Robust carrier phase and frequency tracking for M-QAM demodulation

1. A method for tracking carrier phase and frequency offsets associated with a series of blocks of received symbols modulated using an M quadrature amplitude modulation (M-QAM) symbol map, comprising:

determining coarse carrier phase and frequency offsets of an initial block of the series of blocks by:

creating a grid of discrete candidate phase offset values;

for each candidate phase offset value in the grid:

applying the candidate phase offset value to each of the M-QAM symbols of the initial block;

applying a respective hard decision from the symbol map to each of the candidate phase offset value-applied M-QAM symbols of the initial block; and

computing a figure of merit for the candidate phase offset value based on the candidate phase offset value-applied M-QAM symbols of the initial block and their respective hard decisions;

selecting the discrete candidate phase offset value from the grid having the best figure of merit as an initial phase offset estimate; and

computing an initial frequency offset estimate using the M-QAM symbols of the initial block updated with the initial phase offset estimate, their respective hard decisions, and an approximation of the complex exponential function; and

for each M-QAM symbol of a current block of the series of blocks subsequent to the initial block:

setting a binary trust weight for the M-QAM symbol based on comparison of a computed parameter with a threshold, wherein each binary trust weight has a value of either zero or one; and

using the binary trust weights to compute a phase offset error and a frequency offset error associated with the current block, wherein each of the M-QAM symbols having a zero-valued binary trust weight is non-contributory in the computation of the phase and frequency offset errors.

2. The method of claim 1 ,

wherein said creating a grid of discrete candidate phase offset values comprises:

generating a sequence of values in increments of a step size within a search range; and

wherein the step size is computed based on a selectable fraction of a maximum supported frequency offset by the initial block of received M-QAM symbols.

3. The method of claim 2 ,

wherein the maximum supported frequency offset by the initial block of received M-QAM symbols is determined by a maximum supported phase offset for the M-QAM symbol map and the block length of the initial block of received M-QAM symbols.

4. The method of claim 2 ,

wherein the search range is −45 degrees to +45 degrees.

5. The method of claim 1 ,

wherein said computing a figure of merit comprises:

computing a root mean square (RMS) error across the candidate phase offset value-applied M-QAM symbols relative to their respective hard decisions; and

wherein the best figure of merit comprises the minimum RMS error.

6. The method of claim 1 , further comprising:

refining the initial phase offset estimate using the initial frequency offset estimate, the updated M-QAM symbols, and their respective hard decisions.

7. The method of claim 1 ,

wherein the approximation of the complex exponential function comprises a cubic polynomial;

wherein a domain of approximation of the complex exponential function is restricted to a set of pre-determined values that depends upon a maximum supported carrier frequency offset for the M-QAM symbol map that depends upon the value of M; and

wherein the coefficients of the cubic polynomial are chosen to minimize a mean squared approximation error averaged over the restricted domain of approximation.

8. The method of claim 1 ,

wherein the initial phase and frequency offset estimates are made without reference to blocks of symbols previous in time to the initial block of received M-QAM symbols.

9. The method of claim 1 , further comprising:

for each M-QAM symbol of the current block, computing the parameter based on a first hard decision symbol of the symbol map closest to the received M-QAM symbol and a second hard decision symbol of the symbol map second closest to the M-QAM symbol.

10. The method of claim 9 ,

wherein said computing the parameter is based on a ratio of:

a first distance between the M-QAM symbol and the first hard decision symbol of the symbol map; and

a second distance between the M-QAM symbol and the second hard decision symbol of the symbol map.

11. The method of claim 1 , further comprising:

prior to said using the binary trust weights to compute a phase offset error and a frequency offset error associated with the current block, computing a noise covariance matrix of the previous block using:

received M-QAM symbols of the previous block corrected by predicted phase and frequency offsets of the previous block;

hard decisions on the corrected received M-QAM symbols of the previous block;

the binary trust weights for the corrected received M-QAM symbols of the previous block; and

a size of each block; and

predicting a noise covariance matrix of the current block using the computed noise covariance matrix of the previous block.

12. The method of claim 11 , further comprising:

computing a Kalman gain matrix for the current block using the predicted noise covariance matrix of the current block and a covariance matrix; and

estimating the phase offset and a frequency offset of the current block using:

predicted phase and frequency offsets of the current block;

the computed phase and frequency offset errors of the current block; and

the computed Kalman gain matrix of the current block.

13. The method of claim 1 ,

wherein the phase offset error is computed using an approximation of the complex exponential function;

wherein the approximation of the complex exponential function comprises a cubic polynomial;

wherein a domain of approximation of the complex exponential function is restricted to a set of pre-determined values that depends upon a maximum supported carrier frequency offset for the M-QAM symbol map that depends upon the value of M; and

wherein the coefficients of the cubic polynomial are chosen to minimize a mean squared approximation error averaged over the restricted domain of approximation.

14. The method of claim 1 , further comprising:

performing the operations of claim 9 in a forward direction on received M-QAM symbols of a plurality of blocks of the series, followed by performing the operations of claim 9 in a reverse direction on the received M-QAM symbols of the plurality of blocks to produce transmitted M-QAM symbols for the received M-QAM symbols.

15. The method of claim 1 ,

wherein the blocks of the series are segregated into overlapping blocks; and

wherein a number of symbols of overlap between consecutive blocks is selectable.

16. The method of claim 1 ,

wherein a size of the blocks is selectable.

17. A non-transitory computer-readable medium having instructions stored thereon that when executed by a processor, cause the processor to perform operations of first and second methods, comprising:

the first method is for determining coarse carrier phase and frequency offsets of an initial block of a series of blocks of received symbols modulated using an M quadrature amplitude modulation (M-QAM) symbol map that includes the operations of:

creating a grid of discrete candidate phase offset values;

for each candidate phase offset value in the grid:

applying the candidate phase offset value to each of the M-QAM symbols;

applying a respective hard decision from the symbol map to each of the candidate phase offset value-applied M-QAM symbols; and

computing a figure of merit for the candidate phase offset value based on the candidate phase offset value-applied M-QAM symbols and their respective hard decisions;

selecting the discrete candidate phase offset value from the grid having the best figure of merit as an initial phase offset estimate; and

computing an initial frequency offset estimate using the M-QAM symbols updated with the initial phase offset estimate, their respective hard decisions, and an approximation of the complex exponential function; and

the second method is for tracking carrier phase and frequency offsets associated with subsequent blocks of the series of blocks after the initial block that includes the operations of:

for each M-QAM symbol of a current block of the subsequent blocks:

setting a binary trust weight for the M-QAM symbol based on comparison of a computed parameter with a threshold; and

using the binary trust weights to compute a phase offset error and a frequency offset error associated with the current block, wherein each of the M-QAM symbols having a zero-valued binary trust weight is non-contributory in the computation of the phase and frequency offset errors.

18. The non-transitory computer-readable medium of claim 17 ,

wherein said creating a grid of discrete candidate phase offset values comprises:

generating a sequence of values in increments of a step size within a search range; and

wherein the step size is computed based on a selectable fraction of a maximum supported frequency offset by the initial block of received M-QAM symbols.

19. The non-transitory computer-readable medium of claim 17 , the operations further comprising:

for each M-QAM symbol of the current block, computing the parameter based on a first hard decision symbol of the symbol map closest to the received M-QAM symbol and a second hard decision symbol of the symbol map second closest to the M-QAM symbol.

20. A radio frequency device that tracks carrier phase and frequency offsets associated with a series of blocks of received symbols modulated using an M quadrature amplitude modulation (M-QAM) symbol map, comprising:

a processor configured to:

determine coarse carrier phase and frequency offsets of an initial block of the series of blocks by:

creating a grid of discrete candidate phase offset values;

for each candidate phase offset value in the grid:

applying the candidate phase offset value to each of the M-QAM symbols of the initial block;

applying a respective hard decision from the symbol map to each of the candidate phase offset value-applied M-QAM symbols of the initial block; and

computing a figure of merit for the candidate phase offset value based on the candidate phase offset value-applied M-QAM symbols of the initial block and their respective hard decisions;

selecting the discrete candidate phase offset value from the grid having the best figure of merit as an initial phase offset estimate; and

computing an initial frequency offset estimate using the M-QAM symbols of the initial block updated with the initial phase offset estimate, their respective hard decisions, and an approximation of the complex exponential function; and

for each M-QAM symbol of a current block of the series of blocks subsequent to the initial block:

set a binary trust weight for the M-QAM symbol based on comparison of a computed parameter with a threshold, wherein each binary trust weight has a value of either zero or one; and

use the binary trust weights to compute a phase offset error and a frequency offset error associated with the current block, wherein each of the M-QAM symbols having a zero-valued binary trust weight is non-contributory in the computation of the phase and frequency offset errors.