcting codes for many real world applications in digital communication and magnetic storage. However, because of their distinct properties, LDPC codes decoder/encoder design and implementation are not trivial and design of these codes stills remain a challenging task. How well one attacks this problem directly determines the extent of LDPC application in the real world. As a class of LDPC codes, (3,k)-regular LDPC codes can achieve very good performance and hence are considered in this invention.

[0005] The main challenge when implementing the message passing algorithm for decoding LDPC codes is managing the passing of the messages. The realization of the message passing bandwidth results in very different and difficult challenges depending on whether all the messages are passed in fully parallel or partly parallel manner. By fully exploiting the parallelism of the message passing decoding algorithm, a fully parallel decoder can achieve very high decoding throughput but suffers from prohibitive implementation complexity (see, e.g., A. J. Blanksby and C. J. Howland, “A 690-mW 1-Gb/s 1024-b, rate-½ low-density parity-check code decoder”, IEEE Journal of Solid-State Circuits, vol. 37, pp. 404–412 (March, 2002)). Furthermore, the large number of interconnection may limit the speed performance and increase the power dissipation. Thus the fully parallel design strategy is only suitable to short code length scenarios.

[0006] In partly parallel decoding, the computations associated with a certain number of variable nodes or check nodes are time-multiplexed to a single processor. Meanwhile, since the computation associated with each node is not complicated, the fully parallel interconnection network should be correspondingly transformed to partly parallel ones to achieve both the communication complexity reduction and high-speed partly parallel decoding. Unfortunately, the randomness of the Tanner graph makes it nearly impossible to develop such a transformation. In other words, an arbitrary random LDPC code has little chance to be suited for high-speed partly parallel decoder hardware implementation.

[0007] Furthermore, to perform LDPC encoding, the generator matrix is typically used, which has quadratic complexity in the block length. How to reduce the encoding complexity for the practical coding system implementation is another crucial issue.

[0008] What is needed is new joint code-encoder-decoder design methodology and techniques for designing practical LDPC coding system, that overcomes the limitations of the conventional code first scheme and/or designs.

[0009] The present invention provides a joint code and decoder design approach to construct good (3,k)-regular LDPC codes that exactly fit to partly parallel decoder and efficient encoder implementations. A highly regular partly parallel decoder architecture design is developed. A systematic efficient encoding scheme is presented to significantly reduce the encoder implementation complexity.

[0010] In accordance with the present invention, LDPC coding system is designed using a joint code-encoder-decoder methodology. First a method is developed to explicitly construct a high-girth (girth is the length of a shortest cycle in a graph) (2,k)-regular LDPC code that exactly fits to a high-speed partly parallel decoder. Then this (2,k)-regular LDPC decoder is extended to a (3,k)-regular LDPC partly parallel decoder that is configured by a set of constrained random parameters. This decoder defines a (3,k)-regular LDPC code ensemble from which a good (3,k)-regular LDPC code can be selected based on the criterion of fewer short cycles and computer simulations. Due to certain unique structure properties of such (3,k)-regular LDPC codes, an efficient systematic encoding scheme is developed to reduce the encoding complexity. Since each code in such a code ensemble is actually constructed by randomly inserting certain check nodes into the deterministic high-girth (2,k)-regular LDPC code under the constraint specified by the decoder, the codes in this ensemble more likely do not contain too many short cycles and hence a good code can be easily selected from these codes.

[0011] The (3,k)-regular LDPC code-encoder-decoder design of the present invention can be used to design and implement LDPC coding system for a wide variety real-world applications that require excellent error-correcting performance and high decoding speed/low power consumption, such as deep space and satellite communication, optical links, and magnetic or holographic storage systems, etc. These codes can also be used for fixed and mobile wireless systems, ultra wide-band systems for personal area networks and other applications, and wireless local area networks containing wireless receivers with one or more antennas.

[0012] It is an advantage of the joint code-encoder-decoder design of the present invention that it effectively exploits the LDPC code construction flexibility to improve the overall LDPC coding system implementation performance.

[0013] Further embodiments, features, and advantages of the present invention, as well as the structure and operation of the various embodiments of the present invention are described in detail below with reference to accompanying drawings.

[0022] An LDPC code is defined as the null space of a very sparse M×N parity check matrix, and is typically represented by a bipartite graph (a bipartite graph is one in which the nodes can be partitioned into two disjoint sets) usually called a Tanner graph, between N variable (or message) nodes in one set and M check (or constraint) nodes in another set. An LDPC code is called (j, k)-regular if each variable node has a degree of j and each check node has a degree of k. The construction of an LDPC code (or its Tanner graph) is typically random. LDPC codes can be effectively decoded by the iterative message passing. The structure of the message passing decoding algorithm directly matches the Tanner graph: decoding messages are iteratively computed by all the variable nodes and check nodes and exchanged through the edges between the neighboring nodes. It is well known that the message passing decoding algorithm works well if the underlying Tanner graph does not contain too many short cycles. Thus, the random Tanner graph is typically restricted to be 4-cycle free, which is easy to achieve. However, the construction of random Tanner graphs free of higher order cycles, e.g., 6 and 8, is not trivial.

[0023] Before presenting the message passing algorithm, some definitions should be introduced first: Let H denote the M×N parity check matrix and H_i,j denote the entry of H at the position (i, j). Define the set of bits n that participate in parity check m as N(m)={n:H_m,n=1}, and the set of parity checks m in which bit n participates as M(n)={m:H_m,n=1}. Let N(m)\n denote the set N(m) with bit n excluded, and M(n)\m denote the set M(n) with parity check m excluded.

[0024] Iterative message passing Decoding Algorithm

• [0025] Input: The channel observations p_n⁰=P(x_n=0) and p_n¹=P(x_n=1) =1−p_n⁰, n=1, . . . , N;
• [0026] Output: Hard decision {circumflex over (x)}={{circumflex over (x)}₁, . . . , {circumflex over (x)}_N};
• [0027] Procedure:
  • [0028] (a) Initialization: For each n, compute the channel message

[0029] ${\gamma }_n=\log \frac{p_n^0}{p_n^1}$
and for each (m,n)∈{(i, j)|H_ij=1}, compute

[0030] ${\alpha }_{m,n}=\mathrm{sign}\left({\gamma }_n\right)\log \left(\tanh \left({\gamma }_n2\right)\right),\mathrm{where}\mathrm{sign}\left({\gamma }_n\right)=\{\begin{array}{cc}+1& {\gamma }_n\ge 0\\ -1& {\gamma }_n<0\end{array}.$

• • [0031] (b) Iterative Decoding
    • [0032] Horizontal (or check node processing) step: For each (m, n)∈{(i, j)|H_i,j=1}, compute

[0033] $\begin{array}{cc}{\beta }_{m,n}=\log \left(\tanh \left(\alpha /2\right)\right)\prod _{n^{\prime }\in 𝒩\left(m\right)\backslash n}\mathrm{sign}\left({\alpha }_{m,n^{\prime }}\right),& \mathrm{EQ}.\left(1\right)\end{array}$

• • • [0034]  where

[0035] $\alpha =\sum _{n^{\prime }\in 𝒩\left(m\right)\backslash n}^{}{\alpha }_{m,n^{\prime }}.$

• • • [0036] Vertical (or variable node processing) step: For each (m, n)∈{(i, j)|H_i,j=1}, compute
      α_m,n=sign(γ_m,n)log(tanh(|γ_m,n|/2)),  EQ.(2)
    • [0037]  where

[0038] ${\gamma }_{m,n}={\gamma }_n+\sum _{m^{\prime }\in M\left(n\right)\backslash m}^{}{\beta }_{m^{\prime },n}.$
For each n, update the “pseudo-posterior log-likelihood ratio (LLR)” λ_n as:

[0039] $\begin{array}{cc}{\lambda }_n={\gamma }_n+\sum _{m\in M\left(n\right)}^{}{\beta }_{m,n}.& \mathrm{EQ}.\left(3\right)\end{array}$

• • • [0040] Decision step.
      • [0041] i. Perform hard decision on {λ₁, . . . , λ_N} to obtain {circumflex over (x)}={{circumflex over (x)}₁, . . . , {circumflex over (x)}_N }such that {circumflex over (x)}_n=0 if λ_n>0 and {circumflex over (x)}_n=1 if λ≦0;
      • [0042] ii. If H·{circumflex over (x)}=0, then algorithm terminates, else go to Horizontal step. A failure will be declared if pre-set maximum number of iterations occurs without successful decoding.

[0043] In the above algorithm, α_m,n and β_m,n are called variable-to-check messages and check-to-variable messages, respectively. Each check node computation is realized by a Check Node processing Unit (CNU) to compute the check-to-variable message β_m,n according to EQ. (1), each variable node computation is realized by Variable Node processing Unit (VNU) to compute the variable-to-check message α_m,n and pseudo-posterior LLR λ_n according to EQ. (2) and EQ. (3), respectively, and generate {circumflex over (x)}_n by performing hard decision on λ_n.

• • [0044] Joint(3,k)-Regular LDPC Code and Decoder/Encoder Design

[0045] It is well known that the message passing algorithm for LDPC decoding works well if the underlying Tanner graph is 4-cycle free and does not contain too many short cycles. Thus the essential objective of this joint design approach is to construct LDPC codes that not only fit to practical decoder/encoder implementations but also have large average cycle length in their 4-cycle free Tanner graphs.

[0046] Given a graph G, let g_u denote the length of the shortest cycle that passes through node u in graph G, then

[0047] $\sum _{u\in G}^{}{g}_u/N$
is denoted as girth average of G, where N=|G| is the total node number of G. Girth average can be used as an effective criterion for searching good LDPC code over one code ensemble. FIG. 1 illustrates the schematic diagram of the joint design approach which is briefly described below.

• • [0048] (a) Step 100: Explicitly construct the two matrices, H₁ and H₂, so that {tilde over (H)}=[H₁^T, H₂^T]^T defines a (2,k)-regular LDPC code denoted as C₂;
  • [0049] (b) Step 102: Obtain an elaborated (3,k)-regular LDPC decoder architecture which defines a random (3,k)-regular LDPC code ensemble and each code in this ensemble is a sub-code of C₂;
  • [0050] (c) Step 104: Use the decoder to randomly generate a certain number of (3,k)-regular LDPC codes from which one can select one code with good performance by girth average comparison and computer simulations;
  • [0051] (d) Step 106: Let H denote the parity check matrix of the selected code. Introduce an explicit column permutation π_c to generate an approximate upper triangular matrix H^en=π_c(H) based on which an efficient encoding scheme is obtained.

[0052] A method is developed to construct matrix {tilde over (H)}=[H₁^T, H₂^T]^T which defines a (2, k)-regular LDPC code with girth of 12. Such construction method leads to a very simple decoder architecture and provides more freedom on the code length selection: Given k, any code length that could be factored as L·k² is permitted, where L can not be factored as L=a·b, ∀a,b ∈{0, . . . , k−1}.

[0053] FIG. 2 shows the structures of H₁ and H₂. Each block matrix I_x,y in H₁ is an L×L identity matrix and each block matrix P_x,y in H₂ is obtained by a cyclic shift of an L×L identity matrix. Let T denote the right cyclic shift operator where Tⁱ(U) represents right cyclic shifting matrix U by i columns, then P_x,y=T^u(I) where u=((x−1)·y) mod L and I represents the L×L identity matrix, e.g., let L=5, x=3 and y=4, then u=(x−1)·y mod L=8 mod 5=3, and

[0054] $P_{3,4}=T^3\left(I\right)=\lfloor \begin{array}{ccccc}0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\\ 1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\end{array}\rfloor .$
Clearly, matrix {tilde over (H)}=[H₁^T, H₂^T]^T defines a (2, k)-regular LDPC code with L·k² variable nodes and 2L·k check nodes. Let G denote the corresponding Tanner graph. The proposed (2,k) code has the property that if L can not be factored as L=a·b, where a, b ∈{0, . . . , k−1}, then the girth of G is 12 and there is at least one 12-cycle passing each check node.

[0055] A partly parallel decoder architecture is developed for the (2,k)-regular LDPC code constructed in the above. To facilitate the following description, L·k² variable nodes of the code are arranged into k² variable node (VG) groups, each group VG_x,y (1≦x,y≦k) contains the L variable nodes corresponding to the L columns in {tilde over (H)} going through the block matrix I_x,y and P_x,y. Notice that any two variable nodes in the same group never connect to the same check node and all the k variable nodes connected to the same check node as specified by H₁ (H₂) always come from k groups with the same x-index (y-index). Based on the above observations, a partly parallel decoder architecture can be directly constructed, as shown in FIG. 3.

[0056] This decoder contains k² memory banks, denoted as MEM BANK-(x,y) for 1≦x,y≦k. MEM BANK-(x,y) stores all the p-bit channel messages in RAM (random access memory) I, q-bit variable-to-check and check-to-variable messages in RAMs E₁ and E₂ and hard decisions in RAM C associated with the L variable nodes in VG_x,y. In this decoder, the check-to-variable message β_m,n and variable-to-check message α_m,n associated with each pair of neighboring nodes alternatively occupy the same memory location. The two check-to-variable or variable-to-check messages associated with each variable node are stored in E₁ and E₂ with the same address. This decoder completes each decoding iteration in 2L clock cycles, and in each clock cycle it performs:

• • [0057] (a) In each memory bank, if all the check-to-variable messages β_m,n associated with one variable node become available after previous clock cycle, then
    • [0058] i. Retrieve 1 channel message γ_n from RAM I and 2 check-to-variable messages β_m,n associated with this variable node from RAMs E1 and E2;
    • [0059] ii. VNU computes 2 variable-to-check messages α_m,n and LLR λ_n, and obtains {circumflex over (x)}_n by performing hard decision on λ_n;
    • [0060] iii. Store the 2 variable-to-check messages α_m,n back to RAM E1 and E2 and {circumflex over (x)}_n to RAM C.
  • [0061] (b) Retrieve k² variable-to-check messages α_m,n and hard decisions {circumflex over (x)}_n (obtained in the previous iteration) from the k² memory banks at the addresses provided by AG_i's;
  • [0062] (c) Shuffle the k² variable-to-check messages and hard decision by a one layer shuffle network 300. This shuffle network is configured by the configuration bit c₋₁ leading to a fixed permutation (or re-ordering pattern), denoted by π₋₁, if c₋₁, =1, or to the identity permutation, denoted by Id, if c₋₁=0.
  • [0063] (d) Each CNU_i computes k check-to-variable messages β_m,n and performs the parity check on the k hard decisions {circumflex over (x)}_n;
  • [0064] (e) Unshuffle the k² check-to-variable messages β_m,n and store them back into the k² memory banks at the original locations.

[0065] To realize the connectivity specified by H₁ and H₂ in the 1^st and 2^nd L clock cycles, respectively, this decoder has the following features:

• • [0066] Each Address Generator (AG_x,y) provides memory address to E1 and E2 during the 1^st and 2^nd L clock cycles, respectively. Each AG_x,y is a modulo-L binary counter which is set to initial value D_x,y every L clock cycles, i.e., at r=0, L, where

[0067] $\begin{array}{cc}{D}_{x,y}=\{\begin{array}{cc}0,& r=0,\\ \left(\left(x-1\right)·y\right)\mathrm{mod}L,& r=L.\end{array}& \mathrm{EQ}.\left(4\right)\end{array}$

• • [0068] During the 1^st L clock cycles, the configuration bit c₋₁=1 and π₋₁ permute input data {x₀, . . . , x_k2−1} to {x_π−1(0), . . . , x_{π−1(k2−1)}}, where

[0069] $\begin{array}{cc}{\pi }_{-1}\left(i\right)=\left(i\mathrm{mod}k\right)·k+\lfloor \frac{i}{k}\rfloor .& \mathrm{EQ}.\left(5\right)\end{array}$

• • [0070] During the 2^nd L clock cycles, the configuration bit c₋₁=0 and the shuffle network is bypassed.

[0071] FIG. 4 illustrates the (3,k)-regular LDPC partly parallel decoder obtained by introducing a few new blocks into the (2,k)-regular LDPC decoder. Configured by a set of constrained random parameters, this decoder defines a (3,k)-regular LDPC code ensemble in which each code has L ·k² variable nodes and 3L ·k check nodes.

[0072] In this decoder, a Random Permutation Generator (RPG) 400 and a g-layer shuffle network 402 are inserted between the 1-layer shuffle network (π₋₁ or Id) and all the CNUs. FIG. 5 shows the structure of the g-layer shuffle network: each layer is a single layer shuffle network and configured by c_i leading to a given permutation π_i if c_i=1 (π_i¹), or to the identity permutation (Id=π_i⁰) otherwise. Thus, configured by the g-bit word c=(c_g−1, . . . , C₀)2 generated by RPG, the overall permutation pattern {tilde over (π)} in each clock cycle is the product of g permutations:

[0073] $\tilde{\pi }={\pi }_{g-1}^{c_{g-1}}\mathrm{•\dots •}{\pi }_0^{c_0}.$

[0074] This decoder completes each decoding iteration in 3L clock cycles. CNUs access the messages stored in E1, E2 and E3 in the 1^st, 2^nd and 3^rd L clock cycles, respectively. Denote the 2L·k check nodes associated with the messages stored in E1 and E2 as deterministic check nodes and the other L·k check nodes associated with the messages stored in E3 as random check nodes.

[0075] During the first 2L clock cycles, this decoder bypasses the g-layer shuffle network by setting the output of RPG as a zero vector, and works in the exactly same way as the (2,k)-regular LDPC decoder. In other words, this decoder realizes the connectivity specified by {tilde over (H)}[H₁^T, H₂^T]^T between all variable nodes and the 2L·k deterministic check nodes during the first 2L clock cycles.

[0076] During the last L clock cycles, this decoder realizes the connectivity between all variable nodes and the L·k random check nodes according to the following specifications.

• • [0077] RPG performs as a hash function f: {2L, . . . , 3L−1}→{0, . . . , 2^g−1} and its g-bit output vector c configures the grayer shuffle network;
  • [0078] The configuration bit c₋₁=0 so that the 1-layer shuffle network performs the identity permutation;
  • [0079] AG_x,y provides address to E3 with the counter set to D_x,y=t_x,y at r=2L, where t_x,y ∈{0, . . . , L−1}.

[0080] In order to guarantee that this code ensemble only contains 4-cycle free codes and facilitate the design process, the hash function f and the g-layer shuffle network are generated randomly and the value of each t_x,y is chosen randomly under the following constraints:

• • [0081] (a) Given x, t_x,t1≠t_x,y2, ∀y₂, y₂ ∈{1, . . . , k};
  • [0082] (b) Given y, t_x1,y−t_x2y≢((x₁−x₂)·y)mod L, ∀x₁, x₂ ∈{1, . . . , k}.
    The code ensemble defined by the proposed decoder is referred as implementation-oriented (3,k)-regular LDPC code ensemble. For real applications, a good code is selected from this code ensemble by two steps: first randomly generate a certain number of implementation-oriented (3, k)-regular LDPC codes, then pick few codes with high girth averages and finally select the one leading to the best code performance simulation results.

[0083] One unique property of this work is to use the counter for memory address generation, which largely simplifies the decoder hardware implementation complexity and improves the throughput performance compared with the previous design solution (see, E. Boutillon and J. Castura and F. R. Kschischang, “Decoder-First Code Design”, proceedings of the 2nd International Symposium on Turbo Codes and Related Topics, pp. 459–462 (September 2000)) in which more complex random number generators are used to generate the memory access address.

[0084] The architecture shown in FIG. 4 demands 3L clock cycles to complete one decoding iteration. As an alternative design solution, FIG. 6 shows the principal structure of a partly parallel decoder that requires only 2L clock cycles for one iteration. It mainly contains k² PE Blocks PE_x,y for 1≦x,y≦k, three bi-directional shuffle networks π₁, π₂ and π₃ and 3·k CNU's. Each PE_x,y contains one memory bank RAMs_x,y that stores all the decoding information associated with all the L variable nodes in the variable node group VG_x,y, and contains one VNU to perform the variable node computations for these L variable nodes. Each bi-directional shuffle network π_i realizes the decoding information exchange between all the L·k² variable nodes and the L·k check nodes corresponding to H_i. The k CNU_i,j's for j=1, . . . , k perform the check node computations for all the L·k check nodes corresponding to H_i.

[0085] This decoder completes each decoding iteration in 2L clock cycles, and during the 1^st and 2^nd L clock cycles, it works in check node processing mode and variable node processing mode, respectively. In the check node processing mode, the decoder not only performs the computations of all the check nodes but also completes the decoding information exchange between neighboring nodes. In variable node processing mode, the decoder only performs the computations of all the variable nodes.

[0086] The straightforward encoding scheme for LDPC codes, using the generator matrix, has quadratic complexity in the block length, which is prohibitive with respect to implementation complexity. Based on the specific structure of the parity check matrix of the proposed (3,k)-regular LDPC codes, a systematic approach is proposed for its efficient encoding. The basic idea is: First obtain an approximate upper triangular matrix H^en=π_c(H) by introducing an explicit column permutation π_c, and then obtain {circumflex over (x)} by performing efficient encoding based on H^en, and finally get the codeword x=π_c⁻¹({circumflex over (x)}).

[0087] The parity check matrix of the developed (3,k)-regular LDPC code has the form: H=[H₁^T, H₂^T, H₃^T]^T, where H₁ and H₂ are shown in FIG. 2. The submatrix consisting of all the columns of H which go through the block matrix I_x,y in H₁ is denoted as H^(x,y), e.g., H^(1,2) as shown in FIG. 7. A column permutation π_c is introduced to move each H^(1,x) forward to the position just right to H^(k,1) where x increases from 3 to k successively. Because each P_1,y is an identity matrix, the matrix H^en=π_c(H) has the structure as shown in FIG. 7, based on which one can write matrix H^en in block matrix form as:

[0088] $\begin{array}{cc}{H}^{\mathrm{en}}=\left[\begin{array}{ccc}T& B& D\\ A& C& E\end{array}\right].& \mathrm{EQ}.\left(6\right)\end{array}$

[0089] The efficient encoding is carried out based on the matrix H^en. Let {circumflex over (x)}=({circumflex over (x)}_a, {circumflex over (x)}_b, {circumflex over (x)}_c) be a tentative codeword decomposed according to (6), where {circumflex over (x)}_c contains the information bits of length N−M+r; redundant bits {circumflex over (x)}_a and {circumflex over (x)}_b are of length (2k−1)·L and (k+1)·L−r, respectively. The encoding process is outlined as follows:

• • [0090] (a) Compute y_c=D·{circumflex over (x)}_c and z_c=E·{circumflex over (x)}_c, which is efficient because both D and E are sparse;
  • [0091] (b) Solve T·{circumflex over (x)}′_a=y_c. Since T has the form as shown in FIG. 7, it can be proved that T⁻¹=T. Thus {circumflex over (x)}′_a=T·y_c, which can be easily computed since T is sparse;
  • [0092] (c) Evaluate ŝ=A·{circumflex over (x)}′_a+z_c, which is also efficient since A is sparse;
  • [0093] (d) Compute {circumflex over (x)}_b=G·ŝ, where G=(A·T·B+C)⁻¹. In this step, the complexity is proportional to ((k+1)·L−r)²;
  • [0094] (e) Finally one can obtain {circumflex over (x)}_a by solving T·{circumflex over (x)}_a=B·{circumflex over (x)}_b+y_c. Since T^{{circumflex over (x)}}_a=T·(B·{circumflex over (x)}_b+y_c). This is efficient since both T and B are sparse.

[0095] The real codeword is obtained as x=π_c⁻¹({circumflex over (x)}). The information bits on the decoder side can be easily obtained by performing the column permutation π_c on the decoder output.

[0096] A joint (3,k)-regular LDPC code and decoder/encoder design approach has been presented. By jointly considering the good LDPC code construction and practical decoder/encoder VLSI implementation, this successful attacks the practical (3,k)-regular LDPC coding system design and implementation problems for real-world applications. It will be understood by those skilled in the art that various changes in form and details can be made therein without departing from the spirit and scope of the invention as defined in the appended claims. Thus, the breadth and scope of the present invention should not be limited by any of the above-described exemplary embodiments, but should be defined only in accordance with the following claims and their equivalents.