1. A system in which a codeword from an error correcting code with minimum Hamming distance d has symbols in which the positions of d\u22121 of the parity check symbols may be arbitrarily chosen,
and each of the d\u22121 parity check symbols is produced by solving a parity check equation that is independent of all of the other parity check symbols and formed by multiplying together d\u22121 codewords from the dual code of the error correcting code, symbol by symbol,
starting with a codeword of the dual code that has a non zero symbol in the position of the parity check symbol being solved,
and in which each of the other d\u22122 dual code codewords comprise symbols given by the difference between an incrementally increasing power of a primitive root of unity and the primitive root of unity raised to a power equal to the position of each of the other d\u22122 parity check symbols,
and a total of d\u22121 parity check equations are so formed in parallel and used simultaneously to solve for the d\u22121 parity check symbols which together with the information symbols form an encoded codeword.
2. A system in which a codeword from an error correcting code with minimum Hamming distance d has symbols in which the positions of d\u22121 of the parity check symbols may be arbitrarily chosen,
and one of the d\u22121 parity check symbols is produced by solving a parity check equation that is independent of all of the other parity check symbols and formed by multiplying together d\u22121 codewords from the dual code, symbol by symbol,
and each of the d\u22121 parity check equations resulting from each dual code codeword multiplication is retained in the order they are calculated,
starting with an initial codeword of the dual code that has a non zero symbol in the position of the parity check symbol being solved,
and in which each of the other d\u22122 dual code codewords comprise symbols given by the difference between an incrementally increasing power of a primitive root of unity and the primitive root of unity raised to a power equal to the position of one of the other parity check symbols taken in turn,
and the resulting parity check equation is used to solve the parity check symbol,
and the solved parity check symbol is appended with the information symbols to form a partially solved codeword,
and each of the retained parity check equations are used in reverse order on the partially solved codeword to solve for each of the other parity check symbols in turn with each solved parity check symbol substituted into the partially solved codeword to update the partially solved codeword,
until the parity check equation corresponding to the initial codeword of the dual code is used to solve the last parity check symbol which is substituted into the partially solved codeword to form the encoded codeword.
3. A system in which e correctable symbols of a codeword from an error correcting code are unknown,
and each of the e unknown symbols is determined by solving a parity check equation that is independent of all of the other unknown symbols and formed by multiplying together e codewords from the dual code of the error correcting code, symbol by symbol,
starting with a codeword of the dual code that has a non zero symbol in the position of the unknown symbol being solved,
and in which each of the other e\u22121 dual code codewords comprise symbols given by the difference between an incrementally increasing power of a primitive root of unity and the primitive root of unity raised to a power equal to the position of each of the other e\u22121 unknown symbols,
and a total of e parity check equations are so formed in parallel and used simultaneously to solve for the e unknown symbols which together with the known symbols form the original codeword.
4. A system in which e correctable symbols of a codeword from an error correcting code are unknown,
and a parity check equation for one unknown symbol is produced that is independent of all of the other e\u22121 unknown symbols by multiplying together e codewords from the dual code of the error correcting code, symbol by symbol,
and where each of the e parity check equations resulting from each dual code codeword multiplication are retained in the order they are calculated,
starting with an initial codeword of the dual code that has a non zero symbol in the position of the unknown symbol being solved,
and in which each of the other e\u22121 dual code codewords are formed from the difference between an incrementally increasing power of a primitive root of unity and the primitive root of unity raised to a power equal to the position of each of the other unknown symbols taken in turn,
and the resulting parity check equation is used to solve the unknown symbol,
and the solved unknown symbol is appended with the known symbols to form a partially solved codeword,
and each of the retained parity check equations are used in reverse order on the partially solved codeword to solve for each of the other unknown symbols in turn with each solved symbol substituted into the partially solved codeword to update the partially solved codeword,
until the parity check equation corresponding to the initial codeword of the dual code is used to solve the last unknown symbol which is substituted into the partially solved codeword to form the original codeword.
5. A system according to claim 1 in which the symbols of the error correcting code and dual code are from the field of complex numbers.
6. A system according to claim 2 in which the symbols of the error correcting code and dual code are from the field of complex numbers.
7. A system according to claim 3 in which the symbols of the error correcting code and dual code are from the field of complex numbers.
8. A system according to claim 4 in which the symbols of the error correcting code and dual code are from the field of complex numbers.
9. A system according to claim 1 except that some products of dual code codewords, symbol by symbol, are not calculated as required but are pre-calculated and stored in memory in one or more look up tables,
and the required dual code codewords are accessed from the sets of dual code codewords stored in one or more look up tables.
10. A system according to claim 2 except that some products of dual code codewords, symbol by symbol, are not calculated as required but are pre-calculated and stored in memory in one or more look up tables,
and the required dual code codewords are accessed from the sets of dual code codewords stored in one or more look up tables.
11. A system according to claim 3 except that some products of dual code codewords, symbol by symbol, are not calculated as required but are pre-calculated and stored in memory in one or more look up tables,
and the required dual code codewords are accessed from the sets of dual code codewords stored in one or more look up tables.
12. A system according to claim 4 except that some products of dual code codewords, symbol by symbol, are not calculated as required but are pre-calculated and stored in memory in one or more look up tables,
and the required dual code codewords are accessed from the sets of dual code codewords stored in one or more look up tables.
The claims below are in addition to those above.
All refrences to claim(s) which appear below refer to the numbering after this setence.
1. An apparatus for determining a NOx content, an ammonia slip content, or both a NOx content and an ammonia slip content of a gas stream comprising:
a container;
an ammonia slip catalyst positioned in said container;
said container having a sample inlet upstream of said ammonia slip catalyst for receiving a sample of a gas stream;
a first NOx sensor having a sensor element positioned in said container upstream of said ammonia slip catalyst; and
a second NOx sensor having a sensor element positioned in said container downstream of said ammonia slip catalyst;
wherein said first and said second NOx sensors are each of a type which will quantify both NOx and any ammonia present in said sample as NOx.
2. The apparatus of claim 1 wherein said first and said second NOx sensors are automotive-type NOx sensors.
3. The apparatus of claim 1 wherein;
said container further comprises a sample outlet downstream of said ammonia slip catalyst and
said apparatus further comprises a suction device connected to said sample outlet for drawing said sample into said sample inlet, through said ammonia slip catalyst, and out of said sample outlet.
4. The apparatus of claim 3 wherein said suction device comprises a venturi vacuum element.
5. The apparatus of claim 3 further comprising a sampling tube for delivering said sample to said sample inlet of said container, said sampling tube having a series of sample receiving openings provided therein.
6. The apparatus of claim 5 wherein:
said sample receiving openings each have a diameter of at least \xbc inch,
said series of said sample receiving openings extends from a first to a last of said sample receiving openings, and
said series of said sample receiving openings has from about 2 to about 6 of said sample receiving openings per foot.
7. The apparatus of claim 1 wherein said first NOx sensor also measures an oxygen concentration value for said sample.
8. An apparatus for selective catalytic reduction comprising:
a SCR catalyst;
a flow passageway extending downstream from said SCR catalyst;
a container, at least a portion of said container being positioned in said flow passageway;
an ammonia slip catalyst positioned in said container;
said container having a sample inlet upstream of said ammonia slip catalyst, said sample inlet being in fluid communication with said flow passageway for receiving a gas sample from said flow passageway;
a first NOx sensor having a sensor element positioned in said container upstream of said ammonia slip catalyst; and
a second NOx sensor having a sensor element positioned in said container downstream of said ammonia slip catalyst,
wherein said first and said second NOx sensors are each of a type which will quantify both NOx and any ammonia present in said gas sample as NOx.
9. The apparatus of claim 8 wherein said first and said second NOx sensors are automotive-type NOx sensors.
10. The apparatus of claim 8 wherein:
said container further comprises a sample outlet downstream of said ammonia slip catalyst and said apparatus further comprises a suction device connected to said sample outlet for drawing said gas sample from said gas flow passageway into said sample inlet, through said ammonia slip catalyst, and out of said sample outlet.
11. The apparatus of claim 10 wherein said suction device has a discharge outlet located in said flow passageway.
12. The apparatus of claim 11 wherein said suction device comprises a venturi vacuum element positioned in said flow passageway.
13. The apparatus of claim 12 further comprising a pressurized air line extending into said flow passageway to said venturi vacuum element.
14. The apparatus of claim 10 further comprising a sampling tube for delivering said sample to said sample inlet of said container, said sampling tube having a series of sample receiving openings provided therein and said series of sample receiving openings being positioned in said flow passageway.
15. The apparatus of claim 14 wherein said series of sample receiving openings of said sampling tube extends traversely across at least most of a cross-sectional width, a cross-sectional height, a cross-sectional diagonal dimension, or a cross-sectional diameter of said flow passageway.
16. A method of determining a NOx content, an ammonia slip content, or both a NOx content and an ammonia slip content of a gas stream comprising the steps of:
(a) receiving a sample of said gas stream in an analyzer container having an ammonia slip catalyst therein;
(b) measuring a first NOx content value of said sample in said analyzer container upstream of said ammonia slip catalyst using a first NOx sensor which quantifies both NOx and any ammonia present in said sample as NOx;
(c) conducting said sample through said ammonia slip catalyst; and
(d) measuring a second NOx content value of said sample in said analyzer container downstream of said ammonia slip catalyst using a second NOx sensor which quantifies both NOx and any ammonia present in said sample as NOx.
17. The method of claim 16 wherein said ammonia slip catalyst operates in step (c) to convert at least most, if any, ammonia slip material present in said sample to reaction products comprising nitrogen and water.
18. The method of claim 16 wherein said first and said second NOx sensors are automotive-type NOx sensors.
19. The method of claim 16 further comprising the step of heating said ammonia slip catalyst by contacting at least a portion of said analyzer container with said gas stream.
20. The method of claim 16 wherein:
said analyzer container has a sample outlet downstream of said ammonia slip catalyst and
said sample is drawn into said analyzer container in step (a) and through said ammonia slip catalyst in step (c) using a venturi vacuum element which also pulls said sample out of said sample outlet.
21. The method of claim 20 wherein said venturi vacuum element discharges said sample back into said gas stream.
22. The method of claim 16 further comprising the step of determining a NOx content value, an ammonia slip content value, or both for said sample of said gas stream by comparing said second NOx content value of said sample to said first NOx content value of said sample.
23. The method of claim 16 wherein:
said gas stream is flowing through a flow passageway having a cross-sectional width, a cross-sectional height, a cross-sectional diameter, or other cross-sectional dimension of at least 20 inches and
said sample is drawn into said analyzer container from said flow passageway in step (a) through a sample tube having a series of inlet openings which extend transversely in said flow passageway across at least most of said cross-sectional dimension.
24. A method for selective catalytic reduction of a gas stream comprising the steps of:
(a) adding an ammonia source material to said gas stream at an addition rate;
(b) delivering said gas stream through an SCR catalyst;
(c) obtaining a sample of said gas stream after step (b);
(d) measuring a first NOx content value of said sample using a first NOx sensor which quantifies both NOx and any ammonia present in said sample as NOx;
(e) delivering said sample through an ammonia slip catalyst after step (d);
(f) measuring a second NOx content value of said sample after step (e) using a second NOx sensor which quantifies both NOx and any ammonia present in said sample as NOx; and
(g) controlling or correcting said addition rate of said ammonia source material used in step (a) based at least in part on a comparison of said first and said second NOx content values measured in steps (d) and (f).
25. The method of claim 24 wherein said ammonia slip catalyst operates in step (e) to convert at least most, if any, ammonia slip material present in said sample to other reaction products, said other reaction products comprising nitrogen and water.
26. The method of claim 24 wherein said ammonia source material is urea or ammonia.
27. The method of claim 24 where said first and said second NOx sensors are automotive-type NOx sensors.
28. The method of claim 24 wherein;
said ammonia slip catalyst is located in an analyzer container;
said first NOx sensor has a sensor element located in said analyzer container between said ammonia slip catalyst and an inlet of said analyzer container; and
said second NOx sensor has a sensor element located in said analyzer container between said ammonia slip catalyst and an outlet of said analyzer container.
29. The method of claim 28 further comprising the step of heating said ammonia slip catalyst by contacting at least a portion of said analyzer container with said gas stream.
30. The method of claim 28 wherein said sample is obtained in step (c) by drawing said sample into said inlet of said analyzer container using a suction device connected to said outlet of said analyzer container.
31. The method of claim 30 wherein said suction device comprises a venturi vacuum element which returns said sample to said gas stream.
32. The method of claim 28 wherein:
in step (c), said gas stream is flowing through a flow passageway having a cross-sectional width, a cross-sectional height, a cross-sectional diameter, and other cross-sectional dimension of at least 20 inches and
said sample is obtained in step (c) by drawing said sample into said inlet of said analyzer container through a sample tube having a series of inlet openings which extend transversely in said flow passageway across at least most of said cross-sectional dimension.
33. The method of claim 32 wherein at least a portion of said analyzer container is positioned in said flow passageway.