Method of approximating log-likelihood ratios in QAM digital transmissions and relative QAM soft-output de-mapper
Log-likelihood ratios are approximated for encoded bits modulated with a 2 m -ary QAM constellation. Each symbol of the constellation is identified by a respective string of m bits. The log-likelihood ratio of each bit of the m bits is approximated with a product λ of a respective factor by a respective variable D that depends on a received signal and on communication channel characteristics. The approximating includes determining a value of at least one of the variables D using a parametric nonlinear function of an equalized replica z of a respective received signal.
1. A method of approximating log-likelihood ratios for a plurality of encoded bits, the method comprising:
receiving the plurality of encoded bits modulated with a 2 m -ary QAM constellation, with each symbol of the constellation being identified by a respective string of m bits;
approximating the log-likelihood ratio of each bit of the m bits with a product λ of a respective factor by a respective variable D that depends on a received signal and on communication channel characteristics, the approximating comprising determining a value of at least one of the variables D using a parametric nonlinear function of an equalized replica z of a respective received signal; and
providing de-interleaved modulation symbols based on the approximated log-likelihood ratios.
2. The method of claim 1 , wherein the nonlinear parametric function is a parabolic function of at least one of an in-phase component z l and a quadrature component z Q of the equalized replica z of the respective received signal.
3. The method of claim 2 , wherein m is an even number, with a first half of the m bits being transmitted in phase and with a second half of the m bits being transmitted in quadrature, with the values of the variables D for each bit of the bits modulated in-phase or in quadrature being determined through the following:
establishing a value of the variable for a first bit of the first half of the m bits or of the second half of the m bits as a linear function of an absolute value of the in-phase component z l or of the quadrature component z Q of the equalized replica z of the respective received signal; and
calculating a value of each variable D I,l or D Q,t of the other bits of the first half of the m bits or of the second half of the m bits, respectively, using the parametric parabolic function of an equalized replica z l or z Q of the respective received signal.
4. The method of claim 3 , wherein the first bit is a most significant bit.
5. The method of claim 3 , wherein a position of the first bit is established as a function of a Gray coding that associates the encoded bits to QAM modulation symbols.
6. The method of claim 1 , wherein the respective factor is determined for each bit as a function of the characteristics of a used communication channel and of noise that corrupts signals transmitted through the communication channel.
7. The method of claim 3 , wherein m is 6 or 8 or 10; and the values of the variables for a first bit of the first half of the m bits or of the second half of the m bits are calculated based on the following:
calculating a first auxiliary variable IN f (m/2) or IN e (m/2) equal to or opposite to a real part or to an imaginary part, respectively, of the equalized replica z of the respective received signal;
calculating a second auxiliary variable IN f (m/2−1) or IN e (m/2−1) equal to or opposite to a difference between an absolute value of the first auxiliary variable IN f (m/2) or IN e (m/2), respectively, and a ratio of a power 2 m/2−1 by a first parameter C M ;
calculating a third auxiliary variable IN f (m/2−2) or IN e (m/2−2) opposite to or equal to a difference between an absolute value of the second auxiliary variable IN f (m/2−1) or IN e (m/2−1), respectively, and a ratio of a power 2 m/2−2 by the first parameter C M ;
calculating, for m=8 or m=10, a fourth auxiliary variable IN f (m/2−3) or IN e (m/2−3) opposite to or equal to a difference between an absolute value of the third auxiliary variable IN f (m/2−2) or IN e (m/2−2), respectively, and a ratio of a power 2 m/2−3 by the first parameter c M ; and
calculating for m=10 a fifth auxiliary variable IN f (m/2−4) or IN e (m/2−4) opposite to or equal to a difference between an absolute value of the fourth auxiliary variable IN f (m/2−3) or IN e (m/2−3), respectively, and a ratio of a power 2 m/2−4 by the first parameter C M , the variable for a first bit of the first half of the m bits or of the second half of the m bits being equal to or opposite to a ratio of an (m/2)-th auxiliary variable IN f (.) or IN e (.), respectively, by the first parameter C M .
8. The method of claim 7 , wherein the parametric parabolic function calculates the value of each variable D I,l or D Q,t of the other bits of the first half of the m bits or of the second half of the m bits, respectively, as a product of an l-th or t-th auxiliary variable by a second parameter b and by a sum of a ratio 2/c M of 2 by the first parameter c M with an absolute value of the l-th or t-th auxiliary variable, where D I,l and D Qt are defined as follows:
D I,l =b*IN f ( m/ 2 −l+ 1)*[| IN f ( m/ 2 −l+ 1)|+2 /c M ]
D Q,t =b*IN e ( m/ 2 −t+ 1)*[| IN e ( m/ 2 −t+ 1)|+2 /c M ].
9. The method of claim 8 , wherein the second parameter (b) is equal to ¼.
10. The method of claim 8 , wherein the first parameter c M is established as a function of the number m of bits.
11. The method of claim 8 , wherein the first parameter c M is established as a function of a constellation power.
12. The method of claim 8 , wherein the number m of bits is 6 (64-QAM) and the first parameter c M is equal to SQRT(42).
13. The method of claim 8 , wherein the number m of bits is 8 (256-QAM) and the first parameter c M is equal to SQRT(170).
14. The method of claim 8 , wherein the number m of bits is 10 (1024-QAM) and the first parameter c M is equal to SQRT(682).
15. A de-mapper comprising:
an input to receive a plurality of encoded bits modulated with a 2 m -ary QAM constellation, with each symbol of the constellation being identified by a respective string of m bits;
a circuit to approximate a log-likelihood ratio of each bit of the m bits with a product λ of a respective factor by a respective variable D that depends on a received signal and on communication channel characteristics, the approximating comprising determining a value of at least one of the variables D using a parametric nonlinear function of an equalized replica z of a respective received signal; and
an output to provide de-interleaved modulation symbols based on the approximated log-likelihood ratios.
16. The de-mapper of claim 15 , wherein the nonlinear parametric function is a parabolic function of at least one of an in-phase component z I and a quadrature component z Q of the equalized replica z of the respective received signal.
17. The de-mapper of claim 16 , wherein m is an even number, with a first half of the m bits being transmitted in phase and with a second half of the m bits being transmitted in quadrature, with the values of the variables D for each bit of the bits modulated in-phase or in quadrature being determined through the following:
establishing a value of the variable for a first bit of the first half of the m bits or of the second half of the m bits as a linear function of an absolute value of the in-phase component z I or of the quadrature component z Q of the equalized replica z of the respective received signal; and
calculating a value of each variable D I,I or D Q,t of the other bits of the first half of the m bits or of the second half of the m bits, respectively, using the parametric parabolic function of an equalized replica z I or z Q of the respective received signal.
18. The de-mapper of claim 17 , wherein the first bit is a most significant bit; and wherein a position of the first bit is established as a function of a Gray coding that associates the encoded bits to QAM modulation symbols.
19. The de-mapper of claim 15 , wherein the respective factor is determined for each bit as a function of the characteristics of a used communication channel and of noise that corrupts signals transmitted through the communication channel.
20. The de-mapper of claim 17 , wherein m is 6 or 8 or 10; and the values of the variables for a first bit of the first half of the m bits or of the second half of the m bits are calculated based on the following:
calculating a first auxiliary variable IN f (m/2) or IN e (m/2) equal to or opposite to a real part or to an imaginary part, respectively, of the equalized replica z of the respective received signal;
calculating a second auxiliary variable IN f (m/2−1) or IN e (m/2−1) equal to or opposite to a difference between an absolute value of the first auxiliary variable IN f (m/2) or IN e (m/2), respectively, and a ratio of a power 2 m2−1 by a first parameter C M ;
calculating a third auxiliary variable IN f (m/2−2) or IN e (m/2−2) opposite to or equal to a difference between an absolute value of the second auxiliary variable IN f (m/2−1) or IN e (m/2−1), respectively, and a ratio of a power 2 m/2−2 by the first parameter C M ;
calculating, for m=8 or m=10, a fourth auxiliary variable IN f (m/2−3) or IN e (m/2−3) opposite to or equal to a difference between an absolute value of the third auxiliary variable IN f (m/2−2) or IN e (m/2−2), respectively, and a ratio of a power 2 m/2−3 by the first parameter c M ; and
calculating for m=10 a fifth auxiliary variable IN f (m/2−4) or IN e (m/2−4) opposite to or equal to a difference between an absolute value of the fourth auxiliary variable IN f (m/2−3) or IN e (m/2−3), respectively, and a ratio of a power 2 m/2−4 by the first parameter c M , the variable for a first bit of the first half of the m bits or of the second half of the m bits being equal to or opposite to a ratio of an (m/2)-th auxiliary variable respectively, by the first parameter c M .
21. The de-mapper of claim 20 , wherein the parametric parabolic function calculates the value of each variable D I,l or D Q,t of the other bits of the first half of the m bits or of the second half of the m bits, respectively, as a product of an l-th or t-th auxiliary variable by a second parameter b and by a sum of a ratio 2/c M of 2 by the first parameter c M with an absolute value of the l-th or t-th auxiliary variable, where D I,l and D Qt are defined as follows:
D I,l =b*IN f ( m/ 2 −l+ 1)*[| IN f ( m/ 2 −l+ 1)|+2 /c M ]
D Q,t =b*IN e ( m/ 2 −t+ 1)*[|IN e ( m/ 2 −t+ 1)|+2 /c M ].
22. A receiver comprising:
an analog front end to receive a signal from a communication channel, the signal including a plurality of encoded bits modulated with a 2 m -ary QAM constellation, with each symbol of the constellation being identified by a respective string of m bits;
an automatic gain control and time synchronization block to adjust a level of the received signal, and to generate a first internal signal that is a replica of the received signal with a pre-established level;
a demodulator input with the first internal signal to generate a digital signal y i that represents a replica of transmitted symbols a i distorted by the communication channel and corrupted by noise;
a demapper input with the digital signal y i to generate second digital signals representing a product λ associated to each bit of the distorted and corrupted replica y i of the transmitted symbols, the second digital signals being generated based on approximating the log-likelihood ratio of each bit of the m bits with a product λ of a respective factor by a respective variable D that depends on the received signal and on communication channel characteristics, the approximating comprising determining a value of at least one of the variables D using a parametric nonlinear function of an equalized replica z of a respective received signal; and
a decoder that processes the second digital signals and to decide as a function thereof which bits have been transmitted.
23. The receiver of claim 22 , wherein said automatic gain control and time synchronization block (AGC & TIME SYNC) is to remove a cyclic prefix for eliminating inter-symbol and inter-channel interference; and wherein said demodulator comprises an OFDM demodulator.
24. The receiver of claim 22 , wherein the nonlinear parametric function is a parabolic function of at least one of an in-phase component z l and a quadrature component z Q of the equalized replica z of the respective received signal.
25. The receiver of claim 23 , wherein m is an even number, with a first half of the number m of bits being transmitted in phase and with a second half being transmitted in quadrature, with the values of the variables D for each bit of the bits modulated in-phase or in quadrature being determined through the following:
establishing a value of the variable for a first bit of the first half of the m bits or of the second half of the m bits as a linear function of an absolute value of the in-phase component z I or of the quadrature component z Q of the equalized replica z of the respective received signal; and
calculating a value of each variable D I,l or D Q,t of the other bits of the first half or of the second half, respectively, using the parametric parabolic function of an equalized replica z I or z Q of the respective received signal.
26. The receiver of claim 24 , wherein the first bit is a most significant bit; and wherein a position of the first bit is established as a function of a Gray coding that associates the encoded bits to QAM modulation symbols.
27. The receiver of claim 22 , wherein the respective factor is determined for each bit as a function of the characteristics of a used communication channel and of noise that corrupts signals transmitted through the communication channel.
28. The receiver of claim 24 , wherein m is 6 or 8 or 10; and the values of the variables for a first bit of the first half of the m bits or of the second half of the m bits are calculated based on the following:
calculating a first auxiliary variable IN f (m/2) or IN e (m/2) equal to or opposite to a real part or to an imaginary part, respectively, of the equalized replica z of the respective received signal;
calculating a second auxiliary variable IN f (m/2−1) or IN e (m/2−1) equal to or opposite to a difference between an absolute value of the first auxiliary variable IN f (m/2) or IN e (m/2), respectively, and a ratio of a power 2 m/2−1 by a first parameter C M ;
calculating a third auxiliary variable IN f (m/2−2) or IN e (m/2−2) opposite to or equal to a difference between an absolute value of the second auxiliary variable IN f (m/2−1) or IN e (m/2−1), respectively, and a ratio of a power 2 m/2−2 by the first parameter C M ;
calculating, for m=8 or m=10, a fourth auxiliary variable IN f (m/2−3) or IN e (m/2−3) opposite to or equal to a difference between an absolute value of the third auxiliary variable IN f (m/2−2) or IN e (m/2−2), respectively, and a ratio of a power 2 m/2−3 by the first parameter C M ; and
calculating for m=10 a fifth auxiliary variable IN f (m/2−4) or IN e (m/2−4) opposite to or equal to a difference between an absolute value of the fourth auxiliary variable IN f (m/2−3) or IN e (m/2−3), respectively, and a ratio of a power 2 m/2−4 by the first parameter C M , the variable for a first bit of the first half of the m bits or of the second half of the m bits being equal to or opposite to a ratio of an (m/2)-th auxiliary variable IN f (.) or IN e (.), respectively, by the first parameter C M .
29. The receiver of claim 28 , wherein the parametric parabolic function calculates the value of each variable D I,l or D Q,t of the other bits of the first half of the m bits or of the second half of the m bits, respectively, as a product of an l-th or t-th auxiliary variable by a second parameter b and by a sum of a ratio 2/c M of 2 by the first parameter c M with an absolute value of the l-th or t-th auxiliary variable, where D I,l and D Qt are defined as follows:
D I,l =b*IN f ( m/ 2 −l+ 1)*[| IN f ( m+ 2 −l+ 1)|+2 /c M ]
D Q,t =b*IN e ( m/ 2 −t+ 1)*[| IN e ( m/ 2 −t+ 1)|+2 /c M ].