Steepest descent FFE computation and tracking

Abstract

A nonlinear equalizer for iteratively equalizing a data communication channel, which comprises a transmitter at the input of the channel, for transmitting data and one or more training sequences over the channel; a receiver at the output of the channel, for receiving the data and the one or more training sequences; a sampling circuit for sampling received data; a processor, for processing the samples. The processor is adapted to calculate the derivative of the MSE for each of the FFE taps; calculate the derivative of the variance of the enhanced noise with the FFE taps; iteratively update the FFE coefficients, while during each update, injecting samples of a known training sequence into the channel. During each update, the processor computes the derivative of the output noise variance, by applying convolution between the noise correlation and the current FFE taps; computes the effective channel and the modified effective channel; computes the derivative of the residual ISI, by applying correlation between the original channel h and the modified effective channel; and updates the FFE coefficients, with a step proportional to the opposite of the gradient.

Claims

1. A nonlinear equalizer for iteratively equalizing a data communication channel, comprising: a) a transmitter at an input of said channel, for transmitting data and one or more training sequences over said channel; b) a receiver at an output of said channel, for receiving said data and said one or more training sequences; c) a sampling circuit for sampling received data; d) a processor, for processing samples, said processor is adapted to: d.1) calculate a derivative of a Mean Square Error (MSE) for a filter using Feed Forward Equalization (FFE) taps; d.2) calculate a derivative of a variance of noise enhanced by said filter, using said Feed Forward Equalization (FFE) taps; d.3) iteratively update Feed Forward Equalization (FFE) coefficients of said filter, while during each update, injecting samples of a known training sequence into said channel, wherein during each update, performing the following operations: d.4) computing a derivative (dN) of a noise variance at an output of said equalizer, by applying convolution between a noise correlation (R.sub.WW) and current Feed Forward Equalization (FFE) taps (f.sup.(i1)); d.5) computing an effective channel and a modified effective channel (g.sub.Win); d.6) computing a derivative (dR) of a residual Inter-Symbol Interference (ISI), by applying correlation between an original channel (h) and the modified effective channel (g.sub.Win); and d.7) updating the Feed Forward Equalization (FFE) coefficients, with a step proportional to an opposite of the gradient.

2. The equalizer according to claim 1, in which the equalized data channel is part of: data center connectivity; router connectivity; data center inter connections; optical networks; a metro system; wireless back-haul; or communication links with slow varying channels.

3. The equalizer according to claim 1, wherein the equalizing signals transmitted through the data communication channel are modulated according to a modulation format selected from a group consisting of: Non-Return-To-Zero (NRZ); On-Off Keying (OOK); Optical Dual Binary (ODB); Pulse-Amplitude Modulation (PAM); Quadrature Amplitude Modulation (QAM); Phase-Shift Keying (PSK); Frequency-Shift Keying (FSK); Minimum-Shift Keying (MSK).

4. The equalizer according to claim 1, in which the derivative $\left(\frac{d\left(E\left({.Math.n_{\mathrm{out}}.Math.}^2\right)\right)}{\mathrm{df}\left[p\right]}\right)$ of the variance of the enhanced noise with the taps of the Feed Forward Equalization (FFE) in one or more iterations is a convolution between the input noise correlation and the current Feed Forward Equalization (FFE) coefficients.

5. The equalizer according to claim 1, in which the training sequences are replaced by decisions of a decision algorithm, for decoding the received data.

6. A method of calculating coefficients of an equalizer utilizing Feed Forward Equalization (FFE)/Decision Feedback Equalizer (DFE) for equalizing a data communication channel, comprising: e) generating an analytical estimated channel model; f) applying a real time steepest descent iterative computation and tracking process, based on channel estimation inputs, provided by said estimated analytical channel model; g) enabling real time hardware tracking by using new channel estimation inputs, provided by said analytical estimated channel model, and previous Feed Forward Equalization (FFE)/Decision Feedback Equalizer (DFE) coefficients; and h) converging to a new optimal computation of coefficients by: h.1) calculating the derivative of Mean Square Error (MSE) for each filter, implementing taps of said Feed Forward Equalization (FFE); h.2) calculating the derivative of the variance of the noise enhanced by said filter, using said Feed Forward Equalization (FFE) taps; h.3) iteratively updating the Feed Forward Equalization (FFE) coefficients (f.sup.(i)) while during each update, injecting samples of a known training sequence into said channel, wherein during each update, performing the following operations: h.4) computing the derivative (dN) of a noise variance at the output of said equalizer, by applying convolution between the noise correlation (R.sub.WW) and the current Feed Forward Equalization (FFE) taps (f.sup.(i1); h.5) computing the effective channel and the modified effective channel (g.sub.Win); h.6) computing the derivative (dR) of the residual Inter-Symbol Interference (ISI), by applying correlation between the original channel (h) and the modified effective channel (g.sub.Win); and h.7) updating the Feed Forward Equalization (FFE) coefficients, with a step proportional to the opposite of the gradient.

7. The method according to claim 6, wherein a cost-function for optimization of Feed Forward Equalization (FFE)/Decision Feedback Equalizer (DFE) coefficients for reaching an optimal solution that corresponds to a minimal value of said cost-function is defined according to implementation/application criterion.

8. The method according to claim 7, wherein the implementation/application criterion is selected from the group of: Minimum Mean Square Error (MMSE); Zero-Forcing (ZF); fixed target channel; Minimum Mean Square Error-Unit Energy Constraint (MMSE-UEC).

9. The method according to claim 6, applied to coherent systems.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The above and other characteristics and advantages of the invention will be better understood through the following illustrative and non-limitative detailed description of preferred embodiments thereof, with reference to the appended drawings, wherein:

(2) FIGS. 1a-1f illustrate the convergence of the proposed FFE coefficients calculation algorithm for several randomly chosen channels, from a simple initialization with a small step size;

(3) FIGS. 2a-2d illustrate the performance of one iteration with a large step size; and

(4) FIG. 3 shows the basic block diagram for an FIR filter of length N, for implementing the FFE proposed by the present invention.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

(5) The present invention proposes a system and method for calculating Feed Forward Equalization (FFE) coefficients, using an iterative process, which on one hand, does not require a large number of samples and on the other hand, rapidly converges. The proposed system assumes that the changes in the communication channel are very slow. Therefore, the solution executes only a portion of the analytic computation and assures that after relatively few iterations, the solution (the computation of coefficients) will converge to the fully analytic solution.

(6) The solution proposed by the present invention used the steepest descent (gradient descent) algorithm (an algorithm for finding the nearest local minimum of a function which presupposes that the gradient of the function can be computed), which is an iterative algorithm for finding a local minimum of a function by taking steps proportional to the gradient of the function.

(7) A cost-function for optimization of FFE/DFE coefficients for reaching an optimal solution that corresponds to a minimal value of a defined cost-function is defined according to an implementation/application criterion, which may be MMSE. According to the present invention, the cost-function to be minimized is the MSE at the FFE output. Thus, the FFE coefficients update step is given by:

(8) $\begin{array}{cc}\underset{\_}{f^{\left(i+1\right)}}=\underset{\_}{f^{\left(i\right)}}-.Math.\frac{d\mathrm{MSE}}{d\underset{\_}{f^{\left(i\right)}}}& \left[\mathrm{Eq}.11\right]\end{array}$
Where is a design parameter
f.sup.(i)the FFE coefficients of the current update
f.sup.(i+1)the FFE coefficients of the next update

(9) Zero-Forcing (ZF); fixed target channel and MMSEUnit Energy Constraint (UEC).

(10) The expression for MSE is given by [Eq. 10]. The derivative of the MSE for each of the FFE taps is given by:

(11) $\begin{array}{cc}\frac{\mathrm{dMSE}}{d\underset{\_}{f}}=\frac{d\left(\mathrm{ResIsi}\right)}{d\underset{\_}{f}}+\frac{d\left(E\left({.Math.n_{\mathrm{out}}.Math.}^2\right)\right)}{d\underset{\_}{f}}& \left[\mathrm{Eq}.12\right]\end{array}$

(12) The two derivatives for a specific FFE tap are given by:

(13) $\begin{array}{cc}\frac{d\left(\mathrm{ResIsi}\right)}{\mathrm{df}\left[p\right]}=\frac{d\left\{\underset{l=1}{\overset{L+F-1}{.Math.}}{.Math.g_{\mathrm{Win}}\left[1\right].Math.}^2\right\}}{\mathrm{df}\left[p\right]}=\underset{l=1}{\overset{L+F-1}{.Math.}}2.Math.g_{\mathrm{Win}}\left[l\right]\frac{d{g}_{\mathrm{Win}}\left[l\right]}{\mathrm{df}\left[p\right]}=\underset{l=1}{\overset{L+F-1}{.Math.}}2.Math.g_{\mathrm{Win}}\left[l\right]\frac{dg\left[l\right]}{\mathrm{df}\left[p\right]}& \left[\mathrm{Eq}.13\right]\end{array}$

(14) The derivative of the effective channel with the FFE coefficient are given by:

(15) $\begin{array}{cc}\frac{dg\left[n\right]}{df\left[p\right]}=\frac{d\underset{k=0}{\overset{L-1}{.Math.}}h\left[l\right]f\left[n-1\right]}{df\left[p\right]}=h\left[n-p\right]& \left[\mathrm{Eq}.14\right]\end{array}$
where p is the FFE coefficient index to be updated.

(16) Combining the two results yields:

(17) $\begin{array}{cc}\frac{d\left(\mathrm{ResIsi}\right)}{df\left[p\right]}=\underset{l=1}{\overset{L+F-1}{.Math.}}2.Math.g_{\mathrm{Win}}\left[l\right]h\left[l-p\right]& \left[\mathrm{Eq}.15\right]\end{array}$

(18) It is seen that the derivatives of the residual ISI with the FFE coefficients is the correlation function between the modified effective channel g.sub.Win and the original channel h.

(19) The derivative of the variance of the enhanced noise with the taps of the FFE is given by:

(20) $\begin{array}{cc}\frac{d\left(E\left({.Math.n_{\mathrm{out}}.Math.}^2\right)\right)}{df\left[p\right]}=\frac{\left(\underset{M=0}{\overset{\mathrm{cl}}{.Math.}}{R}_{\mathrm{WW}}\left[M\right]\underset{k=1}{\overset{F-1}{.Math.}}f\left[k\right]f\left[k-M\right]\right)}{df\left[p\right]}=\underset{M=0}{\overset{\mathrm{cl}}{.Math.}}{R}_{\mathrm{WW}}\left[M\right]f\left[p-M\right]& \left[\mathrm{Eq}.16\right]\end{array}$

(21) It can be seen that in this case, the derivative is a simple convolution between the input noise correlation and the current FFE coefficients. The steepest decent algorithm is used to calculate the derivative of the MSE (the target function) as a function of the residual ISI and of the noise, in order to obtain the minimum.

(22) The FFE coefficients are iteratively updated, while during each update, samples of a known training sequence are injected into the channel. FFE coefficients update may be done for example every several millions of symbols. Each training sequence is received at the receiver side and can be accurately identified (since it is known in advance) and is used to update the FFE coefficients. Alternatively, it is possible to use a decision algorithm to decode the received data (rather than using training sequences) and use it as it would have been a known sequence (provided that there were no decoding errors).

(23) During each update, the performed operations are:

(24) Computing the derivative of the output noise variance, by applying convolution between the noise correlation and the current FFE taps:
dN=R.sub.WW*f.sup.(i1)[Eq. 17]

(25) Computing the effective channel g and the modified effective channel g.sub.Win:
g=h*f.sup.(i1)[Eq. 18]
g.sub.Win=g[Eq. 19]
g.sub.Win[cursor]=g[cursor]1[Eq. 20]
g.sub.Win[cursor+1: cursor+Nisi]=0[Eq. 21]
where,
hThe channel impulse response taken from the channel estimation mechanism
R.sub.WWThe noise correlation vector for noise correlation estimator (software)
f.sup.(i1)The FFE coefficients from last update

(26) Computing the derivative of the residual ISI, by applying correlation between the original channel h and the modified effective channel g.sub.Win:
dR=h*g.sub.Win[Eq. 22]
where h is a flipped version of h.

(27) Updating the FFE coefficients, with a step proportional to the opposite of the gradient:
f.sup.(i)=f.sup.(i1).Math.(dN+dR)[Eq. 23]

(28) It can be seen that each update requires 3 vector convolution operations and a final vector subtract operation. The advantage of the proposed method is rapid convergence to the MMSE analytical (and optimal) solution using iterations (rather than a statistical algorithm).

(29) FIGS. 1a-1f illustrate the convergence of the proposed algorithm for several randomly chosen channels, from a simple (trivial) initialization (impulse) with a small step size of 0.1. It can be seen that for the tested random channels, the algorithm rapidly converges to the MMSE analytical solution, as expected. The algorithm shows perfect stability and converges to the MMSE analytical solution.

(30) FIGS. 2a-2d illustrate the performance of one iteration with a large step size of 0.5, after initializing the algorithm with the optimal FFE+some error vector, for different noise levels. It can be seen that the proposed algorithm gives an improvement of several dBs towards the target SNR, even after a single iteration, at tracking mode.

(31) Another advantage is that in all the above examples, the proposed iterative algorithm always improves the equalization or remains as in the last iteration, but never deteriorate the last result (unlike using an adaptive algorithm that may deteriorate).

(32) The FFE may be implemented a Finite Impulse Response (FIR) filter, using a series of delays, multipliers, and adders to create the filter's output. FIG. 3 shows the basic block diagram for an FIR filter of length N, implementing the proposed FFE. The delays result in operating on prior input samples. The f.sub.k values are the coefficients used for multiplication, so that the output at time n is the summation of all the delayed samples multiplied by the appropriate coefficients. The longer the filter (more taps), the more finely the response can be tuned.

(33) The above examples and description have of course been provided only for the purpose of illustration, and are not intended to limit the invention in any way. As will be appreciated by the skilled person, the invention can be carried out in a great variety of ways, employing more than one technique from those described above, all without exceeding the scope of the invention.