Blind estimation of residual chromatic dispersion and carrier frequency offset
Upon receiving a communications signal conveying symbols at a symbol period T, a receiver applies filter coefficients to a digital representation of the communications signal, thereby generating filtered signals characterized by a substantially raised cosine shape in the frequency domain with a roll-off factor α, where components of the filtered signals correspond to angular frequencies ω = - π ( 1 + α ) T … - π ( 1 - α ) T , + π ( 1 - α ) T … + π ( 1 + α ) T . The receiver calculates first-order components from a first phase derivative of the components at a first differential distance, second-order components from a second phase derivative of the first-order components at a second differential distance, and composite second-order components from an average of the second-order components over multiple time intervals. Using the composite second-order components, the receiver calculates at least one of (i) an estimate of residual chromatic dispersion (CD) associated with the filtered signals, and (ii) an estimate of carrier frequency offset (CFO) associated with the filtered signals.
1. A receiver apparatus comprising:
a communication interface configured to receive a communications signal conveying symbols at a symbol period T, wherein T is a positive real number;
circuitry configured to apply filter coefficients to a digital representation of the communications signal, thereby generating filtered signals characterized by a substantially raised cosine shape in the frequency domain with a roll-off factor α, wherein α is a real number satisfying 0<α≤1, and wherein components of the filtered signals correspond to a range of angular frequencies
ω
=
-
π
(
1
+
α
)
T
…
-
π
(
1
-
α
)
T
,
+
π
(
1
-
α
)
T
…
+
π
(
1
+
α
)
T
;
circuitry configured to calculate first-order components from a first phase derivative of the components of the filtered signals at a first differential distance;
circuitry configured to calculate second-order components from a second phase derivative of the first-order components at a second differential distance;
circuitry configured to calculate composite second-order components from an average of the second-order components over a plurality of time intervals; and
circuitry configured to calculate, using the composite second-order components, at least one of (i) an estimate of residual chromatic dispersion (CD) associated with the filtered signals, and (ii) an estimate of carrier frequency offset (CFO) associated with the filtered signals.
2. The receiver apparatus as claimed in claim 1 , further comprising
circuitry configured to calculate updated filter coefficients using the estimate of the residual CD; and
circuitry configured to apply the updated filter coefficients to the digital representation of the communications signal, thereby generating updated filtered signals.
3. The receiver apparatus as claimed in claim 2 , wherein the estimate of the residual CD comprises an initial estimate based on an initial value of the second differential distance, the receiver apparatus further comprising
circuitry configured to calculate a subsequent estimate of the residual CD associated with the updated filtered signals based on a subsequent value of the second differential distance,
wherein the subsequent value of the second differential distance exceeds the initial value of the second differential distance, and
wherein the initial estimate of the residual CD exceeds the subsequent estimate of the residual CD.
4. The receiver apparatus as claimed in claim 1 , further comprising
circuitry configured to calculate third-order components from a third phase derivative of the composite second-order components at a third differential distance; and
circuitry configured to calculate the estimate of the residual CD using an average of the third-order components over a plurality of frequencies.
5. The receiver apparatus as claimed in claim 1 , further comprising
circuitry configured to calculate a timing bin corresponding to a peak magnitude of an inverse Fourier transform of the composite second-order components; and
circuitry configured to calculate the estimate of the residual CD using the timing bin.
6. The receiver apparatus as claimed in claim 1 , further comprising
circuitry configured to perform carrier recovery on the filtered signals using the estimate of the CFO.
7. The receiver apparatus as claimed in claim 1 , further comprising
circuitry configured to calculate either a frequency corresponding to a peak magnitude of the composite second-order components or a frequency corresponding to a center of gravity of the magnitudes of the composite second-order components; and
circuitry configured to calculate the estimate of the CFO using the frequency.
8. The receiver apparatus as claimed in claim 1 , further comprising
circuitry configured to calculate a frequency shift of an expected shape of the composite second-order components that maximizes a correlation between the composite second-order components and the expected shape; and
circuitry configured to calculate the estimate of the CFO using the frequency shift.
9. The receiver apparatus as claimed in claim 1 , further comprising
circuitry configured to calculate an autocorrelation of an inverse Fourier transform of the composite second-order components; and
circuitry configured to calculate the estimate of the CFO using the autocorrelation.
10. The receiver apparatus as claimed in claim 1 , wherein the estimate of the CFO is calculated using only the composite second-order components having magnitudes equal to or greater than a predefined threshold.
11. A method comprising:
receiving a communications signal conveying symbols at a symbol period T, wherein T is a positive real number;
applying filter coefficients to a digital representation of the communications signal, thereby generating filtered signals characterized by a substantially raised cosine shape in the frequency domain with a roll-off factor α, wherein a is a real number satisfying 0<α≤1, and wherein components of the filtered signals correspond to a range of angular frequencies
ω
=
-
π
(
1
+
α
)
T
…
-
π
(
1
-
α
)
T
,
+
π
(
1
-
α
)
T
…
+
π
(
1
+
α
)
T
;
calculating first-order components from a first phase derivative of the components of the filtered signals at a first differential distance;
calculating second-order components from a second phase derivative of the first-order components at a second differential distance;
calculating composite second-order components from an average of the second-order components over a plurality of time intervals; and
using the composite second-order components, calculating at least one of (i) an estimate of residual chromatic dispersion (CD) associated with the filtered signals, and (ii) an estimate of carrier frequency offset (CFO) associated with the filtered signals.
12. The method as claimed in claim 11 , further comprising
calculating updated filter coefficients using the estimate of the residual CD; and
applying the updated filter coefficients to the digital representation of the communications signal, thereby generating updated filtered signals.
13. The method as claimed in claim 12 , wherein the estimate of the residual CD comprises an initial estimate based on an initial value of the second differential distance, the method further comprising
calculating a subsequent estimate of the residual CD associated with the updated filtered signals based on a subsequent value of the second differential distance,
wherein the subsequent value of the second differential distance exceeds the initial value of the second differential distance, and
wherein the initial estimate of the residual CD exceeds the subsequent estimate of the residual CD.
14. The method as claimed in claim 11 , further comprising
calculating third-order components from a third phase derivative of the composite second-order components at a third differential distance; and
calculating the estimate of the residual CD using an average of the third-order components over a plurality of frequencies.
15. The method as claimed in claim 11 , further comprising
calculating a timing bin corresponding to a peak magnitude of an inverse Fourier transform of the composite second-order components; and
calculating the estimate of the residual CD using the timing bin.
16. The method as claimed in claim 11 , further comprising
performing carrier recovery on the filtered signals using the estimate of the CFO.
17. The method as claimed in claim 11 , further comprising
calculating either a frequency corresponding to a peak magnitude of the composite second-order components or a frequency corresponding to a center of gravity of the magnitudes of the composite second-order components; and
calculating the estimate of the CFO using the frequency.
18. The method as claimed in claim 11 , further comprising
calculating a frequency shift of an expected shape of the composite second-order components that maximizes a correlation between the composite second-order components and the expected shape; and
calculating the estimate of the CFO using the frequency shift.
19. The method as claimed in claim 11 , further comprising
calculating an autocorrelation of an inverse Fourier transform of the composite second-order components; and
calculating the estimate of the CFO using the autocorrelation.
20. The method as claimed in claim 11 , wherein the estimate of the CFO is calculated using only the composite second-order components having magnitudes equal to or greater than a predefined threshold.