Residue division multiplexing system and apparatus for discrete-time signals

Claim 2

A transmitter, comprising: a terminal for receiving M subchannel signals X m (z) (m=0 to M−1) of length greater than one, a multiplexing apparatus that is responsive to the received M subchannel signals X m (z) (m=0 to M−1) of length greater than one, to produces a multiplexed signal X(z), the multiplexing apparatus comprising means of obtaining a polynomial X(z) that satisfies X(z)≡X m (z)mod(P m (z)) (m=0 to M−1), wherein P m (z) (m=0 to M−1) are relatively prime polynomials; and means for transmitting the multiplexed signal X(z).

Claim 3

The transmitter of claim 1 , wherein the multiplexing apparatus includes a Fast Fourier Transform processor.

Claim 4

The transmitter of claim 1 , wherein the multiplexing apparatus includes a digital signal processor.

Claim 5

The transmitter of claim 1 , wherein the multiplexing apparatus includes an LSI processor.

Claim 6

The transmitter of claim 1 , wherein the polynomials P m (z) (m=0 to M−1) have the form (1e j2πm/M z −K ) (m=0 to M−1), wherein K is greater than one.

Claim 7

The transmitter of claim 1 , wherein the polynomials P m (z) (m=0 to M−1) have the form (1−e j2πm/M r 0 z −1 )(1−e j2πm/M r 1 z −1 ) . . . (1−e j2πm/M r K−1 z −1 ))(m=0 to M−1), where r i is a non-zero complex number (i=0 to K−1); K greater than one.

Claim 8

The multiplexing apparatus of claim 1 , in which arithmetic is performed on a finite field and (1−α m β 0 z −1 )(1−α m β 1 z −1 ) . . . (1−α m β K−1 z −1 ) is used as P m (z) (m=0 to M−1), wherein β i is a non-zero element (i=0 to K−1), and α is a primitive Mth root of unity in the finite field.

Claim 9

A receiver, comprising means for receiving a receiver side multiplexed signal Y(z), and a demultiplexing apparatus that takes the receiver side multiplexed signal Y(z), and produces receiver side subchannel signals Y m (z) (m=0 to M−1), the demultiplexing apparatus comprising means of computing Y m (z)≡Y(z)mod(P m (z)) (m=0 to M−1), wherein P m (z) (m=0 to M−1) are relatively prime polynomials.

Claim 10

A receiver, comprising means for receiving a receiver side multiplexed signal Y(z), and a demultiplexing apparatus that takes the receiver side multiplexed signal Y(z), and produces receiver side subchannel signals Y m (z) (m=0 to M−1), the demultiplexing apparatus comprising means of computing Y m (z)≡Y(z)mod(P m (z)) (m=0 to M−1), wherein P m (z) (m=0 to M−1) are relatively prime polynomials.

Claim 11

The receiver of claim 8 , wherein said demultiplexing apparatus includes an Inverse Fast Fourier Transform processor.

Claim 12

The receiver of claim 8 , wherein said demultiplexing apparatus includes a digital signal processor.

Claim 13

The receiver of claim 8 , wherein said demultiplexing apparatus includes an LSI processor.

Claim 14

The receiver of claim 8 , wherein the polynomials P m (z) (m=0 to M−1) have the form (1−e j2πm/M z −K ) (m=0 to M−1), wherein K is greater than one.

Claim 15

The receiver of claim 8 , wherein the polynomials P m (z) (m=0 to M−1) have the form (1−e j2πm/M r 0 z −1 )(1−e j2πm/M r 1 z −1 ) . . . (1−e j2πm/M r K−1 z −1 )) (m=0 to M−1), where r i is a non-zero complex number (i=0 to K−1); K greater than one.

Claim 16

The demultiplexing apparatus of claim 8 , in which arithmetic is performed on a finite field, and (1−α m β 0 z −1 )(1−α m β 1 z −1 ) . . . (1−α m β K−1 z −1 ) is used as P m (z) (m=0to M−1), wherein β i is a non-zero element (i=0 to K−1), and α is a primitive Mth root of unity in the finite field.

Claim 17

A multiplexing system for transmitting subchannel signals over a communication channel, the system comprising: a transmitter, including a terminal for receiving M subchannel signals X m (z) (m=0 to M−1) of length greater than one, a multiplexing apparatus, the multiplexing apparatus taking the received M subchannel signals X m (z) (m=0 to M−1) of length greater than one, and producing a multiplexed signal X(z) to be transmitted, the multiplexing apparatus comprising means of obtaining a polynomial X(z) that satisfies X(z)≡X m (z)mod(P m (z)) (m=0 to M−1), wherein P m (z) (m=0 to M−1) are relatively prime polynomials, and means for transmitting the multiplexed signal X(z) over the communication channel; and a receiver, including means for receiving a receiver side multiplexed signal Y(z) from the communication channel, including the signal X(z) transmitted from the transmitter, and a demultiplexing apparatus that takes the received multiplexed signal Y(z), and produces receiver side subchannel signals Y m (z) (m=0 to M−1), the demultiplexing apparatus comprising means of computing Y m (z)≡Y(z)mod(P m (z)) (m=0 to M−1), wherein P m (z) (m=0 to M−1) are relatively prime polynomials.

Claim 18

A multiplexing system for transmitting subchannel signals over a communication channel, the system comprising: a transmitter, including a terminal for receiving M subchannel signals X m (z) (m=0 to M−1) of length greater than one, a multiplexing apparatus, the multiplexing apparatus taking the received M subchannel signals X m (z) (m=0 to M−1) of length greater than one, and producing a multiplexed signal X(z) to be transmitted, the multiplexing apparatus comprising means of obtaining a polynomial X(z) that satisfies X(z)≡X m (z)mod(P m (z)) (m=0 to M−1), wherein P m (z) (m=0 to M−1) are relatively prime polynomials, and means for transmitting the multiplexed signal X(z) over the communication channel; and a receiver, including means for receiving a receiver side multiplexed signal Y(z) from the communication channel, including the signal X(z) transmitted from the transmitter, and a demultiplexing apparatus that takes the received multiplexed signal Y(z), and produces receiver side subchannel signals Y m (z) (m=0 to M−1), the demultiplexing apparatus comprising means of computing Y m (z)≡Y(z)mod(P m (z)) (m=0 to M−1), wherein P m (z) (m=0 to M−1) are relatively prime polynomials.

Claim 19

The multiplexing system of claim 15 , wherein the transmitter includes a modulator and the receiver includes a demodulator, further comprising a linear channel though which the multiplexed signal X(z) transmitted from the modulator to the demodulator.

Claim 20

The multiplexing system of claim 15 , wherein said multiplexing apparatus and said demultiplexing apparatus each include a digital signal processor, respectively implemented as a Fast Fourier Transform processor and an Inverse Fast Fourier Transform processor.

Claim 21

The multiplexing system of claim 15 , wherein said multiplexing apparatus and said demultiplexing apparatus each includes an LSI processor, respectively implemented as a Fast Fourier Transform processor to perform a discrete Fourier Transform and an Inverse Fast Fourier Transform processor to perform an inverse discrete Fourier Transform.

Claim 22

A multiplexing system for transmitting subchannel signals over a communication channel, the system comprising: a transmitter, including a terminal for receiving M subchannel signals X m (z) (m=0 to M−1) of length greater than one, a multiplexing apparatus, the multiplexing apparatus taking the received M subchannel signals X m (z) (m=0 to M−1) of length greater than one, and producing a multiplexed signal X(z) to be transmitted, the multiplexing apparatus comprising means of obtaining a polynomial X(z) that satisfies X(z)≡X m (z)mod(1−e j2πm/M z −K ) (m=0 to M−1), where K is greater than one, and means for transmitting the multiplexed signal X(z); and a receiver, including means for receiving that a receiver side multiplexed signal Y(z), including the multiplexed signal X(z) transmitted from the transmitter, and a demultiplexing apparatus that takes the received multiplexed signal Y(z), and produces receiver side subchannel signals Y m (z) (m=0 to M−1), the demultiplexing apparatus comprising means of computing Y m (z)≡Y(z)mod(1−e j2πm/M z −K ) (m=0 to M−1) where K is greater than one.

Claim 23

A multiplexing system for transmitting subchannel signals over a communication channel, the system comprising: a transmitter, including a terminal for receiving M subchannel signals X m (z) (m=0 to M−1) of length greater than one, a multiplexing apparatus, the multiplexing apparatus taking the received M subchannel signals X m (z) (m=0 to M−1) of length greater than one, and producing a multiplexed signal X(z) to be transmitted, the multiplexing apparatus comprising means of obtaining a polynomial X(z) that satisfies X(z)≡X m (z)mod(1−e j2πm/M z −K ) (m=0 to M−1), where K is greater than one, and means for transmitting the multiplexed signal X(z); and a receiver, including means for receiving that a receiver side multiplexed signal Y(z), including the multiplexed signal X(z) transmitted from the transmitter, and a demultiplexing apparatus that takes the received multiplexed signal Y(z), and produces receiver side subchannel signals Y m (z) (m=0 to M−1), the demultiplexing apparatus comprising means of computing Y m (z)≡Y(z)mod(1−e j2πm/M z −K ) (m=0 to M−1) where K is greater than one.

Claim 24

A demultiplexing apparatus that takes a receiver side multiplexed signal Y(z), and produces receiver side subchannel signals Y m (z) (m=0 to M−1), the demultiplexing apparatus comprising: means of performing K-fold polyphase decomposition of the receiver side multiplexed signal, and producing K polyphase sequences of length M, K an integer greater than 1; means of computing M-point DFT for all the polyphase sequences; means of arranging the DFT output samples in M groups of K samples such that each group contains samples at the same position in the DFT outputs, taking one from each DFT output; and means of producing M polynomials of degree K−1 by assigning samples in each group to coefficients of each polynomial in the order of the polyphase decomposition.

Claim 25

A demultiplexing apparatus that takes a receiver side multiplexed signal Y(z), and produces receiver side subchannel signals Y m (z) (m=0 to M−1), the demultiplexing apparatus comprising: means of performing K-fold polyphase decomposition of the receiver side multiplexed signal, and producing K polyphase sequences of length M, K an integer greater than 1; means of computing M-point DFT for all the polyphase sequences; means of arranging the DFT output samples in M groups of K samples such that each group contains samples at the same position in the DFT outputs, taking one from each DFT output; and means of producing M polynomials of degree K−1 by assigning samples in each group to coefficients of each polynomial in the order of the polyphase decomposition.

Claim 26

The demultiplexing apparatus of in claim 20 , in which arithmetic is performed on an integer residue ring, and the DFT is replaced by the number theoretic transform.

Claim 27

A demultiplexing apparatus according to claim 20 , in which arithmetic is performed on a finite field, and the DFT is replaced by the DFT-like transform on the finite field.

Claim 28

A multiplexing system for transmitting subchannel signals over a communication channel, the system comprising: a transmitter, including means for receiving M subchannel signals X m (z) (m=0 to M−1) including K ordered positions, where K is greater than one, a multiplexer including means of extracting a sample from each of the K positions of each of the M subchannel signals, means of arranging the MK samples in K groups of M samples such that each group contains samples corresponding to the same position in the order of K positions in the subchannel signals, means of arranging the samples of each group in the order of subchannels, and computing M-point inverse DFT of the arranged samples for all of the groups, and means of producing an MK-length signal by performing K-fold polyphase composition of the inverse DFT outputs, and means for transmitting the MK-length signal; and a receiver, including means for receiving a receiver side multiplexed signal Y(z) including the MK-length signal transmitted from the transmitter, and a demultiplexer that produces receiver side subchannel signals Y m (z) (m=0 to M−1), from the receiver side multiplexed signal Y(z), including means of performing K-fold polyphase decomposition of the receiver side multiplexed signal, and producing K polyphase sequences of length M, means of computing M-point DFT for all the polyphase sequences; means of arranging the DFT output samples in M groups of K samples such that each group contains samples at the same position in the DFT outputs, taking one from each DFT output; and means of producing M polynomials of degree K−1 by assigning samples in each group to coefficients of each polynomial in the order of the polyphase decomposition.

Claim 29

A multiplexing system for transmitting subchannel signals over a communication channel, the system comprising: a transmitter, including means for receiving M subchannel signals X m (z) (m=0 to M−1) including K ordered positions, where K is greater than one, a multiplexer including means of extracting a sample from each of the K positions of each of the M subchannel signals, means of arranging the MK samples in K groups of M samples such that each group contains samples corresponding to the same position in the order of K positions in the subchannel signals, means of arranging the samples of each group in the order of subchannels, and computing M-point inverse DFT of the arranged samples for all of the groups, and means of producing an MK-length signal by performing K-fold polyphase composition of the inverse DFT outputs, and means for transmitting the MK-length signal; and a receiver, including means for receiving a receiver side multiplexed signal Y(z) including the MK-length signal transmitted from the transmitter, and a demultiplexer that produces receiver side subchannel signals Y m (z) (m=0 to M−1), from the receiver side multiplexed signal Y(z), including means of performing K-fold polyphase decomposition of the receiver side multiplexed signal, and producing K polyphase sequences of length M, means of computing M-point DFT for all the polyphase sequences; means of arranging the DFT output samples in M groups of K samples such that each group contains samples at the same position in the DFT outputs, taking one from each DFT output; and means of producing M polynomials of degree K−1 by assigning samples in each group to coefficients of each polynomial in the order of the polyphase decomposition.

Claim 30

A multiplexing system for transmitting subchannel signals over a communication channel, the system comprising: a transmitter, including means for receiving M subchannel signals X m (z) (m=0 to M−1) including K ordered positions, where K is greater than one, a multiplexer including means of extracting a sample from each of the K positions of each of the M subchannel signals, means of arranging the MK samples in K groups of M samples such that each group contains samples corresponding to the same position in the order of K positions in the subchannel signals, means of arranging the samples of each group in the order of subchannels, and computing M-point inverse number theoretic transform of the arranged samples for all of the groups, and means of producing MK-length signal by performing K-fold polyphase composition of the inverse number theoretic transform outputs, wherein arithmetic is performed on an integer residue ring, and means for transmitting the MK-length signal; and a receiver, including means for receiving a receiver side multiplexed signal Y(z), including the MK-length transmitted by the transmitter, and a demultiplexer that produces receiver side subchannel signals Y m (z) (m=0 to M−1), from the receiver side multiplexed signal Y(z), including means of performing K-fold polyphase decomposition of the receiver side multiplexed signal, and producing K polyphase sequences of length M, means of computing M-point number theoretic transform for all the polyphase sequences, means of arranging the number theoretic transform output samples in M groups of K samples such that each group contains samples at the same position in the number theoretic transform outputs, taking one from each number theoretic transform output, and means of producing M polynomials of degree K−1 by assigning samples in each group to coefficients of each polynomial in the order of the polyphase decomposition, wherein arithmetic is performed on an integer residue ring.

Claim 31

A multiplexing system for transmitting subchannel signals over a communication channel, the system comprising: a transmitter, including means for receiving M subchannel signals X m (z) (m=0 to M−1) including K ordered positions, where K is greater than one, a multiplexer including means of extracting a sample from each of the K positions of each of the M subchannel signals, means of arranging the MK samples in K groups of M samples such that each group contains samples corresponding to the same position in the order of K positions in the subchannel signals, means of arranging the samples of each group in the order of subchannels, and computing M-point inverse number theoretic transform of the arranged samples for all of the groups, and means of producing MK-length signal by performing K-fold polyphase composition of the inverse number theoretic transform outputs, wherein arithmetic is performed on an integer residue ring, and means for transmitting the MK-length signal; and a receiver, including means for receiving a receiver side multiplexed signal Y(z), including the MK-length transmitted by the transmitter, and a demultiplexer that produces receiver side subchannel signals Y m (z) (m=0 to M−1), from the receiver side multiplexed signal Y(z), including means of performing K-fold polyphase decomposition of the receiver side multiplexed signal, and producing K polyphase sequences of length M, means of computing M-point number theoretic transform for all the polyphase sequences, means of arranging the number theoretic transform output samples in M groups of K samples such that each group contains samples at the same position in the number theoretic transform outputs, taking one from each number theoretic transform output, and means of producing M polynomials of degree K−1 by assigning samples in each group to coefficients of each polynomial in the order of the polyphase decomposition, wherein arithmetic is performed on an integer residue ring.

Claim 32

A multiplexing system for transmitting subchannel signals over a communication channel, the system comprising: a transmitter, including means for receiving M subchannel signals X m (z) (m=0 to M−1) including K ordered positions, where K is greater than one, a multiplexer that performs arithmetic on the finite field, the multiplexer including means of extracting a sample from each of the K positions of each of the M subchannel signals, means of arranging the MK samples in K groups of M samples such that each group contains samples corresponding to the same position in the order of K positions in the subchannel signals, means of arranging the samples of each group in the order of subchannels, and computing M-point inverse of DFT-like transform on a finite field, of the arranged samples for all of the groups, and means of producing MK-length signal by performing K-fold polyphase composition of the inverse of DFT-like transform outputs, and means for transmitting the MK-length signal over the communication channel; and a receiver, including means for receiving a receiver side multiplexed signal Y(z), including the MK-length signal transmitted from the transmitter, and a demultiplexer that produces receiver side subchannel signals Y m (z) (m=0 to M−1), from the receiver side multiplexed signal Y(z), performing arithmetic on the finite field, the demultiplexer including means of performing K-fold polyphase decomposition of the receiver side multiplexed signal, and producing K polyphase sequences of length M, means of computing M-point DFT-like transform on a finite field, for all the polyphase sequences, means of arranging the DFT-like transform output samples in M groups of K samples such that each group contains samples at the same position in the DFT-like transform outputs, taking one from each DFT-like transform output, and means of producing M polynomials of degree K−1 by assigning samples in each group to coefficients of each polynomial in the order of the polyphase decomposition.

Claim 33

A multiplexing system for transmitting subchannel signals over a communication channel, the system comprising: a transmitter, including means for receiving M subchannel signals X m (z) (m=0 to M−1) including K ordered positions, where K is greater than one, a multiplexer that performs arithmetic on the finite field, the multiplexer including means of extracting a sample from each of the K positions of each of the M subchannel signals, means of arranging the MK samples in K groups of M samples such that each group contains samples corresponding to the same position in the order of K positions in the subchannel signals, means of arranging the samples of each group in the order of subchannels, and computing M-point inverse of DFT-like transform on a finite field, of the arranged samples for all of the groups, and means of producing MK-length signal by performing K-fold polyphase composition of the inverse of DFT-like transform outputs, and means for transmitting the MK-length signal over the communication channel; and a receiver, including means for receiving a receiver side multiplexed signal Y(z), including the MK-length signal transmitted from the transmitter, and a demultiplexer that produces receiver side subchannel signals Y m (z) (m=0 to M−1), from the receiver side multiplexed signal Y(z), performing arithmetic on the finite field, the demultiplexer including means of performing K-fold polyphase decomposition of the receiver side multiplexed signal, and producing K polyphase sequences of length M, means of computing M-point DFT-like transform on a finite field, for all the polyphase sequences, means of arranging the DFT-like transform output samples in M groups of K samples such that each group contains samples at the same position in the DFT-like transform outputs, taking one from each DFT-like transform output, and means of producing M polynomials of degree K−1 by assigning samples in each group to coefficients of each polynomial in the order of the polyphase decomposition.

Claim 34

A multiplexing system for transmitting subchannel signals over a communication channel, the system comprising: a transmitter, including a terminal for receiving M subchannel signals X m (z) (m=0 to M−1) of length greater than one, a multiplexing apparatus that is responsive to the received M subchannel signals X m (z) (m=0 to M−1) of length greater than one, to produces a multiplexed signal X(z), the multiplexing apparatus comprising means of obtaining a polynomial X(z) that satisfies X(z)≡X m (z)mod((1−e j2πm/M r 0 z −1 ) (1−e j2πm/M r 1 z −1 ) . . . (1−e j2πm/M r K−1 z −1 )) (m=0 to M−1) wherein r i is a non-zero complex number (i=0 to K−1), and means for transmitting the multiplexed signal X(z) over the communication channel; and a receiver, including means for receiving a receiver side multiplexed signal Y(z), including the multiplexed signal X(z) transmitted from the transmitter, and a demultiplexing apparatus that takes the receiver side multiplexed signal Y(z), and produces receiver side subchannel signals Y m (z) (m=0 to M−1), the demultiplexing apparatus comprising means of computing Y m (z)≡Y(z)mod((1−e j2πm/M r 0 z −1 )(1−e j2πm/M r 1 z −1 ) . . . (1−e j2πm/M r K−1 z −1 ) (m=0 to M−1), wherein r i is a non-zero complex number (i=0 to K−1).

Claim 35

A multiplexing system for transmitting subchannel signals over a communication channel, the system comprising: a transmitter, including a terminal for receiving M subchannel signals X m (z) (m=0 to M−1) of length greater than one, a multiplexing apparatus that is responsive to the received M subchannel signals X m (z) (m=0 to M−1) of length greater than one, to produces a multiplexed signal X(z), the multiplexing apparatus comprising means of obtaining a polynomial X(z) that satisfies X(z)≡X m (z)mod((1−e j2πm/M r 0 z −1 ) (1−e j2πm/M r 1 z −1 ) . . . (1−e j2πm/M r K−1 z −1 )) (m=0 to M−1) wherein r i is a non-zero complex number (i=0 to K−1), and means for transmitting the multiplexed signal X(z) over the communication channel; and a receiver, including means for receiving a receiver side multiplexed signal Y(z), including the multiplexed signal X(z) transmitted from the transmitter, and a demultiplexing apparatus that takes the receiver side multiplexed signal Y(z), and produces receiver side subchannel signals Y m (z) (m=0 to M−1), the demultiplexing apparatus comprising means of computing Y m (z)≡Y(z)mod((1−e j2πm/M r 0 z −1 )(1−e j2πm/M r 1 z −1 ) . . . (1−e j2πm/M r K−1 z −1 ) (m=0 to M−1), wherein r i is a non-zero complex number (i=0 to K−1).

Claim 36

A transmitter, comprising: a terminal for receiving M subchannel signals X m (z) (m=0 to M−1) of length greater than one, a multiplexing apparatus that is responsive to the received M subchannel signals X m (z) (m=0 to M−1) of length K where K is greater than one, the multiplexing apparatus including a Chinese remainder computation unit which obtains a polynomial A(z) that satisfies A(z)≡A l (z)mod((1−e j2π(f+l)/t r 0 d/t z −d/t ) (1−e j2π(f+l)/t r 1 d/t z −d/t ) . . . (1−e j2π(f+l)/t r K−1 d/t z −d/t )) (l=0 to t−1) for given A l (z), wherein r i is a non-zero complex number (i=0 to K−1), and t is a positive integer that divides d, and means of obtaining X(z) as a residue polynomial on mod((1−r 0 M z −M )(1−r 1 M z −M ) . . . (1−r K−1 M z −M m)) by applying the Chinese remainder computation unit recursively, starting with the subchannel signals X m (z) regarding them as residue polynomials on mod((1e j2πm/M r 0 z −1 )(1−e j2πm/M r 1 z −1 ) . . . (1−e j2πm/M r K−1 z −1 )) (m=0 to M−1); and means for transmitting the multiplexed signal X(z).

Claim 37

A transmitter, comprising: a terminal for receiving M subchannel signals X m (z) (m=0 to M−1) of length greater than one, a multiplexing apparatus that is responsive to the received M subchannel signals X m (z) (m=0 to M−1) of length K where K is greater than one, the multiplexing apparatus including a Chinese remainder computation unit which obtains a polynomial A(z) that satisfies A(z)≡A l (z)mod((1−e j2π(f+l)/t r 0 d/t z −d/t ) (1−e j2π(f+l)/t r 1 d/t z −d/t ) . . . (1−e j2π(f+l)/t r K−1 d/t z −d/t )) (l=0 to t−1) for given A l (z), wherein r i is a non-zero complex number (i=0 to K−1), and t is a positive integer that divides d, and means of obtaining X(z) as a residue polynomial on mod((1−r 0 M z −M )(1−r 1 M z −M ) . . . (1−r K−1 M z −M m)) by applying the Chinese remainder computation unit recursively, starting with the subchannel signals X m (z) regarding them as residue polynomials on mod((1e j2πm/M r 0 z −1 )(1−e j2πm/M r 1 z −1 ) . . . (1−e j2πm/M r K−1 z −1 )) (m=0 to M−1); and means for transmitting the multiplexed signal X(z).

Claim 38

A receiver, comprising means for receiving a receiver side multiplexed signal Y(z), and a demultiplexing apparatus that takes the receiver side multiplexed signal Y(z), and produces receiver side subchannel signals Y m (z) (m=0 to M−1), the demultiplexing apparatus including residue computation unit which computes A l (z)≡A(z)mod((1−e j2π(f+l)/t r 0 d/t z −d/t )(1−e j2π(f+l)/t r 1 d/t z −d/t ) . . . (1−e j2π(f+l)/t r K−1 d/t z −d/t )) (l=0 to t−1) for a given polynomial A(z), wherein r i is a non-zero complex number (i=0 to K−1), and t is a positive integer that divides d, and means of obtaining Y m (z) as a residue polynomial on mod((1−e j2πm/M r 0 z −1 )(1−e j2πm/M r 1 z −1 ) . . . (1−e j2πm/M r K−1 z −1 )) (m=0 to M−1) by applying the residue computation unit recursively, starting with the receiver side multiplexed signal Y(z) regarding it as a residue polynomial on mod((1−r 0 M z −M )(1−r 1 M z −M ) . . . (1−r K−1 M z −M )).

Claim 39

A receiver, comprising means for receiving a receiver side multiplexed signal Y(z), and a demultiplexing apparatus that takes the receiver side multiplexed signal Y(z), and produces receiver side subchannel signals Y m (z) (m=0 to M−1), the demultiplexing apparatus including residue computation unit which computes A l (z)≡A(z)mod((1−e j2π(f+l)/t r 0 d/t z −d/t )(1−e j2π(f+l)/t r 1 d/t z −d/t ) . . . (1−e j2π(f+l)/t r K−1 d/t z −d/t )) (l=0 to t−1) for a given polynomial A(z), wherein r i is a non-zero complex number (i=0 to K−1), and t is a positive integer that divides d, and means of obtaining Y m (z) as a residue polynomial on mod((1−e j2πm/M r 0 z −1 )(1−e j2πm/M r 1 z −1 ) . . . (1−e j2πm/M r K−1 z −1 )) (m=0 to M−1) by applying the residue computation unit recursively, starting with the receiver side multiplexed signal Y(z) regarding it as a residue polynomial on mod((1−r 0 M z −M )(1−r 1 M z −M ) . . . (1−r K−1 M z −M )).

Claim 40

A multiplexing system for transmitting subchannel signals over a communication channel, the system comprising: transmitter, including a terminal for receiving M subchannel signals X m (z) (m=0 to M−1) of length K where K greater than one, a multiplexing apparatus that is responsive to the received M subchannel signals X m (z) (m=0 to M−1) of length K, the multiplexing apparatus including a Chinese remainder computation unit which obtains a polynomial A(z) that satisfies A(z)≡A l (z)mod((1−e j2π(f+l)/t r 0 d/t )(1−e j2π(f+l)/t r 1 d/t z −d/t ) . . . (1−e j2π(f+l)/t r K−1 d/t )) (l=0 to t−1) for given A l (z), wherein r i is a non-zero complex number (i=0 to K−1), and t is a positive integer that divides d, and means of obtaining X(z) as a residue polynomial on mod((1−r 0 M z −M )(1−r 1 M z −M ) . . . (1−r K−1 M z −M )) by applying the Chinese remainder computation unit recursively, starting with the subchannel signals X m (z) regarding them as residue polynomials on mod((1−e j2πm/M r 0 z −1 )(1−e j2πm/M r 1 z −1 ) . . . (1−e j2πm/M r K−1 z −1 ))(m=0 to M−1), and means for transmitting the multiplexed signal X(z) over the communication channel; and a receiver, including means for receiving from the communication channel a receiver side multiplexed signal Y(z), including the multiplexed signal X(z), and a demultiplexing apparatus that takes the receiver side multiplexed signal Y(z), and produces receiver side subchannel signals Y m (z) (m=0 to M−1), the demultiplexing apparatus including residue computation unit which computes A l (z)≡A(z)mod((1−e j2π(f+l)/t r 0 d/t z −d/t ) (1−e j2π(f+l)/t r 1 d/t z −d/t ) . . . (1−e j2π(f+l)/t r K−1 d/t z −d/t )) (l=0 to t−1) for a given polynomial A(z), wherein r i is a non-zero complex number (i=0 to K−1), and t is a positive integer that divides d, and means of obtaining Y m (z) as a residue polynomial on mod((1−e j2πm/M r 0 z −1 )(1−e j2πm/M r 1 z −1 ) . . . (1−e j2πm/M r K−1 z −1 )) (m=0 to M−1) by applying the residue computation unit recursively, starting with the receiver side multiplexed signal Y(z) regarding it as a residue polynomial on mod((1−r 0 M z −M )(1−r 1 M z −M ) . . . (1−r K−1 M z −M )).

Claim 41

A multiplexing system for transmitting subchannel signals over a communication channel, the system comprising: transmitter, including a terminal for receiving M subchannel signals X m (z) (m=0 to M−1) of length K where K greater than one, a multiplexing apparatus that is responsive to the received M subchannel signals X m (z) (m=0 to M−1) of length K, the multiplexing apparatus including a Chinese remainder computation unit which obtains a polynomial A(z) that satisfies A(z)≡A l (z)mod((1−e j2π(f+l)/t r 0 d/t )(1−e j2π(f+l)/t r 1 d/t z −d/t ) . . . (1−e j2π(f+l)/t r K−1 d/t )) (l=0 to t−1) for given A l (z), wherein r i is a non-zero complex number (i=0 to K−1), and t is a positive integer that divides d, and means of obtaining X(z) as a residue polynomial on mod((1−r 0 M z −M )(1−r 1 M z −M ) . . . (1−r K−1 M z −M )) by applying the Chinese remainder computation unit recursively, starting with the subchannel signals X m (z) regarding them as residue polynomials on mod((1−e j2πm/M r 0 z −1 )(1−e j2πm/M r 1 z −1 ) . . . (1−e j2πm/M r K−1 z −1 ))(m=0 to M−1), and means for transmitting the multiplexed signal X(z) over the communication channel; and a receiver, including means for receiving from the communication channel a receiver side multiplexed signal Y(z), including the multiplexed signal X(z), and a demultiplexing apparatus that takes the receiver side multiplexed signal Y(z), and produces receiver side subchannel signals Y m (z) (m=0 to M−1), the demultiplexing apparatus including residue computation unit which computes A l (z)≡A(z)mod((1−e j2π(f+l)/t r 0 d/t z −d/t ) (1−e j2π(f+l)/t r 1 d/t z −d/t ) . . . (1−e j2π(f+l)/t r K−1 d/t z −d/t )) (l=0 to t−1) for a given polynomial A(z), wherein r i is a non-zero complex number (i=0 to K−1), and t is a positive integer that divides d, and means of obtaining Y m (z) as a residue polynomial on mod((1−e j2πm/M r 0 z −1 )(1−e j2πm/M r 1 z −1 ) . . . (1−e j2πm/M r K−1 z −1 )) (m=0 to M−1) by applying the residue computation unit recursively, starting with the receiver side multiplexed signal Y(z) regarding it as a residue polynomial on mod((1−r 0 M z −M )(1−r 1 M z −M ) . . . (1−r K−1 M z −M )).

Claim 42

A transmitter, comprising: a terminal for receiving M subchannel signals X m (z) (m=0 to M−1) of length K where K is greater than one, a multiplexing apparatus that is responsive to the received M subchannel signals X m (z) (m=0 to M−1) of length K, and produces a MK-length multiplexed signal X(z) to be transmitted, the multiplexing apparatus including means of computing V i [m]≡X m (z)mod(1e j2πm/M r i z −1 ) (i=0 to K−1, and m=0 to M−1), wherein r i is a non-zero complex number (i=0 to K−1), means of obtaining a polynomial V i (z) that satisfies V i (z)≡V l [m]mod(1−e j2πm/M r i z −1 ) (m=0 to M−1, and i=0 to K−1), and means of obtaining X(z) that satisfies X(z)≡V i (z)mod(1−r i M z −M ) (i=0 to K−1); and means for transmitting the multiplexed signal X(z).

Claim 43

A transmitter, comprising: a terminal for receiving M subchannel signals X m (z) (m=0 to M−1) of length K where K is greater than one, a multiplexing apparatus that is responsive to the received M subchannel signals X m (z) (m=0 to M−1) of length K, and produces a MK-length multiplexed signal X(z) to be transmitted, the multiplexing apparatus including means of computing V i [m]≡X m (z)mod(1e j2πm/M r i z −1 ) (i=0 to K−1, and m=0 to M−1), wherein r i is a non-zero complex number (i=0 to K−1), means of obtaining a polynomial V i (z) that satisfies V i (z)≡V l [m]mod(1−e j2πm/M r i z −1 ) (m=0 to M−1, and i=0 to K−1), and means of obtaining X(z) that satisfies X(z)≡V i (z)mod(1−r i M z −M ) (i=0 to K−1); and means for transmitting the multiplexed signal X(z).

Claim 44

A receiver, comprising: means for receiving a receiver side multiplexed signal Y(z); and a demultiplexing apparatus that takes the receiver side multiplexed signal Y(z), and produces receiver side subchannel signals Y m (z), m=0, 1, . . . , M−1, the demultiplexing apparatus including means of computing U i (z)≡Y(z)mod(1−r i M z −M ) (i=0 to K−1), wherein r i is a non-zero complex number (i=0 to K−1), means of computing U i [m]≡U i (z)mod(1−e j2πm/M r i z −1 ) (m=0 to M−1, and i=0 to K−1), and means of obtaining Y m (z) that satisfies Y m (z)≡U i [m]mod(1−e j2πm/M r i z −1 ) (i=0 to K−1) (m=0 to M−1).

Claim 45

A receiver, comprising: means for receiving a receiver side multiplexed signal Y(z); and a demultiplexing apparatus that takes the receiver side multiplexed signal Y(z), and produces receiver side subchannel signals Y m (z), m=0, 1, . . . , M−1, the demultiplexing apparatus including means of computing U i (z)≡Y(z)mod(1−r i M z −M ) (i=0 to K−1), wherein r i is a non-zero complex number (i=0 to K−1), means of computing U i [m]≡U i (z)mod(1−e j2πm/M r i z −1 ) (m=0 to M−1, and i=0 to K−1), and means of obtaining Y m (z) that satisfies Y m (z)≡U i [m]mod(1−e j2πm/M r i z −1 ) (i=0 to K−1) (m=0 to M−1).

Claim 46

A multiplexing system for transmitting subchannel signals over a communication channel, the system comprising: a transmitter, including a terminal for receiving M subchannel signals X m (z) (m=0 to M−1) of length K where K is greater than one, a multiplexing apparatus that is responsive to the received M subchannel signals X m (z) (m=0 to M−1) of length K, and produces a MK-length multiplexed signal X(z) to be transmitted, the multiplexing apparatus including means of computing V i [m]X m (z)mod(1−e j2πm/M r i z −1 ) (i=0 to K−1, and m=0 to M−1), wherein r i is a non-zero complex number (i=0 to K−1), means of obtaining a polynomial V i (z) that satisfies V i (z)≡V l [m]mod(1−e j2πm/M r i z −1 ) (m=0 to M−1, and means of obtaining X(z) that satisfies X(z)≡V i (z)mod(1−r i M z −M ) (i=0 to K−1), and means for transmitting the multiplexed signal X(z) over the communication channel; and a receiver, including means for receiving from the communication channel a receiver side multiplexed signal Y(z), including the transmitted multiplexed signal X(z), and a demultiplexing apparatus that takes the receiver side multiplexed signal Y(z), and produces receiver side subchannel signals Y m (z), m=0, 1, . . . , M−1, the demultiplexing apparatus including means of computing U i (z)≡Y(z)mod(1−r i M z −M ) (i=0 to K−1), wherein r i is a non-zero complex number (i=0 to K−1), means of computing U i [m]≡U i (z)mod(1−e j2πm/M r i z −1 ) (m=0 to M−1, and i=0 to K−1), and means of obtaining Y m (z) that satisfies Y m (z)≡U i [m]mod(1−e j2πm/M r i z −1 ) (i=0 to K−1) (m=0 to M−1).

Claim 47

A multiplexing system for transmitting subchannel signals over a communication channel, the system comprising: a transmitter, including a terminal for receiving M subchannel signals X m (z) (m=0 to M−1) of length K where K is greater than one, a multiplexing apparatus that is responsive to the received M subchannel signals X m (z) (m=0 to M−1) of length K, and produces a MK-length multiplexed signal X(z) to be transmitted, the multiplexing apparatus including means of computing V i [m]X m (z)mod(1−e j2πm/M r i z −1 ) (i=0 to K−1, and m=0 to M−1), wherein r i is a non-zero complex number (i=0 to K−1), means of obtaining a polynomial V i (z) that satisfies V i (z)≡V l [m]mod(1−e j2πm/M r i z −1 ) (m=0 to M−1, and means of obtaining X(z) that satisfies X(z)≡V i (z)mod(1−r i M z −M ) (i=0 to K−1), and means for transmitting the multiplexed signal X(z) over the communication channel; and a receiver, including means for receiving from the communication channel a receiver side multiplexed signal Y(z), including the transmitted multiplexed signal X(z), and a demultiplexing apparatus that takes the receiver side multiplexed signal Y(z), and produces receiver side subchannel signals Y m (z), m=0, 1, . . . , M−1, the demultiplexing apparatus including means of computing U i (z)≡Y(z)mod(1−r i M z −M ) (i=0 to K−1), wherein r i is a non-zero complex number (i=0 to K−1), means of computing U i [m]≡U i (z)mod(1−e j2πm/M r i z −1 ) (m=0 to M−1, and i=0 to K−1), and means of obtaining Y m (z) that satisfies Y m (z)≡U i [m]mod(1−e j2πm/M r i z −1 ) (i=0 to K−1) (m=0 to M−1).

Claim 48

A multiplexing system for transmitting subchannel signals over a communication channel, the system comprising: a transmitter, including a terminal for receiving M subchannel signals X m (z) (m=0 to M−1) of length greater than one, a multiplexing apparatus that is responsive to the received M subchannel signals X m (z) (m=0 to M−1) of length greater than one, to produces a multiplexed signal X(z), the multiplexing apparatus comprising means of obtaining a polynomial X(z) that satisfies X(z)≡X m (z)mod(P m (z)) (m=0 to M−1), wherein arithmetic is performed on a finite field and wherein P m (z) (m=0 to M−1) are relatively prime polynomials given by and (1−α m β 0 z −1 )(1−α n β 1 Z −1 ) . . . (1−α m β K−1 z −1 ), where β i is a non-zero element (i=0 to K−1), and α is a primitive Mth root of unity in the finite field, and means for transmitting the multiplexed signal X(z) over the communication channel; and a receiver, including means for receiving from the communication channel a receiver side multiplexed signal Y(z), including the transmitted multiplexed signal X(z), and a demultiplexing apparatus that takes the receiver side multiplexed signal Y(z), and produces receiver side subchannel signals Y m (z) (m=0 to M−1), the demultiplexing apparatus comprising means of computing Y m (z)≡Y(z)mod(P m (z)) (m=0 to M−1), wherein arithmetic is performed on a finite field, and wherein P m (z) (m=0 to M−1) are relatively prime polynomials given by (1−α m β 0 z −1 )(1−α m β 1 z −1 ) . . . (1−α m β K−1 z −1 ), where β i is a non-zero element (i=0 to K−1), and α is a primitive Mth root of unity in the finite field.

Claim 49

A multiplexing system for transmitting subchannel signals over a communication channel, the system comprising: a transmitter, including a terminal for receiving M subchannel signals X m (z) (m=0 to M−1) of length greater than one, a multiplexing apparatus that is responsive to the received M subchannel signals X m (z) (m=0 to M−1) of length greater than one, to produces a multiplexed signal X(z), the multiplexing apparatus comprising means of obtaining a polynomial X(z) that satisfies X(z)≡X m (z)mod(P m (z)) (m=0 to M−1), wherein arithmetic is performed on a finite field and wherein P m (z) (m=0 to M−1) are relatively prime polynomials given by and (1−α m β 0 z −1 )(1−α n β 1 Z −1 ) . . . (1−α m β K−1 z −1 ), where β i is a non-zero element (i=0 to K−1), and α is a primitive Mth root of unity in the finite field, and means for transmitting the multiplexed signal X(z) over the communication channel; and a receiver, including means for receiving from the communication channel a receiver side multiplexed signal Y(z), including the transmitted multiplexed signal X(z), and a demultiplexing apparatus that takes the receiver side multiplexed signal Y(z), and produces receiver side subchannel signals Y m (z) (m=0 to M−1), the demultiplexing apparatus comprising means of computing Y m (z)≡Y(z)mod(P m (z)) (m=0 to M−1), wherein arithmetic is performed on a finite field, and wherein P m (z) (m=0 to M−1) are relatively prime polynomials given by (1−α m β 0 z −1 )(1−α m β 1 z −1 ) . . . (1−α m β K−1 z −1 ), where β i is a non-zero element (i=0 to K−1), and α is a primitive Mth root of unity in the finite field.

Claim 50

A transmitter, comprising: a terminal for receiving M subchannel signals X m (z) (m=0 to M−1) of length greater than one, a multiplexing apparatus that is responsive to the received M subchannel signals X m (z) (m=0 to M−1) of length K where K is greater than one, the multiplexing apparatus performing arithmetic on a finite field, and including a Chinese remainder computation unit which obtains a polynomial A(z) that satisfies A(z)≡A l z)mod((1−e j2π(f+l)/t β 0 d/t z −d/t )(1−e j2π(f+l)/t β 1 d/t z −d/t ) . . . (1−e j2π(f+l)/t β K−1 d/t z −d/t )) (l=0 to t−1) for given A l (z), wherein β i is a non-zero element (i=0 to K−1), and t is a positive integer that divides d, and means of obtaining X(z) as a residue polynomial on mod((1−β 0 M z −M )(1−β 1 M z −M ) . . . (1−β K−1 M z −M )) by applying the Chinese remainder computation unit recursively, starting with the subchannel signals X m (z) regarding them as residue polynomials on mod((1−α m β 0 z −1 )(1−α m β 1 z −1 ) . . . (1−α m β K−1 z −1 )) (m=0 to M−1), where α is a primitive Mth root of unity in the finite field; and means for transmitting the multiplexed signal X(z).

Claim 51

A transmitter, comprising: a terminal for receiving M subchannel signals X m (z) (m=0 to M−1) of length greater than one, a multiplexing apparatus that is responsive to the received M subchannel signals X m (z) (m=0 to M−1) of length K where K is greater than one, the multiplexing apparatus performing arithmetic on a finite field, and including a Chinese remainder computation unit which obtains a polynomial A(z) that satisfies A(z)≡A l z)mod((1−e j2π(f+l)/t β 0 d/t z −d/t )(1−e j2π(f+l)/t β 1 d/t z −d/t ) . . . (1−e j2π(f+l)/t β K−1 d/t z −d/t )) (l=0 to t−1) for given A l (z), wherein β i is a non-zero element (i=0 to K−1), and t is a positive integer that divides d, and means of obtaining X(z) as a residue polynomial on mod((1−β 0 M z −M )(1−β 1 M z −M ) . . . (1−β K−1 M z −M )) by applying the Chinese remainder computation unit recursively, starting with the subchannel signals X m (z) regarding them as residue polynomials on mod((1−α m β 0 z −1 )(1−α m β 1 z −1 ) . . . (1−α m β K−1 z −1 )) (m=0 to M−1), where α is a primitive Mth root of unity in the finite field; and means for transmitting the multiplexed signal X(z).

Claim 52

A receiver, comprising means for receiving a receiver side multiplexed signal Y(z), and a demultiplexing apparatus that takes the receiver side multiplexed signal Y(z), and produces receiver side subchannel signals Y m (z) (m=0 to M−1), the demultiplexing apparatus including residue computation unit which computes A l (z)≡A(z)mod((1−e j2π(f+l)/t β 0 d/t z −d/t )(1−e j2π(f+l)/t β 1 d/t z −d/t ) . . . (1−e j2π(f+l)/t β k−1 d/t z −d/t )) (l=0 to t−1) for a given polynomial A(z), wherein β i is a non-zero element (i=0 to K−1), and t is a positive integer that divides d, and means of obtaining Y m (z) as a residue polynomial on mod((1−α m β 0 z −1 )(1−α m β 1 z −1 ) . . . (1α m β K−1 z −1 )) (m=0 to M+1), wherein α is a primitive Mth root of unity in the finite field, by applying the residue computation unit recursively, starting with the receiver side multiplexed signal Y(z) regarding it as a residue polynomial on mod((1−β 0 M z −m )(1−β 1 M z −M ) . . . (1−β K−1 M z −M )).

Claim 53

A receiver, comprising means for receiving a receiver side multiplexed signal Y(z), and a demultiplexing apparatus that takes the receiver side multiplexed signal Y(z), and produces receiver side subchannel signals Y m (z) (m=0 to M−1), the demultiplexing apparatus including residue computation unit which computes A l (z)≡A(z)mod((1−e j2π(f+l)/t β 0 d/t z −d/t )(1−e j2π(f+l)/t β 1 d/t z −d/t ) . . . (1−e j2π(f+l)/t β k−1 d/t z −d/t )) (l=0 to t−1) for a given polynomial A(z), wherein β i is a non-zero element (i=0 to K−1), and t is a positive integer that divides d, and means of obtaining Y m (z) as a residue polynomial on mod((1−α m β 0 z −1 )(1−α m β 1 z −1 ) . . . (1α m β K−1 z −1 )) (m=0 to M+1), wherein α is a primitive Mth root of unity in the finite field, by applying the residue computation unit recursively, starting with the receiver side multiplexed signal Y(z) regarding it as a residue polynomial on mod((1−β 0 M z −m )(1−β 1 M z −M ) . . . (1−β K−1 M z −M )).

Claim 54

A multiplexing system for transmitting subchannel signals over a communication channel, the system comprising: transmitter, including a terminal for receiving M subchannel signals X m (z) (m=0 to M−1) of length greater than one, a multiplexing apparatus that is responsive to the received M subchannel signals X m (z) (m=0 to M−1) of length K where K is greater than one, the multiplexing apparatus performing arithmetic on a finite field, and including a Chinese remainder computation unit which obtains a polynomial A(z) that satisfies A(z)≡A l (z)mod((1−e j2π(f+l)/t β 0 d/t z −d/t )(1−e j2π(f+l)/t β 1 d/t z −d/t ) . . . (1−e j2π(f+l)/t β K−1 d/t z −d/t )) (l=0 to t−1) for given A l (z), wherein β i is a non-zero element (i=0 to K−1), and t is a positive integer that divides d, and means of obtaining X(z) as a residue polynomial on mod((1−β 0 M z −M )(1−β 1 M z −M ) . . . (1−β K−1 M z −M )) by applying the Chinese remainder computation unit recursively, starting with the subchannel signals X m (z) regarding them as residue polynomials on mod((1−α m β 0 z −1 )(1−α m β 1 z −1 ) . . . (1−α m β K−1 z −1 )) (m=0 to M−1), where α is a primitive Mth root of unity in the finite field, and means for transmitting the multiplexed signal X(z) over a communication channel; and a receiver, including means for receiving a receiver side multiplexed signal Y(z), including the multiplexed signal X(z), from the communication channel, and a demultiplexing apparatus that takes the receiver side multiplexed signal Y(z), and produces receiver side subchannel signals Y m (z) (m=0 to M−1), the demultiplexing apparatus including residue computation unit which computes A l (z)≡A(z)mod((1−e j2π(f+l)/t β 0 d/t z −d/t )(1−e j2π(f+l)/t β 1 d/t z −d/t ) . . . (1−e j2π(f+l)/t β K−1 d/z z −d/t )) (l=0 to t−1) for a given polynomial A(z), wherein β i is a non-zero element (i=0 to K−1), and t is a positive integer that divides d, and means of obtaining Y m (z) as a residue polynomial on mod((1−α m β 0 z −1 )(1−α m β 1 z −1 ) . . . (1−α m β K−1 z −1 )) ((m=0 to M−1), wherein α is a primitive Mth root of unity in the finite field, by applying the residue computation unit recursively, starting with the receiver side multiplexed signal Y(z) regarding it as a residue polynomial on mod((1−β 0 M z −M )(1−β 1 M z −M ) . . . (1−β K−1 M z −M )).

Claim 55

A multiplexing system for transmitting subchannel signals over a communication channel, the system comprising: transmitter, including a terminal for receiving M subchannel signals X m (z) (m=0 to M−1) of length greater than one, a multiplexing apparatus that is responsive to the received M subchannel signals X m (z) (m=0 to M−1) of length K where K is greater than one, the multiplexing apparatus performing arithmetic on a finite field, and including a Chinese remainder computation unit which obtains a polynomial A(z) that satisfies A(z)≡A l (z)mod((1−e j2π(f+l)/t β 0 d/t z −d/t )(1−e j2π(f+l)/t β 1 d/t z −d/t ) . . . (1−e j2π(f+l)/t β K−1 d/t z −d/t )) (l=0 to t−1) for given A l (z), wherein β i is a non-zero element (i=0 to K−1), and t is a positive integer that divides d, and means of obtaining X(z) as a residue polynomial on mod((1−β 0 M z −M )(1−β 1 M z −M ) . . . (1−β K−1 M z −M )) by applying the Chinese remainder computation unit recursively, starting with the subchannel signals X m (z) regarding them as residue polynomials on mod((1−α m β 0 z −1 )(1−α m β 1 z −1 ) . . . (1−α m β K−1 z −1 )) (m=0 to M−1), where α is a primitive Mth root of unity in the finite field, and means for transmitting the multiplexed signal X(z) over a communication channel; and a receiver, including means for receiving a receiver side multiplexed signal Y(z), including the multiplexed signal X(z), from the communication channel, and a demultiplexing apparatus that takes the receiver side multiplexed signal Y(z), and produces receiver side subchannel signals Y m (z) (m=0 to M−1), the demultiplexing apparatus including residue computation unit which computes A l (z)≡A(z)mod((1−e j2π(f+l)/t β 0 d/t z −d/t )(1−e j2π(f+l)/t β 1 d/t z −d/t ) . . . (1−e j2π(f+l)/t β K−1 d/z z −d/t )) (l=0 to t−1) for a given polynomial A(z), wherein β i is a non-zero element (i=0 to K−1), and t is a positive integer that divides d, and means of obtaining Y m (z) as a residue polynomial on mod((1−α m β 0 z −1 )(1−α m β 1 z −1 ) . . . (1−α m β K−1 z −1 )) ((m=0 to M−1), wherein α is a primitive Mth root of unity in the finite field, by applying the residue computation unit recursively, starting with the receiver side multiplexed signal Y(z) regarding it as a residue polynomial on mod((1−β 0 M z −M )(1−β 1 M z −M ) . . . (1−β K−1 M z −M )).

Claim 56

A transmitter, comprising: a terminal for receiving M subchannel signals X m (z) (m=0 to M−1) of length K where K is greater than one, a multiplexing apparatus that performs arithmetic on a finite field and is responsive to the received M subchannel signals X m (z) (m=0 to M−1) of length K, to produce a MK-length multiplexed signal X(z), the multiplexing apparatus including means of computing V i [m]≡X m (z)mod(1−α m/M β i z −1 ) (i=0 to K−1, and m=0 to M−1), where β i is a non-zero element (i=0 to K−1), means of obtaining a polynomial V i (z) that satisfies V i (z)≡V l [m]mod(1−α m/M β i z −1 ) (m=0 to M−1, and i=0 to K−1), where α is a primitive Mth root of unity in the finite field, and means of obtaining X(z) that satisfies X(z)≡V i (z)mod(1−β i M z −M ) (i=0 to K−1); and means for transmitting the multiplexed signal X(z).

Claim 57

A transmitter, comprising: a terminal for receiving M subchannel signals X m (z) (m=0 to M−1) of length K where K is greater than one, a multiplexing apparatus that performs arithmetic on a finite field and is responsive to the received M subchannel signals X m (z) (m=0 to M−1) of length K, to produce a MK-length multiplexed signal X(z), the multiplexing apparatus including means of computing V i [m]≡X m (z)mod(1−α m/M β i z −1 ) (i=0 to K−1, and m=0 to M−1), where β i is a non-zero element (i=0 to K−1), means of obtaining a polynomial V i (z) that satisfies V i (z)≡V l [m]mod(1−α m/M β i z −1 ) (m=0 to M−1, and i=0 to K−1), where α is a primitive Mth root of unity in the finite field, and means of obtaining X(z) that satisfies X(z)≡V i (z)mod(1−β i M z −M ) (i=0 to K−1); and means for transmitting the multiplexed signal X(z).

Claim 58

A receiver, comprising: means for receiving a receiver side multiplexed signal Y(z); and a demultiplexing apparatus that performs arithmetic on a finite field and takes the receiver side multiplexed signal Y(z), and produces receiver side subchannel signals Y m (z), m=0, 1, . . . , M−1, the demultiplexing apparatus including means of computing U i (z)≡Y(z)mod(1−β i M z −M ) (i=0 to K−1), wherein β i is a non-zero element (i=0 to K−1), means of computing U i [m]≡U i (z)mod(1−α m/M β i z −1 ) (m=0 to M−1, and i=0 to K−1), where α is a primitive Mth root of unity in the finite field, and means of obtaining Y m (z) that satisfies Y m (z)≡U i [m]mod(1−α m/M β i z −1 ) (i=0 to K−1) (m=0 to M−1).

Claim 59

A receiver, comprising: means for receiving a receiver side multiplexed signal Y(z); and a demultiplexing apparatus that performs arithmetic on a finite field and takes the receiver side multiplexed signal Y(z), and produces receiver side subchannel signals Y m (z), m=0, 1, . . . , M−1, the demultiplexing apparatus including means of computing U i (z)≡Y(z)mod(1−β i M z −M ) (i=0 to K−1), wherein β i is a non-zero element (i=0 to K−1), means of computing U i [m]≡U i (z)mod(1−α m/M β i z −1 ) (m=0 to M−1, and i=0 to K−1), where α is a primitive Mth root of unity in the finite field, and means of obtaining Y m (z) that satisfies Y m (z)≡U i [m]mod(1−α m/M β i z −1 ) (i=0 to K−1) (m=0 to M−1).

Claim 60

A multiplexing system for transmitting subchannel signals over a communication channel, the system comprising: a transmitter, including a terminal for receiving M subchannel signals X m (z) (m=0 to M−1) of length K where K is greater than one, a multiplexing apparatus that performs arithmetic on a finite field and is responsive to the received M subchannel signals X m (z) (m=0 to M−1) of length K, to produce a MK-length multiplexed signal X(z), the multiplexing apparatus including means of computing V i [m]≡X m (z)mod(1−α m/M β i z −1 ) (i=0 to K−1, and m=0 to M−1), where β i is a non-zero element (i=0 to K−1), means of obtaining a polynomial V i (z) that satisfies V i (z)≡V l [m]mod(1−α m/M β i z −1 ) (m=0 to M−1, and i=0 to K−1), where α is a primitive Mth root of unity in the finite field, and means of obtaining X(z) that satisfies X(z)≡V i (z)mod(1−β i M z −M ) (i=0 to K−1), and means for transmitting the multiplexed signal X(z) over the communication channel; and a receiver, including means for receiving a receiver side multiplexed signal Y(z), including the multiplexed signal X(z), from the communication channel, and p 2 a demultiplexing apparatus that performs arithmetic on a finite field and takes the receiver side multiplexed signal Y(z), and produces receiver side subchannel signals Y m (z), m=0, 1, . . . , M−1, the demultiplexing apparatus including means of computing U i (z)≡Y(z)mod(1−β i M z −M ) (i=0 to K−1), wherein β i is a non-zero element (i=0 to K−1), means of computing U i [m]≡U i (z)mod(1−α m/M β i z −1 ) (m=0 to M−1, and i=0 to K−1), where α is a primitive Mth root of unity in the finite field, and means of obtaining Y m (z) that satisfies Y m (z)≡U i [m]mod(1−α m/M β i z −1 ) (i=0 to K−1) (m=0 to M−1).

Claim 61

A multiplexing system for transmitting subchannel signals over a communication channel, the system comprising: a transmitter, including a terminal for receiving M subchannel signals X m (z) (m=0 to M−1) of length K where K is greater than one, a multiplexing apparatus that performs arithmetic on a finite field and is responsive to the received M subchannel signals X m (z) (m=0 to M−1) of length K, to produce a MK-length multiplexed signal X(z), the multiplexing apparatus including means of computing V i [m]≡X m (z)mod(1−α m/M β i z −1 ) (i=0 to K−1, and m=0 to M−1), where β i is a non-zero element (i=0 to K−1), means of obtaining a polynomial V i (z) that satisfies V i (z)≡V l [m]mod(1−α m/M β i z −1 ) (m=0 to M−1, and i=0 to K−1), where α is a primitive Mth root of unity in the finite field, and means of obtaining X(z) that satisfies X(z)≡V i (z)mod(1−β i M z −M ) (i=0 to K−1), and means for transmitting the multiplexed signal X(z) over the communication channel; and a receiver, including means for receiving a receiver side multiplexed signal Y(z), including the multiplexed signal X(z), from the communication channel, and p 2 a demultiplexing apparatus that performs arithmetic on a finite field and takes the receiver side multiplexed signal Y(z), and produces receiver side subchannel signals Y m (z), m=0, 1, . . . , M−1, the demultiplexing apparatus including means of computing U i (z)≡Y(z)mod(1−β i M z −M ) (i=0 to K−1), wherein β i is a non-zero element (i=0 to K−1), means of computing U i [m]≡U i (z)mod(1−α m/M β i z −1 ) (m=0 to M−1, and i=0 to K−1), where α is a primitive Mth root of unity in the finite field, and means of obtaining Y m (z) that satisfies Y m (z)≡U i [m]mod(1−α m/M β i z −1 ) (i=0 to K−1) (m=0 to M−1).

Claim 62

A multiplexing apparatus that takes M subchannel signals X m (z) (m=0 to M−1) each including K ordered positions, where K is greater than one, and produces a MK-length multiplexed signal X(z) to be transmitted, the multiplexing apparatus comprising: means of extracting a sample from each of the K positions of each of the M subchannel signals; means of arranging the MK samples in K groups of M samples such that each group contains samples corresponding to the same position in the order of K positions in the subchannel signals; means of arranging the samples of each group in the order of subchannels, and computing M-point inverse DFT of the arranged samples for all of the groups; and means of producing an MK-length signal by performing K-fold polyphase composition of the inverse DFT outputs.

Claim 63

A multiplexing apparatus that takes M subchannel signals X m (z) (m=0 to M−1) each including K ordered positions, where K is greater than one, and produces a MK-length multiplexed signal X(z) to be transmitted, the multiplexing apparatus comprising: means of extracting a sample from each of the K positions of each of the M subchannel signals; means of arranging the MK samples in K groups of M samples such that each group contains samples corresponding to the same position in the order of K positions in the subchannel signals; means of arranging the samples of each group in the order of subchannels, and computing M-point inverse DFT of the arranged samples for all of the groups; and means of producing an MK-length signal by performing K-fold polyphase composition of the inverse DFT outputs.

Claim 64

The multiplexing apparatus of claim 40 , in which arithmetic is performed on an integer residue ring, and the inverse DFT is replaced by the inverse number theoretic transform.

Claim 65

A multiplexing apparatus as described in claim 40 , in which arithmetic is performed on a finite field, and the inverse DFT is replaced by the inverse of DFT-like transform on the finite field.