Method for reference signal generation in the presence of frequency offsets in a communications station with spatial processing

• US 6,690,747 B2
• Filed: 06/06/2001
• Issued: 02/10/2004
• Est. Priority Date: 10/11/1996
• Status: Expired due to Term

• Current Assignee: Intel Corporation
• Original Assignee: ArrayComm LLC (Ygomi LLC)
• Inventors: Trott, Mitchell D, Parish, David M., Youssefmir, Michael, Rosenfeld, Dov, Petrus, Paul, Chiodini, Alain M.
• Primary Examiner(s): Ghebretinsae, Temesghen
• Application Number:
  US09/877,807
• Publication Number:
  US 20010031022A1
• Time in Patent Office: 979 Days
• Field of Search: 375/324, 375/326, 375/130, 370/310, 343/879, 343/893, 342/368, 342/378
• US Class Current: 375/324
• CPC Class Codes: H04B 7/0851 using training sequences or error signal minimizing error signal H04B7/0854
• CPC Class Codes: H04L 1/06 using space diversity
• CPC Class Codes: H04L 2027/003 at baseband only
• CPC Class Codes: H04L 2027/0046 Open loops
• CPC Class Codes: H04L 2027/0095 in a preamble or similar structure
• CPC Class Codes: H04L 27/2332 using a non-coherent carrier

Abstract

A method for generating a reference signal from a modulated signal transmitted to a communications station that includes an array of antenna elements and spatial processing means including: separating from the signals received at the antenna elements a copy signal corresponding to the signal transmitted by a particular remote station using an initial spatial weight vector corresponding to the particular remote station; determining from the terminal copy signal a reference signal having substantially the same frequency offset and time alignment as the received antenna signals; and computing a new spatial weight vector by optimizing a cost function, the cost function using the received antenna signals and the reference signal. For demodulation, the method further includes extracting the symbols of the modulated signal.

Claims

1. A method for producing a reference signal from a modulated signal, comprising:

weighting a set of received antenna signals to form a copy signal corresponding to a remote station, the weighting using a spatial weight vector corresponding to the remote station, the copy signal being in the form of copy signal samples; and

determining samples of the reference signal by, for each of a set of sample points;

constructing an ideal signal sample from the copy signal at the same sample point, the ideal signal sample having a phase determined from the copy signal at the sample point, with the phase of the ideal signal sample at an initial symbol point set to be an initial ideal signal phase;

relaxing the phase of the ideal signal sample towards the copy signal sample phase to produce the phase of the reference signal; and

producing the reference signal having the phase of the reference signal produced by relaxing the phase of the ideal signal towards the copy signal sample phase.

2. The method in claim 1 wherein the spatial weight vector is determined from the received antenna signals and from the reference signal.

3. The method in claim 1 wherein the set of received antenna signals is received at a communications station, the communications station including an array of antennas and a spatial processor.

4. The method in claim 3 wherein the spatial processor includes mechanisms for weighting the set of received antenna signals by a set of corresponding receive weights.

5. The method in claim 4 wherein each distinct received antenna signal is derived from the signal received at a corresponding antenna of the array.

6. The method in claim 1 wherein the modulated signal is modulated at symbol points by a modulation scheme that has a finite symbol alphabet.

7. The method in claim 6 wherein the finite symbol alphabet includes symbols that have different phases.

8. The method of claim 1 wherein the phase of the ideal signal is determined in constructing an ideal signal sample from the copy signal at the same sample point, sample by sample, the phase of the ideal signal sample at any sample point being determined from the phase of the reference signal at the previous sample point for which said phase is determined, and from a decision based on the copy signal.

9. The method of claim 1 wherein the initial symbol point is a first valid symbol point in a burst of samples of received antenna signals, and determining samples of the reference signal comprises determining the samples of the reference signal in the forward time direction.

10. The method of claim 1 wherein the initial symbol point is the last valid symbol point in a burst of samples of received antenna signals, and determining samples of the reference signal comprises determining the samples of the reference signal in the backwards time direction.

11. The method of claim 1 wherein relaxing the phase of the ideal signal sample towards the phase of the copy signal sample corresponds to adding a filtered version of the difference between the copy signal phase and ideal signal phase.

12. The method of claim 1 wherein relaxing the phase of the ideal signal sample towards the phase of the copy signal sample corresponds to forming the reference signal sample by adding to the ideal signal sample a filtered version of the difference between the copy signal and ideal signal.

13. The method of claim 11 wherein the filter is a zero order filter consisting of multiplication by a constant and wherein the phase of the reference signal sample is equal to the phase of the ideal signal added to the multiplication of the difference between the phases of the copy signal sample and the ideal signal sample with a constant.

14. The method of claim 11 wherein the filter is a linear discrete time filter with a transfer function denoted H(z) in the Z-domain with input to the filter being the difference between the phases of the copy signal sample and the ideal signal sample.

15. The method of claim 13 wherein the quantity of the difference between the phases of the copy signal sample and the ideal signal sample is phase unwrapped.

16. The method of claim 13 wherein the quantity of the difference between the phases of the copy signal sample and the ideal signal sample is constrained to be in the range −

π

to +π

.

17. The method of claim 14 wherein the quantity of the difference between the phases of the copy signal sample and the ideal signal sample is phase unwrapped.

18. The method of claim 14 wherein the quantity of the difference between the phases of the copy signal sample and the ideal signal sample is constrained to be in the range −

π

to +π

.

19. The method of claim 12 wherein the filter is a zero order filter consisting of multiplication by a constant so that the reference signal sample is equal to the ideal signal added to the multiplication of the difference between the copy signal sample and the ideal signal sample with a constant.

20. The method of claim 12 wherein the filter is a linear discrete time filter with a transfer function denoted H(z) in the Z-domain with input to the filter being the difference between the copy signal sample and the ideal signal sample.

21. The method of claim 14 wherein the filter is a first order filter having a transfer function $H$

(z)=γ

+β

z-11-δ



 



z-1

where γ

, β

, and δ

are parameters.

22. The method of claim 20 wherein the filter is a first order filter having a transfer function $H$

(z)=γ

+β

z-11-δ



 



z-1

where γ

, β

, and δ

are parameters.

23. The method of claim 13 further comprising correcting the phase of the reference signal sample by an amount dependent on the difference in phase between the previously determined reference signal sample and the previously determined copy signal sample prior to producing the reference signal.

24. The method of claim 19 further comprising correcting the phase of the reference signal sample by an amount dependent on the difference between the previously determined reference signal sample and the previously determined copy signal sample prior to producing the reference signal.

25. The method of claim 6 wherein the modulation scheme is phase shift keying.

26. The method of claim 25 wherein the modulation scheme is differential phase shift keying.

27. The method of claim 6 wherein the modulation scheme is quadrature amplitude modulation, hereafter QAM.

28. A method, comprising:

separating from a set of received antenna signals a copy signal corresponding to a remote station by using a first spatial weight vector corresponding to the remote station;

determining from the copy signal a reference signal; and

computing a second spatial weight vector by optimizing a cost function, the cost function using the received antenna signals and the reference signal.

29. The method in claim 28 wherein the set of received antenna signals is received at a communications station, the communications station including an array of antennas and a spatial processor.

30. The method in claim 29 wherein the spatial processor includes mechanisms for weighting the set of received antenna signals by a set of corresponding receive weights.

31. The method in claim 30 wherein each distinct received antenna signal is derived from the signal received at a corresponding antenna of the array.

32. The method in claim 28 wherein the set of received antenna signals includes a modulated signal transmitted from the remote station, the modulated signal modulated at symbol points by a modulation scheme that has a finite symbol alphabet.

33. The method in claim 32 wherein the finite symbol alphabet includes symbols that have different phases.

34. The method of claim 28 wherein the reference signal has substantially the same frequency offset and time alignment as the received antenna signals.

35. The method of claim 28, wherein the set of received antenna signals includes a modulated signal transmitted from the remote station, the modulated signal modulated at symbol points by a modulation scheme that has a finite symbol alphabet, the method further including extracting the symbols of the modulated signal.

36. The method of claim 28 further including performing timing alignment on the received antenna signals prior to separating from the received antenna signals the copy signal.

37. The method of claim 28 further including performing frequency offset correction on the received antenna signals prior to separating from the received antenna signals the copy signal.

38. The method of claim 36 further including performing frequency offset correction on the received antenna signals prior to separating from the received antenna signals the copy signal.

39. The method of claim 28 further comprising separating from the received antenna signals the copy signal at least once, using the second spatial weight vector previously computed instead of the first spatial weight vector.

40. The method of claim 28 further comprising repeating, at least once, determining from the copy signal the reference signal.

41. The method of claim 28 further including:

estimating a frequency offset and a timing misalignment of the copy signal; and

correcting the copy signal for frequency offset and timing misalignment to form a corrected copy signal,

wherein determining from the copy signal the reference signal comprises;

synthesizing a corrected reference signal that has substantially the same frequency offset and timing alignment as the corrected copy signal; and

applying frequency offset and time misalignment to the corrected reference signal to form a frequency offset and time misaligned reference signal having the same frequency offset and time misalignment as the received antenna signals.

42. The method of claim 28 further including:

estimating a timing misalignment of the copy signal; and

correcting the copy signal for timing misalignment to form a timing aligned copy signal,

wherein determining from the copy signal the reference signal comprises;

synthesizing a timing aligned reference signal that has substantially the same timing alignment as the timing-aligned copy signal; and

applying time misalignment to the timing aligned reference signal to form a timing misaligned reference signal having substantially the same time alignment as the received antenna signals.

43. The method of claim 28 further including:

estimating the frequency offset of the copy signal, and

correcting the copy signal for frequency offset to form a frequency offset corrected copy signal,

wherein determining from the copy signal the reference signal comprises;

synthesizing a frequency offset corrected reference signal that has substantially the same frequency offset as the frequency offset corrected copy signal; and

applying frequency offset to the frequency offset corrected reference signal to form a frequency offset reference signal having substantially the same frequency offset as the received antenna signals.

44. The method of claim 28 wherein determining from the copy signal the reference signal includes, for each of a set of sample points:

constructing an ideal signal sample from the copy signal at the same sample point, the ideal signal sample having a phase determined from the copy signal at the sample point, with the phase of the ideal signal sample at an initial symbol point set to be an initial ideal signal phase;

relaxing the phase of the ideal signal sample towards the copy signal sample phase to produce the phase of the reference signal; and

producing the reference signal having the phase of the reference signal produced by relaxing the phase of the ideal signal towards the copy signal sample phase.

45. The method of claim 44 wherein the phase of an ideal signal is determined in constructing an ideal signal sample from the copy signal at the same sample point, sample by sample, the phase of the ideal signal sample at any sample point being determined from the phase of the reference signal at the previous sample point for which said phase is determined, and from a decision based on the copy signal.

46. The method of claim 44 wherein relaxing the phase of the ideal signal sample towards the phase of the copy signal corresponds to adding a filtered version of the difference between the copy signal phase and ideal signal phase.

47. The method of claim 44 wherein relaxing the phase of the ideal signal sample towards the phase of the copy signal corresponds to forming the reference signal sample by adding to the ideal signal sample a filtered version of the difference between the copy signal and ideal signal.

48. The method of claim 43 wherein synthesizing the frequency offset corrected reference signal includes, for each of a set of sample points:

constructing an ideal signal sample from the frequency offset corrected copy signal at the same sample point, the ideal signal sample having a phase determined from the frequency offset corrected copy signal at the sample point, with the phase of the ideal signal sample at an initial symbol point set to be an initial ideal signal phase;

relaxing the phase of the ideal signal sample towards the frequency offset corrected copy signal sample phase to produce the phase of the frequency offset corrected reference signal; and

producing the frequency offset corrected reference signal having the phase of the frequency offset corrected reference signal produced by relaxing the phase of the ideal signal towards the frequency offset copy signal sample phase.

49. The method of claim 48 wherein the phase of the ideal signal is determined in constructing an ideal signal sample from the frequency offset corrected copy signal at the same sample point, sample by sample, the phase of the ideal signal sample at any sample point being determined from the phase of the frequency offset corrected reference signal at the previous sample point for which said phase is determined, and from a decision based on the frequency offset corrected copy signal.

50. The method of claim 48 wherein relaxing the phase of the ideal signal sample towards the phase of the frequency offset corrected copy signal corresponds to adding a filtered version of the difference between the frequency offset corrected copy signal phase and ideal signal phase.

51. The method of claim 48 wherein relaxing the phase of the ideal signal sample towards the phase of the frequency offset corrected copy signal corresponds to forming the reference signal sample by adding to the ideal signal sample a filtered version of the difference between the frequency offset corrected copy signal and ideal signal.

52. The method of claim 42 wherein synthesizing the timing aligned reference signal includes, for each of a set of sample points:

constructing an ideal signal sample from the timing aligned copy signal at the same sample point, the ideal signal sample having a phase determined from the timing aligned copy signal at the sample point, with the phase of the ideal signal sample at an initial symbol point set to be an initial ideal signal phase;

relaxing the phase of the ideal signal sample towards the timing aligned copy signal sample phase to produce the phase of the timing aligned reference signal; and

producing the timing aligned reference signal having the phase of the timing aligned reference signal produced by relaxing the phase of the ideal signal sample towards the timing aligned copy signal sample phase.

53. The method of claim 52 wherein the phase of the ideal signal is determined in constructing an ideal signal sample from the timing aligned copy signal at the same sample point, sample by sample, the phase of the ideal signal sample at any sample point being determined from the phase of the timing aligned reference signal at the previous sample point for which said phase is determined, and from a decision based on the timing aligned copy signal.

54. The method of claim 52 wherein relaxing the phase of the ideal signal sample towards the phase of the timing aligned copy signal corresponds to adding a filtered version of the difference between the timing aligned copy signal phase and ideal signal phase.

55. The method of claim 52 wherein relaxing the phase of the ideal signal sample towards the phase of the timing aligned copy signal corresponds to forming the reference signal sample by adding to the ideal signal sample a filtered version of the difference between the timing aligned copy signal and ideal signal.

56. The method of claim 41 wherein synthesizing the corrected reference signal includes:

coherently demodulating the corrected copy signal to form signal symbols; and

re-modulating the signal symbols to form the corrected reference signal having substantially the same timing alignment and frequency offset as the corrected copy signal.

57. The method of claim 43 wherein synthesizing the frequency offset corrected reference signal includes:

coherently demodulating the frequency offset corrected copy signal to form signal symbols; and

re-modulating the signal symbols to form the frequency offset corrected reference signal having substantially the same frequency offset as the frequency offset corrected copy signal.

58. The method of claim 41 wherein estimating the frequency offset includes:

applying a nonlinearity to a set of samples determined from the copy signal;

taking a DFT; and

determining the shift that when applied to an interpolation function causes the shifted interpolation function to best fit the DFT result, the resulting determined shift being a multiple of the estimated frequency offset.

59. The method of claim 43 wherein estimating the frequency offset includes:

applying a nonlinearity to a set of samples determined from the copy signal;

taking a DFT; and

determining the shift that when applied to an interpolation function causes the shifted interpolation function to best fit the DFT result, the resulting determined shift being a multiple of the estimated frequency offset.

60. An apparatus comprising:

a spatial processor coupled to a weight formation processor to separate a copy signal corresponding to a remote station from a set of received antenna signals using a first spatial weight vector;

a reference signal generator coupled to the spatial processor to determine a reference signal from the copy signal;

the weight formation processor coupled to the reference signal generator to provide the first spatial weight vector and to compute a second spatial weight vector using the received signal corresponding to the remote station and the reference signal.

61. The apparatus in claim 60 wherein the apparatus is a communications station, the communications station further including an array of antennas.

62. The apparatus in claim 60 wherein the spatial processor weighs the set of received antenna signals by a set of corresponding receive weights.

63. The apparatus in claim 60 wherein the determined reference signal has substantially the same frequency offset and time alignment as the received antenna signals.

64. The apparatus in claim 60 further including a timing alignment processor coupled to the spatial processor and weight formation processor to perform timing alignment on the received antenna signals, said spatial processor and weight formation processor using the time aligned received antenna signals.

65. The apparatus in claim 60 further including a frequency offset correction processor coupled to the spatial processor and weight formation processor to perform frequency offset correction on the received antenna signals, said spatial processor and weight formation processor using the frequency-offset corrected received antenna signals.

66. The apparatus in claim 64 further including a frequency offset correction processor coupled to the spatial processor and weight formation processor to perform frequency offset correction on the received antenna signals, said spatial processor and weight formation processor using the frequency-offset corrected received antenna signals.

67. An apparatus comprising:

a means for separating a copy signal corresponding to a particular remote station from a set of received antenna signals using a spatial weight vector;

a means for generating a reference signal from the copy signal; and

a means for generating the spatial weight vector using the received signal corresponding to a particular remote station and the reference signal.

68. The apparatus in claim 67 further including a means for aligning the timing of the received antenna signals.

69. The apparatus in claim 67 further including a means for correcting the frequency offset of the received antenna signals.

70. The apparatus in claim 68 further including a means for correcting the frequency offset of the received antenna signals.

71. An article of manufacture, comprising:

a machine accessible medium, the machine accessible medium providing instructions that, when executed by a machine, cause the machine to generate a reference signal from a modulated signal by;

separating from the received antenna signals a copy signal corresponding to the particular remote station by using an initial spatial weight vector corresponding to the particular remote station;

determining from the copy signal a reference signal; and

computing a new spatial weight vector by optimizing a cost function, the cost function using the received antenna signals and the reference signal.

72. The article of claim 71, wherein the instructions that, when executed, further cause the machine to extract the symbols of the modulated signal.

73. The article of claim 71, wherein the instructions that, when executed, further cause the machine to perform timing alignment on the received antenna signals prior to separating from the received antenna signals the copy signal.

74. The article of claim 71, wherein the instructions that, when executed, further cause the machine to perform frequency offset correction on the received antenna signals prior to separating from the received antenna signals the copy signal.

75. The article of claim 73, wherein the instructions that, when executed, further cause the machine to perform frequency offset correction on the received antenna signals prior to separating from the received antenna signals the copy signal.

76. The article of claim 71, wherein the instructions that, when executed, further cause the machine to repeat separating from the received antenna signals a copy signal at least once, using the second new spatial weight vector previously computed instead of the initial spatial weight vector.

77. The article of claim 71, wherein the instructions that, when executed, further cause the machine to repeat, at least once, determining from the copy signal the reference signal.

78. The article of claim 71, wherein the instructions that, when executed, further cause the machine to:

estimate a frequency offset and a timing misalignment of the copy signal; and

correct the copy signal for frequency offset and timing misalignment to form a corrected copy signal,

and wherein the instructions that, when executed, cause the machine to determine from a copy signal the reference signal further include instructions that, when executed, cause the machine to;

synthesize a corrected reference signal that has substantially the same frequency offset and timing alignment as the corrected copy signal; and

apply a frequency offset and time misalignment to the corrected reference signal to form a frequency offset and time misaligned reference signal having the same frequency offset and time misalignment as the received antenna signals.

79. The article of claim 71, wherein the instructions that, when executed, further cause the machine to:

estimate a timing misalignment of the copy signal; and

correct the copy signal for timing misalignment to form a timing aligned copy signal,

and wherein the instructions that, when executed, cause the machine to determine from a copy signal the reference signal further include instructions that, when executed, cause the machine to;

synthesize a timing aligned reference signal that has substantially the same timing alignment as the timing-aligned copy signal; and

apply a time misalignment to the timing aligned reference signal to form a timing misaligned reference signal having substantially the same time alignment as the received antenna signals.

80. The article of claim 71, wherein the instructions that, when executed, further cause the machine to:

estimate the frequency offset of the copy signal, and

correct the copy signal for frequency offset to form a frequency offset corrected copy signal,

and wherein the instructions that, when executed, cause the machine to determine from a copy signal the reference signal further include instructions that, when executed, cause the machine to;

synthesize a frequency offset corrected reference signal that has substantially the same frequency offset as the frequency offset corrected copy signal; and

apply a frequency offset to the frequency offset corrected reference signal to form a frequency offset reference signal having substantially the same frequency offset as the received antenna signals.

81. The article of claim 71, wherein the instructions that, when executed, further cause the machine to determine the reference signal for each of a set of sample points by:

constructing an ideal signal sample from the copy signal at the same sample point, the ideal signal sample having a phase determined from the copy signal at the sample point, with the phase of the ideal signal sample at an initial symbol point set to be an initial ideal signal phase;

relaxing the phase of the ideal signal sample towards the copy signal sample phase to produce the phase of the reference signal; and

producing the reference signal having the phase of the reference signal produced by relaxing the phase of the ideal signal sample towards the copy signal sample phase.

82. The article of claim 81, wherein the instructions that, when executed, further cause the machine, in constructing an ideal signal sample from the copy signal at the same sample point, to determine sample by sample, the phase of the ideal signal sample at any sample point from the phase of the reference signal at the previous sample point for which said phase is determined and a decision based on the copy signal.

83. The article of claim 81, wherein the instructions that, when executed, cause the machine to relax the phase of the ideal signal sample towards the phase of the copy signal, comprise instructions that, when executed, cause the machine to add a filtered version of the difference between the copy signal phase and ideal signal phase.

84. The article of claim 81, wherein the instructions, when executed, further cause the machine to relax the phase of the ideal signal sample towards the phase of the copy signal by adding to the ideal signal sample a filtered version of the difference between the copy signal and ideal signal to form the reference signal sample.

85. The article of claim 80, wherein the instructions that, when executed, further cause the machine to:

construct an ideal signal sample from the frequency offset corrected copy signal at the same sample point, the ideal signal sample having a phase determined from the frequency offset corrected copy signal at the sample point, with the phase of the ideal signal sample at an initial symbol point set to be an initial ideal signal phase;

relax the phase of the ideal signal sample towards the frequency offset corrected copy signal sample phase to produce the phase of the frequency offset corrected reference signal; and

produce the frequency offset corrected reference signal having the phase of the frequency offset corrected reference signal produced by relaxing the phase of the ideal signal sample towards the frequency offset corrected copy signal.

86. The article of claim 85, wherein the instructions that, when executed, further cause the machine, in constructing an ideal signal sample from the frequency offset corrected copy signal at the same sample point, to determine sample by sample, the phase of the ideal signal sample at any sample point from the phase of the frequency offset corrected reference signal at the previous sample point for which said phase is determined and a decision based on the frequency offset corrected copy signal.

87. The article of claim 85, wherein the instructions that, when executed, cause the machine to relax the phase of the ideal signal sample towards the phase of the frequency offset corrected copy signal, comprise instructions that, when executed, cause the machine to add a filtered version of the difference between the frequency offset corrected copy signal phase and ideal signal phase.

88. The article of claim 85, wherein the instructions that, when executed, cause the machine to relax the phase of the ideal signal sample towards the phase of the frequency offset corrected copy signal, comprise instructions that, when executed cause the machine to form the reference signal sample by adding to the ideal signal sample a filtered version of the difference between the frequency offset corrected copy signal and ideal signal.

89. The article of claim 79, wherein the instructions that, when executed, further cause the machine to, for each of a set of sample points:

construct an ideal signal sample from the timing aligned copy signal at the same sample point, the ideal signal sample having a phase determined from the timing aligned copy signal at the sample point, with the phase of the ideal signal sample at an initial symbol point set to be an initial ideal signal phase;

relax the phase of the ideal signal sample towards the timing aligned copy signal sample phase to produce the phase of the timing aligned reference signal; and

produce the timing aligned reference signal having the phase of the timing aligned reference signal produced by relaxing the phase of the ideal signal sample towards the timing aligned copy signal.

90. The article of claim 89, wherein the instructions that, when executed, further cause the machine to, in constructing an ideal signal sample from the timing aligned copy signal at the same sample point, sample by sample, determine the phase of the ideal signal sample at any sample point from the phase of the timing aligned reference signal at the previous sample point for which said phase is determined and a decision based on the timing aligned copy signal.

91. The article of claim 89, wherein the instructions that, when executed, cause the machine to relax the phase of the ideal signal sample towards the phase of the timing aligned copy signal, comprise instructions that, when executed, cause the machine to add a filtered version of the difference between the timing aligned copy signal phase and ideal signal phase.

92. The article of claim 89, wherein the instructions that, when executed, cause the machine, to relax the phase of the ideal signal sample towards the phase of the timing aligned copy signal, comprise instructions that, when executed, cause the machine to form the reference signal sample by adding to the ideal signal sample a filtered version of the difference between the timing aligned copy signal and ideal signal.

93. The article of claim 78, wherein the instructions that, when executed, further cause the machine to:

coherently demodulate the corrected copy signal to form signal symbols; and

re-modulate the signal symbols to form the corrected reference signal having substantially the same timing alignment and frequency offset as the corrected copy signal.

94. The article of claim 80, wherein the instructions, when executed, further cause the machine to:

coherently demodulate the frequency offset corrected copy signal to form signal symbols; and

re-modulate the signal symbols to form the frequency offset corrected reference signal having substantially the same frequency offset as the frequency offset corrected copy signal.

95. The article of claim 78, wherein the instructions that, when executed, further cause the machine to estimate the frequency offset by:

applying a nonlinearity to a set of samples determined from the copy signal;

taking a DFT; and

determining the shift that when applied to an interpolation function causes the shifted interpolation function to best fit the DFT result, the resulting determined shift being a multiple of the estimated frequency offset.

96. The article of claim 80, wherein the instructions that, when executed, further cause the machine to estimate the frequency offset by:

applying a nonlinearity to a set of samples determined from the copy signal;

taking a DFT; and

determining the shift that when applied to an interpolation function causes the shifted interpolation function to best fit the DFT result, the resulting determined shift being a multiple of the estimated frequency offset.