Integer representation of relative timing between desired output samples and corresponding input samples
In general, this disclosure describes techniques for changing a sampling frequency of a digital signal. In particular, the techniques provide a more accurate way to determining a relative timing between a desired output sample and a corresponding input sample using a non-approximated integer representation of the relative timing. The relative timing between the desired output sample and corresponding input sample may be represented using a first component that identifies a latest input sample of the digital signal used to generate intermediate samples, a second component that identifies an intermediate sample, and a third component that identifies a timing difference between the desired output sample and the intermediate sample. Each of the components may be recursively updated using non-approximated integer values.
1. A method comprising representing a relative timing between a desired output sample and a corresponding input sample of a digital signal using a first component that identifies a latest input sample of the digital signal used to generate related intermediate samples, a second component that identifies an intermediate sample, and a third component that identifies a timing difference between the desired output sample and the intermediate sample, wherein each of the components is represented as a non-approximated integer.
2. The method of claim 1 , further comprising updating the third component according to the equation:
θ( n )=(θ( n −1)+ d frac )−└(θ( n −1)+ d frac )/ f o ┘f o
where, θ(n−1) is a previous value of the third component, d frac is an increment value, f o is a desired output sampling frequency, and └(θ(n−1)+d frac )/f o ┘ is a floor function that returns the largest integer that is less than or equal to (θ(n−1)+d frac )/f o .
3. The method of claim 2 , further comprising computing the increment value d frac according to the equation
d
frac
=
Nf
i
-
⌊
Nf
i
f
o
⌋
f
o
,
where N is an up-sampling factor, f i is an input sampling frequency, and
⌊
Nf
i
f
o
⌋
is a floor function that returns the largest integer that is less than or equal to
Nf
i
f
o
.
4. The method of claim 2 , further comprising setting a first carryover value ξ frac (n) associated with the third component to one when θ(n−1)+d frac is greater than or equal to f o .
5. The method of claim 4 , further comprising updating the second component β m (n) according to the equation:
β
m
(
n
)
=
(
β
m
(
n
-
1
)
+
⌊
Nf
l
f
o
⌋
+
ξ
frac
(
n
)
)
-
⌊
(
β
m
(
n
-
1
)
+
⌊
Nf
i
f
o
⌋
+
ξ
frac
(
n
)
)
/
N
⌋
N
where, β m (n−1) is the a previous value of the second component and
⌊
β
m
(
n
-
1
)
+
⌊
Nf
i
f
o
⌋
+
ξ
frac
(
n
)
⌋
is a floor function that returns the largest integer that is less than or equal to
β
m
(
n
-
1
)
+
⌊
Nf
l
f
o
⌋
+
ξ
frac
(
n
)
.
6. The method of claim 5 , further comprising setting a second carryover value to one when the second component is greater than or equal to the up-sampling factor N.
7. The method of claim 6 , further comprising incrementing the first component when the second carryover value is equal to one.
8. The method of claim 7 , wherein the first component and the second component are associated with a first intermediate sample, further comprising computing a first component and a second component associated with a second intermediate sample as a function of the first component and the second component of the first intermediate sample.
9. The method of claim 8 , further comprising computing the second component associated with the second intermediate sample β k (n) according to the equation:
β k ( n )=(β m ( n )+1)−└(β m ( n )+1)/ N┘N
where, └(β m (n)+1)+1)/N┘ is a floor function that returns the largest integer that is less than or equal to (β m (n)+1)/N.
10. The method of claim 9 , further comprising setting a third carryover value to one when the second component associated with the second intermediate sample is greater than or equal to the up-sampling factor N.
11. The method of claim 10 , further comprising incrementing the first component associated with the second intermediate sample when the third carryover value is equal to one.
12. The method of claim 8 , wherein the first intermediate sample comprises a previous intermediate sample and the second intermediate sample comprises a next intermediate sample.
13. The method of claim 8 , wherein the first intermediate sample comprises a next intermediate sample and the second intermediate sample comprises a previous intermediate sample.
14. The method of claim 2 , further comprising:
computing an interpolation coefficient using the equation θ(n)/f o ; and
performing interpolation to obtain the desired output sample as a function of the interpolation coefficient.
15. The method of claim 13 , further comprising:
pre-calculating 1/f o ; and
setting a quality factor to be equal to d+k, where k is the largest positive number such that 2 k <f o and d is the number of bits used to represent 1/f o as a fixed point number.
16. An apparatus comprising, a phase tracking module that represents a relative timing between a desired output sample and a corresponding input sample of a digital signal using a first component that identifies a latest input sample of the digital signal used to generate related intermediate samples, a second component that identifies an intermediate sample, and a third component that identifies a timing difference between the desired output sample and the intermediate sample, wherein each of the components is represented as a non-approximated integer.
17. The apparatus of claim 16 , wherein the phase tracking module updates the third component according to the equation:
θ( n )=(θ( n− 1)+ d frac )−└(θ( n− 1)+ d frac )/ f o ┘f o
where, θ(n−1) is a previous value of the third component, d frac is an increment value, f 0 is a desired output sampling frequency, and └(θ(n−1)+d frac )/f o ┘ is a floor function that returns the largest integer that is less than or equal to (θ(n−1)+d frac )/f o .
18. The apparatus of claim 17 , wherein the phase tracking module computes the increment value d frac according to the equation
d
frac
=
Nf
i
-
⌊
Nf
i
f
o
⌋
f
o
,
where N is an up-sampling factor, f i is an input sampling frequency, and
⌊
Nf
i
f
o
⌋
is a floor function that returns the largest integer that is less than or equal to
Nf
i
f
o
.
19. The apparatus of claim 17 , wherein the phase tracking module sets a first carryover value ξ frac (n) associated with the third component to one when θ(n−1)+d frac is greater than or equal to f o .
20. The apparatus of claim 19 , wherein the phase tracking module updates the second component β m (n) according to the equation:
β
m
(
n
)
=
(
β
m
(
n
-
1
)
+
⌊
Nf
l
f
o
⌋
+
ξ
frac
(
n
)
)
-
⌊
(
β
m
(
n
-
1
)
+
⌊
Nf
i
f
o
⌋
+
ξ
frac
(
n
)
)
/
N
⌋
N
where, β m (n−1) is a previous value of the second component and
⌊
β
m
(
n
-
1
)
+
⌊
Nf
i
f
o
⌋
+
ξ
frac
(
n
)
⌋
is a floor function that returns the largest integer that is less than or equal to
β
m
(
n
-
1
)
+
⌊
Nf
l
f
o
⌋
+
ξ
frac
(
n
)
.
21. The apparatus of claim 20 , wherein the phase tracking module sets a second carryover value to one when the second component is greater than or equal to the up-sampling factor N.
22. The apparatus of claim 21 , wherein the phase tracking module increments the first component when the second carryover value is equal to one.
23. The apparatus of claim 22 , wherein the first component and the second component are associated with a first intermediate sample, further comprising computing a first component and a second component associated with a second intermediate sample as a function of the first component and the second component of the first intermediate sample.
24. The apparatus of claim 23 , wherein the phase tracking module computes the second component associated with the second intermediate sample β k (n) according to the equation:
β k ( n )=(β m ( n )+1)−└(β m ( n )+1)/ N┘N,
where └(β m (n)+1)/N┘ is a floor function that returns the largest integer that is less than or equal to (β m (n)+1)/N.
25. The apparatus of claim 24 , wherein the phase tracking module sets a third carryover value to one when the second component associated with the second intermediate sample is greater than or equal to the up-sampling factor N.
26. The apparatus of claim 25 , wherein the phase tracking module increments a first component associated with the second intermediate sample when the third carryover value is equal to one.
27. The apparatus of claim 23 , wherein the first intermediate sample comprises a previous intermediate sample and the second intermediate sample comprises a next intermediate sample.
28. The apparatus of claim 23 , wherein the first intermediate sample comprises a next intermediate sample and the second intermediate sample comprises a previous intermediate sample.
29. The apparatus of claim 19 , further comprising an interpolation module that computes an interpolation coefficient using the equation θ(n)/f o and performs interpolation to obtain the desired output sample as a function of the interpolation coefficient.
30. The apparatus of claim 29 , wherein the down-sampling module pre-calculates 1/f o and sets a quality factor to be equal to d+k, where k is the largest positive number such that 2 k <f o and d is the number of bits used to represent 1/f o as a fixed point number.
31. A computer-readable medium comprising instructions that upon execution cause a processor to represent a relative timing between a desired output sample and a corresponding input sample of a digital signal using a first component that identifies a latest input sample of the digital signal used to generate related intermediate samples, a second component that identifies an intermediate sample, and a third component that identifies a timing difference between the desired output sample and the intermediate sample, wherein each of the components is represented as a non-approximated integer.
32. The computer-readable medium of claim 31 , further comprising instructions that upon execution cause the processor to:
update the third component according to the equation:
θ( n )=(θ( n− 1)+ d frac )−└(θ( n− 1)+ d frac )/ f o ┘f o
where, θ(n−1) is a previous value of the third component, d frac is an increment value, f 0 is a desired output sampling frequency, and └(θ(n−1)+d frac )/f o ┘ is a floor function that returns the largest integer that is less than or equal to (θ(n−1)+d frac )/f o ; and
set a first carryover value ξ frac (n) associated with the third component to one when θ(n−1)+d frac is greater than or equal to f o .
33. The computer-readable medium of claim 32 , further comprising instructions that upon execution cause the processor to compute the increment value d frac according to the equation
d
frac
=
Nf
i
-
⌊
Nf
i
f
o
⌋
f
o
,
where N is an up-sampling factor, f i is an input sampling frequency, and
⌊
Nf
i
f
o
⌋
is a floor function that returns the largest integer that is less than or equal to
Nf
i
f
o
.
34. The computer-readable medium of claim 32 , further comprising instructions that upon execution cause the processor to:
update the second component β m (n) according to the equation:
β
m
(
n
)
=
(
β
m
(
n
-
1
)
+
⌊
Nf
l
f
o
⌋
+
ξ
frac
(
n
)
)
-
⌊
(
β
m
(
n
-
1
)
+
⌊
Nf
i
f
o
⌋
+
ξ
frac
(
n
)
)
/
N
⌋
N
where, β m (n−1) is a previous value of the second component and
⌊
β
m
(
n
-
1
)
+
⌊
Nf
i
f
o
⌋
+
ξ
frac
(
n
)
⌋
is a floor function that returns the largest integer that is less than or equal to
β
m
(
n
-
1
)
+
⌊
Nf
i
f
o
⌋
+
ξ
frac
(
n
)
;
set a second carryover value one when the second component is greater than or equal to the up-sampling factor N; and
increment the first component when the second carryover value is equal to one.
35. The computer-readable medium of claim 34 , wherein the first component and the second component are associated with a first intermediate sample, further comprising instructions that upon execution cause the processor to:
update a second component associated with a second intermediate sample β k (n) according to the equation:
β k ( n )=(β m ( n )+1)−└β m ( n )+1)/ N┘N,
where └(β m (n)+1)/N┘ is a floor function that returns the largest integer that is less than or equal to (β m (n)+1)/N;
set a third carryover value to one when the second component associated with the second intermediate sample is greater than or equal to the up-sampling factor N; and
increment the first component associated with the second intermediate sample when the third carryover value associated with the second component is equal to one.
36. The computer-readable medium of claim 35 , wherein the first intermediate sample comprises a previous intermediate sample and the second intermediate sample comprises a next intermediate sample.
37. The computer-readable medium of claim 35 , wherein the first intermediate sample comprises a next intermediate sample and the second intermediate sample comprises a previous intermediate sample.
38. The apparatus of claim 16 , wherein said apparatus is a wireless communications device.
39. An apparatus comprising, a means for representing a relative timing between a desired output sample and a corresponding input sample of a digital signal using a first component that identifies a latest input sample of the digital signal used to generate related intermediate samples, a second component that identifies an intermediate sample, and a third component that identifies a timing difference between the desired output sample and the intermediate sample, wherein each of the components is represented as a non-approximated integer.
40. The apparatus of claim 39 , wherein the phase tracking module updates the third component according to the equation:
θ( n )=(θ( n− 1)+ d frac )−└(θ( n− 1)+ d frac )/ f o ┘ f o .
where, θ(n−1) is a previous value of the third component, d frac is an increment value, f o is a desired output sampling frequency, and └(θ(n−1)+d frac )/f o ┘ is a floor function that returns the largest integer that is less than or equal to (θ(n−1)+d frac )/f o .
41. The apparatus of claim 39 , wherein said means for representing a relative timing comprises a phase tracking module.
42. The apparatus of claim 39 , wherein said apparatus comprises a wireless communication device.
43. An apparatus comprising means for representing a relative timing between a desired output sample and a means for corresponding input sample of a digital signal using a first component that identifies a latest input sample of the digital signal used to generate related intermediate samples, a second component that identifies an intermediate sample, and a third component that identifies a timing difference between the desired output sample and the intermediate sample, wherein each of the components is represented as a non-approximated integer.
44. The apparatus of claim 43 , further comprising means for updating the third component according to the equation:
θ( n )=(θ( n− 1)+ d frac )−└(θ( n− 1)+ d frac )/ f o ┘ f o .
where, θ( n −1) is a previous value of the third component, d frac is an increment value, f o is a desired output sampling frequency, and └(θ( n −1)+ d frac / f o ┘ is a floor function that returns the largest integer that is less than or equal to (θ( n −1)+ d frac / f o .
45. The apparatus of claim 44 , further comprising means for computing the increment value d frac according to the equation
d
frac
=
Nf
i
-
⌊
Nf
i
f
o
⌋
f
o
,
where N is an up-sampling factor, f i is an input sampling frequency, and
⌊
Nf
i
f
o
⌋
is a floor function that returns the largest integer that is less than or equal to
Nf
i
f
o
.
46. The apparatus of claim 44 , further comprising means for setting a first carryover value ξ frac ( n ) associated with the third component to one when θ( n −1)+ d frac is greater than or equal to f o .
47. The apparatus of claim 46 , further comprising means for updating the second component β m ( n ) according to the equation:
β
m
(
n
)
=
(
β
m
(
n
-
1
)
+
⌊
Nf
i
f
o
⌋
+
ξ
frac
(
n
)
)
-
⌊
(
β
m
(
n
-
1
)
+
⌊
Nf
i
f
o
⌋
+
ξ
frac
(
n
)
)
/
N
⌋
N
.
where, β m ( n −1) is the a previous value of the second component and
⌊
β
m
(
n
-
1
)
+
⌊
Nf
i
f
o
⌋
+
ξ
frac
(
n
)
⌋
is a floor function that returns the largest integer that is less than or equal to
β
m
(
n
-
1
)
+
⌊
Nf
l
f
o
⌋
+
ξ
frac
(
n
)
.
48. The apparatus of claim 47 , further comprising means for setting a second carryover value to one when the second component is greater than or equal to the up-sampling factor N.
49. The apparatus of claim 48 , further comprising means for incrementing the first component when the second carryover value is equal to one.
50. The apparatus of claim 49 , wherein the first component and the second component are associated with a first intermediate sample, further comprising means for computing a first component and a second component associated with a second intermediate sample as a function of the first component and the second component of the first intermediate sample.
51. The apparatus of claim 50 , further comprising means for computing the second component associated with the second intermediate sample β k ( n ) according to the equation: β k ( n )=(β m ( n )+1)−└(β m ( n )+1)/N┘N. where, └(β m ( n )+1)/N┘is a floor function that returns the largest integer that is less than or equal to (β m ( n )+1)/N.
52. The apparatus of claim 51 , further comprising means for setting a third carryover value to one when the second component associated with the second intermediate sample is greater than or equal to the up-sampling factor N.
53. The apparatus of claim 52 , further comprising means for incrementing the first component associated with the second intermediate sample when the third carryover value is equal to one.
54. The apparatus of claim 50 , wherein the first intermediate sample comprises a previous intermediate sample and the second intermediate sample comprises a next intermediate sample.
55. The apparatus of claim 50 , wherein the first intermediate sample comprises a next intermediate sample and the second intermediate sample comprises a previous intermediate sample.
56. The apparatus of claim 44 , further comprising:
means for computing an interpolation coefficient using the equation θ( n )/ f o ; and
means for performing interpolation to obtain the desired output sample as a function of the interpolation coefficient.
57. The apparatus of claim 55 , further comprising:
means for pre-calculating 1/ f o ; and
means for setting a quality factor to be equal to d+k, where k is the largest positive number such that 2 k < f 0 and d is the number of bits used to represent 1/ f o as a fixed point number.