Sampler reference level, DC offset, and AFE gain adaptation for PAM-N receiver

Abstract

In a PAM-N receiver, sampler reference levels, DC offset and AFE gain may be jointly adapted to achieve optimal or near-optimal boundaries for the symbol decisions of the PAM-N signal. For reference level adaptation, the hamming distances between two consecutive data samples and their in-between edge sample are evaluated. Reference levels for symbol decisions are adjusted accordingly such that on a data transition, an edge sample has on average, equal hamming distance to its adjacent data samples. DC offset may be compensated to ensure detectable data transitions for reference level adaptation. AFE gains may be jointly adapted with sampler reference levels such that the difference between a reference level and a pre-determined target voltage is minimized.

Claims

1. A receiver circuit, comprising: a first error detection circuit to detect based on a sequence of input symbols, a first sequence of error values between a first reference level and a first desired decision boundary; a first combining circuit to combine, over a time window, the first sequence of error values to generate a first cumulative error value; a first reference level computation circuit to adjust the first reference level based on the first cumulative error value to generate a first adjusted reference level; and a decoder circuit to generate output symbols based on a comparison of the sequence of input symbols to the first adjusted reference level.

2. The receiver circuit of claim 1, further comprising: a second error detection circuit to detect based on the sequence of input symbols, a second sequence of error values between a second reference level and a second desired decision boundary; a second combining circuit to combine, over the time window, the second sequence of error values to generate a second cumulative error value; a second reference level computation circuit to adjust the second reference level based on the second cumulative error value to generate a second adjusted reference level; and wherein the decoder circuit is further configured to generate the output symbols based on a comparison of the sequence of input symbols to the second adjusted reference level.

3. The receiver circuit of claim 1, wherein the first sequence of error values are each selected from +1, −1, and 0.

4. The receiver circuit of claim 1, wherein the first error detection circuit is configured to receive a first decoded data symbol at a first sample time, an edge symbol at an edge sample time following the first sample time, and a second decoded data symbol at a second sample time following the edge sample time, and to determine an error value in the first sequence of error values based on the first decoded data symbol, the second decoded data symbol, and the edge symbol.

5. The receiver circuit of claim 4, wherein the first error detection circuit is configured to determine the error value based in part on whether an even or odd number of symbols exist between the first decoded data symbol and the second decoded data symbol.

6. The receiver circuit of claim 1, wherein the first combining circuit is configured to increment the first cumulative error value for each of the first sequence of error values having positive values, and the first combining circuit is configured to decrement the first cumulative error value for each of the first sequence of error values having negative values.

7. The receiver circuit of claim 1, wherein the first combining circuit is configured to set the first cumulative error value based on a majority vote of the first sequence of error values.

8. The receiver circuit of claim 1, wherein the first reference level computation circuit is configured to adjust the first reference level in a direction opposite the first cumulative error value.

9. The receiver circuit of claim 1, wherein the receiver circuit comprises a PAM-N receiver, where N is a number of reference levels.

10. A method for operating a receiver circuit, comprising: detecting based on a sequence of input symbols, by a first error detection circuit, a first sequence of error values between a first reference level and a first desired decision boundary; combining over a time window, by a first combining circuit, the first sequence of error values to generate a first cumulative error value; adjusting, by a first reference level computation circuit, the first reference level based on the first cumulative error value to generate a first adjusted reference level; and generating, by a decoder circuit, output symbols based on a comparison of the sequence of input symbols to the first adjusted reference level.

11. The method of claim 10, further comprising: detecting based on the sequence of input symbols, by a second error detection circuit, a second sequence of error values between a second reference level and a second desired decision boundary; combining over the time window, by a second combining circuit, the second sequence of error values to generate a second cumulative error value; adjusting, by a second reference level computation circuit, the second reference level based on the second cumulative error value to generate a second adjusted reference level; and wherein generating the output symbols is further based on a comparison of the sequence of input symbols to the second adjusted reference level.

12. The method of claim 10, wherein the first sequence of error values are each selected from +1, −1, and 0.

13. The method of claim 10, detecting the first sequence of error values comprises: receiving a first decoded data symbol at a first sample time, an edge symbol at an edge sample time following the first sample time, and a second decoded data symbol at a second sample time following the edge sample time; and determining an error value in the first sequence of error values based on the first decoded data symbol, the second decoded data symbol, and the edge symbol.

14. The method of claim 13, wherein determining the error value is further based on whether an even or odd number of symbols exist between the first decoded data symbol and the second decoded data symbol.

15. The method of claim 10, wherein combining the first sequence of error values comprises: incrementing the first cumulative error value for each of the first sequence of error values having positive values; and decrementing the first cumulative error value for each of the first sequence of error values having negative values.

16. The method of claim 10, wherein combining the first sequence of error values comprises: setting the first cumulative error value based on a majority vote of the first sequence of error values.

17. The method of claim 10, wherein adjusting the first reference level comprises: adjusting the first reference level in a direction opposite the first cumulative error value.

18. The method of claim 10, wherein the receiver circuit comprises a PAM-N receiver, where N is a number of reference levels.

19. A receiver circuit comprising: first error detection means for detecting based on a sequence of input symbols, a first sequence of error values between a first reference level and a first desired decision boundary; first combining means for combining over a time window, the first sequence of error values to generate a first cumulative error value; first reference level adjusting means for adjusting the first reference level based on the first cumulative error value to generate a first adjusted reference level; and decoding means for generating output symbols based on a comparison of the sequence of input symbols to the first adjusted reference level.

20. The receiver circuit of claim 19, further comprising: second error detection means for detecting based on the sequence of input symbols, a second sequence of error values between a second reference level and a second desired decision boundary; second combining means for combining over a time window, the second sequence of error values to generate a second cumulative error value; second reference level adjusting means for adjusting the second reference level based on the second cumulative error value to generate a second adjusted reference level; and wherein the decoding means further generates the output symbols based on a comparison of the sequence of input symbols to the second adjusted reference level.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The teachings of the embodiments herein can be readily understood by considering the following detailed description in conjunction with the accompanying drawings.

(2) FIG. 1 is an embodiment of a PAM-N receiver.

(3) FIG. 2 is an eye diagram illustrating even transitions in a PAM-4 receiver with ideally calibrated reference voltages.

(4) FIG. 3 is an eye diagram illustrating even transitions in a PAM-4 receiver with non-ideally calibrated reference voltages.

(5) FIG. 4 is an eye diagram illustrating odd transitions in a PAM-4 receiver with ideally calibrated reference voltages.

(6) FIG. 5 is an eye diagram illustrating even transitions in a PAM-4 receiver with non-ideally calibrated reference voltages.

(7) FIG. 6 is an eye diagram illustrating pseudo-even transitions in a PAM-4 receiver.

(8) FIG. 7 is an embodiment of a reference level adaptation circuit for a PAM-N receiver.

(9) FIG. 8 is an eye diagram illustrating a signal in a PAM-4 receiver with misaligned DC offset and/or gain.

(10) FIG. 9 is an embodiment of a PAM-N receiver with jointly adapting DC offset, gain, and reference voltages.

(11) FIG. 10 is a flowchart illustrating an embodiment of a process for jointly adapting DC offset, gain, and reference voltages in a PAM-N receiver.

(12) FIG. 11A is an embodiment of an error detection circuit for a first reference voltage in a PAM-4 receiver.

(13) FIG. 11B is an embodiment of an error detection circuit for a second reference voltage in a PAM-4 receiver.

(14) FIG. 11C is an embodiment of an error detection circuit for a third reference voltage in a PAM-4 receiver.

(15) FIG. 12 is an embodiment of a PAM-4 reference level adaptation circuit for a PAM-4 receiver.

DETAILED DESCRIPTION OF EMBODIMENTS

(16) In a PAM-N receiver, sampler reference levels, DC offset and AFE gain may be jointly adapted to achieve optimal or near-optimal boundaries for the symbol decisions of the PAM-N signal. For reference level adaptation, the hamming distances between two consecutive data samples and their in-between edge sample are evaluated. Data transitions are detected as being, for example, even data transitions crossing an even number of decision regions or odd data transitions crossing an odd number of decision regions. Reference levels for symbol decisions are adjusted accordingly such that on a data transition, an edge sample has on average, equal hamming distance to its adjacent data samples. DC offset may be compensated to ensure detectable data transitions for reference level adaptation. Here, DC offset is first coarsely adjusted by balancing the distribution of symbols around a baseline voltage, and then finely adapted by aligning the center reference level with the baseline voltage. AFE gains may be jointly adapted with sampler reference levels such that the difference between a reference level and a pre-determined target voltage is minimized.

(17) FIG. 1 illustrates an embodiment of a PAM-N receiver 100. The receiver 100 comprises an analog front end (AFE) 110, a DC offset adjustment circuit 120, an AFE gain amplifier 130, an analog-to-digital converter 140, an ADC to PAM-N decoder 150, a clock recovery circuit 160, and a calibration circuit 170. The analog front end 110 includes components such as one or more amplifiers, one or more filters, etc. to shape an analog input signal r(t) for processing. The DC offset adjustment circuit 120 applies a DC offset to the analog signal from the analog front end 110 based on an offset control signal 122 from the calibration circuit 170. The AFE gain amplifier 130 amplifies the offset analog signal based on a gain signal 132 from the calibration circuit 170. The ADC 140 samples the analog signal using a data clock (CLK_DATA) 142 and an edge clock (CLK_EDGE) 144 to generate data symbols and edge symbols. The ADC 140 slices the input signal into one of N symbols based on N−1 reference voltages V.sub.0, . . . V.sub.N-2 146 that represent decision boundaries between adjacent symbols. In one embodiment, the edge clock 144 is 180 degrees out of phase from the data clock 142 so that an edge sample is generated at the midway point in between two consecutive data symbols. The ADC to PAM-N decoder 150 decodes the digital samples to generate data symbols d(k) and edge symbols e(k). Generally, the ADC output has more bits than the number of bits of a PAM-N symbol. The ADC to PAM-N decoder 150 converts an ADC output data into its corresponding PAM-N symbol.

(18) The clock recovery circuit 160 receives the data symbol d(k) and edge symbol e(k) and generates the data clock 142 and the edge clock 144. The calibration circuit 170 furthermore receives the data symbols d(k) and edge symbols e(k) and calibrates the reference voltages 146, the gain control signal 132, and the DC offset control signal 122 based on the symbols.

(19) FIG. 2 is an eye diagram illustrating examples of “even transitions” between consecutive data symbols d(k−1), d(k) received by a PAM-N receiver 100. In FIG. 2, the waveforms show the pre-sliced amplitudes of the symbols. In this example, the PAM-N receiver 100 comprises a PAM-4 receiver having three reference voltages V.sub.0, V.sub.1, V.sub.2 that provide decision boundaries between four possible symbols A.sub.0, A.sub.1, A.sub.2, A.sub.3. In an even transition, an even number of symbols or decision regions are crossed during the transition. For example, in a PAM-4 receiver, the following transitions comprise even transitions: A.sub.0.fwdarw.A.sub.1, A.sub.1.fwdarw.A.sub.2, A.sub.2.fwdarw.A.sub.3, A.sub.3.fwdarw.A.sub.2, A.sub.2.fwdarw.A.sub.1, A.sub.1.fwdarw.A.sub.0, A.sub.0.fwdarw.A.sub.1.fwdarw.A.sub.2.fwdarw.A.sub.3, and A.sub.3.fwdarw.A.sub.2.fwdarw.A.sub.1.fwdarw.A.sub.0.

(20) FIG. 2 shows the ideal locations for each of the reference voltages. For each even transition, the crossings of the reference voltage (shown in dashed circles) of even transitions are set to maximize the opening of the eye such that the pre-sliced signal crosses the voltage reference at the midpoint between data sample times.

(21) An “in-range transition” occurs when the initial voltage of a reference level is within the amplitude range of a transition (even if the reference level is non-optimal). For example, in FIG. 3, V.sub.2 is non-ideal in terms of symbol decision but it is within the amplitude range of the transition. As shown in FIG. 3, d(k−1) and d(k) are the decoded PAM-N symbols of two consecutive data samples at discrete time (k−1) and k, respectively. Let d.sub.max, i.e., d.sub.max=max[d(k−1), d(k)] denote the symbol which has larger amplitude, d.sub.min, i.e., d.sub.min=min[d(k−1), d (k)] denote the symbol which has smaller amplitude. If a transition is an even transition, then it has a corresponding reference level V.sub.n such that the hamming distance between d.sub.max and the symbol immediately above V.sub.n equals to the hamming distance between d.sub.min and the symbol immediately below V.sub.n, i.e.,
h.sub.e.sup.max=h.sub.e.sup.min
where the hamming distance
h.sub.e.sup.max=H(d.sub.max,A.sub.n+1)
is the distance between d.sub.max and A.sub.n+1 in terms of the number of symbols, and the hamming distance
h.sub.e.sup.min=H(d.sub.min,A.sub.n)
is the distance between d.sub.min and A.sub.n in terms of the number of symbols.

(22) A misaligned reference level V.sub.n(k) at discrete time k can be adjusted by minimizing the difference between the reference level and the expectation of amplitude (i.e., the pre-sliced values) of edge samples of even transitions, i.e.,

(23) $\begin{array}{cc}\underset{V_n}{\min }{E\left[e_a\left(k\right)-V_n\right]}^2\mathrm{subject}\mathrm{to}\text{:}{h}_e^{\max }=h_e^{\min }& \left(1\right)\end{array}$
where e.sub.a(k) is the amplitude of an edge sample (prior to slicing) such that its phase is a half user interval (UI) away from its adjacent data symbols d(k−1) and d(k) as shown in FIG. 3. The constraint (h.sub.e.sup.max=h.sub.e.sup.min) ensures that only even transitions are used for the adjustment of reference level based on Eq. (1).

(24) Let μ, i.e., 0<μ<1 denote a constant for the control of the step size of reference level adjustment, and E.sub.n(k) denote the sign of the error between reference level V.sub.n(k) and its corresponding optimum decision boundary at discrete time k. An iterative solution of the problem in Eq. (1) is given by

(25) $\begin{array}{cc}{V}_n\left(k+1\right)=V_n\left(k\right)-\mu E_n^d\left(k\right)\mathrm{where}& \left(2\right)\\ E_n\left(k\right)=\{\begin{array}{cc}1& \mathrm{if}\left[\left(h_e^{\max }\text{==}{h}_e^{\min }\right).Math.\left(h_{\mathrm{edge}}^{\max }>h_{\mathrm{edge}}^{\min }\right)\right]\\ -1& \mathrm{if}\left[\left(h_e^{\max }\text{==}{h}_e^{\min }\right).Math.\left(h_{\mathrm{edge}}^{\max }<h_{\mathrm{edge}}^{\min }\right)\right]\\ 0& \mathrm{else}\end{array}& \left(3\right)\end{array}$
and hamming distances
h.sub.edge.sup.max=H{max[(d(k−1),d(k)],e(k)}
h.sub.edge.sup.min=H{min[(d(k−1),d(k)],e(k)}

(26) FIG. 4 is an eye diagram illustrating examples of “odd data transitions” between consecutive data symbols d(k−1), d(k) received by a PAM-N receiver 100, where the waveforms show the pre-sliced amplitude of the signals. In the odd data transitions, an odd number of symbols or decision regions are crossed in a transition between consecutive data symbols. For example, in a PAM-4 receiver, the following transitions comprise odd transitions: A.sub.0.fwdarw.A.sub.1.fwdarw.A.sub.2, A.sub.1.fwdarw.A.sub.2.fwdarw.A.sub.3, A.sub.3.fwdarw.A.sub.2.fwdarw.A.sub.1, and A.sub.2.fwdarw.A.sub.1.fwdarw.A.sub.0.

(27) Ideally, the reference voltages are set so that the crossings (those in the dashed circles in FIG. 4) of odd transitions lie at the midpoint between two adjacent reference levels. If a transition is an odd transition, then it has a center symbol A.sub.n such that the center symbol has equal hamming distances to d.sub.min and d.sub.max, i.e.,
h.sub.o.sup.max=h.sub.o.sup.min
where hamming distances h.sub.o.sup.max and h.sub.o.sup.min are given by
h.sub.o.sup.max=H(d.sub.max,A.sub.n)
and
h.sub.o.sup.min=H(d.sub.min,A.sub.n)

(28) For example, the transition 500 shown in FIG. 5 has a center symbol A.sub.2. Its hamming distances to d.sub.min and d.sub.max are equal to 1.

(29) If reference levels V.sub.n−1(k) and V.sub.n(k), are the lower and upper boundaries of the center symbol on an odd transition, its offset can be detected and adjusted by minimizing the expectation of the difference between the decoded symbol of an edge symbol, i.e., e(k), and the center symbol of the odd transition, i.e.,

(30) $\begin{array}{cc}\underset{V_{n-1},V_n}{\min }{E\left[e\left(k\right)-A_n\right]}^2\mathrm{subject}\mathrm{to}{h}_o^{\max }=h_o^{\min }& \left(4\right)\end{array}$

(31) An iterative solution of the optimization problem in Equation (4) can be derived as

(32) $\begin{array}{cc}{V}_{n-1}\left(k+1\right)=V_{n-1}\left(k\right)-\mu E_{n-1}\left(k\right)\mathrm{and}& \left(5\right)\\ V_n\left(k+1\right)=V_n\left(k\right)-\mu E_n\left(k\right)\mathrm{where}& \left(6\right)\\ E_{n-1}\left(k\right)=\{\begin{array}{cc}1& \mathrm{if}\left[\left(h_o^{\max }\text{==}{h}_o^{\min }\right).Math.\left(h_{\mathrm{edge}}^{\max }>h_{\mathrm{edge}}^{\min }\right)\right]\\ 0& \mathrm{else}\end{array}& \left(7\right)\\ E_n\left(k\right)=\{\begin{array}{cc}-1& \mathrm{if}\left[\left(h_o^{\max }\text{==}{h}_o^{\min }\right).Math.\left(h_{\mathrm{edge}}^{\max }<h_{\mathrm{edge}}^{\min }\right)\right]\\ 0& \mathrm{else}\end{array}& \left(8\right)\end{array}$

(33) V.sub.n−1 and V.sub.n are the lower decision boundary and upper decision boundary of the center symbol A.sub.n on an odd transition, respectively.

(34) An odd transition is detected as an even transition in case a reference level is beyond the range of an odd transition. For example, in FIG. 6, the transition 600 should correspond to an odd transition which has three transmitted symbols A.sub.2.fwdarw.A.sub.1.fwdarw.A.sub.0, but is detected as an even transition such as A.sub.2.fwdarw.A.sub.1 due to misaligned reference level V.sub.0 which is out of the transition range. Let V.sub.high(k) and V.sub.low(k) denote the two reference levels which are immediately outside the amplitude range of a transition being detected. They can be updated as

(35) $\begin{array}{cc}{V}_{\mathrm{high}}\left(k+1\right)=V_{\mathrm{high}}\left(k\right)-\mu E_{\mathrm{high}}\left(k\right)& \left(9\right)\\ V_{\mathrm{low}}\left(k+1\right)=V_{\mathrm{low}}\left(k\right)-\mu E_{\mathrm{low}}\left(k\right)\mathrm{where}& \left(10\right)\\ E_{\mathrm{high}}\left(k\right)=\{\begin{array}{cc}1& \mathrm{if}\left\{\left[\mathrm{mod}\left(h_d,2\right)\text{==}1\right].Math.\left(h_{\mathrm{edge}}^{\max }\le h_{\mathrm{edge}}^{\min }\right)\right\}\\ 0& \mathrm{else}\end{array}& \left(11\right)\\ E_{\mathrm{low}}\left(k\right)=\{\begin{array}{cc}-1& \mathrm{if}\left\{\left[\mathrm{mod}\left(h_d,2\right)\text{==}1\right].Math.\left(h_{\mathrm{edge}}^{\max }\ge h_{\mathrm{edge}}^{\min }\right)\right\}\\ 0& \mathrm{else}\end{array}& \left(12\right)\end{array}$
and hamming distance
h.sub.d=H[d(k−1),d(k)]

(36) An embodiment of a reference level adaptation circuit 700 that may be part of the calibration circuit 170 is illustrated in FIG. 7. The reference level adaptation circuit 700 comprises a lookup up table (LUT) 710, a reference level error detector circuit 720 and a reference level computation circuit 730. The lookup table 710 receives consecutive data symbols d(k−1), d(k) and performs a lookup in a lookup table to determine the maximum symbol and the minimum symbol d.sub.min of the two symbols. The reference level error detector circuit 720 includes an error detector circuit 722-0, 722-1, . . . 722-(N−2) corresponding to each reference voltage. Each error detector 722 generates an error signal E(k) based on the minimum symbol d.sub.min, the maximum symbol d.sub.max, and the edge symbol e(k) between the consecutive data symbols d(k−1), d(k). The error detector 722 determines, based on a detected type of transition (e.g., even transition, odd transition, pseudo-even transition), an error between a current reference voltage V(k) and the optimal decision boundary. In one embodiment, the error signal E(k) represents the sign of the error (e.g., positive, negative, or zero). The reference level computation circuit 730 receives the error signals E(k) for each reference voltage and generates updated reference voltages V(k) based on the error. In one embodiment, the reference level computation circuit 730 comprises a reference level adjustment circuit 740 corresponding to each reference voltage. Each computation circuit includes a multiplier 732, a summation circuit 734, a multiplexer 736, and a delay circuit 738. The multiplier circuit 732 multiplies the error signal by a gain value (−μ) as described in equations (2), (5)/(6) and (9)/(10) above. When the multiplexer 736 is set to select the output from the summation circuit 734, the summation circuit 734 and delay circuit T 738 operate as an integrator to integrate the output from the multiplier 732 to smooth the adjustment of the reference voltage. The reference level preload signal 738 comprises a predefined initial value that may be set differently for each of the reference voltages. The multiplexer 736 is configured to select this initial value to set the reference voltages when the receiver 100 is first initialized.

(37) FIG. 8 is a waveform diagram showing an extreme case in which data transitions cannot be detected due to misalignment between the initial reference levels and an incoming signal. As seen in this diagram, the incoming signal is entirely between reference voltage V.sub.0 and reference voltage V.sub.1, thus resulting in every symbol being detected as A.sub.1. To compensate for this case, a DC offset is first corrected by balancing symbol distribution near the center reference level such that the incoming signal is overlapping with center reference level. Then, the AFE gain and DC offset can be dynamically adjusted to ensure proper operation of the PAM-N receiver.

(38) In one embodiment, the DC offset can be adapted by fixing the center reference level at zero volts and then offsetting the incoming signal by a varying amount until the difference in the numbers of decoded symbol above center reference level and below center reference level falls into a pre-defined range.

(39) The offset voltage for DC offset correction at discrete time k is given by

(40) $\begin{array}{cc}w\left(k+1\right)=w\left(k\right)-{\mu }_{\mathrm{dc}}\left[\underset{n=\frac{N}{2}}{\overset{N-1}{.Math.}}c\left(A_n\right)-\underset{n=0}{\overset{\frac{N}{2}-1}{.Math.}}c\left(A_n\right)\right]& \left(13\right)\end{array}$
where 0<μ.sub.dc<1 is a scaling factor which is used to control the step size of DC offset compensation and c(A.sub.n) represents the number of symbol decision A.sub.n in an evaluation window.

(41) Normally, DC offset adaptation using the above approach is not required. It is only used to handle applications with an extreme corner case where the incoming signal has no overlap with the initial reference levels. In such a case, data transitions are not detectable. After the initial coarse DC offset adaptation, data transitions of the received signal become detectable and can be used for sampler reference level adaptation. For PAM-N modulation where N is an even number, DC offset can be compensated by offsetting the incoming signal such that the adapted center reference level is zero volts. For PAM-N modulation where N is an odd number, DC offset can be compensated by offsetting the incoming signal such that the two reference levels near zero volts have opposite-polarity voltages.

(42) The offset voltage for DC offset correction while jointly adapting with reference levels is given by
w(k+1)=w(k)−μ.sub.dcV.sub.DC(k)  (14)
where V.sub.DC(k) is calculated based on adapted sampler reference levels at discrete time k, i.e.,

(43) $\begin{array}{cc}{V}_{\mathrm{DC}}\left(k\right)=\{\begin{array}{cc}{V}_{\frac{N-2}{2}}\left(k\right)& \mathrm{if}N\mathrm{is}\mathrm{an}\mathrm{even}\mathrm{number}\\ \frac{\left[V_{\frac{N-1}{2}}\left(k\right)+V_{\frac{N-3}{2}}\left(k\right)\right]}{2}& \mathrm{if}N\mathrm{is}\mathrm{an}\mathrm{odd}\mathrm{number}\end{array}& \left(15\right)\end{array}$

(44) AFE gain can be adapted by optimizing gain settings such that the difference between the adapted reference level and a pre-determined target level is minimized. It can be derived that the least-mean-square solution of AFE gain at discrete time k is
g(k+1)=g(k)−μ.sub.g[V.sub.N-2(k)−V.sub.0(k)−V.sub.T]  (15)
where μ.sub.g is a constant for the control of the step size of gain adjustment, V.sub.T represents the desired difference between the maximum PAM-N reference level V.sub.N-2 and minimum reference PAM-N level V.sub.0.

(45) DC offset compensation changes the baseline of the incoming signal, and thus reference levels are re-adapted whenever DC offset compensation changes. In addition, symbol decision boundaries and DC offset may change with AFE gains. Thus joint adaptation of reference levels, DC offset, and AFE gain is desired.

(46) An example calibration circuit 170 for a PAM-N receiver 100 that provides joint reference level, DC offset, and AFE gain adaptation is shown in FIG. 9. The calibration circuit 170 comprises a reference level adaptation circuit 700 (such as the reference level adaptation circuit 700 shown in FIG. 7), an adaptation finite state machine (FSM) 972, a variable gain amplifier (VGA) control circuit 976, a DC offset correction circuit 974 and digital-to-analog (DAC) converters 978, 980. The reference level adaptation circuit 700 comprises a lookup table 710, reference level error detector circuit 720, and a reference level computation circuit 730 that collectively operate to generate a set of digital reference voltages V.sub.0, . . . , V.sub.N-2 based on the data symbols d(k) and edge symbols e(k) as described above with reference to FIG. 7. The DAC 980 converts the digital reference voltages to analog signals used by the ADC 140. The adaptation FSM 972 determines a DC offset voltage V.sub.DC based on the data symbols d(k) and edge symbols e(k) as described in Eq. (15) above. The DC offset correction circuit 974 then determines a digital DC offset correction w(k) to apply to the incoming signal (via DAC 978 and summation circuit 120) based on Eq. (14) above. During initialization, in the extreme case in which data transitions cannot be detected due to misalignment between the initial reference levels and incoming signal (as shown in FIG. 8), the DC offset correction circuit 974 may instead generate a digital DC offset correction signal w(k) using the technique described with respect to Eq. (13) above. The DAC 978 converts the DC offset correction signal w(k) to an analog signal to be combined with the incoming signal via summation circuit 120. The adaptation FSM 972 also passes the lowest and highest reference voltages, V.sub.0, V.sub.N-2 respectively, to the VGA control circuit 976. The VGA control circuit 976 generates a gain control signal g(k) to control the gain of applied by the AFE gain circuit 130 as described in Eq. (15) above.

(47) FIG. 10 illustrates an embodiment of a process performed by the calibration circuit 170 shown in FIG. 9. The process starts 1002 and the calibration circuit 170 performs 1004 an initial coarse DC offset correction to improve the balance of symbol distribution if needed as explained above with reference to Eq. (13). If at decision block 1006, the symbol distribution is not balanced near a center voltage (e.g., within a predefined threshold of zero volts), the process loops back to step 1004 to continue adjusting the DC offset correction. If at decision block 1006, the symbol distribution is sufficiently balanced, the process continues. After initial DC calibration, the calibration circuit 170 performs 1008 reference level adaptation based on the data symbols d(k) and edge symbols e(k) as described above. Based on the adapted reference levels, the calibration circuit 170 performs 1010 DC offset correction and performs 1012 VGA adaptation. The calibration circuit 170 determines 1012 if the difference between the highest reference voltage V.sub.N-2 and the lowest reference voltage V.sub.0 is within a threshold V.sub.max of the desired voltage difference V.sub.T. If this criteria is not met, the process loops back to step 1008 to perform additional adjustment to the reference voltages, DC offset, and VGA. Otherwise, the process ends 1014.

(48) FIGS. 11A-11C illustrate example implementations of error detector circuits 722 for reference voltages V.sub.0, V.sub.1, and V.sub.2 respectively of an example PAM-4 receiver. In these example embodiments, the error detector circuits 722 each generate an error signal (E.sub.0(k), E.sub.1(k), E.sub.2(k) respectively) used to adjust the reference voltages (V.sub.0, V.sub.1, and V.sub.2 respectively) based on even transitions, odd transitions, and pseudo-even transitions that meet specified transition criteria. Particularly, in the illustrated embodiments, a digital comparator logic circuit 1102 compares the digital values of different pairs of symbols d.sub.max, d.sub.min, e(k) and outputs a logic high signal responsive to matches. A set of logic gates 1104 (e.g., AND gates and OR gates) generate a two-bit select signal (Neg., Pos.) that control a multiplexer 1106 to output either a 0, −1, or 1 as error signal E(k) corresponding to a pair of consecutive data symbols d(k−1), d(k) and edge symbol e(k). For even transitions, the error detector circuits 722 output a positive error signal if the edge symbol is equal to the symbol immediately below the first reference voltage of the transition and output a negative error signal if the edge symbol is equal to the symbol immediately above the reference voltage. In other words, error signals based on even transitions are given by:

(49) $\begin{array}{cc}{E}_0\left(k\right)=\{\begin{array}{cc}-1& \mathrm{if}\left[\left(e\left(k\right)\text{==}{A}_1\right).Math.\left(d_{\max }\text{==}{A}_1\right).Math.\left(d_{\min }\text{==}{A}_0\right)\right]\\ 1& \mathrm{if}\left[\left(e\left(k\right)\text{==}{A}_0\right).Math.\left(d_{\max }\text{==}{A}_1\right).Math.\left(d_{\min }\text{==}{A}_0\right)\right]\\ 0& \mathrm{else}\end{array}& \left(16\right)\\ E_1\left(k\right)=\{\begin{array}{cc}-1& \mathrm{if}\left[\left(e\left(k\right)\text{==}{A}_2\right).Math.\left(d_{\max }\text{==}{A}_2\right).Math.\left(d_{\min }\text{==}{A}_1\right)\right]\\ 1& \mathrm{if}\left[\left(e\left(k\right)\text{==}{A}_1\right).Math.\left(d_{\max }\text{==}{A}_2\right).Math.\left(d_{\min }\text{==}{A}_1\right)\right]\\ -1& \mathrm{if}\left[\left(e\left(k\right)\text{==}{A}_2\right).Math.\left(d_{\max }\text{==}{A}_3\right).Math.\left(d_{\min }\text{==}{A}_0\right)\right]\\ 1& \mathrm{if}\left[\left(e\left(k\right)\text{==}{A}_1\right).Math.\left(d_{\max }\text{==}{A}_3\right).Math.\left(d_{\min }\text{==}{A}_0\right)\right]\\ 0& \mathrm{else}\end{array}& \left(17\right)\\ E_2\left(k\right)=\{\begin{array}{cc}-1& \mathrm{if}\left[\left(e\left(k\right)\text{==}{A}_3\right).Math.\left(d_{\max }\text{==}{A}_3\right).Math.\left(d_{\min }\text{==}{A}_2\right)\right]\\ 1& \mathrm{if}\left[\left(e\left(k\right)\text{==}{A}_2\right).Math.\left(d_{\max }\text{==}{A}_3\right).Math.\left(d_{\min }\text{==}{A}_2\right)\right]\\ 0& \mathrm{else}\end{array}& \left(18\right)\end{array}$
where E.sub.0(k), E.sub.1(k) and E.sub.2(k) are the estimated errors of sampler reference levels V.sub.0(k), V.sub.1(k) and V.sub.2(k), respectively.

(50) As described above, odd transitions involve crossing a center symbol equidistant from the first and second data symbols d(k−1), d(k) in the transition bound by an upper reference voltage and a lower reference voltage. For odd transitions, the error detector circuits 722 output a positive error signal for the minimum reference voltage if the amplitude of an edge symbol is less than the amplitude of the middle symbol, and output a negative error signal for the maximum reference voltage if the edge symbol is greater than the middle symbol. In other words, error signals based on odd transitions are given by:

(51) $\begin{array}{cc}{E}_0\left(k\right)=\{\begin{array}{cc}1& \mathrm{if}\left[\left(e\left(k\right)\text{==}{A}_0\right).Math.\left(d_{\max }\text{==}{A}_2\right).Math.\left(d_{\min }\text{==}{A}_0\right)\right]\\ 0& \mathrm{else}\end{array}\mathrm{and}& \left(19\right)\\ E_2\left(k\right)=\{\begin{array}{cc}-1& \mathrm{if}\left[\left(e\left(k\right)\text{==}{A}_3\right).Math.\left(d_{\max }\text{==}{A}_3\right).Math.\left(d_{\min }\text{==}{A}_1\right)\right]\\ 0& \mathrm{else}\end{array}& \left(20\right)\end{array}$
here V.sub.0(k) and V.sub.2(k) are the low-voltage and high-voltage PAM-4 reference levels, respectively. In alternative embodiments, non-zero error signals may be additionally generated based on other odd transitions not included in Eqs. (19)-(20) above. In the illustrated implementation, only a select set of odd transitions are used to simplify the architecture while still providing enough information to enable the reference voltages to be adjusted within an acceptable tolerance of their optimal values. In other embodiments, more or fewer types of odd transitions (including ignoring odd transitions completely) may be accounted for depending on the desired tradeoff.

(52) To account for pseudo-even transitions, the error detector circuits 722 generates negative error signals for the lowest reference voltage V.sub.0 if the edge symbol is equal to d.sub.min, and generate a positive error signal for the highest reference voltage V.sub.2 if the edge symbol is equal to d.sub.max. In other words, error signals based on pseudo-even transitions are given by:

(53) 0$\begin{array}{cc}{E}_0\left(k\right)=\{\begin{array}{cc}-1& \mathrm{if}\left[\left(e\left(k\right)\text{==}{A}_1\right).Math.\left(d_{\max }\text{==}{A}_2\right).Math.\left(d_{\min }\text{==}{A}_1\right)\right]\\ 0& \mathrm{else}\end{array}\mathrm{and}& \left(21\right)\\ E_2\left(k\right)=\{\begin{array}{cc}1& \mathrm{if}\left[\left(e\left(k\right)\text{==}{A}_2\right).Math.\left(d_{\max }\text{==}{A}_2\right).Math.\left(d_{\min }\text{==}{A}_1\right)\right]\\ 0& \mathrm{else}\end{array}& \left(22\right)\end{array}$

(54) In alternative embodiments, non-zero error signals may be additionally generated based on other pseudo-even transitions not included in Eqs. (21)-(22) above. In the illustrated implementation, only a select set of pseudo-even transitions are used to simplify the architecture while still providing enough information to enable the reference voltages to be adjusted within an acceptable tolerance of their optimal values. In other embodiments, more or fewer types of pseudo-even transitions (including ignoring pseudo-even transitions completely) may be accounted for depending on the desired tradeoff.

(55) A combined reference level adaptation based on equations (16-22) of various transitions is given by

(56) $\begin{array}{cc}{V}_n\left(k+1\right)=V_n\left(k\right)-\mu E_n\left(k\right)\mathrm{where}n=0,1,2\mathrm{and}& \left(23\right)\\ E_0\left(k\right)=\{\begin{array}{cc}-1& \mathrm{if}\left[\left(e\left(k\right)\text{==}{A}_1\right).Math.\left(d_{\max }\text{==}{A}_1\right).Math.\left(d_{\min }\text{==}{A}_0\right)\right]\\ 1& \mathrm{if}\left[\left(e\left(k\right)\text{==}{A}_0\right).Math.\left(d_{\max }\text{==}{A}_1\right).Math.\left(d_{\min }\text{==}{A}_0\right)\right]\\ 1& \mathrm{if}\left[\left(e\left(k\right)\text{==}{A}_0\right).Math.\left(d_{\max }\text{==}{A}_2\right).Math.\left(d_{\min }\text{==}{A}_0\right)\right]\\ -1& \mathrm{if}\left[\left(e\left(k\right)\text{==}{A}_1\right).Math.\left(d_{\max }\text{==}{A}_2\right).Math.\left(d_{\min }\text{==}{A}_1\right)\right]\\ 0& \mathrm{else}\end{array}& \left(24\right)\\ E_1\left(k\right)=\{\begin{array}{cc}-1& \mathrm{if}\left[\left(e\left(k\right)\text{==}{A}_2\right).Math.\left(d_{\max }\text{==}{A}_2\right).Math.\left(d_{\min }\text{==}{A}_1\right)\right]\\ 1& \mathrm{if}\left[\left(e\left(k\right)\text{==}{A}_1\right).Math.\left(d_{\max }\text{==}{A}_2\right).Math.\left(d_{\min }\text{==}{A}_1\right)\right]\\ -1& \mathrm{if}\left[\left(e\left(k\right)\text{==}{A}_2\right).Math.\left(d_{\max }\text{==}{A}_3\right).Math.\left(d_{\min }\text{==}{A}_0\right)\right]\\ 1& \mathrm{if}\left[\left(e\left(k\right)\text{==}{A}_1\right).Math.\left(d_{\max }\text{==}{A}_3\right).Math.\left(d_{\min }\text{==}{A}_0\right)\right]\\ 0& \mathrm{else}\end{array}& \left(25\right)\\ E_2\left(k\right)=\{\begin{array}{cc}-1& \mathrm{if}\left[\left(e\left(k\right)\text{==}{A}_3\right).Math.\left(d_{\max }\text{==}{A}_3\right).Math.\left(d_{\min }\text{==}{A}_2\right)\right]\\ 1& \mathrm{if}\left[\left(e\left(k\right)\text{==}{A}_2\right).Math.\left(d_{\max }\text{==}{A}_3\right).Math.\left(d_{\min }\text{==}{A}_2\right)\right]\\ -1& \mathrm{if}\left[\left(e\left(k\right)\text{==}{A}_3\right).Math.\left(d_{\max }\text{==}{A}_3\right).Math.\left(d_{\min }\text{==}{A}_1\right)\right]\\ 1& \mathrm{if}\left[\left(e\left(k\right)\text{==}{A}_2\right).Math.\left(d_{\max }\text{==}{A}_2\right).Math.\left(d_{\min }\text{==}{A}_1\right)\right]\\ 0& \mathrm{else}\end{array}& \left(26\right)\end{array}$

(57) FIG. 12 illustrates an embodiment of a reference level adaptation circuit for a PAM-4 receiver using the error detection circuits described above. In this embodiment, the data symbols and edge symbols are de-serialized and parallel processed through M lanes. For example, the first lane for the adaptation of V.sub.0(k) detects E.sub.0.sup.0(k) based on input signals d(k−1), d(k) and e(k). The (M−1) lane for the adaptation of V.sub.0(k) detects E.sub.0.sup.M-1(k) based on input signals d(k−M), d(k−M+1) and e(k−M+1). Other lanes are processed similarly. The reference level error detected from different lanes, i.e., E.sub.0.sup.m(k), for m=0, 1, . . . , M−1, are then passed to a combining circuit 1202-0 (e.g., an adder or a majority voter circuit) to form a cumulative reference level error s.sub.0(k) for the computation of reference level V.sub.0(k). For example, in one embodiment, the combining circuit 1202-0 increments a cumulative error value for each positive error value it receives, and decrements the cumulative error value for each negative error value it receives. The combining circuit 1202-0 then periodically updates the cumulative error signal s.sub.0(k). Because the error values are de-serialized in the combining circuit 1202, the update period for the cumulative error signal s.sub.0(k) is generally substantially longer than a sampling period for obtaining the decoded data symbols. The remaining reference voltages are similarly generated.

(58) In alternative embodiments, reference level adaptation may be based on even transitions only (e.g., Eqs. (16)-(18)). Although robust reference level adaptation can still be achieved using only even transitions, the reference levels are likely to converge more quickly to their optimal or near-optimal levels if odd transitions and pseudo-even transitions are also used. In another alternative embodiment, reference level adaptation may be based on a combination of even and pseudo-even transitions only (e.g., Eqs. (16)-(18) and (21)-(22)) or a combination of even and odd transitions only (e.g., Eqs. (16)-(20)). Furthermore, in other alternatives embodiments, the principles described herein may be extended to PAM-8, PAM-16, or other types of receivers.

(59) Upon reading this disclosure, those of ordinary skill in the art will appreciate still alternative structural and functional designs and processes for the described embodiments, through the disclosed principles of the present disclosure. Thus, while particular embodiments and applications of the present disclosure have been illustrated and described, it is to be understood that the disclosure is not limited to the precise construction and components disclosed herein. Various modifications, changes and variations which will be apparent to those skilled in the art may be made in the arrangement, operation and details of the method and apparatus of the present disclosure herein without departing from the scope of the disclosure as defined in the appended claims.