AU4037393A

AU4037393A - Optimum estimation of data and parameters for communication systems

Info

Publication number: AU4037393A
Application number: AU40373/93A
Authority: AU
Inventors: Timothy Clifton Giles; Jason Beaufort Scholz
Original assignee: Commonwealth of Australia
Current assignee: Commonwealth of Australia
Priority date: 1992-04-27
Filing date: 1993-04-27
Publication date: 1993-11-29
Anticipated expiration: 2013-04-27
Also published as: AU668166B2

Description

OPTIMUM ESTIMATION OF DATA AND PARAMETERS FOR

COMMUNICATION SYSTEMS

TECHNICAL FIELD

This invention relates to a method and apparatus for generalised optimum detection and estimation of parameters defining a communication link. In particular, the invention provides a method for best estimating data, noise and/or channel parameters of a communication link in the presence of arbitrary distortion and arbitrary noise. Arbitrary noise includes interference, noise probability density functions and burst noise. Arbitrary distortion includes inter-symbol interference, fading and non-linearities.

BACKGROUND ART

The principle of Maximum A posteriori Probability (MAP) involves maximizing a probability function or "likelihood function". From a statistical point-of-view this is the best way to design a device which estimates a set of parameters subject to random disturbances, since we maximize the probability of estimating that parameter correctly. If the parameters have a uniform distribution, this problem reduces to Maximum Likelihood (ML) estimation. This principle forms the basis of many theories in the fields of radar, communications and sonar. The theory of Maximum Likelihood estimation for Gaussian random processes has been well established over a great many years. This is because, for many situations, this form of distribution is most important. One example of this is a technique for optimum detection of data in the presence of additive white Gaussian noise and inter-symbol interference known as maximum likelihood sequence estimation (MLSE). A commonly used method of implementing

MLSE is the Viterbi algorithm (Viterbi, 1967). The Viterbi algorithm usinig the usual Euclidean distance metre produces sub-optimum estimates for non-Gaussian noise and interference environments.

In the field of communications over recent years the emphasis on systems which will perform better over a wider range of environmental conditions has led to a realisation that performance must also be optimised for non-Gaussian forms of disturbance. Non-Gaussian disturbances include random noise, interference and signal distortions.

Examples of non-Gaussian noise includes impulse noise due to lightning discharges and electrical equipment interference, (eg ignition system). Non-Gaussian interference results from Carrier Waves (CW), co-channel interference, voice, radar, computers, etc. Distortions arise as a result of the propagation environment and system non-linearities. Optimum estimators have not generally been found for these distortions.

DISCLOSURE OF THE INVENTION

An object of this invention is to provide a general framework within which parameters describing a communications system can be estimated. A further object is to provide methods of estimating these parameters and apparatus for obtaining such estimates. The methods aim to produce improved

communications performance in terms of reduced probability of data errors and increased speed compared to present communication links. In this invention, MAP estimation is approached from a communications perspective. A very general channel model is proposed, which incorporates fading, Inter-Symbol Interference (ISI) and arbitrary noise type which includes interference. Simpler forms of estimator are posed by ignoring effects which are irrelevant in the final equations. In one form the MAP and ML estimator structures are derived on the basis of the following general model of a communications link with fading, ISI and additive noise or interference:

Where: r_i = Received signal sample i.

x_i = Transmitted data sample.

h_ij = Fading, correlation (ISI) term. Note that this is in general time-varying.

n_i = Noise and/or interference sample,

θ = Set of parameters describing the noise an/or interference. Therefore, according to one form of the invention, though it need not be the only or indeed the broadest form, there is proposed a method of analyzing a communication link comprising the steps of :

receiving a signal;

measuring one or more samples r_i of the received signal;

analyzing the received sample or samples in terms of a theoretical model having parameters h, x, θ in the form :-

wherein h is a parameter describing the communication channel, n are the noise samples, θ is a set of parameters describing the noise and x represents transmitted data in the form of symbols;

to produce optimum estimates of the parameters h, x, θ defining the

communications link.

An optimum detector for the data signal maximizes the probability of correct detection over all possible combinations of data. In general however, we may consider the general MAP case of joint maximization over all x, all channel correlations h, and all noise parameters, θ:

Where

and argmax [f (z ) ] means "find the value of z which corresponds to the maximum value of f (z) .

By initially considering the maximization over all parameters, simpler forms of estimation of any combination of parameters are possible by later removing unwanted terms by integration, as will be shown. According to Bayes Rule:

x, h and θ are independent , so: p(x,h,θ ) = P_x(x)P_h(h)P_θ (θ ) (4) where P_x(x) denotes the discrete probability distribution of x.

P_h(x) denotes the discrete probability distribution of h.

P_θ (θ) denotes the discrete probability distribution of θ. Since p (r) is not a function of x, h or θ, (2) becomes:

Where θ = a set of parameters describing the noise or interference process. If the noise or interference samples are independent then where Pn(n_ilθ) = the probability density function (pdf) of the noise/interference process.

Substituting equation (6a) and (6b) into (5) gives

This is the form of a MAP method of jointly estimating the data x, the channel h and the noise parameters θ.

This maximisation may be equivalently performed using the logarithm since the logarithm is a monotonic increasing function.

The full joint estimator of parameters x, h, and θ as given in (7) or (8) may be simplified if it is intended to estimate fewer parameters. Three approaches to this are: [1] Remove some parameter(s) by integration (or summation)

[2] Remove some parameter(s) by external (separate) estimation of those parameters and substituting the values into the equation to obtain the final desired estimate.

[3] Remove some parameter(s) by combined integration (or summation) and substitution of external estimates.

Note that in the process of performing integration (or summation) that :

1. The probability p_h (h) must be carefully defined or the equations will not be solvable.

2. The probability p_θ (θ) can be defined as a constant (uniform distribution) if no prior knowledge is available.

3. The probability p_x (x) is a constant (uniform distribution) for an uncoded system.

For a coded system :

In one preferred form a joint estimate of any two parameters is obtained by integrating over the third parameter. Thus under approach [1] above each parameter of equation (7) or (8) can be integrated out so that the optimization occurs over fewer variables. This is described for the simplest form (equation (7)) in the following :

To find a joint estimate of x and θ; h may be eliminated by integration :

To find a joint estimate of x and h integrate over all noise parameters, θ, as i in equation (10). In practice the integration over all noise parameters is performed as a summation.

To find a joint estimate of the channel h and noise pdf θ, sum over all possible data sequences x :

The summation of product terms over all x may be simplified to an iterative procedure. This procedure is to sum the product terms so that at any i there is only one term for each state (x_i, x_i-1, x_i-2 , . . . , x_i-L) . To find the terms for the (i+1)^th period, multiply and sum the terms using the (i+i)^th probability.

In another preferred form an estimate of a single parameter is obtained by integrating over the other two parameters. Thus further simplifications are possible by secondary integration or summation. A data detector is produced by integrating over all h and θ :

To estimate the noise pdf model, sum over x and integrate over h

To estimate the channel, sum over all x and θ

Integrating over two parameters to obtain an estimate of the remaining parameter may be a computationally expensive approach in some cases.

In a further preferred form of the invention a joint estimate of any two parameters is obtained by utilizing an external estimate of the remaining parameter. As indicated by approach [2] above, the full joint MAP estimator may be simplified by externally estimating a parameter or set of parameters and substituting these external estimates into equation (7) or (8), to determine the reduced estimator.

An external channel estimator for h may be used, giving a simplified joint estimator of x and θ :

An estimator for x and h, using external estimate of the noise parameters:

An estimator of h and θ, using external estimates of the data :

In a yet further preferred form an estimate of a single parameter is obtained by utilizing external estimates of the remaining parameters.

To estimate the data x, alone :

To estimate the noise parameter θ alone :

To estimate the channel h, alone

In a still further preferred form combined integration and utilisation of external estimates is used to obtain estimates of a single parameter. This is according to approach [3] above. To estimate the data x only, could integrate over all channels h and use external estimation of the noise parameter θ :

Or sum over all noise pdf's and use external estimation of the channel h:

According to a yet further form of the invention there is proposed a method of obtaining an estimate of a symbol transmitted through a communications link comprising the steps of :

receiving a signal;

measuring at repetitive points in time a plurality of samples of said signal;

comparing the signal samples to a plurality of expected symbol values for the received signal and calculating a first index indicative of the comparison;

calculating a second index from the first index and an estimate of parameters describing noise;

selecting the largest second index from the plurality of second index values corresponding to the plurality of expected symbol values wherein the largest second index value determines the best estimate of the transmitted symbol. In preference the first index is of the form

where r_i is the received signal sample;

x_i-j is the expected symbol value;

is an estimate of the impulse response of the link; and

j represents the channel delay.

In preference the second index is of the form

where 6 is an estimate of the parameters describing noise and/or interference. These parameters can be used to define the probability density function (pdf) by using the techniques disclosed in co-pending PCT application number PCT/AU90/00581 titled ERROR RATE MONITOR.

In preference the estimate of the noise probability density function is obtained externally.

To estimate the channel h only, could sum over all data x and use external estimation of the noise parameter, θ:

Or, could sum over all possible noise parameters and use external estimation of the data x:

In a still further form the invention may be said to reside in a method of estimating a channel associated with a communications link comprising the steps of :

receiving a signal;

measuring at repetitive points in time a plurality of samples of said signal; calculating for each signal sample a first index indicative of the difference between the received signal sample and the expected signal sample;

calculating a second index from the first index by summing the first index over all possible transmitted symbols;

selecting the largest second index from the plurality of second indices corresponding to the plurality of possible channels; wherein

the largest second index value corresponds to the best estimate of the channel.

In preference the first index is of the form

where r_t is the received signal sample;

is the estimate of symbol value; and

is an estimate of the channel impulse response of the link.

In preference the second index is of the form

To estimate the noise parameter θ only, could sum over all possible data and use external estimation of the channel h :

Or integrate over all channels h and use external estimation of the data x

Therefore, in another form of the invention there is proposed a method of estimating parameters describing noise associated with a communications link comprising the steps of :

receiving a signal;

easuring at repetitive points in time a plurality of samples of said signal; calculating for each signal sample a first index indicative of the noise associated with the signal samples;

selecting the largest second index from the plurality of second indices corresponding to the plurality of possible noise probability functions; wherein the largest second index value corresponds to the best estimate of the noise probability density function.

In preference the first index is of the form

where η is the received signal sample;

x_i-j is the estimate of symbol value; and

is an estimate of the channel impulse response of the link.

In preference the second index is of the form

In an alternate form of the invention the first index is of the form

and the second index is of the form

β =logp_n(n_i⃒θ). According to a further form of the invention there is proposed an apparatus for estimating a transmitted symbol for use in a communication link including an arbitrarily distributed additive white noise channel comprising a receiver, a comparison means, processing means and a decision means;

said receiver being adapted to repetitively sample a received signal to obtain a plurality of samples of transmitted symbols corrupted with noise and distortion;

said comparison means being adapted to subtract expected symbol values from the sampled received signal to produce an index value, n_j for all possible expected symbols;

the processing means being adapted to calculate

where θ is obtained from an external noise parameter estimator and M is the number of samples obtained; and

said decision means being adapted to select the largest value of β

corresponding to the maximum likelihood symbol thereby resulting in the optimum estimate of the transmitted symbol.

According to a still further form of the invention there is proposed an apparatus for estimating a transmitted symbol for use in a communication link including an arbitrarily distributed additive white noise channel and inter-symbol interference comprising a receiver, a processing means, a noise parameters estimating means and a channel estimating means;

said receiver being adapted to repetitively sample a received signal to obtain a plurality of samples of transmitted symbols corrupted by noise and distortion; said channel estimating means being adapted to provide as an input to the processing means an estimate of the channel impulse response;

said noise parameters estimating means being adapted to provide as an input to the processing means an estimate of the noise parameters;

said processing means adapted to utilize a maximum likelihood sequence estimator to determine the maximum likelihood transmitted symbol from the received signal sample and the channel estimate wherein the maximum likelihood sequence estimator utilizes a metric of the form

In a yet further form there is proposed an apparatus for use in a communications link including an arbitrarily distributed additive white noise channel and inter-symbol interference comprising a receiver and a processing means;

said receiver being adapted to repetitively sample a received signal to obtain a plurality of samples of transmitted symbols corrupted by noise and distortion; said processing means adapted to utilize a maximum likelihood sequence estimator to determine the maximum likelihood transmitted symbol from the received signal sample and the channel estimate wherein the maximum likelihood sequence estimator utilizes a metric of the form

In another form of the invention there is proposed an apparatus for

determining the transmitted modulation scheme in a communications link comprising a receiver, a processing means, a noise parameter estimating means and a decision means;

said receiver being adapted to repetitively sample a received signal to obtain a plurality of samples of a transmitted signal representing a symbol;

said noise parameters estimating means being adapted to provide as an input to the processing means an estimate of parameters describing noise ;

said processing means being adapted to calculate

where is obtained from the noise estimating means and z corresponds to each possible modulation scheme; and

said decision means being adapted to select the largest β_z thereby

determining the maximum likelihood estimate of the modulation scheme. BRIEF DESCRIPTION OF THE DRAWINGS

The above discussion explains the method of obtaining estimates of the parameters describing a communication link under a variety of conditions and assumptions. In order to assist in the understanding of the invention a number of preferred embodiments will now be described with reference to the attached drawings in which :

FIG 1 shows a first embodiment of the present invention to obtain an estimation of the noise probability density function; FIG 2 shows an example of one possible noise probability density function;

FIG 3 shows a prior art arrangement to optimise the detection of symbols without inter-symbol interference present but in the presence of Gaussian noise; FIG 4 shows a second embodiment of the present invention to obtain improved detection under similar conditions to FIG 3 but in the more general case of non-Gaussian noise;

FIG 5 shows a comparison of the performance of the apparatus of FIG 3 and FIG 4; FIG 6 shows a prior art arrangement to optimise the detection of symbols in the presence of inter-symbol interference and Gaussian noise;

FIG 7 shows a third embodiment of the present invention to obtain improved detection under similar conditions to FIG 6 but with non-Gaussian noise; FIG 8 shows a comparison of the performance of the apparatus of FIG 6 and FIG 7;

FIG 9 shows a fourth embodiment of the present invention applied to the problem of modulation identification; and

FIG 10 shows three modulation schemes used in a fourth embodiment; FIG 11 shows the performance of the apparatus of FIG 10. BEST MODE FOR CARRYING OUTTHE INVENTION

Referring now to the figures in detail, the first embodiment deals with a method of estimating the noise probability density function (pdf). Equation (25) provides the theoretical model applicable to this embodiment. For simplicity the following assumptions are made :

1. All pdf models are equally likely so Pθ(θ)=1/Lfor L models;

2. Binary data transmission so P_x(x=0) = 1/2 and P_x(x=1) = 1/2;

3. No inter-symbol interference so for j=0 or 0 otherwise.

Thus

taking logarithms gives

If the noise is Gaussian the only parameter (θ) required to describe the noise is the variance σn² so equation (A2) becomes

FIG 1 shows a schematic of a detector embodying this method. A signal r ( t) is sampled to produce a plurality of signal samples r_i. A comparion means compares the measured signal sample ri to the expected symbols x₁ and x₂ to produce noise estimates n_L1 and ή_i2. A processing means 2 calculates an index value of the form

for each of 1 possible variance values. The summation over i reflects the sequence of samples taken of each signal r(t). A decision means chooses between the possible pdf models by selecting the largest index value.

FIG 2 illustrates the nature of the pdf used in the embodiment of FIG 1. The pdf used is the sum of pdf's associated with each possible symbol. Another approach is to consider each symbol's pdf independently but this ignores the section of the actual pdf indicated by the dotted line and therefore results in a poorer estimate of the transmitted symbol.

FIG 3 shows a prior art method of optimising detection of data on a .ommunication link. The reception end of the link comprises a receiver, a matched filter, a sampler and some form of decision making means to output an estimate of the received data. Since the communication channel introduces noise and interference the received signal is rarely an exact replica of the transmitted signal and it is therefore necessary to calculate an estimate of the transmitted signal.

The prior art samples. the decision variable at 3 to calculate an estimate of the most likely symbol to match the received signal. The invention samples the received signal at 4 and calculates a maximum a posterior probability estimate of the received symbol.

The second embodiment is for the fundamental case of signalling over a link with arbitrary noise and no inter-symbol interference (ISI). In all digital demodulators a conventional matched filter or correlator detector is used (which is based on the premise of Gaussian noise). A replacement detector is described which equals or outperforms the conventional detector in all cases.

Consider the additive noise channel. We take Msamples per symbol for the detection process. Then, for one symbol x, the received samples are: r₁ = x + θ₁

r₂ = x + θ₂

. . .

r_M = x + θ_M (B1)

Rewriting (B1) in terms of the noise: θ_j = r _j - x {j=1, 2 , . . . ,M} (B2)

If the noise samples are independent :

Before considering the optimum case, we examine the usual correlator or matched-filter form of detector. If _pk(θ_j ) is Gaussian then substituting (B2) into (B3) gives :

The likelihood function may be used in the form given by equation (18), with further simplifications since h_j = 1 for j = 0 and h_j = 0 otherwise.

Furthermore, data decisions can be made on a symbol by symbol basis since there is no ISI. Then, the ML estimator for data is:

Or equivalently: Substituting into equation (B7) for p_n(k_i) with equation (B5) gives:

Simplifying (since σ is not a function of x):

Further simplification by expansion of the squared term gives:

Since rj² is not a function of x it may be ignored. Also, the x² term may be ignored since the value is (usually) the same regardless of which symbol was transmitted ie. energy of all signals is the same. The detection process is thus a correlator (or matched filter) between the received signal samples and each of the expected data symbols. We choose the data symbol which maximises this correlation:

A detector according to this description is shown in FIG 3. The received signal r(t) is sampled 5 at a rate MT producing M samples of each symbol i, r_ij. These samples r _j are mixed 6 with the expected symbols x_jk to produce Y_ijk=x_jkr_ij which is summed over all M samples. The values∑Y_ijk are compared for each possible x and the maximum value is selected thereby determining the maximum likelihood symbol estimate x^.

Now, if the noise is non-Gaussian this correlator or matched filter is no longer optimum, and we need to use the original equation (B7).

As an example, consider the case of mixed AWGN and impulse noise. The impulse noise model to be used here is bi-variate Gaussian distributed in the complex plane (ie has Rayleigh distributed magnitude and uniform distributed phase). Noise impulses are typically infrequent but with very great magnitude with respect of the received signal, effectively swamping background AWGN. The resulting noise pdf may be modelled thus in terms of its real and imaginary components:

P_kr (n_rj ) = αp_b (n_{r j} ) + (1 - α)p_q (n_rj )

(B12) p_ks (n_sj ) = αp_b(n_sj ) + (1 - α)p_q(n_sj )

Where α = Probability of an impulse (uniform distribution).

z_{r j} = Real part of random variable z, sample j.

= r_rj-x_rj

z_{s j} = Imaginary part of random variable z, sample j.

= r_sj-x_sj

P_b () = Background noise pdf.

p_q () = Impulse noise pdf.

Since real and imaginary components are independent, the combined pdf is: p_k (θ) =p_kr (θ_r) p_ks (θ_s ) (B15)

The new test becomes:

FIG 4 shows a schematic of a detector embodying this detection method. The received signal r(t) is sampled 5 at a rate of MT to produce signal samples r_ij. Expected symbols χ_jk are subtracted 11 from signal samples r_ij to produce θ_ijk=r_ij-X_jk which are processed in processor 12 by

An estimate of the noise parameters is obtained from an external noise parameter estimator 13. The maximum β is selected thereby identifying the maximum likelihood transmitted symbol. FIG 5a shows the performance for M=2 samples per symbol versus the signal to noise (background) ratio. The performance of either detector ("pdf-directed" or conventional correlator/matched filter) for an AWGN channel only, is shown in the lower curve. This represents the ultimate theoretical performance limit if AWGN only were present. The upper-most curve shows the performance of the conventional detector in the presence of impulse noise and AWGN. The simulation was for randomly-occurring impulses for 1% of the time (α=0.01). The impulse to signal ratio is 20db (impulse variance 20dB greater than the signal). The centre curve shows the performance of the pdf-directed detector for perfect noise pdf estimation and represents the ultimate performance attainable in the presence of impulse noise. The pdf-directed detector outperforms the conventional detector at all signal to noise ratios.

FIG 5b shows the performance for a mixture of 10% impulse noise (α=0.1). This figure again shows the impressive performance improvement of using the pdf-directed detector. Furthermore, for the case of M=4 samples per symbol, the performance improvement is greater. In fact, the larger the number of samples per symbol that need to be taken, the more spectacular the

performance improvement.

The second embodiment is for an ISI channel with additive arbitrary noise or interference. If a channel is known to introduce ISI and has additive white noise, a maximum Likelihood Sequence Estimator (MLSE) is the optimum means of detecting the transmitted symbols. This is also the case of partialresponse systems which introduce controlled amounts of ISI at the transmitter in order to achieve greater bandwidth usage. The Viterbi algorithm is commonly used to practically implement the MLSE. When this algorithm is used, a minimum distance metric is employed to trace through the trellis_.of possible decisions. This metric is a result of the assumption of Additive White Gaussian Noise (AWGN). If non-Gaussian noise or interference is present, the use of the minimum distance metric no longer yields a Maximum

Likelihood (ML) estimate. For this case, a more general metric is required, to provide better performance.

FIG 6 shows a detector suitable for the Gaussian case. The transmitted symbol x_i is corrupted by the channel. Sample r_i are taken of the received signal. The received samples r_i and an external estimate of the channel impulse response are input to a processor calculating the most likely sequence, typically implemented using the Viterbi algorithm. The output is the estimates of the transmitted symbol sequence x_i. (i = 1 , 2, .... N)

FIG 7 shows the modified detector for use in the non-Gaussian case. The processor utilizes any sequence estimating algorithm except for using a different data decision metric (derived below) and requiring the input of an external estimate of the noise probability density function p

Consider that the impulse response (channel tap gains) is estimated externally using a channel estimator and the actual noise model is estimated externally. If we also assume that the a priori probability for all noise models θ and channel mddels h are equally likely equation (18) reduces to:

Let

Consider the case where the probability density function of the noise is Gaussian. i.e.

Substituting equation (C3) into (C1):

Maximizing over x, the σ_n term not being a function of x is ignored. Substituting for z_i from (C2), this reduces to determining a minimum squared distance as given in the following expression: ie. argmax[-x] = argmin[-l-x]

This is the most famous form of the MLSE. The Viterbi algorithm calculates this minimum squared distance and uses a culling procedure to thereby estimate the data sequence x_i. This equation is optimum for Gaussian noise only. In order to cope with non-Gaussian noise we can use the usual MLSE algorithm, except with a new metric; to find the maximum over all x of :

Instead of the usual (optimum for AWGN) distance metrics:

In order to demonstrate the performance improvements possible in a non-Gaussian noise environment, the impulse noise model used in the previous simulation was again used. A severe ISI Channel was chosen whose impulse response was (1, 2, 1) warranting the use of sequence estimation. FIG 8a shows the performance of either detector in the lower curve for the case of AWGN (background noise) only. The upper-most curve shows the

performance of the Viterbi algorithm using the conventional metric in the presence of mixed AWGN and impulse noise. The centre curve shows the performance of the pdf-directed detector, which uses Viterbi's algorithm but with the new metric. The performance improvement is significant. This result is further supported in FIG 8b which shows performance for a channel with severe (10%) impulse content.

A fourth embodiment shows the ability to identify one of a set of possible transmitted modulation schemes. This example was inspired by the desire to minimize the amount of transmission overhead on an adaptive

communications link system. Instead of including several bits in a

transmission packet to tell the receiver what modulation scheme (constellation packing) was being sent, the technique is used to estimate which

constellation is actually being received in the shortest amount of time and without sending overhead information and without loss of data.

Consider the additive noise channel. The situation is that one of a set of possible modulations may be received. For illustration consider binary phase shift keying BPSK, quarternary phase shift keying QPSK or octernary phase shift keying 8PSK. Let: x_b Correspond to all possible sequences for BPSK

x_q Correspond to all possible sequences for QPSK x_o Correspond to all possible sequences for 8PSK

An estimator of the constellation type x_b, x_q, x_o would choose that

constellation type which maximizes the log likelihood for the particular

Z-{x_b,x_q,x_o} :

Where L ( z) is the number of constellation points in modulation scheme z.

Consider that the noise pdf is estimated externally and applied to the equation. If the noise is Gaussian, estimating the noise pdf is reduced to estimating the variance only. Furthermore, if we "ignore" the data by summing over possible data, we are left with the following implementation:

1. Calculate χ for each case M=2 (binary), M=4 (quaternary and M=8

(octernary), as follows:

2. The Maximum Likelihood (most likely) modulation scheme (z) is then the one for which the largest x_z was produced.

FIG 9 shows schematically the implementation of the method for 1 sample per symbol. The received signal sample r_i is used to estimate the noise probability density function p_θ. The received signal and the noise are input to a processing means which calculates χ₂ for each possible constellation type z. The maximum value is chosen thus determining the best estimate of the constellation type being used.

The performance of this scheme was simulated for the constellation points shown in FIG 10. The results of this simulation are shown in FIG 11 for coherent detection and N=50 samples. The embodiments described above are specific examples of the application of the invention to solve particular problems and are not meant to exclude other applications obvious to those skilled in the relevant art.

The inventors envisage application of the invention to the problem of optimizing symbol timing recovery, block synchronisation, optimum channel estimation (including the estimation of fading), non-linear channels, interferers and correlated noise.

Claims

1. A method of analyzing a communication link comprising the steps of : receiving a signal;

measuring one or more samples ri of the received signal;

wherein h is a parameter describing the communication channel, θ are parameters describing noise and x represents transmitted data in the form of symbols;

to produce optimum estimates of the parameters h, x, θ defining the

communications link.

2. A Maximum A posteriori Probability method of analyzing a

communication link by jointly estimating a parameter x representing

transmitted data in the form of symbols, a parameter h describing a

communication channel and a parameter θ describing noise, comprising the steps of :

receiving a signal;

measuring one or more samples r^ of the received signal;

analyzing the received sample or samples in terms of an algorithm in the form,

to produce optimum estimates of the parameters x, h and θ defining the communications link.

3. A method of analyzing a communication link by jointly estimating x representing transmitted data in the form of symbols and θ parameters describing noise while eliminating the effects of a communication channel comprising the steps of :

receiving a signal;

measuring one or more samples r_i of the received signal;

analyzing the received sample or samples in terms of an algorithm having parameters h, x, θ in the form :-

wherein h is a parameter describing the communication channel;

to produce optimum estimates of the parameters x and θ defining the communications link.

4. The method of claim 3 wherein an external estimate of the channel parameter h is used to simplify the algorithm to the form,

5. A method of analyzing a communication link by jointly estimating x representing transmitted data in the form of symbols and h a parameter describing the communication channel while eliminating the effects of noise comprising the steps of :

receiving a signal;

measuring one or more samples ri of the received signal;

wherein θ is a parameter describing noise;

to produce optimum estimates of the parameters x and h defining the communications link.

6. The method of claim 5 wherein an external estimate of a noise probability density function pø is used to simplify the algorithm to the form,

7. A method of analyzing a communication link by jointly estimating θ a parameter describing noise and h a parameter describing a communication channel by summing over all possible data sequences

comprising the steps of :

receiving a signal;

measuring one or more samples ri of the received signal; analyzing the received sample or samples in terms of an algorithm having parameters h, x, θ in the form :-

wherein x represents transmitted data in the form of symbols;

to produce optimum estimates of the parameters θ and h defining the communications link.

8. The method of claim 7 wherein an external estimate x of the data x is used to simplify the algorithm to the form,

9. A method of estimating x representing data in the form of symbols transmitted through a communication link comprising the steps of :

receiving a signal;

measuring one or more samples r_i of the received signal;

wherein h is a parameter describing the communication channel and θ is a parameter describing noise and wherein the received signal samples are integrated over all possible noise models and all possible channel models; to produce optimum estimates of the data x transmitted over the

communications link.

10. The method of claim 9 wherein the step of integrating over all possible noise models is replaced by the step of obtaining an external estimate pø of the noise probability function describing noise and the algorithm is simplified to :

11. The method of claim 9 wherein the step of integrating over all possible channels is replaced by the step of obtaining an external estimate of the channel and the algorithm is simplified to :

12. The method of claim 9 wherein the steps of integrating over all possible noise models and all possible channel models are replaced by the step of obtaining an external estimate ft of the channel an external estimate of the noise probability function describing noise and the algorithm is simplified to :

13. A method of obtaining an estimate of a symbol transmitted through a communications link comprising the steps of :

receiving a signal;

measuring at repetitive points in time a plurality of samples r_i of said signal; comparing the signal samples to a plurality of expected symbol values for the received signal and calculating a first index indicative of the comparison; calculating a second index from the first index and an estimate of a noise probability density function;

selecting the largest second index from the plurality of second index values corresponding to the plurality of expected symbol values wherein the largest second index determines the best estimate of the transmitted symbol.

14. The method of claim 13 wherein the first index is of the form,

and the second index is of the form

15. A method of estimating θ a parameter describing noise associated with a communications link comprising the steps of :

receiving a signal;

measuring one or more samples r_i of the received signal;

wherein h is a parameter describing the communication channel and x represents transmitted data in the form of symbols and wherein the received signal samples are integrated over all possible data sequences and over all possible channel models;

to produce optimum estimates of a noise probability density function model defining the communications link.

16. The method of claim 15 wherein the step of integrating over all possible data sequences is replaced by the step of obtaining an external estimate x of x representing transmitted data in the form of symbols and the algorithm is simplified to :

17. The method of claim 15 wherein the step of integrating over all possible channels is replaced by the step of obtaining an external estimate ft of the channel and the algorithm is simplified to :

18. The method of claim 15 wherein the steps of integrating over all possible data sequences and all possible channel models are replaced by the steps of obtaining an external estimate ft of the channel and an external estimate of the transmitted and the algorithm is simplified to :

19. A method of estimating the probability density function of noise associated with a communications link comprising the steps of :

receiving a signal;

measuring at repetitive points in time a plurality of samples of said signal; calculating for each signal sample a first index indicative of the noise associated with the signal samples;

calculating a second index from the first index by summing the first index over all possible transmitted symbols; selecting the largest second index-from the plurality of second indices

corresponding to the plurality of possible noise probability functions;

wherein the largest second index value corresponds to the best estimate of the noise probability density function.

20. The method of claim 19 wherein the first index is of the form,

where r_i is the received signal sample;

x_i-j is the estimate of symbol value;

is an estimate of the impulse response of the link; and

j represents channel delay,

and wherein the second index is of the form

21. A method of estimating h a parameter describing a communication channel of a communication link comprising the steps of :

receiving a signal;

measuring one or more samples r_i of the received signal;

wherein θ is a parameter describing noise and x represents transmitted data in the form of symbols and wherein the received signal samples are integrated over all possible data sequences and all possible noise models;

to produce optimum estimates of a channel probability density function of the communications link.

22. The method of claim 21 wherein the step of integrating over all possible noise models is replaced by the step of obtaining an extemal estimate § of the parameters describing noise and the algorithm is simplified to :

23. The method of claim 21 wherein the step of integrating over ail possible data sequences is replaced by the step of obtaining an external estimate x of x representing transmitted data in the form of symbols and the algorithm is simplified to :

24. The method of claim 21 wherein the steps of integrating over all possible noise models and all possible channel models are replaced by the step of obtaining an extemal estimate x of the transmitted data and an external estimate of the parameters describing noise and the algorithm is simplified to :

25. A method of estimating a channel associated with a communications link comprising the steps of :

receiving a signal;

the largest second index value corresponds to the best estimate of the channel.

26. The method of claim 25 wherein the first index is of the form

where η is the received signal sample;

is the estimate of symbol value; and

h_j is an estimate of the channel impulse response of the link and wherein the second index is of the form

27. An apparatus for estimating a transmitted symbol for use in a communication link including an arbitrarily distributed additive white noise channel comprising a receiver, a comparison means, processing means and a decision means; said receiver being adapted to repetitively sample a received signal to obtain a plurality of samples of transmitted symbols corrupted with noise and distortion;

the processing means being adapted to calculate

where is obtained from an extemal noise parameter estimator and M is the number of samples obtained; and

said decision means being adapted to select the largest value of β

28. An apparatus for estimating a transmitted symbol for use in a communication link including an arbitrarily distributed additive white noise channel and inter-symbol interference comprising a receiver, a processing means, a noise parameters estimating means and a channel estimating means;

29. An apparatus for use in a communications link including an arbitrarily distributed additive white noise channel and inter-symbol

interference comprising a receiver and a processing means;

30. An apparatus for determining the transmitted modulation scheme in a communications link comprising a receiver, a processing means, a noise parameter estimating means and a decision means;

said processing means being adapted to calculate

where $ is obtained from the noise estimating means and z corresponds to each possible modulation scheme; and

said decision means being adapted to select the largest β_z thereby

determining the maximum likelihood estimate of the modulation scheme.