Filter structure for iterative signal processing

• US 7,486,726 B2
• Filed: 07/23/2004
• Issued: 02/03/2009
• Est. Priority Date: 05/02/2002
• Status: Active Grant

• Current Assignee: Cohda Wireless Pty Ltd., University of South Australia (University of Adelaide)
• Original Assignee: Cohda Wireless Pty Ltd., University of South Australia (University of Adelaide)
• Inventors: Jakas, Stephen Peter, Rasmussen, Lars Kildehoj, Alexander, Paul Dean, Grant, Alexander James
• Primary Examiner(s): Payne; David C
• Assistant Examiner(s): Nguyen; Leon-Viet Q
• Application Number:
  US10/897,886
• Publication Number:
  US 20040264561A1
• Time in Patent Office: 1,656 Days
• Field of Search: 375/232
• US Class Current: 375/232
• CPC Class Codes: H04L 25/0202 Channel estimation
• CPC Class Codes: H04L 25/0204 of multiple channels
• CPC Class Codes: H04L 25/03171 Arrangements involving maximum a posteriori probability [MAP] detection
• CPC Class Codes: H04L 27/2657 Carrier synchronisation
• CPC Class Codes: H04L 27/2662 Symbol synchronisation
• CPC Class Codes: H04L 27/2675 Pilot or known symbols

Abstract

The present invention relates to improved multiple access communications. In one form, the invention relates to an improved signal processing method and apparatus for an iterative method of determining the reception of a signal in a multi user packet based wireless OFDM (Orthogonal Frequency Division Multiplexing) communication system. In other forms the present invention provides recursive filtering for joint iterative decoding in a variety of systems and functions such as linear multiple access channel decoders, iterative equalisation, iterative joint channel estimation and detection/decoding, iterative space-time processing, iterative multi user interference cancellation and iterative demodulation. In one particular form the present invention provides an iterative decoding circuit for a wireless multiuser communications receiver comprising a first signal processing means for receiving at least one received signal, said first signal processing means comprising at least two linear iterative filters such that the first linear iterative filter provides an estimate of a selected received signal to an estimated signal output and a second linear iterative filter provides estimates of at least one other received signal, delayed by one iteration cycle, to an input of said first linear iterative filter, a second signal processing means for receiving the estimated signal output of the first linear iterative filter and providing a further received signal estimate to the input of the first signal processing means in a succeeding iteration cycle of the decoding circuit.

Claims

1. An iterative signal processing arrangement having:

one or more pairs of first and second signal processing components, the pairs of components being in iterative configuration, each of the first signal processing components having as input one or more received signals dependent upon one or more transmitted signals, wherein for each said signal processing component pair the output of said first signal processing component is an estimate of a characteristic of a selected transmitted signal based on the current and one or more previous input signals received by said first signal processing component, which is input to said corresponding second signal processing component that provides a further estimate of said selected transmitted signal to the output of said second signal processing component, the outputs of all said second signal processing components of respective pairs are input to each said first signal processing components of all said pairs in a succeeding iteration cycle,

wherein said first signal processing components consists of;

at least two linear iterative filters wherein

a first of said linear iterative filters outputs an estimate of a selected characteristic of a selected one of said transmitted signals to said second signal processing component, and

a second of said iterative filters having the same inputs as said first linear iterative filter provides an estimate of a characteristic of a selected of one or more transmitted signals and then delays by one iteration cycle said estimate and outputs said delayed estimate to an input of said first linear iterative filter.

2. An iterative signal processing arrangement according to claim 1, wherein said first linear iterative filter provides a minimum least squared estimate of said selected transmitted signal subject to predetermined statistical models of said signals.

3. An iterative signal processing arrangement according to claim 1, wherein said second linear iterative filter provides a minimum least squared estimate of said selected transmitted signal subject to predetermined statistical models of said signals.

4. An iterative signal processing arrangement according to claim 1, a wherein said first and second linear iterative filters provide a minimum least squared estimate of said selected transmitted signal subject to predetermined statistical models of said signals.

5. An iterative signal processing arrangement according to claim 4, wherein said first and second linear iterative filters further consist of a switch the input of which at a first iteration is all received signals, and the input for subsequent iterations is the output of all second signal processing components wherein the output of said switch is input to a first summing device, and said first linear iterative filter receives as input the output of said first summing device which is input to a filter having taps that are recursively updated based on receiving one or more said received signals, the output of said first linear iterative filter is input to a second summing device the output of which becomes the output of said first signal processing component as well as being input to a first single iteration delay device the output of which is input to said second summing device, while said second linear iterative filter receives as input the output of said first summing device which is input to a second linear iterative filter having taps which are recursively updated based on receiving one or more said received signals the output of said second linear iterative filter is input to a third summing device the output of which is input to a second single iteration delay device, the output of which is input to said third summing device, the output of which is negated and input to said first summing device.

6. An iterative signal processing arrangement according to claim 5, wherein said filters in said first and a second linear iterative filters are of the type that conform to the following mathematical expression using the following assumptions,

A1:

The received signal is described as r=Sx+n, where S is the constraint matrix, containing all the linear channel constraints, x is a vector containing all transmitted information symbols and n is circularly symmetric complex Gaussian with covariance matrix cov n=σ

²I, and where the noise variance σ

² and the constraint matrix S are known,

A2;

The interleaved code symbol estimates of the interfering users {circumflex over (x)}_k⁽ⁿ⁾ which is a vector containing all the signal estimates at iteration n for all users except user k, coming out of said corresponding signal processing component 2 can be modelled as {circumflex over (x)}_k⁽ⁿ⁾=x_k+{circumflex over (v)}_k⁽ⁿ⁾ where x_k is the transmitted symbol for user k and {circumflex over (v)}_k⁽ⁿ⁾ is the corresponding estimated noise sample which is uncorrelated with x, which is a vector containing the transmitted symbols for all users, and also uncorrelated over time and iterations, but not over users at a given iteration, that is <

x, {circumflex over (v)}_k⁽ⁿ⁾>

=0, <

{circumflex over (v)}_k⁽ⁿ⁾, {circumflex over (v)}_k⁽ⁿ⁾>

=0 for n≠

m, where n and m denote different iteration numbers, and the estimated noise correlation for user k and j at iteration n is defined as <

{circumflex over (v)}_k⁽ⁿ⁾, {circumflex over (v)}_j⁽ⁿ⁾>

=q_kj,

Define the estimated noise covariance matrix Q_k⁽ⁿ⁾=<

{circumflex over (v)}_k⁽ⁿ⁾, {circumflex over (v)}_k⁽ⁿ⁾>

, with elements determined as shown above,

Let c_k⁽ⁿ⁾ be the auxiliary vector that contains all signals received from user k at iteration n and all previous iterations, according to the following recursively defined vector of observables as input to the said linear iterative filter denoted by Λ

_k⁽ⁿ⁾,

$c_k^{\left(n\right)}=\{\begin{array}{cc}r& n=1\\ \left(\begin{array}{c}{c}_k^{\left(n-1\right)}\\ {\hat{x}}_k^{\left(n-1\right)}\end{array}\right)& n=2,3,\mathrm{\dots <br><br>}\end{array}$

Under A1 and A2, the linear minimum mean square error estimate of said signal x_k given said signal c_k⁽ⁿ⁾ is given by the output x_k⁽ⁿ⁾ of the recursive filter which is an updated estimate of the transmitted signal for user k at iteration n, defined as follows;

m_k⁽ⁿ⁾=−

w_k⁽ⁿ⁾(I+Q_k<sup>(n−

1)</sup>−

W_k⁽ⁿ⁾)<sup>−

1</sup>

M_k⁽ⁿ⁾=(I−

W_k⁽ⁿ⁾)(I+Q_k<sup>(n−

1)</sup>−

W_k⁽ⁿ⁾)<sup>−

1</sup>

where for user k at iteration n m_k⁽ⁿ⁾ is the said first linear iterative filter, M_k⁽ⁿ⁾ is the said second linear iterative filter, I is an identity matrix with ones on the diagonal and zeros everywhere else, w_k⁽ⁿ⁾ is a recursive, complex auxiliary vector and W_k⁽ⁿ⁾ is a first recursive, complex auxiliary matrix, respectively, the recursive update equations for n=3,4, . . . are as follows;

w_k⁽ⁿ⁾=w_k<sup>(n−

1)</sup>[I−

(H_k<sup>(n−

1</sup>)<sup>−

1</sup>(I−

W_k<sup>(n−

1)</sup>)]<sup>−

1</sup>

W_k⁽ⁿ⁾=W_k<sup>(n−

1)</sup>+(I−

W_k<sup>(n−

1)</sup>)(H_k<sup>(n−

1)</sup>)<sup>−

1</sup>(I−

W_k<sup>(n−

1)</sup>)

H_k<sup>(n−

1)</sup>−

I+Q_k<sup>(n−

2)</sup>−

W_k<sup>(n−

1)</sup>

where H_k<sup>(n−

1)</sup> is a second recursive, complex auxiliary matrix;

the initial conditions with x_k⁽⁰⁾=0 and x_k⁽⁰⁾=0 are m_k⁽¹⁾=s_k^t(SS^t+σ

²I)<sup>−

1</sup>, M_k⁽¹⁾=S_k^t(SS^t+σ

²I)<sup>−

1</sup> for n=1 and w_k⁽²⁾=s_k^t(SS^t+I)<sup>−

1</sup>S_k, W_k⁽²⁾=S_k^t(SS^t+σ

²I)<sup>−

1</sup>S_k for n=2, where s_k is the linear constraint for user k, s_k^t denotes the complex conjugate transpose of said vector s_k,S_k is the constraint matrix with column k deleted and S_k denotes the complex conjugate transpose of vector S_k.

7. An iterative filter signal processing arrangement according to claim 1, wherein the output of said first signal processing component is de-interleaved and the output of said second signal processing component is interleaved.

8. An iterative signal processing arrangement according to claim 1, wherein the characteristic of a selected transmitted signal is a discrete time series representation of a said selected transmitted signal and where the output of said first signal processing component is a minimum least squared estimate of said selected transmitted signal subject to predetermined statistical models of said signals.

9. An iterative signal processing arrangement according to claim 1, wherein said first and second linear iterative filters provide a minimum least squared estimate of said selected transmitted signal subject to different predetermined statistical models of said signals.

10. An iterative decoding circuit for a wireless multiuser communications receiver comprising:

a first signal processing means for receiving at least one received signal, said first signal processing means comprising at least two linear iterative filters such that;

the first linear iterative filter provides an estimate of a selected received signal to an estimated signal output and;

a second linear iterative filter provides estimates of at least one other received signal, delayed by one iteration cycle, to an input of said first linear iterative filter;

a second signal processing means for receiving the estimated signal output of the first linear iterative filter and providing a further received signal estimate to the input of the first signal processing means in a succeeding iteration cycle of the decoding circuit.

11. An iterative decoding circuit according to claim 10 wherein the linear filters function in accordance with at least one predetermined recursive Bayesian expression.

12. An iterative decoding circuit according to claim 11 wherein the predetermined recursive expression comprises the following recursive Bayesian estimation using the following assumptions:

A1;

The received signal is described as r=Sx+n, where S is the constraint matrix, containing all the linear channel constraints, x is a vector containing all transmitted information symbols and n is circularly symmetric complex Gaussian with covariance matrix cov n=σ

²I, and where the noise variance σ

² and the constraint matrix S are known,

A2;

The interleaved code symbol estimates of the interfering users {circumflex over (x)}_k⁽ⁿ⁾ which is a vector containing all the signal estimates at iteration n for all users except user k, coming out of said corresponding signal processing component 2 can be modelled as {circumflex over (x)}_k⁽ⁿ⁾=x_k+{circumflex over (v)}_k⁽ⁿ⁾ where x_k is the transmitted symbol for user k and {circumflex over (v)}_k⁽ⁿ⁾ is the corresponding estimated noise sample which is uncorrelated with x, which is a vector containing the transmitted symbols for all users, and also uncorrelated over time and iterations, but not over users at a given iteration, that is <

x, {circumflex over (v)}_k⁽ⁿ⁾>

=0, <

{circumflex over (v)}_k⁽ⁿ⁾, {circumflex over (v)}_k^(m)>

=0 for n≠

m, where n and m denote different iteration numbers, and the estimated noise correlation for user k and j at iteration n is defined as <

{circumflex over (v)}_k⁽ⁿ⁾, {circumflex over (v)}_j⁽ⁿ⁾>

=q_kj;

define the estimated noise covariance matrix Q_k⁽ⁿ⁾=<

{circumflex over (v)}_k⁽ⁿ⁾, {circumflex over (v)}_k⁽ⁿ⁾>

, with elements determined as shown above;

let c_k⁽ⁿ⁾ be the auxiliary vector that contains all signals received from user k at iteration n and all previous iterations, according to the following recursively defined vector of observables as input to the said linear iterative filter denoted by Λ

_k⁽ⁿ⁾,

$c_k^{\left(n\right)}=\{\begin{array}{cc}r& n=1\\ \left(\begin{array}{c}{c}_k^{\left(n-1\right)}\\ {\hat{x}}_k^{\left(n-1\right)}\end{array}\right)& n=2,3,\mathrm{\dots <br><br>}\end{array}$

Under A1 and A2, the linear minimum mean square error estimate of said signal x_k given said signal c_k⁽ⁿ⁾ is given by the output {tilde over (x)}_k⁽ⁿ⁾ of the recursive filter which is an updated estimate of the transmitted signal for user k at iteration n, defined as follows;

{tilde over (x)}_k⁽ⁿ⁾={tilde over (x)}_k<sup>(n−

1)</sup>+m_k⁽ⁿ⁾({circumflex over (x)}_k<sup>(n−

1)</sup>−

{tilde over (x)}_k<sup>(n−

1)</sup>)

{tilde over (x)}_k⁽ⁿ⁾={tilde over (x)}_k<sup>(n−

1)</sup>+M_k⁽ⁿ⁾({circumflex over (x)}_k<sup>(n−

1)</sup>−

{tilde over (x)}_k<sup>(n−

1)</sup>)

m_k⁽ⁿ⁾=−

w_k⁽ⁿ⁾(I+Q_k<sup>(n−

1)</sup>−

W_k⁽ⁿ⁾)<sup>−

1</sup>

M_k⁽ⁿ⁾=(I−

W_k⁽ⁿ⁾)(I+Q_k<sup>(n−

1)</sup>−

W_k⁽ⁿ⁾)<sup>−

1</sup>

where for user k at iteration n m_k⁽ⁿ⁾ is the said first linear iterative filter, M_k⁽ⁿ⁾ is the said second linear iterative filter, I is an identity matrix with ones on the diagonal and zeros everywhere else, w_k⁽ⁿ⁾ is a recursive, complex auxiliary vector and W_k⁽ⁿ⁾ is a first recursive, complex auxiliary matrix, respectively, the recursive update equations for n=3,4, . . . are as follows;

w_k⁽ⁿ⁾=w_k<sup>(n−

1)</sup>[I−

(H_k<sup>(n−

1</sup>)<sup>−

1</sup>(I−

W_k<sup>(n−

1)</sup>)]<sup>−

1</sup>

W_k⁽ⁿ⁾=W_k<sup>(n−

1)</sup>+(I−

W_k<sup>(n−

1)</sup>)(H_k<sup>(n−

1)</sup>)<sup>−

1</sup>(I−

W_k<sup>(n−

1)</sup>)

H_k<sup>(n−

1)</sup>−

I+Q_k<sup>(n−

2)</sup>−

W_k<sup>(n−

1)</sup>

where H_k<sup>(n−

1)</sup> is a second recursive, complex auxiliary matrix;

the initial conditions with {tilde over (x)}_k⁽⁰⁾=0 and x_k⁽⁰⁾=0 are m_k⁽¹⁾=s_k^t(SS^t+σ

²I)<sup>−

1</sup>, M_k⁽¹⁾=S_k^t(SS^t+σ

²I)<sup>−

1</sup> for n=1 and w_k⁽²⁾=s_k^t(SS^t+I)<sup>−

1</sup>S_k, W_k⁽²⁾=S_k^t(SS^t+σ

²I)<sup>−

1</sup>S_k for n=2, where s_k is the linear constraint for user k, s_k^t denotes the complex conjugate transpose of said vector s_k, S_k is the constraint matrix with column k deleted and S_k denotes the complex conjugate transpose of vector S_k.

13. A computer program product comprising:

a computer usable medium having computer readable program code and computer readable system code embodied on said medium for communicating in a multiple access communication network, said computer program product comprising;

computer readable code within said computer usable medium for performing the method steps of a method of communicating in a multiple access network by iteratively receiving OFDM packets, the method comprising the following steps;

a) sample a receiver input signal consisting of signals from one or more antenna;

b) add the input signal with one of a plurality of prior stored received packet sample estimates to determine a packet sample hypothesis;

c) determine an information bit estimate from the sample hypothesis for storage in an information bit estimates list;

d) determine an updated received packet sample estimate from the sample hypothesis for updating the plurality of prior stored estimates;

e) subtract the updated sample estimate from the sample hypothesis to determine a noise hypothesis and provide the noise hypothesis as the receiver input signal;

f) repeat steps a) to e) until at least one or more complete packets are accumulated in the information bit estimates list.

14. An iterative signal processing arrangement having:

one or more pairs of first and second signal processing components, the pairs of components being in iterative configuration, each of the first signal processing components having as input one or more received signals dependent upon one or more transmitted signals, wherein for each said signal processing component pair the output of said first signal processing component is an estimate of a characteristic of a selected transmitted signal based on the current and one or more previous input signals received by said first signal processing component, which is input to said corresponding second signal processing component that provides a further estimate of said selected transmitted signal to the output of said second signal processing component, the outputs of all said second signal processing components of respective pairs are input to each said first signal processing components of all said pairs in a succeeding iteration cycle,

wherein said first signal processing components consists of;

at least two linear iterative filters wherein

a first of said linear iterative filters outputs an estimate of a selected characteristic of a selected one of said transmitted signals to said second signal processing component, and

a second of said iterative filters having the same inputs as said first linear iterative filter provides an estimate of a characteristic of a selected of one or more transmitted signals and then delays by one iteration cycle said estimate and outputs said delayed estimate to an input of said first linear iterative filter,

wherein said first and second linear iterative filters provide a minimum least squared estimate of said selected transmitted signal subject to predetermined statistical models of said signals, and

wherein said first and second linear iterative filters further consist of a switch the input of which at a first iteration is all received signals, and the input for subsequent iterations is the output of all second signal processing components wherein the output of said switch is input to a first summing device, and said first linear iterative filter receives as input the output of said first summing device which is input to a filter having taps that are recursively updated based on receiving one or more said received signals, the output of said first linear iterative filter is input to a second summing device the output of which becomes the output of said first signal processing component as well as being input to a first single iteration delay device the output of which is input to said second summing device, while said second linear iterative filter receives as input the output of said first summing device which is input to a second linear iterative filter having taps which are recursively updated based on receiving one or more said received signals the output of said second linear iterative filter is input to a third summing device the output of which is input to a second single iteration delay device, the output of which is input to said third summing device, the output of which is negated and input to said first summing device.

15. An iterative signal processing arrangement according to claim 14, wherein said first linear iterative filter provides a minimum least squared estimate of said selected transmitted signal subject to predetermined statistical models of said signals.

16. An iterative signal processing arrangement according to claim 14, wherein said second linear iterative filter provides a minimum least squared estimate of said selected transmitted signal subject to predetermined statistical models of said signals.

17. An iterative signal processing arrangement according to claim 14, wherein said filters in said first and a second linear iterative filters are of the type that conform to the following mathematical expression using the following assumptions,

A1:

The received signal is described as r=Sx+n, where S is the constraint matrix, containing all the linear channel constraints, x is a vector containing all transmitted information symbols and n is circularly symmetric complex Gaussian with covariance matrix cov n=σ

²I, and where the noise variance σ

² and the constraint matrix S are known,

A2;

The interleaved code symbol estimates of the interfering users {circumflex over (x)}_k⁽ⁿ⁾ which is a vector containing all the signal estimates at iteration n for all users except user k , coming out of said corresponding signal processing component 2 can be modelled as {circumflex over (x)}_k⁽ⁿ⁾=_k+{circumflex over (v)}_k⁽ⁿ⁾ where x_k is the transmitted symbol for user k and {circumflex over (v)}_k⁽ⁿ⁾ is the corresponding estimated noise sample which is uncorrelated with x, which is a vector containing the transmitted symbols for all users, and also uncorrelated over time and iterations, but not over users at a given iteration, that is <

x, {circumflex over (v)}_k⁽ⁿ⁾>

=0, <

{circumflex over (v)}_k⁽ⁿ⁾, {circumflex over (v)}_k⁽ⁿ⁾>

=0 for n≠

m, where n and m denote different iteration numbers, and the estimated noise correlation for user k and j at iteration n is defined as <

{circumflex over (v)}_k⁽ⁿ⁾, {circumflex over (v)}_j⁽ⁿ⁾>

=q_kj;

define the estimated noise covariance matrix Q_k⁽ⁿ⁾=<

{circumflex over (v)}_k⁽ⁿ⁾, {circumflex over (v)}_k⁽ⁿ⁾>

, with elements determined as shown above;

let c_k⁽ⁿ⁾ be the auxiliary vector that contains all signals received from user k at iteration n and all previous iterations, according to the following recursively defined vector of observables as input to the said linear iterative filter denoted by Λ

_k⁽ⁿ⁾,

$c_k^{\left(n\right)}=\{\begin{array}{cc}r& n=1\\ \left(\begin{array}{c}{c}_k^{\left(n-1\right)}\\ {\hat{x}}_k^{\left(n-1\right)}\end{array}\right)& n=2,3,\mathrm{\dots <br><br>}\end{array}$

Under A1 and A2, the linear minimum mean square error estimate of said signal x_k given said signal c_k⁽ⁿ⁾ is given by the output x_k⁽ⁿ⁾ of the recursive filter which is an updated estimate of the transmitted signal for user k at iteration n, defined as follows.

m_k⁽ⁿ⁾=−

w_k⁽ⁿ⁾(I+Q_k<sup>(n−

1)</sup>−

W_k⁽ⁿ⁾)<sup>−

1</sup>

M_k⁽ⁿ⁾=(I−

W_k⁽ⁿ⁾)(I+Q_k<sup>(n−

1)</sup>−

W_k⁽ⁿ⁾)<sup>−

1</sup>

where for user k at iteration n m_k⁽ⁿ⁾ is the said first linear iterative filter, M_k⁽ⁿ⁾ is the said second linear iterative filter, I is an identity matrix with ones on the diagonal and zeros everywhere else, w_k⁽ⁿ⁾ is a recursive, complex auxiliary vector and W_k⁽ⁿ⁾ is a first recursive, complex auxiliary matrix, respectively, the recursive update equations for n=3,4, . . . are as follows;

w_k⁽ⁿ⁾=w_k<sup>(n−

1)</sup>[I−

(H_k<sup>(n−

1</sup>)<sup>−

1</sup>(I−

W_k<sup>(n−

1)</sup>)]<sup>−

1</sup>

W_k⁽ⁿ⁾=W_k<sup>(n−

1)</sup>+(I−

W_k<sup>(n−

1)</sup>)(H_k<sup>(n−

1)</sup>)<sup>−

1</sup>(I−

W_k<sup>(n−

1)</sup>)

H_k<sup>(n−

1)</sup>−

I+Q_k<sup>(n−

2)</sup>−

W_k<sup>(n−

1)</sup>

where H_k<sup>(n−

1)</sup> is a second recursive, complex auxiliary matrix;

the initial conditions with x_k⁽⁰⁾=0 and x_k⁽⁰⁾=0 are m_k⁽¹⁾=s_k^t(SS^t+σ

²I)<sup>−

1</sup>, M_k⁽¹⁾=S_k^t(SS^t+σ

²I)<sup>−

1</sup> for n=1 and w_k⁽²⁾=s_k^t(SS^t+I)<sup>−

1</sup>S_k, W_k⁽²⁾=S_k^t(SS^t+σ

²I)<sup>−

1</sup>S_k for n=2, where s_k is the linear constraint for user k, s_k^t denotes the complex conjugate transpose of said vector s_k, S_k is the constraint matrix with column k deleted and S_k denotes the complex conjugate transpose of vector S_k.

18. An iterative signal processing arrangement according to claim 14, wherein the output of said first signal processing component is de-interleaved and the output of said second signal processing component is interleaved.

19. An iterative signal processing arrangement according to claim 14, wherein the characteristic of a selected transmitted signal is a discrete time series representation of a said selected transmitted signal and where the output of said first signal processing component is a minimum least squared estimate of said selected transmitted signal subject to predetermined statistical models of said signals.

20. An iterative signal processing arrangement according to claim 14, wherein said first and second linear iterative filters provide a minimum least squared estimate of said selected transmitted signal subject to different predetermined statistical models of said signals.

21. An iterative decoding circuit for a wireless multiuser communications receiver comprising:

a first signal processing means for receiving at least one received signal, said first signal processing means comprising at least two linear iterative filters such that;

the first linear iterative filter provides an estimate of a selected received signal to an estimated signal output and;

a second linear iterative filter provides estimates of at least one other received signal, delayed by one iteration cycle, to an input of said first linear iterative filter;

a second signal processing means for receiving the estimated signal output of the first linear iterative filter and providing a further received signal estimate to the input of the first signal processing means in a succeeding iteration cycle of the decoding circuit,

wherein the linear filters function in accordance with at least one predetermined recursive Bayesian expression, and

wherein the predetermined recursive expression comprises the following recursive Bayesian estimation using the following assumptions

A1;

The received signal is described as r=Sx+n, where S is the constraint matrix, containing all the linear channel constraints, x is a vector containing all transmitted information symbols and n is circularly symmetric complex Gaussian with covariance matrix cov n=σ

²I, and where the noise variance σ

² and the constraint matrix S are known;

A2;

The interleaved code symbol estimates of the interfering users {circumflex over (x)}_k⁽ⁿ⁾ which is a vector containing all the signal estimates at iteration n for all users except user k, coming out of said corresponding signal processing component 2 can be modelled as {circumflex over (x)}_k⁽ⁿ⁾=x_k+{circumflex over (v)}_k⁽ⁿ⁾ where x_k is the transmitted symbol for user k and {circumflex over (v)}_k⁽ⁿ⁾ is the corresponding estimated noise sample which is uncorrelated with x, which is a vector containing the transmitted symbols for all users, and also uncorrelated over time and iterations, but not over users at a given iteration, that is <

x, {circumflex over (v)}_k⁽ⁿ⁾>

=0, <

{circumflex over (v)}_k⁽ⁿ⁾, {circumflex over (v)}_k^(m)>

=0 for n≠

m, where n and m denote different iteration numbers, and the estimated noise correlation for user k and j at iteration n is defined as <

{circumflex over (v)}_k⁽ⁿ⁾, {circumflex over (v)}_j⁽ⁿ⁾>

=q_kj;

define the estimated noise covariance matrix Q_k⁽ⁿ⁾=<

{circumflex over (v)}_k⁽ⁿ⁾, {circumflex over (v)}_k⁽ⁿ⁾>

, with elements determined as shown above;

let c_k⁽ⁿ⁾ be the auxiliary vector that contains all signals received from user k at iteration n and all previous iterations, according to the following recursively defined vector of observables as input to the said linear iterative filter denoted by Λ

_k⁽ⁿ⁾,

$c_k^{\left(n\right)}=\{\begin{array}{cc}r& n=1\\ \left(\begin{array}{c}{c}_k^{\left(n-1\right)}\\ {\hat{x}}_k^{\left(n-1\right)}\end{array}\right)& n=2,3,\mathrm{\dots <br><br>}\end{array}$

Under A1 and A2, the linear minimum mean square error estimate of said signal x_k given said signal c_k⁽ⁿ⁾ is given by the output {tilde over (x)}_k⁽ⁿ⁾ of the recursive filter which is an updated estimate of the transmitted signal for user k at iteration n, defined as follows;

{tilde over (x)}_k⁽ⁿ⁾={tilde over (x)}_k<sup>(n−

1)</sup>+m_k⁽ⁿ⁾({circumflex over (x)}_k<sup>(n−

1)</sup>−

{tilde over (x)}_k<sup>(n−

1)</sup>)

{tilde over (x)}_k⁽ⁿ⁾={tilde over (x)}_k<sup>(n−

1)</sup>+M_k⁽ⁿ⁾({circumflex over (x)}_k<sup>(n−

1)</sup>−

{tilde over (x)}_k<sup>(n−

1)</sup>)

m_k⁽ⁿ⁾=−

w_k⁽ⁿ⁾(I+Q_k<sup>(n−

1)</sup>−

W_k⁽ⁿ⁾)<sup>−

1</sup>

M_k⁽ⁿ⁾=(I−

W_k⁽ⁿ⁾)(I+Q_k<sup>(n−

1)</sup>−

W_k⁽ⁿ⁾)<sup>−

1</sup>

where for user k at iteration n m_k⁽ⁿ⁾ is the said first linear iterative filter, M_k⁽ⁿ⁾ is the said second linear iterative flute, I is an identity matrix with ones on the diagonal and zeros everywhere else, w_k⁽ⁿ⁾ is a recursive, complex auxiliary vector and W_k⁽ⁿ⁾ is a first recursive, complex auxiliary matrix, respectively, the recursive update equations for n=3,4, . . . are as follows;

w_k⁽ⁿ⁾=w_k<sup>(n−

1)</sup>[I−

(H_k<sup>(n−

1)−

1</sup>(I−

W_k<sup>(n−

1)</sup>)]<sup>−

1</sup>

W_k⁽ⁿ⁾=W_k<sup>(n−

1)</sup>+(I−

W_k<sup>(n−

1)</sup>)(H_k<sup>(n−

1)</sup>)<sup>−

1</sup>(I−

W_k<sup>(n−

1)</sup>)

H_k<sup>(n−

1)</sup>−

I+Q_k<sup>(n−

2)</sup>−

W_k<sup>(n−

1)</sup>

where H_k<sup>(n−

1)</sup> is a second recursive, complex auxiliary matrix;

the initial conditions with {tilde over (x)}_k⁽⁰⁾=0 and x_k⁽⁰⁾=0 are m_k⁽¹⁾=s_k^t(SS^t+σ

²I)<sup>−

1</sup>, M_k⁽¹⁾=S_k^t(SS^t+σ

²I)<sup>−

1</sup> for n=1 and w_k⁽²⁾=s_k^t(SS^t+I)<sup>−

1</sup>S_k, W_k⁽²⁾=S_k^t(SS^t+σ

²I)<sup>−

1</sup>S_k for n=2, where s_k is the linear constraint for user k, s_k^t denotes the complex conjugate transpose of said vector s_k,S_k is the constraint matrix with column k deleted and S_k denotes the complex conjugate transpose of vector S_k.