# Patent

## Title
Method for reducing the computational complexity of viterbi decoder

## Abstract

A method for reducing the computational complexity of a Viterbi decoder, which is suitable for all code rates of a convolutional code applied by the Viterbi decoder. The method dramatically reduces the branch metric computation to thus reduce the complexity of implementing the Viterbi decoder, without affecting the capability of error correction. Upon the best mode, the Viterbi decoder can reduce the required branch metric computation to 1⁄4k of the original computation.

## Background
BACKGROUND OF THE INVENTION
1. Field of the Invention
The invention relates to a method for reducing the computational complexity of a Viterbi decoder and, more particularly, to a method for reducing the computational complexity of a Viterbi decoder at all code rates of a convolutional code.
2. Description of Related Art
In current communication systems, a transmitter transmits a radio signal to a remote receiver. When the radio signal passes through a wireless channel so as to cause a fading and result in errors. Accordingly, the transmitted signal is typically performed with a convolutional encoding before being modulated to the RF signal. After demodulating the received radio signal, the remote receiver performs the convolutional decoding to thereby reconstruct the transmitted signal. Such a way can reduce the channel fading effect and noise interference. The convolutional encoding is referred to as a channel encoding, which is performed by a channel encoder. In general, a convolutional code has three parameters: the number of input bits k, the number of output bits n, and constraint length. The input and output bit parameters k and n are typically represented by the code rate k/n, where k, n are a positive integer. The constraint length indicates the number of memory element of the convolutional code. The convolutional decoding uses a Viterbi algorithm in order to provide an optimal coding gain. Therefore, the current communication systems mostly implements the Viterbi decoder to perform the convolutional decoding. The convolutional code can enhance the capability of error correction by increasing the constraint length. However, the decoding computation at the receiving side is increased with the increased constraint length. To overcome this, U.S. Pat. No. 5,539,757 granted to Cox, et al. for an “Error correction systems with modified Viterbi decoding” has disclosed a method to relatively reduce the computation of the Viterbi decoder by efficiently using the relation of the branches. However, the method can be applied only at a specific code rate, 1/n, which cannot satisfy a user with a desire for a code rate k/n other than the code rate 1/n.
SUMMARY OF THE INVENTION
The object of the invention is to provide a method for reducing the computational complexity of a Viterbi decoder, which can be used in the Viterbi decoder at all code rates of a convolutional code and effectively reduce the computation required for a convolutional decoding.
To achieve the object, there is provided a method for reducing the computational complexity of a Viterbi decoder. The method comprises the steps of: (A) decomposing a trellis diagram with a convolutional code at a code rate k/n into 2v−k radix-2k butterfly units, where k, n, v are a positive integer, and k indicates a number of input bits, n indicates a number of output bits, and v indicates a constraint length; (B) decomposing each radix-2k−r butterfly unit into four radix-2k−r−1 butterfly units; (C) determining a branch symmetry among the radix-2k−r−1 butterfly units based on a generator sequence for the convolutional code; (D) determining if r is equal to k, and when r is not equal to k, adding r by one and executing step (B); and (E) using the branch symmetry to reduce a branch metric computation required by a convolutional decoding at the code rate k/n.
Other objects, advantages, and novel features of the invention will become more apparent from the following detailed description when taken in conjunction with the accompanying drawings.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a schematic diagram of a convolutional encoder with a convolutional code at a code rate k/n in accordance with the invention;
FIG. 2 is a trellis diagram of a convolutional code at a code rate k/n in accordance with the invention;
FIG. 3 is a diagram of radix-2k butterfly units in accordance with the invention;
FIGS. 4(A) and (B) are original and simplified diagrams of a radix-2k−r butterfly unit in the rth level in accordance with the invention;
FIG. 5 is a schematic diagram of four radix-2k−1 butterfly units in the first level in accordance with the invention;
FIG. 6 is a schematic diagram of four radix-2k−r butterfly units in the rth level in accordance with the invention;
FIG. 7 is a schematic diagram of four radix-21 butterfly units in the (k−1)th level in accordance with the invention;
FIG. 8 is a schematic diagram of one radix-21 butterfly unit in accordance with the invention;
FIG. 9 is a table of a generator sequence and branch symmetries among the butterfly units in accordance with the invention;
FIG. 10 is a flowchart of a method for reducing the computational complexity of a Viterbi decoder in accordance with the invention; and
FIG. 11 is a schematic trellis diagram obtained by applying the method of FIG. 10 to an example of code rate k/n=3⁄4 and constraint length v=4 for decomposition in accordance with the invention.



DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT
The method for reducing the computational complexity of a Viterbi decoder, which is suitable for all code rates of a convolutional code which is decoded by the Viterbi decoder. The method uses the butterfly structure to gather the corresponding branches in the trellis diagram to a same butterfly unit, and finds the symmetries of all branches in the trellis diagram in accordance with the generator sequence. Accordingly, the branch metric values for partial branches are required in computation, and the remainders can be obtained in accordance with the branch symmetries. Thus, the reduced computation is achieved. Upon the best mode as using the method, the Viterbi decoder can reduce the required branch metric computation to 1⁄4k of the original computation.
As shown in FIG. 1, the encoder with the convolutional code at the code rate k/n has k+v registers, for v is a positive integer and indicates the constraint length or the number of state bits. In a unit of time, the k bits input data denoted by u1u2 . . . uk are input to the encoder. There are n modulo-2 adders to generate n output bits y1y2 . . . yn. In this case, the relation between the input bit and the output bit is determined by the generators. If the generator for the i-th output bit yi is gi=[g1i g2i . . . gk+vi], the i-th output bit yi can be defined by the following equation:

yi=g1i·u1⊕ . . . ⊕gki·uk⊕gk+1i·s1⊕ . . . ⊕gk+vi·sv,   (1)

where i is a positive integer smaller than or equal to n, and ⊕ indicates a modulo-2 adder.

In addition, a trellis diagram is another approach that is commonly used to show the convolutional code with the code rate k/n. In the trellis diagram, each branch shows a set of output bits, and each state is defined by referring to the state bits in the register, which is represented by the following equation:

S=s1s2 . . . sv−1sv.   (2)

As shown in FIG. 2, the trellis diagram in each stage has N=2v state values, and the states are numbered in decimal, where sv indicates a most significant bit (MSB) while s1 indicates a least significant bit (LSB). If notation j indicates the current stage state set containing the states numbered 1 to N, state j′ indicates a next stage state set with respect to the current stage state set j, and the sets j and j′ have an N state respectively. The oldest k input bits are removed from the registers as k input bits enter in the registers every time, thereby generating a state transition, i.e., the state is changed from the state denoted as s1s2 . . . sv of the set j into the state denoted as u1 . . . uks1 . . . sv−k of the set j′. As shown in FIG. 2, each state has 2k state transitions, and each transition has a corresponding branch. Accordingly, for a Viterbi decoding, each set has 2v+k branch metric values to be computed. However, the number to be computed presents an exponential growth as the constraint length v is increased. To overcome this, the invention finds the branch symmetries of the branches in the trellis diagram to thereby simplify the computation and achieve the purpose of reducing the computational amount required for the Viterbi decoding.

Because an effective approach to find the symmetries of the branches in the trellis diagram does not exist, the invention discloses a butterfly structure to overcome the aforementioned problem.
Referring again to FIG. 2, it is observed that the 2k branches emitted from a same source state have a certain relation because the content of the register for the 2k branches emitted is determined only by the different input bits, i.e., the relation of the branches can be found from the bits u1 . . . uk. In addition, the 2k branches entering in a same destination state also have a certain relation because the content of the register for the 2k branches is determined only by the bits sv−k+1 . . . sv of the source states, i.e., the relation of the branches can be determined by the bits of the source states.
All relative states and branches are collected to form a butterfly unit, as shown in FIG. 3. Namely, the trellis diagram of FIG. 2 can be decomposed into 2v−k butterfly units. The butterfly unit shown in FIG. 3 is defined as a radix-2k butterfly unit that has 2k states. Each of the 2k states is joining 2k branches.
All states in the radix-2k butterfly unit have the same bits s1 . . . sv−k, which is notated by xx in the figures for the convenience. Accordingly, equation (1) can be rewritten into:

yi=g1i·u1⊕g2i·u2⊕Λ⊕gki·uk⊕Xi⊕gv+1i·sv−k+1⊕Λ⊕gk+vi·sv,   (3)

where Xi is generated from the same portion s1 . . . sv−k, i.e., the same component presented in every branch,

Xi=gk+1i·s1⊕gk+2i·s2⊕Λ⊕gvi·sv−k.   (4)

Therefore, upon the relation of the branches of FIG. 3, the trellis diagram of FIG. 2 can be decomposed into 2v−k radix-2k butterfly units. Namely, the corresponding branches are all gathered to a same radix-2k butterfly unit, and the branch symmetry is obtained by referring to the states joining the branches in every radix-2k butterfly unit. The relation of the branches is not affected because Xi indicates the same portion of the bits in every branch. From equation (3), it can show that the relation of the branches is influenced only by the 2k members, g1i,g2iΛgki,gv+1i,gv+2iΛgk+vi, in the generator. Namely, for every bit on the branches, there are 22k symmetries in total. Because a branch symmetry is obtained from the intersection of bit symmetries, a great number of bit symmetries can cause the difficulty of obtaining the branch symmetry. The invention can continuously and regularly decompose a radix-2k butterfly unit in every level to thereby reduce the influence of the generator on the branch symmetries in the butterfly unit.
In order to simplify the following illustrations, the butterfly unit in every level is illustrated by a simplified diagram. FIGS. 4(A) and (B) show original and simplified diagrams of a radix-2k−r butterfly unit in the r-th level. As shown in FIG. 4(A), each of the radix-2k−r butterfly units in the r-th level has 2k−r states, and each state joins 2k−r branches. The butterfly unit in every level hereinafter is illustrated in a simplified form.
First decomposition divides the radix-2k butterfly unit into four radix-2k−1 butterfly units based on the bits (u1,sv) of the states in the radix-2k butterfly unit, and distributes all branches, which are the same at the bits (u1,sv) in the states, to a same radix-2k−1 butterfly unit. As shown in FIG. 5, the branch relation in a same radix-2k−1 butterfly unit is not influenced by the bits (u1,sv). In addition, the states joining the branches among the four radix-2k−1 butterfly units are the same except at the bits (u1,sv), and thus the relation of the branches of the four radix-2k−1 butterfly units is determined only by the bits (u1,sv). Besides, the symmetry type of the four radix-2k−1 butterfly units is influenced only by the members (gi1,gik+v) of the generators that are mapped by the bits (u1,sv). For example, if gi1=0 and gik+v=0, the symmetry of the four radix-2k−1 butterfly units is B1k−1=B2k−1=B3k−1=B4k−1. If gi1=0 and gik+v=1, the symmetry of the four radix-2k−1 butterfly units is B1k−1=B2k−1=B3k−1′=B4k−1′, for B3k−1′ and B3k−1 are complementary, and B4k−1′ and B4k−1 are complementary.
Similarly, for a butterfly unit in the r-th level, a radix-2k−r+1 butterfly unit is decomposed into four radix-2k−r butterfly units by settling the bits (u1,sv−r+1) of the states, as shown in FIG. 6. In addition, the relation of the four radix-2k−r butterfly units is determined by the members (gir,gik+v−r+1) of the generators. Accordingly, a last radix-22 butterfly unit is decomposed into four radix-21 butterfly units, as shown in FIG. 7, and the relation of the four radix-21 butterfly units is determined by the members (gik−1,gik+v+2). Finally, as shown in FIG. 8, each of the four radix-21 butterfly units remains only four branches, and the states joining the four branches are different only at the bits (uk,sv−k+1). Therefore, the relation of the four branches can be determined by the members (gik,giv+1) of the generators.
When the relation of the four branches in the radix-21 butterfly unit is considered as four butterfly units {B10,B20,B30,B40} obtained by the k-th decomposition, it is known that the members of the generators influencing the symmetries in the levels have the same influence on the symmetries of the butterfly units in the levels, which can be generally shown in the table of FIG. 9. As shown in FIG. 9, no matter what the generator sequence is, the butterfly units in the levels can have 1⁄4 symmetries. In the invention, the 1/x symmetry indicates that only one of the x branches is necessarily subjected to the branch metric computation, and in accordance with the symmetry, the branch metric values for the remaining (x−1) branches can be obtained. Therefore, the (1⁄4)k symmetry owned by the bits of each branch can be found by means of such a butterfly structure. It is noted that such a symmetry only indicates the symmetry owned by a single bit of the branches in the butterfly units in the levels. The complete symmetry of the branches is obtained by intersecting the symmetries of the bits corresponding to the branches. Thus, the branch symmetry is influenced by the similarity of bit symmetries. The bits of the branches have more similar symmetry, and the symmetry of the branches in the trellis diagram is more preferred.
FIG. 10 is a flowchart of a method for reducing the computational complexity of a Viterbi decoder in accordance with the invention. The method includes the steps as follows.
Step A: a trellis diagram with a convolutional code at code rate k/n is decomposed into 2v−k butterfly units with radix-2k. Each butterfly unit, as shown in FIG. 3, has 2k states, and each state joins 2k branches. The values of the states contain a same portion s1 . . . sv−k to thereby distribute the corresponding branches to a same radix-2k butterfly unit.
Step B: each radix-2k−r butterfly unit is decomposed into four radix-2k−r−1 butterfly units in order to reduce the influence of the generators on the branch symmetries.
Step C: each bit symmetry is determined in accordance with the generators for the convolutional code, and the bit symmetries respectively of the n bits are intersected to thus obtain the branch symmetry of the four radix-2k−r−1 butterfly units.
Step D: it determines if r=k; when r is not equal to k, it indicates that a single radix-2k−r−1 butterfly unit can be further decomposed. Namely, a symmetry among the radix-2k−r butterfly units is determined only by the members (gri, gk+v−r+1i) of the generators corresponding to the bits (ur,sv−r+1). In this case, r=r+1, and step B is executed for the further butterfly unit decomposition. When r=k, it indicates that the last level is decomposed completely.
Step E: the branch symmetry is used to reduce the branch metric computation required for decoding the convolutional code at the code rate k/n.
FIG. 11 is a schematic trellis diagram obtained by applying the method of FIG. 10 to an example of code rate k/n=3⁄4 and constraint length v=4 for decomposition in accordance with the invention. As shown in FIG. 11, a trellis diagram corresponding to the convolutional code at state set j has 16 state values numbered 0-15 in total. Each of the state values has eight state value transitions. The trellis diagram can be divided into two radix-23 butterfly units. Each butterfly unit has eight state values, and each state value has four state value transitions. Accordingly, if the generators are g1=[1111101], g2=[1111001], g3=[1010111] and g4=[1110101], when r=1, the bit symmetries among the radix-2k−1 butterfly units are shown as follows.

    
    
        Bit y1 (g11=1, g71=1): B21=B22′=B23′=B24;
        Bit y2 (g12=1, g72=1): B21=B22′=B23′=B24;
        Bit y3 (g13=1, g73=1): B21=B22′=B23′=B24; and
        Bit y4 (g14=1, g74=1): B21=B22′=B23′=B24.
    
    


The four bits have the same symmetry so that the branch symmetry B21=B22′=B23′=B24 is obtained by intersecting the bit symmetries. When r=2, the bit symmetries among the radix-2k−2 butterfly units are shown as follows.

    
    
        Bit (g21=1, g61=0): B11=B12′=B13=B14′;
        Bit (g22=1, g62=0): B11=B12′=B13=B14′;
        Bit (g23=0, g63=1): B11=B12=B13′=B14′; and
        Bit y4 (g24=1, g64=0): B11=B12′=B13=B14′.
    
    


The four bits have different symmetries so that only the branch symmetry B11=B14′, B12=B13′ is obtained after the intersection. When r=3, the bit symmetries among the radix-2k−3 butterfly units are shown as follows.

    
    
        Bit y1 (g31=1, g51=0): B01=B02′=B03′=B04;
        Bit y2 (g32=1, g52=0): B01=B02′=B03=B04′;
        Bit y3 (g33=0, g53=1): B01=B02′=B03′=B04; and
        Bit y4 (g34=1, g54=0): B01=B02′=B03′=B04.
    
    


The four bits have different symmetries so that only the branch symmetry B01=B02′, B03=B04′ is obtained after the intersection.
The branch symmetries owned by the butterfly units in the three levels are combined so as to provide the convolutional code with a 1/16 symmetry. Accordingly, on implementing a Viterbi decoder with the convolutional code, the required branch metric computation is reduced by the butterfly structure to 1/16 of the original computation, and the remaining can be obtained in accordance with the branch symmetries. For example, as shown in FIG. 11, the trellis diagram has 16 states in total, each state having eight branches, and in this case, it is required that total 128 branch metric values are originally computed for every level of the trellis diagram. Accordingly, the invention uses the decomposition of the butterfly structure to first divide the trellis diagram into two radix-23 butterfly units. The four radix-22 butterfly units contained in each radix-23 butterfly units needs only to compute the branch metric values of the branch B12, the others can be obtained through the symmetry B12=B22′=B32′=B42. In addition, as to the branch B12, only the branches B11 and B31 in the radix-2 butterfly unit are computed, without computing all branch metric values of the branch B12, and the remaining can be obtained through the symmetry B11=B41′, B21=B31′. Similarly, as to the branches B11 and B31, only the branches B10 and B30 in the radix-1 butterfly unit are computed, and the remaining can be obtained through the symmetry B10=B20′, B30=B40. Therefore, the entire radix-23 butterfly unit has 64 branches, but only four branches, i.e., B10 and B30 corresponding to the branches B11 and B31 respectively, are computed. Accordingly, the branch metric computation can be effectively reduced to 1/16 of the original. Likewise, the computation is performed on the other radix-23 butterfly unit.
In view of the foregoing, it is known that the method of the invention for reducing the computational complexity of a Viterbi decoder can effectively reduce the complexity of decoding the convolutional code at the code rate k/n, thereby achieving the purpose of the invention.
Although the present invention has been explained in relation to its preferred embodiment, it is to be understood that many other possible modifications and variations can be made without departing from the spirit and scope of the invention as hereinafter claimed.

## Claims

1. A method for reducing the computational complexity of a Viterbi decoder included in a channel decoder of a receiver in a communication system, comprising the steps of:
(A) decomposing a trellis diagram with a convolutional code at a code rate k/n into 2v−k radix-2k butterfly units, where k, n, v are positive integers, and k indicates a number of input bits, n indicates a number of output bits, and v indicates a constraint length;
(B) for a butterfly unit in the r-th level, decomposing each radix-2k−r butterfly unit into four radix-2k−r−l butterfly units;
(C) determining a branch symmetry among the radix-2k−r−l butterfly units based on a generator sequence for the convolutional code;
(D) determining if r is equal to k, and when r is not equal to k, adding one to r and executing step (B); and
(E) using the branch symmetry to reduce a branch metric computation required by a convolutional decoding at the code rate k/n,
wherein steps (A) to (E) are performed by said receiver.

2. The method as claimed in claim 1, wherein in step (C) the branch symmetry is obtained by intersecting bit symmetries, and each bit symmetry is determined in accordance with the generator sequence for the convolutional code.