Low density parity check encoder having length of 16200 code rate of 3/15, and low density parity check encoding method using the same

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First Claim
1. A method of transmitting a broadcast signal, the method comprising:
 initializing first memory configured to store an LDPC codeword having a length of 16200 and a code rate of 3/15, and second memory;
generating the LDPC codeword corresponding to information bits by performing accumulation of at least one of bits from the first memory at parity bit addresses of the second memory, the parity bit addresses being updated using numbers of a sequence, the sequence being equivalent to a parity check matrix (PCM);
generating a transmission signal based on the LDPC codeword, the LDPC codeword including parity bits for correcting errors occurring over a physical channel; and
transmitting, using an antenna, the transmission signal over the physical channel,wherein the sequence is expressed as follows;
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Abstract
A low density parity check (LDPC) encoder, an LDPC decoder, and an LDPC encoding method are disclosed. The LDPC encoder includes first memory, second memory, and a processor. The first memory stores an LDPC codeword having a length of 16200 and a code rate of 3/15. The second memory is initialized to 0. The processor generates the LDPC codeword corresponding to information bits by performing accumulation with respect to the second memory using a sequence corresponding to a parity check matrix (PCM).
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4 Claims
 1. A method of transmitting a broadcast signal, the method comprising:
initializing first memory configured to store an LDPC codeword having a length of 16200 and a code rate of 3/15, and second memory; generating the LDPC codeword corresponding to information bits by performing accumulation of at least one of bits from the first memory at parity bit addresses of the second memory, the parity bit addresses being updated using numbers of a sequence, the sequence being equivalent to a parity check matrix (PCM); generating a transmission signal based on the LDPC codeword, the LDPC codeword including parity bits for correcting errors occurring over a physical channel; and transmitting, using an antenna, the transmission signal over the physical channel, wherein the sequence is expressed as follows;  View Dependent Claims (2, 3, 4)
1 Specification
This application is a continuation of and claims priority of U.S. application Ser. No. 14/496,356 filed Sep. 25, 2014, which claims the benefit of Korean Patent Application Nos. 1020140106174 and 1020140120009, filed Aug. 14, 2014 and Sep. 11, 2014, respectively, which are hereby incorporated by reference herein in their entirety.
1. Technical Field
The present disclosure relates generally to a low density parity check (LDPC) code that is used to correct errors occurring over a wireless channel, and, more particularly, to an LDPC code that is applicable to a digital broadcasting system.
2. Description of the Related Art
Current terrestrial television (TV) broadcasting generates cochannel interference across an area within a distance that is three times a service radius, and thus the same frequency cannot be reused in the area within the distance that is three times the service radius. An area in which the same frequency cannot be reused is called a white space. Spectrum efficiency significantly deteriorates due to the occurrence of a white space.
Accordingly, there arises a need for the development of a transmission technology that facilitates the elimination of a white space and the reuse of a frequency with an emphasis on reception robustness in order to improve spectrum efficiency.
In response to this, the paper “Cloud Transmission: A New SpectrumReuse Friendly Digital Terrestrial Broadcasting Transmission System” published on September of 2012 in IEEE Transactions on Broadcasting, Vol. 58, No. 3 proposes a terrestrial cloud transmission technology that facilitates reuse, does not generate a white space, and makes the construction and operation of a single frequency network easy.
Using this terrestrial cloud transmission technology, a broadcasting station can transmit the same nationwide content or locally different content over a single broadcasting channel. However, for this purpose, a receiver should receive one or more terrestrial cloud broadcast signals in an area in which signals transmitted from different transmitters overlap each other, that is, an overlap area, over a single frequency network, and then should distinguish and demodulate the received terrestrial cloud broadcast signals. That is, the receiver should demodulate one or more cloud broadcast signals in a situation in which cochannel interference is present and the timing and frequency synchronization between transmitted signals are not guaranteed.
Meanwhile, Korean Patent Application Publication No. 20130135746 entitled “Low Density Parity Check Code for Terrestrial Cloud Transmission” discloses an LDPC code that is optimized for terrestrial cloud transmission and exhibits excellent performance at low code rate (<0.5).
However, Korean Patent Application Publication No. 20130135746 is directed to a code length completely different from an LDPC code length used in the DVB broadcast standard, etc., and does not teach a specific LDPC encoding method.
At least one embodiment of the present invention is directed to the provision of a new LDPC codeword having a length of 16200 and a code rate of 3/15, which is capable of being used for general purposes.
At least one embodiment of the present invention is directed to the provision of an LDPC encoding technique that is capable of efficiently performing LDPC encoding using a sequence having a number of rows equal to a value that is obtained by dividing the sum of the length of the systematic part of an LDPC codeword, that is, 3240, and the length of the first parity part of the LDPC codeword, that is, 1080, by 360.
In accordance with an aspect of the present invention, there is provided an LDPC encoder, including first memory configured to store an LDPC codeword having a length of 16200 and a code rate of 3/15; second memory configured to be initialized to 0; and a processor configured to generate the LDPC codeword corresponding to information bits by performing accumulation with respect to the second memory using a sequence corresponding to a parity check matrix (PCM).
The accumulation may be performed at parity bit addresses that are updated using the sequence corresponding to the PCM.
The LDPC codeword may include a systematic part corresponding to the information bits and having a length of 3240, a first parity part corresponding to a dual diagonal matrix included in the PCM and having a length of 1080, and a second parity part corresponding to an identity matrix included in the PCM and having a length of 11880.
The sequence may have a number of rows equal to the sum of a value obtained by dividing a length of the systematic part, that is, 3240, by a circulant permutation matrix (CPM) size corresponding to the PCM, that is, 360, and a value obtained by dividing a length of the first parity part, that is, 1080, by the CPM size.
The sequence may be represented by the following Sequence Table:
The accumulation may be performed while the rows of the sequence are being repeatedly changed by the CPM size of the PCM.
In accordance with an aspect of the present invention, there is provided an LDPC encoding method, including initializing first memory configured to store an LDPC codeword having a length of 16200 and a code rate of 3/15, and second memory; and generating the LDPC codeword corresponding to information bits by performing accumulation with respect to the second memory using a sequence corresponding to a PCM.
The accumulation may be performed at parity bit addresses that are updated using the sequence corresponding to the PCM.
The LDPC codeword may include a systematic part corresponding to the information bits and having a length of 3240, a first parity part corresponding to a dual diagonal matrix included in the PCM and having a length of 1080, and a second parity part corresponding to an identity matrix included in the PCM and having a length of 11880.
The sequence may have a number of rows equal to the sum of a value obtained by dividing a length of the systematic part, that is, 3240, by a circulant permutation matrix (CPM) size corresponding to the PCM, that is, 360, and a value obtained by dividing a length of the first parity part, that is, 1080, by the CPM size.
The sequence may be represented by the above Sequence Table.
In accordance with still another aspect of the present invention, there is provided an LDPC decoder, including a receiving unit configured to receive an LDPC codeword encoded using a sequence corresponding to a PCM and is represented by the above Sequence Table; and a decoding unit configured to restore information bits from the received LDPC codeword by performing decoding corresponding to the PCM.
The above and other objects, features and advantages of the present invention will be more clearly understood from the following detailed description taken in conjunction with the accompanying drawings, in which:
Embodiments of the present invention will be described in detail below with reference to the accompanying drawings. Repeated descriptions and descriptions of wellknown functions and configurations that have been deemed to make the gist of the present invention unnecessarily obscure will be omitted below. The embodiments of the present invention are intended to fully describe the present invention to persons having ordinary knowledge in the art to which the present invention pertains. Accordingly, the shapes, sizes, etc. of components in the drawings may be exaggerated to make the description obvious.
Embodiments of the present invention will be described in detail below with reference to the accompanying drawings.
Referring to
The transmitter 10 generates an nbit codeword by encoding k information bits using an LDPC encoder 13. The codeword is modulated by the modulator 15, and is transmitted via an antenna 17. The signal transmitted via the wireless channel 20 is received via the antenna 31 of the receiver 30, and, in the receiver 30, is subjected to a process reverse to the process in the transmitter 10. That is, the received data is demodulated by a demodulator 33, and is then decoded by an LDPC decoder 35, thereby finally restoring the information bits.
It will be apparent to those skilled in the art that the abovedescribed transmission and reception processes have been described within a minimum range required for a description of the features of the present invention and various processes required for data transmission may be added.
In the following, the specific processes of encoding and decoding that are performed using an LDPC code in the LDPC encoder 13 or LDPC decoder 35 and the specific configurations of encoding and decoding devices, such as the LDPC encoder 13 and the LDPC decoder 35, are described. The LDPC encoder 13 illustrated in
Referring to
That is, at step S210, an nbit codeword is generated by encoding k information bits using the LDPC encoder.
In this case, step S210 may be performed as in an LDPC encoding method illustrated in
Furthermore, in the broadcast signal transmission and reception method, the encoded data is modulated at step S220.
That is, at step S220, the encoded nbit codeword is modulated using the modulator.
Furthermore, in the broadcast signal transmission and reception method, the modulated data is transmitted at step S230.
That is, at step S230, the modulated codeword is transmitted over a wireless channel via the antenna.
Furthermore, in the broadcast signal transmission and reception method, the received data is demodulated at step S240.
That is, at step S240, the signal transmitted over the wireless channel is received via the antenna of the receiver, and the received data is demodulated using the demodulator.
Furthermore, in the broadcast signal transmission and reception method, the demodulated data is subjected to LDPC decoding at step S250.
That is, at step S250, the information bits are finally restored by performing LDPC decoding using the demodulator of the receiver.
In this case, step S250 corresponds to a process reverse to that of the LDPC encoding method illustrated in
An LDPC code is known as a code very close to the Shannon limit for an additive white Gaussian noise (AWGN) channel, and has the advantages of asymptotically excellent performance and parallelizable decoding compared to a turbo code.
Generally, an LDPC code is defined by a lowdensity parity check matrix (PCM) that is randomly generated. However, a randomly generated LDPC code requires a large amount of memory to store a PCM, and requires a lot of time to access memory. In order to overcome these problems, a quasicyclic LDPC (QCLDPC) code has been proposed. A QCLDPC code that is composed of a zero matrix or a circulant permutation matrix (CPM) is defined by a PCM that is expressed by the following Equation 1:
In this equation, J is a CPM having a size of L×L, and is given as the following Equation 2. In the following description, L may be 360.
Furthermore, J^{i }is obtained by shifting an L×L identity matrix I (J^{0}) to the right i (0≤i<L) times, and J^{∞} is an L×L zero matrix. Accordingly, in the case of a QCLDPC code, it is sufficient if only index exponent i is stored in order to store J^{i}, and thus the amount of memory required to store a PCM is considerably reduced.
Referring to
where I_{L×L }is an identity matrix having a size of L×L.
That is, the matrix B may be a bitwise dual diagonal matrix, or may be a blockwise dual diagonal matrix having identity matrices as its blocks, as indicated by Equation 3. The bitwise dual diagonal matrix is disclosed in detail in Korean Patent Application Publication No. 20070058438, etc.
In particular, it will be apparent to those skilled in the art that when the matrix B is a bitwise dual diagonal matrix, it is possible to perform conversion into a Quasicyclic form by applying row or column permutation to a PCM including the matrix B and having a structure illustrated in
In this case, N is the length of a codeword, and K is the length of information.
The present invention proposes a newly designed QCLDPC code in which the code rate thereof is 3/15 and the length of a codeword is 16200, as illustrated in the following Table 1. That is, the present invention proposes an LDPC code that is designed to receive information having a length of 3240 and generate an LDPC codeword having a length of 16200.
Table 1 illustrates the sizes of the matrices A, B, C, D and Z of the QCLDPC code according to the present invention:
The newly designed LDPC code may be represented in the form of a sequence (progression), an equivalent relationship is established between the sequence and matrix (parity bit check matrix), and the sequence may be represented, as follows:
An LDPC code that is represented in the form of a sequence is being widely used in the DVB standard.
According to an embodiment of the present invention, an LDPC code presented in the form of a sequence is encoded, as follows. It is assumed that there is an information block S=(s_{0}, s_{1}, . . . , s_{K−1}) having an information size K. The LDPC encoder generates a codeword Λ=(λ_{0}, λ_{1}, λ_{2}, . . . , λ_{N−1}) having a size of N=K+M_{1}+M_{2 }using the information block S having a size K. In this case, M_{1}=g, and M_{2}=N−K−g. Furthermore, M_{1 }is the size of parity bits corresponding to the dual diagonal matrix B, and M_{2 }is the size of parity bits corresponding to the identity matrix D. The encoding process is performed, as follows:
Initialization:
λ_{i}=s_{i }for i=0,1, . . . ,K−1
p_{j}=0 for j=0,1, . . . ,M_{1}+M_{2}−1 (4)
First information bit λ_{0 }is accumulated at parity bit addresses specified in the 1st row of the sequence of the Sequence Table. For example, in an LDPC code having a length of 16200 and a code rate of 3/15, an accumulation process is as follows:
p_{8}=p_{8}⊕λ_{0 }p_{372}=p_{372}⊕λ_{0 }p_{841}=p_{841}⊕λ_{0 }p_{4522}=p_{4522}⊕λ_{0 }p_{5253}=p_{5253}⊕λ_{0 }
p_{7430}=p_{7430}⊕λ_{0 }p_{8542}=p_{8542}⊕λ_{0 }p_{9822}=p_{9822}⊕λ_{0 }p_{10550}=p_{10550}⊕λ_{0 }
p_{11896}=p_{11896}⊕λ_{0 }p_{11988}=p_{11988}⊕λ_{0 }
where the addition ⊕ occurs in GF(2).
The subsequent L−1 information bits, that is, λ_{m}, m=1, 2, . . . , L−1, are accumulated at parity bit addresses that are calculated by the following Equation 5:
(x+m×Q_{1})mod M_{1 }if x<M_{1 }
M_{1}+{(x−M_{1}+m×Q_{2})mod M_{2}} if x≥M_{1} (5)
where x denotes the addresses of parity bits corresponding to the first information bit λ_{0}, that is, the addresses of the parity bits specified in the first row of the sequence of the Sequence Table, Q_{1}=M_{1}/L, Q_{2}=M_{2}/L, and L=360. Furthermore, Q_{1 }and Q_{2 }are defined in the following Table 2. For example, for an LDPC code having a length of 16200 and a code rate of 3/15, M_{1}=1080, Q_{1}=3, M_{2}=11880, Q_{2}=33 and L=360, and the following operations are performed on the second bit λ_{1 }using Equation 5:
p_{11}=p_{11}⊕λ_{1 }p_{375}=p_{375}⊕λ_{1 }p_{844}=p_{844}⊕λ_{1 }p_{4555}=p_{4555}⊕λ_{1 }p_{5286}=p_{5286}⊕λ_{1 }
p_{7463}=p_{7463}⊕λ_{1 }p_{8575}=p_{8575}⊕λ_{1 }p_{9855}=p_{9855}⊕λ_{1 }p_{10583}=p_{10583}⊕λ_{1 }
p_{11929}=p_{11929}⊕λ_{1 }p_{12021}=p_{12021}⊕λ_{1 }
Table 2 illustrates the sizes of M_{1}, Q_{1}, M_{2 }and Q_{2 }of the designed QCLDPC code:
The addresses of parity bit accumulators for new 360 information bits from λ_{L }to 2_{2L−1 }are calculated and accumulated from Equation 5 using the second row of the sequence.
In a similar manner, for all groups composed of new L information bits, the addresses of parity bit accumulators are calculated and accumulated from Equation 5 using new rows of the sequence.
After all the information bits from λ_{0 }to λ_{K−1 }have been exhausted, the operations of the following Equation 6 are sequentially performed from i=1:
p_{i}=p_{i}⊕p_{i−1 }for i=0,1, . . . ,M_{1}−1 (6)
Thereafter, when a parity interleaving operation, such as that of the following Equation 7, is performed, parity bits corresponding to the dual diagonal matrix B are generated:
λ_{K+L·t+s}=p_{Q}_{1}_{·s+t }for 0≤s<L, 0≤t<Q_{1} (7)
When the parity bits corresponding to the dual diagonal matrix B have been generated using K information bits parity bits λ_{0}, λ_{1}, . . . , λ_{K−1}, parity bits corresponding to the identity matrix D are generated using the M_{1 }generated parity bits λ_{K}, λ_{K+1}, . . . , λ_{K+M}_{1}_{−1}.
For all groups composed of L information bits from λ_{K }to λ_{K+M}_{1}_{−1}, the addresses of parity bit accumulators are calculated using the new rows (starting with a row immediately subsequent to the last row used when the parity bits corresponding to the dual diagonal matrix B have been generated) of the sequence and Equation 5, and related operations are performed.
When a parity interleaving operation, such as that of the following Equation 8, is performed after all the information bits from λ_{K }to λ_{K+M}_{1}_{−1 }have been exhausted, parity bits corresponding to the identity matrix D are generated:
λ_{K+M}_{1}_{+L·t+s}=p_{M}_{1}_{+Q}_{2}_{·s+t }for 0≤s<L, 0≤t<Q_{2} (8)
Referring to
The memory 310 is memory that is used to store an LDPC codeword having a length of 16200 and a code rate of 3/15.
The memory 320 is memory that is initialized to 0.
The memory 310 and the memory 320 may correspond to λ_{i }(i=0, 1, . . . , N−1) and p_{j }(j=0, 1, . . . , M_{1}+M_{2}−1), respectively.
The memory 310 and the memory 320 may correspond to various types of hardware for storing sets of bits, and may correspond to data structures, such as an array, a list, a stack and a queue.
The processor 330 generates an LDPC codeword corresponding to information bits by performing accumulation with respect to the memory 320 using a sequence corresponding to a PCM.
In this case, the accumulation may be performed at parity bit addresses that are updated using the sequence of the above Sequence Table.
In this case, the LDPC codeword may include a systematic part λ_{0}, λ_{1}, . . . , λ_{K−1 }corresponding to the information bits and having a length of 3240 (=K), a first parity part λ_{K}, λ_{K+1}, . . . , λ_{K+M}_{1}_{−1 }corresponding to a dual diagonal matrix included in the PCM and having a length of 1080 (=M_{1}=g), and a second parity part λ_{K+M}_{1}, λ_{K+M}_{1}_{+1}, . . . , λ_{K+M}_{1}_{+M}_{2}_{−1 }corresponding to an identity matrix included in the PCM and having a length of 11880 (=M_{2}).
In this case, the sequence may have a number of rows equal to the sum (3240/360+1080/360=12) of a value obtained by dividing the length of the systematic part, that is, 3240, by a CPM size L corresponding to the PCM, that is, 360, and a value obtained by dividing the length M_{1 }of the first parity part, that is, 1080, by 360.
As described above, the sequence may be represented by the above Sequence Table.
In this case, the memory 320 may have a size corresponding to the sum M_{1}+M_{2 }of the length M_{1 }of the first parity part and the length M_{2 }of the second parity part.
In this case, the parity bit addresses may be updated based on the results of comparing each x of the previous parity bit addresses specified in respective rows of the sequence with the length M_{1 }of the first parity part.
That is, the parity bit addresses may be updated using Equation 5. In this case, x may be the previous parity bit addresses, m may be an information bit index that is an integer larger than 0 and smaller than L, L may be the CPM size of the PCM, Q_{1 }may be M_{1}/L, M_{1 }may be the size of the first parity part, Q_{2 }may be M_{2}/L, and M_{2 }may be the size of the second parity part.
In this case, it may be possible to perform the accumulation while repeatedly changing the rows of the sequence by the CPM size L (=360) of the PCM, as described above.
In this case, the first parity part λ_{K}, λ_{K+1}, . . . , λ_{K+M}_{1}_{−1 }may be generated by performing parity interleaving using the memory 310 and the memory 320, as described in conjunction with Equation 7.
In this case, the second parity part λ_{K+M}_{1}, λ_{K+M}_{1}_{+1}, . . . , λ_{K+M}_{1}_{+M}_{2}_{−1 }may be generated by performing parity interleaving using the memory 310 and the memory 320 after generating the first parity part λ_{K}, λ_{K+1}, . . . , λ_{K+M}_{1}_{−1 }and then performing the accumulation using the first parity part λ_{K}, λ_{K+1}, . . . , λ_{K+M}_{1}_{−1 }and the sequence, as described in conjunction with Equation 8.
Referring to
The receiving unit 410 receives an LDPC codeword that has been encoded using a sequence that corresponds to a PCM and is represented by the above Sequence Table.
The decoding unit 420 restores information bits from the received LDPC codeword by performing decoding corresponding to the PCM.
In this case, the sequence may be used to update the parity bit addresses of the memory, and the parity bit addresses are used for accumulation that is performed to generate parity bits corresponding to the LDPC codeword.
In this case, the LDPC codeword may include a systematic part λ_{0}, λ_{1}, . . . , λ_{K−1 }corresponding to the information bits, a first parity part λ_{K}, λ_{K+1}, . . . , λ_{K+M}_{1}_{−1 }corresponding to a dual diagonal matrix included in the PCM, and a second parity part λ_{K+M}_{1}, λ_{K+M}_{1}_{+1}, . . . , λ_{K+M}_{1}_{+M}_{2}_{−1 }corresponding to an identity matrix included in the PCM.
In this case, the parity bit addresses may be updated based on the results of comparing each x of the previous parity bit addresses specified in respective rows of the sequence with the length M_{1 }of the first parity part.
That is, the parity bit addresses may be updated using Equation 5. In this case, x may be the previous parity bit addresses, m may be an information bit index that is an integer larger than 0 and smaller than L, L may be the CPM size of the PCM, Q_{1 }may be M_{1}/L, M_{1 }may be the size of the first parity part, Q_{2 }may be M_{2}/L, and M_{2 }may be the size of the second parity part.
Referring to
In this case, step S510 may be performed using Equation 4.
Furthermore, in the LDPC encoding method according to this embodiment of the present invention, an LDPC codeword corresponding to information bits is generated by performing accumulation with respect to the second memory using a sequence corresponding to a PCM at step S520.
In this case, the accumulation may be performed at parity bit addresses that are updated using the sequence corresponding to the PCM.
In this case, the LDPC codeword may include a systematic part λ_{0}, λ_{1}, . . . , λ_{K−1 }corresponding to the information bits and having a length of 3240 (=K), a first parity part λ_{K}, λ_{K+1}, . . . , λ_{K+M}_{1}_{−1 }corresponding to a dual diagonal matrix included in the PCM and having a length of 1080 (=M_{1}=g), and a second parity part λ_{K+M}_{1}, λ_{K+M}_{1}_{+1}, . . . , λ_{K+M}_{1}_{+M}_{2}_{−1 }corresponding to an identity matrix included in the PCM and having a length of 11880 (=M_{2}).
In this case, the sequence may have a number of rows equal to the sum (3240/360+1080/360=12) of a value obtained by dividing the length of the systematic part, that is, 3240, by a CPM size L corresponding to the PCM, that is, 360, and a value obtained by dividing the length M_{1 }of the first parity part, that is, 1080, by 360.
As described above, the sequence may be represented by the above Sequence Table.
In this case, the parity bit addresses may be updated based on the results of comparing each x of the previous parity bit addresses specified in respective rows of the sequence with the length M_{1 }of the first parity part.
That is, the parity bit addresses may be updated using Equation 5. In this case, x may be the previous parity bit addresses, m may be an information bit index that is an integer larger than 0 and smaller than L, L may be the CPM size of the PCM, Q_{1 }may be M_{1}/L, M_{1 }may be the size of the first parity part, Q_{2 }may be M_{2}/L, and M_{2 }may be the size of the second parity part.
In this case, it may be possible to perform the accumulation while repeatedly changing the rows of the sequence by the CPM size L (=360) of the PCM, as described above.
In this case, the first parity part λ_{K}, λ_{K+1}, . . . , λ_{K+M}_{1}_{−1 }may be generated by performing parity interleaving using the memory 310 and the memory 320, as described in conjunction with Equation 7.
In this case, the second parity part λ_{K+M}_{1}, λ_{K+M}_{1}_{+1}, . . . , λ_{K+M}_{1}_{+M}_{2}_{−1 }may be generated by performing parity interleaving using the memory 310 and the memory 320 after generating the first parity part λ_{K}, λ_{K+1}, . . . , λ_{K+M}_{1}_{−1 }and then performing the accumulation using the first parity part λ_{K}, λ_{K+1}, . . . , λ_{K+M}_{1}_{−1 }and the sequence, as described in conjunction with Equation 8.
The graph illustrated in
At least one embodiment of the present invention has the advantage of providing a new LDPC codeword having a length of 16200 and a code rate of 3/15, which is capable of being used for general purposes.
At least one embodiment of the present invention has the advantage of providing an LDPC encoding technique that is capable of efficiently performing LDPC encoding using a sequence having a number of rows equal to a value that is obtained by dividing the sum of the length of the systematic part of an LDPC codeword, that is, 3240, and the length of the first parity part of the LDPC codeword, that is, 1080, by 360.
Although the specific embodiments of the present invention have been disclosed for illustrative purposes, those skilled in the art will appreciate that various modifications, additions and substitutions are possible without departing from the scope and spirit of the invention as disclosed in the accompanying claims.