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Recursive Least Square Equalizer, Complex (CRLSE)

Recursive Least Square Equalizer, Complex (CRLSE)

Property	Description	Units	Default	Range/Type
NTAPS	The number of filter coefficients	None	4	(-Inf, Inf)/Integer
DELTA	Inverse correlation matrix initialization factor	None	0.0005	(-Inf, Inf)/Real
LAMBDA	Forgetting factor of the RLS algorithm	None	1	[0, 1]/Real
RIN1	Input1 impedance	Ohm	Inf	(0, Inf]/Real
RIN2	Input2 impedance	Ohm	Inf	(0, Inf]/Real
RIN3	Input3 impedance	Ohm	Inf	(0, Inf]/Real
RIN4	Input4 impedance	Ohm	Inf	(0, Inf]/Real
ROUT1	Output1 impedance	Ohm	0	[0, Inf)/Real
ROUT2	Output2 impedance	Ohm	0	[0, Inf)/Real
Ports
Input1	Real part of the complex input signal (real)
Input2	Imaginary part of the complex input signal (real)
Input3	Real part of the error signal (real)
Input4	Imaginary part of the error signal (real)
Output1	Real part of the complex output signal (real)
Output2	Imaginary part of the complex output signal (real)

Notes

This model updates the filter coefficients of the equalizer based on the complex input and error signals (i.e., the difference between the output of the equalizer and the actual desired output). The update is based on the recursive least square algorithm [1], [2].

Let X(n) and h(n) denote the input signal vector and the vector of the complex filter coefficients respectively at time instant n. Each vector is assumed to be of length NTAPS (i.e., number of filter taps). In addition, let K(n) denote the NTAPS x 1 complex Kalman gain vector and let the NTAPS x NTAPS inverse of the complex correlation matrix of the input signal be denoted by P(n).

The recursive least square algorithm is given by the following 5 steps:

1. Compute the filter output:

y(n) = trans(X(n)) * h(n-1)

2. Compute the error:

e(n) = d(n) - y(n), where d(n) is the desired output

3. Compute the NTAPS x 1 Kalaman gain vector:

K(n) = [P(n-1) * conj(X(n))] / [LAMBDA + trans(X(n)) * P(n-1) * conj(X(n))]

4. Update the inverse of the complex correlation matrix:

P(n) = (1/LAMBDA) [P(n-1) - K(n) * trans(X(n)) * P(n-1)]

5. Update the coefficients of the complex filter:

h(n) = h(n-1) + K(n) * e(n)

The following initial conditions are always assumed:

P(-1) = (1/DELTA) * I, where DELTA is a small positive number and I is the NTAPS x NTAPS identity matrix. e(-1) = 0, and h(-1) = 0.

Netlist Form

CRLSE:NAME n1 n2 n3 n4 n5 n6 NTAPS=val DELTA=val LAMBDA=val [RIN1=val] [RIN2=val] [RIN3=val] [RIN4=val] [ROUT1=val] [ROUT2=val]

Netlist Example

CRLSE:1 1 2 3 4 5 6 NTAPS=6 DELTA=.005 LAMBDA=.999

References

1. J. G. Proakis, Digital Communications, McGraw-Hill, 1989.

2. J. G. Proakis and D. G. Manolakis, Digital Signal Processing, Macmillan, 1988.

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