Constellation selection threshold adaptation for slicer

Abstract

System and method of adapting thresholds for constellation selection based on statistic distributions of received data symbols. To determine an adapted threshold, an expected ratio of received symbols with values in a certain range is preset based on an expected statistic distribution of data symbols across the multiple constellations. A first and a second ratios are defined based on the expected ratio, the first ratio being the expected ratio minus an error ratio and the second ratio being the expected ratio plus the error ratio. A first value is determined which makes the received symbols in a firs range to constitute the first ratio of a set of slicer inputs. A second value is determined which makes the received symbols in the second range to constitute the second ratio of a set of slicer outputs. The adapted threshold is then obtained based on the first and the second value.

Claims

1. A method of dynamically determining a threshold of a slicer, the method comprising: accessing a first plurality of inputs for supply to said slicer; comparing each of said first plurality of inputs with a first value; based on said comparing with said first value, determining a first count ratio of first inputs to said first plurality of inputs, wherein each of said first inputs is comprised in said first plurality of inputs and has a value encompassed in a first range that is defined by said first value; and determining said threshold of said slicer based on said first value and said first count ratio.

2. The method of claim 1, further comprising: accessing a second plurality of inputs for supply to said slicer; comparing each of said second plurality of inputs with a second value; and based on said comparing with said second value, determining a second count ratio of second inputs to said second plurality of inputs, wherein each of said second inputs is comprised in said second plurality of inputs and has a value encompassed in a second range that is defined by said second value; wherein said determining said threshold of said slicer is further based on said second value and said second count ratio.

3. The method of claim 2, wherein said determining said threshold comprises averaging said first value and said second value.

4. The method of claim 2 further comprising determining said first value and said second value, wherein said determining said first value comprises adjusting said first value until said first count ratio equals a nominal ratio minus a preset value, and wherein said determining said second value comprises adjusting said second value until said second count ratio equals said nominal ratio plus said preset value.

5. The method of claim 4, wherein said slicer is configured to use Pulse Amplitude Modulation (PAM) comprising N constellation levels, wherein N is an integer greater than 1, wherein said nominal ratio equals $\frac{i}{N},$ wherein i is an integer greater than 0 and smaller than N, and wherein said preset value is equal to or less than 5.

6. The method of claim 4, wherein said slicer is configured to use Quadrature Amplitude Modulation (QAM) comprising 2×N constellation levels, wherein N is an integer greater than 1, wherein said nominal ratio equals $\frac{i}{N},$ wherein i is an integer greater than 0 and smaller than N, and wherein said preset value is equal to or less than 5.

7. The method of claim 2, wherein said first plurality of inputs and said second plurality of inputs are output from an equalizer in a receiver.

8. The method of claim 2, wherein said first range encompasses any value below said first value, and wherein said second range encompasses any value below said second value.

9. A device comprising: a slicer configured to: produce a first output value responsive to an input greater than a threshold, and produce a second output value responsive to an input smaller than said threshold; and a threshold adaptation unit coupled to said slicer, the threshold adaptation unit comprising: a comparator configured to compare each of a first plurality of inputs with a first value, wherein said first plurality of inputs are supplied to said slicer; and a first counter configured to, responsive to comparison decisions output from said comparator, produce a number of first inputs comprised in said first plurality of inputs and each having a value encompassed in a first range, wherein said first range is defined by said first value; wherein the threshold adaptation unit is configured to: determine a first count ratio of said number of first inputs to a total number of said first plurality of inputs, and determine an adapted threshold of said slicer based on said first value and said first count ratio.

10. The device of claim 9, wherein said threshold adaptation unit is further configured to access a second plurality of inputs for supply to said slicer; said comparator compares each of said second plurality of inputs with a second value; and wherein said threshold adaptation unit further comprises a second counter configured to, based on said comparison with said second value, determine a second count ratio of second inputs to said second plurality of inputs, wherein each of said second inputs is comprised in said second plurality of inputs and has a value encompassed in a second range that is defined by said second value, wherein said adapted threshold is determined also based on said second value and said second count ratio.

11. The device of claim 10, wherein said threshold adaptation unit is further configured to average said first value and said second value to generate said adapted threshold for said slicer.

12. The device of claim 10, wherein said threshold adaptation unit is further configured to: determine said first value by varying said first value until said first count ratio equals a nominal ratio minus a preset value; and determine said second value by varying said second value until said second count ratio equals said nominal ratio plus said preset value.

13. The device of claim 12, wherein said slicer is further configured to use Pulse Amplitude Modulation (PAM) comprising N constellation levels, wherein N is an integer greater than 1, wherein said first output value and said second output value are selected from said N constellation levels, and wherein said nominal ratio equals $\frac{i}{N},$ wherein i is an integer greater than 0 and smaller than N, and wherein said preset value is equal to or less than 5.

14. The device of claim 12, wherein said slicer is further configured to use Quadrature Amplitude Modulation (QAM) comprising 2×N constellation levels, wherein N is an integer greater than 1, wherein said nominal ratio equals $\frac{i}{N},$ wherein i is an integer greater than 0 and smaller than N, and wherein said preset value is equal to or less than 5.

15. The device of claim 10, wherein said first range encompasses any value below said first value, and wherein said second range encompasses any value below said second value.

16. A receiver comprising: an analog-to-digital converter (ADC) configured to convert received analog signals to digital signals; an equalizer coupled to said ADC and configured to: generate equalized signals responsive to said digital signals, and send said equalized signals as inputs to a slicer; said slicer coupled to said equalizer and configured to: produce a first output value responsive to an input greater than a threshold, and produce a second output value responsive to an input smaller than said threshold; and a threshold adaptation unit coupled to said slicer and configured to: access a first plurality of inputs for supply to said slicer, compare each of said first plurality of inputs with a first value, based on comparison with said first value, determine a first count ratio of first inputs to said first plurality of inputs, wherein each of said first inputs is comprised in said first plurality of inputs and has a value encompassed in a first range that is defined by said first value, access a second plurality of inputs for supply to said slicer, compare each of said second plurality of inputs with a second value, based on comparison with said second value, determine a second count ratio of second inputs to said second plurality of inputs, wherein each of said second inputs is comprised in said second plurality of inputs and has a value encompassed in a second range that is defined by said second value, determine an adapted threshold based on said first value, said second value, said first count ratio, and said second count ratio, and send said adapted threshold to said slicer.

17. The receiver of claim 16, wherein said threshold adaptation unit is further configured to determine said adapted threshold by averaging said first value and said second value.

18. The receiver of claim 16, wherein said threshold adaptation unit is further configured to: determine said first value by adjusting said first value until said first count ratio equals a nominal ratio minus a preset value, and determine said second value by adjusting said second value until said second count ratio equals said nominal ratio plus said preset value.

19. The receiver of claim 16, wherein said slicer is configured to use one of Pulse Amplitude Modulation (PAM) comprising N constellation levels and Quadrature Amplitude Modulation (QAM) comprising 2×N constellation levels, wherein N is an integer greater than 1, wherein said nominal ratio equals $\frac{i}{N},$ wherein i is an integer greater than 0 and smaller than N, and wherein said preset value is equal to or less than 5.

20. The receiver of claim 16, wherein said first range encompasses any value below said first value, and wherein said second range encompasses any value below said second value.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) Embodiments of the present invention will be better understood from a reading of the following detailed description, taken in conjunction with the accompanying drawing figures in which like reference characters designate like elements.

(2) FIG. 1 illustrates exemplary data ranges used to determine optimal adapted thresholds of constellation selection based on an expected statistic distribution of data symbols across multiple constellations in accordance with an embodiment of the present disclosure.

(3) FIG. 2 illustrates the configuration of an exemplary threshold adaptation unit capable of dynamically adapting constellation selection thresholds used in a slicer in accordance with an embodiment of the present disclosure.

(4) FIG. 3A is a flow chart depicting an exemplary process of determining adapted thresholds for constellation selection in accordance with an embodiment of the present disclosure.

(5) FIG. 3B is a flow chart depicting an exemplary process of determining a first value or a second value associate with a threshold in accordance with an embodiment of the present disclosure

(6) FIG. 4 illustrates exemplary data ranges used to determine optimal adapted constellations based on an expected statistic distribution of data symbols across multiple constellations in accordance with an embodiment of the present disclosure.

(7) FIG. 5 illustrates the configuration of an exemplary constellation adaptation unit capable of dynamically determining optimal adapted constellations in accordance with an embodiment of the present disclosure.

(8) FIG. 6 is a flow chart depicting an exemplary process of determining optimal adapted constellations for demodulation in accordance with an embodiment of the present disclosure.

(9) FIG. 7 illustrates a network signal transmission system that includes constellation adaptation logic and constellation selection threshold adaptation logic at the receiver in accordance with an embodiment of the present disclosure.

(10) FIG. 8 shows a set of simulated results comparing the symbol error rates (SERs) resulting from using default slicer thresholds and the SERs resulting from optimal adapted thresholds obtained in accordance with an embodiment of the present disclosure.

(11) FIG. 9 shows another set of simulated results comparing the SERs resulting from default slicer thresholds and SERs resulting from the optimal adapted thresholds obtained in accordance with an embodiment of the present disclosure.

CONSTELLATION SELECTION THRESHOLD ADAPTATION FOR SLICER

(12) Embodiments of the present disclosure provide a threshold adaptation mechanism of determining optimal adapted thresholds for constellation selection by a slicer at a receiver based on statistic distributions of the data symbols across multiple constellations in a modulation scheme. More specifically, to determine an optimal adapted threshold, threshold adaptation logic coupled to, or included in, the slicer is configured with an expected ratio of received symbols that have values in a certain range, e.g., below the optimal threshold that has shifted from the nominal threshold due to nonlinearity. This optimal threshold is to be discovered as the adapted threshold. The expected ratio can be determined based on an expected statistic distribution of data symbols across the multiple constellations. A first ratio and a second ratio are defined based on the expected ratio, the first ratio equal to the expected ratio minus an error ratio and the second ratio equal to the expected ratio plus the error ratio. The threshold adaptation logic then determines a first value that can make the received symbols in a first range (e.g., below the first value) to constitute the first ratio of a set of slicer inputs, and determine a second value that can make the received symbols in the second range (e.g., below the second value) to constitute the second ratio of a set of slicer outputs. The adapted threshold is then obtained based on the first and the second value.

(13) Embodiments of the present disclosure also provide a constellation adaptation mechanism of determining adapted constellations of a slicer at a receiver based on statistic distributions of the data symbols across the multiple constellations. More specifically, to determine an optimal constellation that offsets from the nominal constellation, constellation adaptation logic is provided with an expected ratio of received symbols that have values in a certain range, e.g., below the optimal constellation value that has shifted from the nominal value due to nonlinearity. This optimal constellation is to be discovered as the adapted constellation. The expected ratio can be determined based on an expected statistic distribution of the slicer inputs across the multiple constellations. The constellation adaption logic compares a first value with a set of received symbols and keeps count of the symbols in the certain range that is defined by the first value (e.g., below the first value) based on the comparison results. The count ratio of the number of inputs in the first range to the total number of the set of inputs is also derived. The first value is adjusted until the ratio reaches the expected ratio. The first value is then designated as an optimal adapted constellation to be used by the slicer.

(14) Embodiments herein are described in detail with reference to 4-level PAM slicers. However, it will be appreciated that the present disclosure is not limited to any particular modulation scheme or any specific number of constellations in a modulation scheme. The constellation adaption and threshold adaptation mechanisms provided herein can be used in any suitable demodulation devices besides slicers.

(15) Statistically speaking, data symbols as modulated and transmitted at the transmitter are distributed across the multiple constellation levels substantially in certain known percentages or ratios. For example, a large number of symbols are distributed substantially evenly across the N nominal constellations as a result of modulation. During signal transmission, the received symbols are altered in an unpredictable fashion because nonlinearity and amplitude compression can cause different noise levels for constellations and different constellation offsets. However, the collective symbol distribution with respect to the multiple constellations is expected to carry over the receiver side despite impairment on the signals during transmission. Particularly, for a large, number of received symbols at the receiver, the symbols falling in a particular data range that is defined by the actual constellations or the actual thresholds is expected to constitute a known ratio of the overall received symbols regardless of signal nonlinearity or amplitude compression.

(16) FIG. 1 illustrates exemplary data ranges used to determine optimal adapted thresholds of constellation selection based on an expected statistic distribution of data symbols across multiple constellations in accordance with an embodiment of the present disclosure. Constellation selection may be performed at a slicer. In this example, data symbols are modulated and demodulated according to the PAM-4 scheme, the nominal constellations C(1)˜C(4) being [−3, −1, +1, +3] respectively. Accordingly, a slicer uses 3 thresholds TH(1)˜TH(3) to decide which constellation is assigned to a received symbol, the nominal thresholds being [−2, 0, +2] respectively.

(17) As dictated by the modulation process at the transmitter side, the data symbols are distributed across the 4 constellations evenly. Accordingly, the data symbols falling below the optimal adapted TH (1) should all be associated with the constellation C(1) and are expected to constitute 25% of the overall received symbols. The data symbols falling between the optimal adapted TH(1) and the optimal adapted TH(2) should all be associated with the constellation C(2), and therefore the data symbols falling below the optimal adapted TH(2) are expected to constitute 50% of the overall received symbols. By the same token, the data symbols falling below the optimal adapted TH(3) should constitute 75% of the overall received symbols.

(18) To find an optimal TH(i) (i=1, 2 and 3), two values TH(i)1 and TH(i)2 are first determined based on the definitions that (1) the symbols below TH(i)1 take up

(19) $\left(100\times \frac{i}{N}-P\right)/100\left(\mathrm{alternatively}\mathrm{expressed}\mathrm{as}\frac{i}{N}-\frac{P}{100}\right)$
of the overall received symbols, and (2) the symbols below TH(i)2 take up

(20) $\left(100\times \frac{i}{N}+P\right)/100\left(\mathrm{alternatively}\mathrm{expressed}\mathrm{as}\frac{i}{N}+\frac{P}{100}\right)$
of the overall received symbols. In this case, N equals 4 and

(21) $\frac{P}{100}$
is a programmable error ratio mat is substantially smaller than

(22) $\frac{i}{N},$
e.g., P=5 or less.

(23) As illustrated, with respect to TH(1), TH((1)1 is defined as a value that (25−P) % symbols are below it and TH(1)2 is defined as a value that (25+P) % symbols are below it. With respect to TH(2), TH(2)1) is defined as a value that (50−P) % symbols are below it and TH(2)2 is defined as a value that (50+P) % symbols are below it. With respect to TH(3), TH(3)1 is defined as a value that (75−P) % symbols are below it and TH(3)2 is defined as a value that (75+P) % symbols are below it. TH(i) can then be derived as TH(i)=mean (TH(i)1, TH(i)2). That is, TH(1)=mean (TH(1)1, TH(1)2); TH(2)=mean (TH(2)1, TH(2)2); and TH(3)=mean (TH(3)1, TH(3)2).

(24) It will be appreciated that the present disclosure is not limited to any specific data ranges and the corresponding expected ratios selected for determining adapted thresholds. For example, in some embodiments, an optimal adapted threshold TH(i) can be determined based on the expected ratios of the symbols that are above a first value TH(i)1 and a second value TH(i)2. In some other embodiments, more than two data ranges can be defined and used to generate an optimal adapted threshold. In still some other embodiments, a single data range can be used to find the optimal adapted threshold. For example, 50% of the symbols are expected to be below an optimal adapted threshold TH(2). The error ratio may be set to a lower value if the difference between two thresholds is too high. In some embodiments, different P values may be used for discovering different thresholds.

(25) FIG. 2 illustrates the configuration of an exemplary threshold adaptation unit 200 capable of dynamically adapting constellation selection thresholds used in a slicer 210 in accordance with an embodiment of the present disclosure. The slicer 210 includes decision circuitry configured to map each input to one of the multiple constellations by comparing the input with the constellation thresholds. For example, each input is a sampled digital data output from an equalizer. The slicer 210 can be implemented in any manner that is well known in the art. The slicer inputs are also fed to the threshold adaptation unit 200 which uses a comparator 201 to compare each input with a “TH” value, e.g., stored in a register. Applied in the example shown in FIG. 1, The TH value is either TH(i)1 or TH(i)2. The threshold adaptation unit 200 includes Counter 1 205 for keeping count of all the inputs subject to comparison with the TH value. Counter 2 204 is coupled to the output of the comparator 201 and keeps count for the inputs that fall in the particular value range defined by the TH based on the comparison results.

(26) Using the example shown in FIG. 1, to discover an optimal adapted threshold TH(1), the TH register 202 is first set to be an initial TH(1)1 value. The total number of inputs to the slicer is programmed to be 2.sup.M which can be tracked by the Counter 1 205. Each time the comparator outputs a result indicating that the current input r is smaller than the TH, the Counter 2 204 increments by 1. For each window of 2.sup.M inputs, the TH adjustment logic 203 determines the ratio of inputs below TH value by calculating

(27) $\frac{\mathrm{counter}2}{\mathrm{counter}1}.$
If the ratio is not equal to the expected ratio (25−P) %, the TH adjustment logic 203 adjusts the TH value accordingly. Another window of 2.sup.M inputs are then compared with the new TH value to obtain the ratio of inputs below the new TH by

(28) $\frac{\mathrm{counter}2}{\mathrm{counter}1}.$
This process is repeated for multiple windows of 2.sup.M inputs until finding a TH value that makes

(29) $\frac{\mathrm{counter}2}{\mathrm{counter}1}=\left(25-P\right)\%.$
This TH value is then designated as the TH(1)1. In the same manner, by varying the TH value to obtain

(30) $\frac{\mathrm{counter}2}{\mathrm{counter}1}=\left(25+P\right)\%,$
TH(1)2 can be determined. The TH adjustment logic 203 can then generate the optimal adapted TH(1) by averaging the determined TH(1)1 and TH(1)2.

(31) The threshold adaptation unit 200 can be used to sequentially determine the TH(1)1, TH(1)2, TH(2)1, TH(2)2, TH(3)1, TH(3)2, etc. In some other embodiments, the threshold adaptation unit 200 includes duplicate circuits shown in FIG. 2 which can generate different TH(i)X (X=1 or 2) values in parallel. The resultant adapted thresholds are supplied to the slicer for making constellation selection decisions accordingly for the subsequent inputs. The threshold adaptation process may be performed periodically as well as based on needs.

(32) The TH adjustment logic 203 may be implemented in software, firmware, hardware or a combination thereof. The counters 204 and 205 may have a resolution of 32 bits for instance. For a bit-error rate of 10.sup.−6, the number of sampled inputs of one symbol that yields on sample at the threshold is 4×10.sup.6. For 100 samples at the threshold, the total number of samples is 100×4×10.sup.6, which is equivalent to 29 bits.

(33) In accordance with embodiments of the present disclosure, a set of optimal adapted thresholds for use of constellation selection can be dynamically determined and adapted to signal non-linearity and amplitude compression in a statistical approach. As a result, data demodulation and data recovery can be advantageously performed with reduced error rates at the receiver. Further, the threshold adaptation does not involve computationally intensive calculations and can be implemented by using simple circuitry, e.g., including a comparator and counters. Thus, design and development costs and operational power consumption can be advantageously reduced, compared with the conventional approaches.

(34) FIG. 3A is a flow chart depicting an exemplary process 300 of determining adapted thresholds for constellation selection in accordance with an embodiment of the present disclosure. Process 300 may be performed by a threshold adaptation unit coupled to, or included in, a slicer as shown in FIG. 2 for example. At 301, the index of threshold i is initialized, where i<N, and N represents the number of constellations defined in the modulation scheme. Particularly, N is 4 for PAM-4. At 302, i increments to 1. At 303, a first value TH(i)1 associated with TH(i) is determined. TH(i)1 defines a certain data range (e.g., <TH(i)1) and is associated with a first preset count ratio (e.g.,

(35) $\frac{i}{N}-\frac{P}{100}).$
The optimal TH(i)1 is defined as one that causes the inputs falling in the data range to meet the first preset count ratio. For example, for i=3, the data range associated with the first value TH(3)1 encompasses any value smaller than TH(3)1 and the preset count ratio for this range is (75−P) %.

(36) At 304, a second value TH(i)2 associated with TH(i) is determined. TH(i) defines another data range (e.g., <TH(i)) and is associated with another preset count ratio. (e.g.,

(37) 0$\frac{i}{N}+\frac{P}{100}).$
An optimal TH(i) is defined as one that causes the inputs falling in this data range to meet the second preset count ratio. For example, for i=3, the data range associated with the first value TH(3)2 encompasses any value below TH(3)2 and the preset count ratio, in this range is (75+P) %. At 305, the average of TH(i) 1 and TH(i)2 is assigned as the optimal adapted TH(i), e.g., TH(i)=mean (TH(i)1, TH(i)2). The process 302˜305 is repeated to determine each optimal threshold.

(38) FIG. 3B is a flow chart depicting an exemplary process of determining a first value or a second value TH(i)X (X=1 or 2) associate with a threshold TH(i) in accordance with an embodiment of the present disclosure. At 351, a plurality of inputs of a slicer are accessed. For example, the total number of inputs used for each TH(i)X variation can be preset to a window of 2.sup.M inputs. At 352, each of the plurality of inputs is compared with the current TH(i)X value (as set in the TH register 202 in FIG. 2). At 353, the count ratio of the number of inputs below TH(i)X (counter 2) to the total number of inputs subject to the comparison (counter 1) is determined. At 354, it is determined if

(39) $\frac{\mathrm{counter}2}{\mathrm{counter}1}$
equals the preset count ratio associated with the TH(i)X. If not, the TH(i)X value is adjusted at 355 and the process 351-354 are repeated for the adjusted TH(i)X value. The process 351-355 is repeated until

(40) $\frac{\mathrm{counter}2}{\mathrm{counter}1}$
equals the preset count ratio. At 356, the final TH(i)X is assigned as the optimal TH(i)X to be used for deriving TH(i).

(41) It will be appreciated that the present disclosure is applicable to various other suitable modulation schemes. For example, for a quadrature amplitude modulation (QAM) with 2N constellations, the same threshold adaptation process shown in FIGS. 3A and 3B can be performed on an individual constellation dimension as well to discover optimal adapted thresholds and adapted constellations. The data ranges shown in FIG. 1 and the associated preset count ratios may also be applied to an individual constellation dimension in a QAM-2N scheme.

(42) According to another aspect of the present disclosure, constellation levels can be dynamically adapted based on the statistic distribution of the received symbols with respect to the multiple constellations. Collectively speaking, the inputs in a certain data range and mapped to a specific constellation are expected to constitute a certain ratio (expected ratio) of the total inputs as dictated by the statistic distribution. FIG. 4 illustrates exemplary data ranges used to determine optimal adapted constellations based on an expected statistic distribution of data symbols across multiple constellations in accordance with an embodiment of the present disclosure. In this example, data symbols are modulated and demodulated according to PAM-4 scheme, the nominal constellations C(1)˜C(4) being [−3, −1, +1, +3] respectively. Accordingly, a slicer uses 3 thresholds TH(1)˜TH(3) to decide which constellation is assigned to a received symbol, the nominal thresholds being [−2, 0, +2] respectively.

(43) As dictated by the modulation process at the transmitter side, the data symbols are distributed across the 4 constellations evenly. Accordingly, the inputs to the slicer falling below the optimal adapted C(1) are expected to constitute 12.5% (=0+12.5%) of the overall inputs. The inputs falling below the optimal adapted C(2) are expected to constitute 37.5% (=25%+12.5%) of the overall inputs. By the same token, the inputs falling below the optimal adapted C(3) should constitute 62.5% (=50%+12.5%)) of the overall inputs, and the data symbols falling below the optimal adapted C(4) should constitute 87.5% (=75%+12.5%) of the overall inputs. This is equivalent to median of the points that belong to a constellation. There is no assumption of Gaussian noise.

(44) It will be appreciated that the present disclosure is not limited to any specific data ranges and the corresponding expected ratios selected for determining optimal adapted constellations. For example, in some embodiments, an optimal adapted constellation can be determined based on the expected ratios of the symbols that are above the constellation level. In some other embodiments, more than one data ranges can be defined and used to generate an optimal adapted constellation, in a similar manner in the threshold adaptation process described above.

(45) FIG. 5 illustrates the configuration of an exemplary constellation adaptation unit 500 capable of dynamically determining optimal adapted constellations in accordance with an embodiment of the present disclosure. The slicer 510 includes decision circuitry configured to map each input to one of the multiple constellations by comparing the input with the constellation thresholds. The slicer 510 can be implemented in any manner that is well known in the art. The slicer inputs are also fed to the constellation adaptation unit 500 which uses a comparator 501 to compare each input with a “C(j)” value, e.g., stored in a register 502. The constellation adaptation unit 500 includes Counter 1 505 for keep count of all the inputs used in a constellation adaptation process. The Counter 2 504 is coupled to the output of the comparator 501 and keeps count for the inputs that falls in the particular value range defined by Ca) based on the comparison results.

(46) Using the example shown in FIG. 4, to discover an optimal adapted threshold C(1), the C(j) register is first set to be an initial value. The total number of inputs to the slicer is programmed to be 2.sup.M which can be tracked by the Counter 1 505. Each time the comparator outputs a result that the input r is smaller than C(j), the Counter 2 204 increments by 1. For each window of 2.sup.M inputs, the C(j) adjustment logic 503 determines the ratio of inputs below C(j) by calculating

(47) $\frac{\mathrm{counter}2}{\mathrm{counter}1}.$
If the ratio is not equal to the expected ratio 12.5%, the C(j) adjustment logic 503 adjusts the C(j) value accordingly. Another window of 2.sup.M inputs are then compared with the new C(j) value to obtain the ratio of inputs below the C(j) value in the register 502 based on

(48) $\frac{\mathrm{counter}2}{\mathrm{counter}1}.$
This process is repeated for multiple windows of 2.sup.M inputs until finding a C(j) value that results in

(49) $\frac{\mathrm{counter}2}{\mathrm{counter}1}=12.5\%.$
This C(j) value is then designated as the optimal adapted C(1). By the same token, by varying the C(j) value to obtain

(50) $\frac{\mathrm{counter}2}{\mathrm{counter}1}=37.5\%,$
C(2) can be determined. The same process is performed to generated the adapted C(3) and C(4) by using preset count ratios of 62.5% and 87.5%, respectively.

(51) The constellation adaptation unit 500 can be used to sequentially determine the C(1)˜C(4). The constellation adaption unit 500 can also be used to perform a threshold adaption process as described with reference to FIGS. 1-3B. In some other embodiments, the constellation adaptation unit 500 includes duplicate circuits shown in FIG. 5 which can generate different, levels of adapted constellations in parallel. The adapted constellations are supplied to the slicer for making constellation selection decisions accordingly for the subsequent inputs. The constellation adaptation process may be performed periodically as well as based on needs.

(52) The constellation adjustment logic 503 may be implemented in firmware, software, hardware or a combination thereof. The counters may have a resolution of 32 bits for instance. For a bit-error rate of 10.sup.−6, the number of sampled inputs of one symbol that yields on sample at the threshold is 4×10.sup.6. For 100 samples at the threshold, the total number of samples is 100×4×10.sup.6, which is equivalent to 29 bits.

(53) In some embodiments, to prevent interaction with equalization gain at the receiver, the gain of the 4 optimal constellations can be maintained at a fixed value. For example, Σ.sub.j=1.sup.4|C(j)|=Constant, where Constant=sum (abs[−3, −1, +1, +3]).

(54) FIG. 6 is a flow chart depicting an exemplary process 600 of determining optimal adapted constellations for demodulation in accordance with an embodiment of the present disclosure. Process 600 may be performed by a constellation adaptation unit coupled to a slicer as shown in FIG. 5 for example. At 601, the index of constellation j is initialized, where j≤N, and N represents the number of constellations defined in the modulation scheme. At 602, j increments by 1. At 603, a plurality of inputs of a slicer are accessed. For example, the total number of inputs used for each C(j) variation can be preset to a window of 2.sup.M inputs.

(55) At 604, each of the plurality of inputs is compared with the current C(j) value. At 605, a total number of inputs subject to the comparison (counter 1) is determined. At 606, the number of inputs below the current C(j) value is determined (counter 2). At 606, it is determined if

(56) $\frac{\mathrm{counter}2}{\mathrm{counter}1}$
equals the preset count ratio associated with the C(j), which is

(57) $\left(\frac{j}{N}-\frac{1}{2N}\right).$
If not, the C(j) value is adjusted at 608, and the foregoing 602˜607 are repeated for the adjusted C(j) value. The foregoing 602˜608 is repeated until

(58) $\frac{\mathrm{counter}2}{\mathrm{counter}1}$
equals the preset count ratio

(59) 0$\left(\frac{j}{N}-\frac{1}{2N}\right).$
At 609, the final C(j) value is assigned as the optimal adapted constellation C(j) which is supplied to the slicer for use.

(60) FIG. 7 illustrates a network signal transmission system 700 that includes constellation adaptation logic 739 and constellation selection threshold adaptation logic 738 at the receiver 730 in accordance with an embodiment of the present disclosure. In a simplified form, the system 700 includes an optical transmitter 710, an optical fiber cable 740, an optical receiver 720 and an electrical receiver 730. The optical transmitter 710 has a driver 711 and a Mach-Zehnder interferometer (MZI) 712 and operates to receive data modulated according to PAM-4. The modulated data is sent for transportation through the optical fiber cable 740. The optical receiver 720 has a photodetector (PD) 721 and a transimpedance amplifier (TIA) 722 and operates to receive data from the optical fiber cable 740. The electrical receiver 730 receives the signals from the optical receiver 720 and performs data and clock recovery. The electrical receiver 730 includes an analog-to-digital converter (ADC) 731, an equalizer 734, a slicer 735, a timing recovery (TR) system 737 and a phase interpolator 736. The slicer 735 is coupled to, or integrates, the threshold adaptation logic 738 and the constellation adaptation logic 739 that can generate optimal adapted thresholds and optimal adapted constellations that vary over time due to nonlinearity or other signal distortions. The threshold adaptation logic 738 and constellation adaptation logic 739 can be used in any other suitable signal transmission and processing systems. In some embodiments, a single circuit as shown in FIG. 2 or 5 can be implemented to perform both the threshold adaptation and constellation adaptation using different configurations of data ranges, TH/C(j) values and the expected count ratios.

(61) FIG. 8 shows a set of simulated results comparing the symbol error rates (SERs) resulting from using default slicer thresholds and the SERs resulting from optimal adapted thresholds obtained in accordance with an embodiment of the present disclosure. The data presented in diagrams 810, 820 and 830 are obtained using different input Signal-Noise-Ratio (SNRs) of constellations in PAM-4. In each plot, the SNRs for all 4 constellations are identical, 16 dB in 810, 18 dB in 820 and 20 dB in 830. Each data plot shows SER variation as a function of the amount of constellation #2 shift by using the default slicer thresholds (811, 821 and 831) and by using the optimal adapted thresholds (812, 822 and 832). Comparing the two curves in each plot, it demonstrates that using the optimal adapted thresholds can significantly reduce SER, and the amount of SER reduction increases with constellation shift.

(62) FIG. 9 shows another set of simulated results comparing the symbol error rates (SERs) resulting from default slicer thresholds and SERs resulting from the optimal adapted thresholds obtained in accordance with an embodiment of the present disclosure. The data presented in diagrams 910 and 920 are obtained using different sets of input SNRs of constellations in PAM-4. For plot 910, the input SNRs for the 4 constellations are [18, 18, 18, 14] dB. For plot 920, the input SNRs for the 4 constellations are [20, 20, 20, 16] dB. Each data plot shows SER variation as a function of constellation #2 shift by using the default slicer. thresholds (911 and 921) and by using the optimal adapted thresholds (912 and 922). Comparing the two curves in each plot, it demonstrates that using the optimal adapted thresholds can significantly reduce SER, and the SER reduction effect increases with constellation shift.