WO2022074639A2

WO2022074639A2 - Communication system

Info

Publication number: WO2022074639A2
Application number: PCT/IB2021/061215
Authority: WO
Inventors: Mathieu GOUTAY; Jakob Hoydis; Faycal AIT AOUDIA
Original assignee: Nokia Technologies Oy
Priority date: 2020-10-06
Filing date: 2021-12-02
Publication date: 2022-04-14
Also published as: WO2022074639A3; FI20205980A9; FI20205980A1

Abstract

An apparatus method and computer progress is described comprising: receiving, at a receiver of a transmission system, one or more pilot signals and one or more data symbols, wherein the transmission system comprises a transmitter, a channel and the receiver, wherein the transmitter includes a transmitter algorithm for converting one or more transmission bits into one or more transmission symbols for transmission over the channel; generating, at a channel estimation module of the receiver, a channel estimate based, at least in part, on the one or more received pilot signals, wherein the channel estimation module is trained using machine learning principles; generating, using a channel error covariance module, a corrected channel error covariance matrix of the channel, wherein the channel error covariance module is trained using machine learning principles; generating, using an equalisation module of the receiver an estimate of the one or more transmission symbols based on the received data symbols, the channel estimate and the corrected channel error covariance matrix of the channel; and generating, using a demapping module of the receiver, a soft estimate of the one or more transmission bits.

Description

Communication system Field This specification relates to training in communication systems, such as communication systems having trainable parameters. Background End-to-end communication systems comprising a transmitter, a channel and a receiver in which the transmitter and/or the receiver have trainable parameters are known. Although a number of algorithms for using and training such systems are known, there remains a need for further developments in this field. Summary In a first aspect, this specification describes an apparatus (e.g. a receiver of a transmission system) comprising means for performing: receiving, at a receiver of a transmission system, one or more pilot signals and one or more data symbols, wherein the transmission system comprises a transmitter, a channel and the receiver, wherein the transmitter includes a transmitter algorithm for converting one or more transmission bits into one or more transmission symbols for transmission over the channel; generating, at a channel estimation module of the receiver, a channel estimate based, at least in part, on the one or more received pilot signals (other channel state information, such as one or more signal-to-noise ratio of the transmission, the pilot pattern, RSSI etc. may also be considered), wherein the channel estimation module is trained using machine learning principles; generating, using a channel error covariance module, a corrected channel error covariance matrix of the channel, wherein the channel error covariance module is trained using machine learning principles; generating, using an equalisation module of the receiver (which equalization module may be trainable, for example using machine learning principles) an estimate of the one or more transmission symbols based on the received data symbols, the channel estimate and the corrected channel error covariance matrix of the channel; and generating, using a demapping module of the receiver, a soft estimate of the one or more transmission bits. The apparatus may further comprise means configured to perform training the channel error covariance module. The soft estimate may take the form of log-likelihood ratios (LLRs). Such LLRs may provided to a decoder, which decoder may generate hard decisions that form estimates of the one or more transmission bits. The channel estimation module may be configured to generate an estimate of a covariance matrix of channel estimate error based, at least in part, the one or more received pilot signals, wherein the channel estimation module is trained using machine learning principles. The channel error covariance module may be configured to generate the corrected channel error covariance matrix of the channel by refining the estimate of the covariance matrix of channel estimate error generated by the channel estimation module. In an alternative embodiment, the channel error covariance matrix may be generated directly by the channel error covariance module. Some example embodiment further comprise means configured to perform: generating, using a variance module, an equivalent channel estimated variance, wherein the variance module is trained using machine learning principles, wherein the demapping module of the received generates the soft estimate of the one or more transmission bits based, at least in part, on the estimated transmission symbols and the equivalent channel estimated variance. Some example embodiment further comprise means configured to perform: training the variance module. Some example embodiment further comprise means configured to perform: training said channel estimation module to generate said channel estimate. Generating said channel estimate may comprise generating independent time, frequency and spatial correlation matrices. Generating said channel estimate may comprise determining decay parameters of an exponential decay model for the time, frequency and spatial correlation matrices respectively. The said means for training said channel estimation module may be configured to perform: training a plurality of neural networks. Further, the plurality of neural networks may comprise a first, second and third neural networks configured to process time, frequency and spatial relationships respectively. The means configured to perform generating the channel estimate may be further configured to perform: interpolation and/or extrapolation of time-related pilot signal data following by processing by said first neural network; interpolation and/or extrapolation of frequency-related pilot signal data following by processing by said second neural network; and interpolation and/or extrapolation of spatial-related pilot signal data following by processing by said third neural network. The channel error covariance module and/or the covariance module may be implemented using neural networks. In a second aspect, this specification describes an apparatus (e.g. an apparatus for training a receiver) comprising means for performing: initialising (e.g. randomly or pseudo randomly) a first set of trainable parameters of a channel estimation module of a receiver of a transmission system, wherein the transmission system comprises a transmitter, a channel and the receiver, wherein the transmitter includes a transmitter algorithm for converting one or more transmission bits into one or more transmission symbols and the receiver includes an equalisation module and a demapping module for converting one or more received symbols into one or more received bits; initialising (e.g. randomly or pseudo randomly) a second set of trainable parameters; receiving, at the receiver, one or more pilot signals and one or more data symbols transmitted by the transmitter over the channel; generating, at a channel estimation module of the receiver, a channel estimate based, at least in part, on the one or more received pilot signals, wherein the channel estimation module includes the first set of trainable parameters; generating a corrected channel error covariance matrix of the channel using a channel error covariance module, wherein the channel error covariance module comprises the second set of trainable parameters; generating, using the equalisation module of the receiver, an estimate of the one or more transmission symbols based on the received data symbols, the channel estimate and the corrected channel error covariance matrix of the channel estimate; generating, using the demapping module of the receiver, a soft estimate of the one or more transmission bits based on the estimated transmission symbols and the equivalent channel estimated covariance; generating a loss, using a loss function, based on the one or more transmission bits (which may be known at the receiver/trainer) and the estimate of the one or more transmission bits; updating (e.g. using stochastic gradient descent) the first and second sets of trainable parameters to minimise the loss function; and repeating the receiving, generating and updating until a first condition is reached. The said loss function may be related to one or more of block error rate, bit error rate, mutual information, categorical cross-entropy and binary cross-entropy. Some example embodiment further comprise means configure to perform: generating an equivalent channel estimated variance using a variance module, wherein the covariance module comprises a third set of trainable parameters. The first condition may comprise a defined performance level and/or a defined number of iterations. The channel estimation module of the receiver may comprise a channel estimation neural network. The channel error covariance module and/or the covariance module may be implemented using neural networks. In the first or the second aspect, the said means may comprise: at least one processor; and at least one memory including computer program code, the at least one memory and the computer program configured, with the at least one processor, to cause the performance of the apparatus. In a third aspect, this specification describes a method comprising: receiving, at a receiver of a transmission system, one or more pilot signals and one or more data symbols, wherein the transmission system comprises a transmitter, a channel and the receiver, wherein the transmitter includes a transmitter algorithm for converting one or more transmission bits into one or more transmission symbols for transmission over the channel; generating, at a channel estimation module of the receiver, a channel estimate based, at least in part, on the one or more received pilot signals, wherein the channel estimation module is trained using machine learning principles; generating, using a channel error covariance module, a corrected channel error covariance matrix of the channel, wherein the channel error covariance module is trained using machine learning principles; generating, using an equalisation module of the receiver an estimate of the one or more transmission symbols based on the received data symbols, the channel estimate and the corrected channel error covariance matrix of the channel; and generating, using a demapping module of the receiver, a soft estimate of the one or more transmission bits. The soft estimate may take the form of log-likelihood ratios (LLRs). Such LLRs may provided to a decoder, which may generate hard decisions that form estimates of the one or more transmission bits. The channel estimation module may generate an estimate of a covariance matrix of channel estimate error based, at least in part, the one or more received pilot signals, wherein the channel estimation module is trained using machine learning principles. The channel error covariance module may generate the corrected channel error covariance matrix of the channel by refining the estimate of the covariance matrix of channel estimate error generated by the channel estimation module. In an alternative embodiment, the channel error covariance matrix may be generated directly by the channel error covariance module. Some example embodiment further comprise: generating, using a variance module, an equivalent channel estimated variance, wherein the variance module is trained using machine learning principles, wherein the demapping module of the received generates the soft estimate of the one or more transmission bits based, at least in part, on the estimated transmission symbols and the equivalent channel estimated variance. Some example embodiment further comprise training the variance module. Some example embodiment further comprise training said channel estimation module to generate said channel estimate. Generating said channel estimate may comprise generating independent time, frequency and spatial correlation matrices. Generating said channel estimate may comprise determining decay parameters of an exponential decay model for the time, frequency and spatial correlation matrices respectively. The method may comprise training a plurality of neural networks. Further, the plurality of neural networks may comprise a first, second and third neural networks configured to process time, frequency and spatial relationships respectively. Generating the channel estimate may comprise: interpolation and/or extrapolation of time-related pilot signal data following by processing by said first neural network; interpolation and/or extrapolation of frequency-related pilot signal data following by processing by said second neural network; and interpolation and/or extrapolation of spatial-related pilot signal data following by processing by said third neural network. In the fourth aspect, this specification describes a method comprising: initialising (e.g. randomly or pseudo randomly) a first set of trainable parameters of a channel estimation module of a receiver of a transmission system, wherein the transmission system comprises a transmitter, a channel and the receiver, wherein the transmitter includes a transmitter algorithm for converting one or more transmission bits into one or more transmission symbols and the receiver includes an equalisation module and a demapping module for converting one or more received symbols into one or more received bits; initialising (e.g. randomly or pseudo randomly) a second set of trainable parameters; receiving, at the receiver, one or more pilot signals and one or more data symbols transmitted by the transmitter over the channel; generating, at a channel estimation module of the receiver, a channel estimate based, at least in part, on the one or more received pilot signals, wherein the channel estimation module includes the first set of trainable parameters; generating a corrected channel error covariance matrix of the channel using a channel error covariance module, wherein the channel error covariance module comprises the second set of trainable parameters; generating, using the equalisation module of the receiver, an estimate of the one or more transmission symbols based on the received data symbols, the channel estimate and the corrected channel error covariance matrix of the channel estimate; generating, using the demapping module of the receiver, a soft estimate of the one or more transmission bits based on the estimated transmission symbols and the equivalent channel estimated covariance; generating a loss, using a loss function, based on the one or more transmission bits (which may be known at the receiver/trainer) and the estimate of the one or more transmission bits; updating (e.g. using stochastic gradient descent) the first and second sets of trainable parameters to minimise the loss function; and repeating the receiving, generating and updating until a first condition is reached. The said loss function may be related to one or more of block error rate, bit error rate, mutual information, categorical cross-entropy and binary cross-entropy. The first condition may comprise a defined performance level and/or a defined number of iterations. Some example embodiment further comprise: generating an equivalent channel estimated variance using a variance module, wherein the covariance module comprises a third set of trainable parameters. In a fifth aspect, this specification describes computer-readable instructions which, when executed by computing apparatus, cause the computing apparatus to perform (at least) any method as described with reference to the third or fourth aspects. In a sixth aspect, this specification describes a computer-readable medium (such as a non-transitory computer-readable medium) comprising program instructions stored thereon for performing (at least) any method as described with reference to the third or fourth aspects. In a seventh aspect, this specification describes an apparatus comprising: at least one processor; and at least one memory including computer program code which, when executed by the at least one processor, causes the apparatus to perform (at least) any method as described with reference to the third or fourth aspects. In an eighth aspect, this specification describes a computer program comprising instructions for causing an apparatus to perform at least the following: receiving, at a receiver of a transmission system, one or more pilot signals and one or more data symbols, wherein the transmission system comprises a transmitter, a channel and the receiver, wherein the transmitter includes a transmitter algorithm for converting one or more transmission bits into one or more transmission symbols for transmission over the channel; generating, at a channel estimation module of the receiver, a channel estimate based, at least in part, on the one or more received pilot signals, wherein the channel estimation module is trained using machine learning principles; generating, using a channel error covariance module, a corrected channel error covariance matrix of the channel, wherein the channel error covariance module is trained using machine learning principles; generating, using an equalisation module of the receiver an estimate of the one or more transmission symbols based on the received data symbols, the channel estimate and the corrected channel error covariance matrix of the channel; and generating, using a demapping module of the receiver, a soft estimate of the one or more transmission bits. In a ninth aspect, this specification describes a computer program comprising instructions for causing an apparatus to perform at least the following: initialising (e.g. randomly or pseudo randomly) a first set of trainable parameters of a channel estimation module of a receiver of a transmission system, wherein the transmission system comprises a transmitter, a channel and the receiver, wherein the transmitter includes a transmitter algorithm for converting one or more transmission bits into one or more transmission symbols and the receiver includes an equalisation module and a demapping module for converting one or more received symbols into one or more received bits; initialising (e.g. randomly or pseudo randomly) a second set of trainable parameters; receiving, at the receiver, one or more pilot signals and one or more data symbols transmitted by the transmitter over the channel; generating, at a channel estimation module of the receiver, a channel estimate based, at least in part, on the one or more received pilot signals, wherein the channel estimation module includes the first set of trainable parameters; generating a corrected channel error covariance matrix of the channel using a channel error covariance module, wherein the channel error covariance module comprises the second set of trainable parameters; generating, using the equalisation module of the receiver, an estimate of the one or more transmission symbols based on the received data symbols, the channel estimate and the corrected channel error covariance matrix of the channel estimate; generating, using the demapping module of the receiver, a soft estimate of the one or more transmission bits based on the estimated transmission symbols and the equivalent channel estimated covariance; generating a loss, using a loss function, based on the one or more transmission bits (which may be known at the receiver/trainer) and the estimate of the one or more transmission bits; updating (e.g. using stochastic gradient descent) the first and second sets of trainable parameters to minimise the loss function; and repeating the receiving, generating and updating until a first condition is reached. In a tenth aspect, this specification describes an apparatus comprising means (such as an input module) for receiving, at a receiver of a transmission system, one or more pilot signals and one or more data symbols, wherein the transmission system comprises a transmitter, a channel and the receiver, wherein the transmitter includes a transmitter algorithm for converting one or more transmission bits into one or more transmission symbols for transmission over the channel; means (such as a channel estimation module) for generating a channel estimate based, at least in part, on the one or more received pilot signals, wherein the channel estimation module is trained using machine learning principles; means (such as a channel error covariance module) for generating a corrected channel error covariance matrix of the channel, wherein the channel error covariance module is trained using machine learning principles; means (such as an equalization module) for generating an estimate of the one or more transmission symbols based on the received data symbols, the channel estimate and the corrected channel error covariance matrix of the channel; and means (such as a demapping module) for generating a soft estimate of the one or more transmission bits. In an eleventh aspect, this specification describes an apparatus comprising means (such as a training module) for initialising a first set of trainable parameters of a channel estimation module of a receiver of a transmission system, wherein the transmission system comprises a transmitter, a channel and the receiver, wherein the transmitter includes a transmitter algorithm for converting one or more transmission bits into one or more transmission symbols and the receiver includes an equalisation module and a demapping module for converting one or more received symbols into one or more received bits; means (such as the training module) for initialising a second set of trainable parameters; means (such as an input of the receiver of the transmission system) one or more pilot signals and one or more data symbols transmitted by the transmitter over the channel; means (such as a channel estimation module) for generating a channel estimate based, at least in part, on the one or more received pilot signals, wherein the channel estimation module includes the first set of trainable parameters; means (such as a neural network) for generating a corrected channel error covariance matrix of the channel using a channel error covariance module, wherein the channel error covariance module comprises the second set of trainable parameters; means (such as an equalisation module) for generating an estimate of the one or more transmission symbols based on the received data symbols, the channel estimate and the corrected channel error covariance matrix of the channel estimate; means (such as a demapping module) for generating a soft estimate of the one or more transmission bits based on the estimated transmission symbols and the equivalent channel estimated covariance; generating a loss, using a loss function, based on the one or more transmission bits (which may be known at the receiver/trainer) and the estimate of the one or more transmission bits; means (such as the training module) for updating the first and second sets of trainable parameters to minimise the loss function; and means (such as the training module) for repeating the receiving, generating and updating until a first condition is reached. Brief description of the drawings Example embodiments will now be described, by way of non-limiting examples, with reference to the following schematic drawings: FIGS.1 and 2 are block diagrams of an example end-to-end communication systems in accordance with example embodiments; FIG.3 is a plot showing pilot signals transmitted in accordance with an example embodiment; FIGS.4 and 5 are flow charts showing algorithms in accordance with example embodiments; FIG.6 shows a resource block in a multicarrier SISO system in accordance with an example embodiment. FIG.7 is a plot showing amplitudes of a matrix in accordance with an example embodiment; FIGS.8 and 9 are block diagrams of channel estimators in accordance with example embodiments; FIGS.10 and 11 are block diagrams of systems in accordance with example embodiments; FIGS.12 and 13 are flow charts showing algorithms in accordance with example embodiments; FIG.14 shows an example neural network that may be used in one or more example embodiments; FIG.15 is a plot showing simulation results in accordance with example embodiments; FIG.16 is a block diagram of components of a system in accordance with an example embodiment; and FIGS.17A and 17B show tangible media, respectively a removable non-volatile memory unit and a compact disc (CD) storing computer-readable code which when run by a computer perform operations according to example embodiment. Detailed description The scope of protection sought for various embodiments of the invention is set out by the independent claims. The embodiments and features, if any, described in the specification that do not fall under the scope of the independent claims are to be interpreted as examples useful for understanding various embodiments of the invention. In the description and drawings, like reference numerals refer to like elements throughout. FIG.1 is a block diagram of an example end-to-end communication system, indicated generally by the reference numeral 10, in accordance with an example embodiment. The system 10 includes a transmitter 12, a channel 14 and a receiver 16. Viewed at a system level, the system 10 converts data (b) received at the input to the transmitter 12 into transmit symbols (x) for transmission over the channel 14 and the receiver 16 generates an estimate of the transmitted data from symbols

received from the

channel 14. The transmitter 12 may include a modulator (e.g. using orthogonal frequency division multiplexing (OFDM)) that converts the data symbols (b) into the transmit symbols (x) in accordance with a modulation scheme. The transmit symbols are then transmitted over the channel 14 and received at the receiver 16 as received symbols (γ). The receiver may include a demodulator that converts the received symbols (γ) into the estimate of the originally transmitted data symbols

FIG.2 is a block diagram of an example end-to-end communication system, indicated generally by the reference numeral 20, in accordance with an example embodiment. The system 20 is an example implementation of the system 10. The system 20 includes a transmitter 22, a channel 23, a channel estimation module 24, an equalization module 25 and a demapping module 26. The transmitter 22 and the channel 23 are similar to the transmitter 12 and channel 14 described above. The channel estimation module 24, equalization module 25 and demapping module 26 collectively form a receiver similar to the receiver 16 described above. Channel estimation is a key element of a receiver that operates at the output of a fading channel. Many communication systems use orthogonal frequency division multiplexing (OFDM) to simultaneously encode data in the frequency domain on adjacent subcarriers. This transmission scheme has many advantages, among which a simplified channel estimation and equalization. For a single-input multiple-output (SIMO) system, in which a single antenna transmitter transmits to a multi-antenna receiver, the channel can be modelled as:

where x ∈ ℂ is the symbol sent by the transmitter which is computed from the bit vector to be sent

is the corrupted received signal, and N_r is the number of receiving antennas. is a vector of complex white Gaussian noise with variance and

is the channel vector, which incorporates channel effects such as large and small scale fading. The receiver task is to recover the transmitted bits from the received signal y_x. In that aim, the channel estimation module 24 computes an estimate of the channel vector

and an estimate

of the covariance matrix of the channel estimation error

In some communication systems, channel estimation is performed based on known pilot symbols ^ transmitted by the transmitter. Following channel estimation, the equalization module 25 converts the SIMO channel into an equivalent single-input single-output (SISO) channel by computing an estimate of the transmitted symbol x. This equivalent channel is assumed to incorporate only additive noise with an estimated variance Finally, the demapping module 26

computes log-likelihood ratios (LLRs) for the transmitted bits from the transmitted

symbol estimate

As shown in FIG.2, in addition to data symbols x, the transmitter 22 sends pilot signals p to the channel estimation module 24 of the receiver via the channel 23. Pilot signals may be sent for each user device, in order to estimate the channel between a base station and each user device served by that base station. FIG.3 is a plot, indicated generally by the reference numeral 30, showing a grid of data symbols and pilot symbols transmitted in an OFDM system in accordance with an example embodiment. The pattern shown in the plot 30 includes white cells representing data symbols x and black symbols representing pilot signals p. FIG.4 is a flow chart showing an algorithm, indicated generally by the reference numeral 40, in accordance with an example embodiment. The algorithm 40 may be implemented by the channel estimation module 24 based on pilot symbols p ∈ ℂ that are known to the receiver that are transmitted over the channel 23. The algorithm 40 starts at operation 42, where the channel estimation module 24 computes estimates of the channel at pilot positions, from the received symbols y_p at pilot positions. At operation 44, estimates of the channel

are generated at the positions that correspond to data symbols (i.e., where no pilot symbols were sent) using the estimations . The operation 44 may be implement using a pre-defined

interpolation/extrapolation scheme. Note that this approach is inevitably suboptimal, as more accurate channel estimation would require considering the precise statistics of the channel. However, the channel statistics vary for each user and environment (as they depend on the speed, delay spread, power delay profile, etc.) and therefore such approaches are difficult to implement in practice. FIG.5 is a flow chart showing an algorithm, indicated generally by the reference numeral 50, in accordance with an example embodiment. The algorithm 50 starts at operation 52, where correlation matrices are learnt that can be used to generate channel estimates at data positions based on channel estimates at pilot positions. Assuming that time, frequency, and spatial correlation are independent (Kronecker structure), the operation 52 may involve learning learn three different correlation matrices: a time correlation matrix, a frequency correlation matrix, and a spatial correlation matrix. As described further below, a practical machine learning (ML)-based channel estimation scheme may be used for estimating the relevant correlation matrices. At operation 54 of the algorithm 50, estimates of the channel at pilot positions are

generated, from the received symbols at pilot positions (as in the operation 42

described above). At operation 56 of the algorithm 50, the correlation matrices generated in the operation 52 are applied to the channel estimates generated in the operation 54, thereby implementing operation 44 of the algorithm described above. In a variant of the operation 52, a less complex variant may be implemented by assuming that the frequency, time and spatial correlation matrices follow an exponential decay model. These models rely on decay parameters a_F, a_T, a_s, which are the only parameters that need to be learned, thereby reducing the number of parameters that need to be optimized. In the following, example embodiments relating to a multi-carrier single-input singleoutput (SISO) system are described, where only estimation over the frequencies is considered. The single-input multiple-output (SIMO) setup is then considered.

FIG. 6 shows a resource block, indicated generally by the reference numeral 6o, in a multicarrier SISO system in accordance with an example embodiment. The resource block 6o shows transmitted pilot signals.

Let us first consider a resource block in a multicarrier SISO system. This corresponds to 12 resource elements on adjacent frequencies on a single time step. If we take the time step at a single time point (e.g. t = 3), we obtain the corresponding vector as shown in

FIG. 6.

The SISO channel can be modelled as: y = hx + n for the data symbols with x e (C, y e (C, n e (C, and h e C.

We denote by

• y_F e C¹² the vector of received symbols

• p_F e C¹² the pilot vector, which contains the sent pilot at every pilot position and 0 at every other position • P_F = diag(p_F) the diagonal pilot matrix, where the diagonal elements are the one of the vector p _f

• R_F the square frequency correlation matrix

• o_n the standard deviation of the channel noise n

• I the 12x12 identity matrix

As described above, the algorithm 50 starts at operation 52, where the correlation matrices are learnt.

One estimator that computing channel estimates is an LMMSE estimator, which computes a channel estimate

where L_F may be defined by

An issue with this estimator is that it requires knowledge of the frequency correlation matrix R_F, which may not be known in practice. One could learn the matrix R_F. As the matrix is Hermitian complex, and of size 12x12, there are

parameters to be learned. Although learning R_F should provide better performance, some computational instabilities can arise from the matrix inversion in (3), that may hinder convergence. This issue can be at least partially addressed by assuming that the underlying fading process follows an exponential decay model with a decay parameter α_F . The corresponding correlation matrix is constructed as follows: where

denotes the element of R . FIG.7 is a plot, indicated generally by the

_F

reference numeral 70, showing amplitudes of such a matrix for α_F = 0.9. The value of the parameter α_F is usually set to 0.9 or 0.99. By seeing the transmitter, channel, and receiver as an end-to-end system that is trainable, it is possible to learn α_F. This technique has the benefit of having only one trainable parameter, enabling fast training and convergence on any system. Another possibility is to use a neural network (or similar module). Here, the channel estimates at each pilot position is estimated with

where (⋅)^∗ denotes the conjugate (assuming |p|² = 1). Once the vector of channel estimation

is computed for all 6 pilot elements, the refined channel estimation for all 12 elements is computed by:

where NN₄(. ) is a neural network, and ℝ2ℂ and ℂ2ℝ converts a complex into two real number and vice versa. The channel estimate can also be pre-processed before being inputted to the neural network. For example, interpolated in frequency may be applied so that the neural network input vector has the same size as the neural network output. This allows the neural network to be pilot-position agnostic, meaning that a single neural network can be used with any pilot placement. Another benefit of this approach is that the neural network can also be fed with channel state information (CSI), such as the signal to noise ratio (SNR) of the transmission. FIG.8 is a block diagram of a channel estimator, indicated generally by the reference numeral 80, in accordance with an example embodiment. The channel estimator 80 provides SISO channel estimation using neural networks. Specifically, the channel estimator 80 comprises a frequency interpolator 82, a complex-to-real converter 84, a neural network 86 and a real-to-complex converter 88. The channel estimator 80 receives channel estimates at each pilot position

, for example in the form of a 6x1x1 matrix (i.e.6 frequency positions for one time step at one receiving antenna). The frequency interpolator 82 outputs are larger matrix (e.g. a 12x1x1 matrix). The converter 84 convers the symbols into real numbers (having real and imaginary parts), for example providing a 24x1x1 matrix that is provided to the neural network 86 for processing. The output of the neural network is converted back into symbols by the converter 88, which converter provides the channel estimate .

The channel estimator 80 can therefore be used to impement the operation 54 of the algorithm 50. From our experimentations, a neural network with very low complexity can achieve high performance gains. For example, we have experimented with a neural network made of a unique layer and linear activation, i.e, equivalent to a linear operation, and achieved significant gains over the baseline. The embodiments described above relate to a multi-carrier single-input single-output (SISO) system. To scale-up to SIMO systems, we assume that the frequency, time, and spatial correlations are independent. Therefore, the approaches outlined above are applied successively to these three dimensions. This process is usually called “Wiener filtering”, and based on knowledge of both the time correlation matrix R_T and the spatial correlation matrix R_s in addition to the frequency correlation R_F.

As for the SISO case, the trainable parameters are the correlation matrices R_S, RF, and R_T. Assuming an exponential power decay underlying processes for time, frequency, and space, only the three decay parameters a_F, a_T, and a_s need to be learned. The channel estimator is depicted in FIG. 9.

FIG. 9 is a block diagram of a channel estimator, indicated generally by the reference numeral 90, in accordance with an example embodiment. The channel estimator 90 provides SIMO channel estimation using neural networks.

The channel estimator 90 comprises a first estimator 92, a second estimator 94 and a third estimator 96. Each estimator maybe an LMMSE estimator, as described above.

As noted above, in the example system 90, the received samples are processed in the following order: frequency, time, and then space. Thus, the first estimator 92 provides a channel estimate (W_F) based on processing in the frequency domain, the second estimator 94 provides a channel estimate (H_F:T) based on processing in the frequency and time domains, and the third estimator 96 provides a channel estimate (H_F,T,S) based on processing in the frequency, time and spatial domains.

FIG. 10 is a block diagram of a system, indicated generally by the reference numeral too, in accordance with an example embodiment. As in the SISO embodiments describe above, three successive small neural networks, each operating on a single dimension, can be used instead of LMMSE channel estimation. This is the configuration shown in the system too

We use the notation (i,j, k) for denoting the I^th subcarrier (frequency), j^th time step, and k^th receiving antennas. We denote by x^ the vector of all symbols x on a same subcarrier at time step j and receiving antenna k.

First, the symbols received at a given frequency can be processed in parallel for every time steps and antenna. For each time step j and antenna k, we compute:

We denote by the concatenation of the vectors over all time steps j

and antennas k.

A similar process is applied to the time dimension, for every frequency i and receiving antenna k. If we denote the vector containing all estimates from at a same

time for the frequency i and receiving antenna k, we compute:

The refined channel estimates concatenation of over all

frequencies i and receiving antennas k.

Finally, estimation is carried out over the space dimension for every frequency i and time step j. If we denote by the vector containing all estimates from on a

same antenna and for the frequency i and receiving antenna k :

The final channel estimate is the concatenation of for all

frequency i and receiving antenna k. The task of the three successive neural networks can be seen as refining an initial rough estimate. As those neural networks only operate on one dimension, their input and output size are very small:

In the system 100, the

layers are not shown for clarity.

denotes the channel estimates at every pilot position. In the system 100, it is assumed that N_r =

2 and that the pilot pattern contains two pilot in time and frequency, shown as separate elements. The elements are darkened each time the channel estimates are refined. If, at the base station, the antennas are placed on a 2-dimensional grid, the spatial correlation can be further divided into a horizontal and a vertical spatial correlation. In that case, the equation (10) can be re-written to use both the horizontal and spatial correlation separately. In the same way, the NN_k visible in the system 100 can be replaced with two NN_k,l and NN_k,m placed one after the other. The embodiments described above relate to SISO and SIMO arrangements. The principles described herein can be extended to multiple-input multple-output (MIMO) systems, as described further below. In a multi-user MIMO (MU-MIMO) system, where N_u single-antenna users transmit to a receiver comprising multiple antennas, the channel can be modelled as:

where is the vector containing the symbols sent by the

transmitting antennas mapped from the bit vectors is

the vector containing the signal received from the N_r receiving antennas, and

is the vector of complex additive white Gaussian noise ( is the matrix

containing the channel coefficients for all users and receive antenna. In this case, the receiver similarly estimates the channel matrix

and estimated channel error covariance matrix s, equalizes the signal , and find an estimate of the sent bits

Because the users use orthogonal pilots, i.e., not interfering with each other's, channel estimation is MU-MIMO is similar to channel estimation in SIMO: channel estimation for each user can be carried-out using the process depicted in the previous section. As described above with reference to the system 20, the role of the equalization module 25 is to recover an estimate of the sent symbols from the received signal

. To do so, the equalizer needs the estimation of the channel error covariance matrix

. In the MU-MIMO case, the corresponding equalization module (the module 112 in FIG.11 described below) converts the channel into N_u equivalent AWGN channel (one channel per user) with estimated noise standard deviations . Here, the

subscript

refers to the u^th user. When channel estimation is not perfect, both the channel error covariance matrix

and the equivalent estimated channels noise standard deviations are impossible to know precisely.

FIG.11 is a block diagram of a system, indicated generally by the reference numeral 110, in accordance with an example embodiment. The system 110 is a receiver that may be used in a MU-MIMO system in accordance with example embodiments. The system 110 comprises a plurality of channel estimation modules 111a to 111n, an equalisation module 112 and a plurality of demapping modules 113a to 113n. The channel estimation modules, equalisation module and demapping modules of the system 110 are similar to the channel estimation module 24, equalization module 25 and demapping module 26 of the system 20 described above. Thus: • Each channel estimation module 111a to 111n computes an estimate

of the relevant channel vector and an estimate

of the covariance matrix of the relevant channel estimation error

• The equalization module 112 computes an estimate

of the relevant transmitted symbol x. This equivalent channel is assumed to incorporate only additive noise with an estimated variance

• The demapping modules 113a to 113n computes log-likelihood ratios (LLRs) for the transmitted bits

from the transmitted symbol estimate

The system 110 further comprises a channel error covariance correction modules 114a to 114n and equivalent channel estimated variance correction modules 115a to 115n, which may be implemented using neural networks, as discussed below. The correction modules 114 and 115 may be used to refine the estimates of

respectively as follows:

where NN_^ is a single NN that is used in parallel for every user.

has a particular structure, e.g.

, then the NN can correct only the scalar . In this case,

has a single input and a single output, making it of low complexity

where is is a NN that is used in parallel for every user. As

takes as input a real number and output a real number, it is of low complexity. One can also correct all standard deviations at once by setting

In both cases, the subscript “corr” in the system 110 indicates that the vector or matrices are "corrected" by the relevant neural network. It should be noted that although the channel error covariance correction modules 114a to 114n are described above as refining an estimate of

generated by the respective channel estimation module 111 to 111n, this is not essential to all example embodiments. For example, the channel error covariance correction modules 114a to 114n could generate the initial estimates and refine those estimates (i.e. the function of generating the original estimate could be within the channel estimation module or within the channel error covariance correction module). Similarly, although the variance correction modules 115a to 115n are described above as refining variance estimates generated by the equalisation module 112, this is not essential to all example embodiments. For example, the variance correction modules 115a to 115n could generate the initial variance estimates and refine those estimates (i.e. the function of generating the original estimate could be within the equalization module or within the variance correction module). FIG.12 is a flow chart showing an algorithm, indicated generally by the reference numeral 120, in accordance with an example embodiment. The algorithm 120 starts at operation 121 where pilot signals and data symbols are received at a receiver of a transmission system (such as the transmission system 20). As shown in FIG.2, the received pilot signals are labelled by yp and the received data symbols are labelled by y_x. At operation 122, a channel estimate

and optionally an estimate of a covariance matrix of channel estimate error are generated at a channel estimation module of the receiver. The channel estimate and the covariance matrix estimate are generated based on received pilot signals. The channel estimation module is trained using machine learning principles and may be implemented using a neural network. As discussed in detail above, the estimation of the channel for resource elements carrying data

from the estimation of the channel for resource elements carrying pilot signals

may be implemented in a number of ways. For example, three different correlation matrices may be learned: a time correlation matrix, a frequency correlation matrix, and a spatial correlation matrix, which correlation matrices are then used to estimate . A less complex variant of this approach discussed above assumed

that the frequency, time and spatial correlation matrices follow an exponential decay model. These models rely on decay parameters

which are the only parameters that need to be learned, drastically reducing the number of parameters that need to be optimized. A second approach leverages small neural networks (NNs) to estimate

, and stochastic gradient descend (SGD) to learn a set of NN parameters that are suited for the underlying frequency, time, and spatial correlations. At operation 123, a corrected channel error covariance matrix of the channel is

generated using a channel error covariance module (for example using the modules 114a to 114n of the system 110). Each channel error covariance module may be trained using machine learning principles and may be implemented using a neural network. If a channel error covariance matrix is generated in the operation 122, then the operation 123 may corrected that covariance matrix. In an alternative embodiment, the corrected channel error covariance matrix is generated in a single operation. At operation 124, an estimate of the one or more transmission symbols (^) is generated using an equalisation module (such as the equalisation module 112) of the receiver based on the received data symbols (received in operation 121), the channel estimate (generated in operation 122) and the corrected channel error covariance matrix of the channel (generated in operation 123). The equalisation module is trained using machine learning principles and may be implemented using a neural network. At operation 125, an equivalent channel estimated variance

is generated using a variance module (such as the equivalent channel estimated variance modules 115a to 115n). The variance module is trained using machine learning principles and may be implemented using a neural network. At operation 126, a soft estimate of the one or more transmission bits is generated using a demapping module of the receiver, based on the estimated transmission symbols and the equivalent channel estimated variance. As discussed above, the outputs of the demapping module may take the form of log-likelihood ratios (LLRs); these may be referred to as soft decisions. Such LLRs may provided to a decoder, which may generate hard decisions that form estimates of the one or more transmission bits. As discussed further below, various modules of the system may be trained. For example, the channel estimation module (such as the modules 111a to 111n) may be trained in order to generate said channel estimate. Moreover, the channel estimation module may be configured to perform: generating independent time, frequency and spatial correlation matrices, each having trainable parameters. The trainable parameters of the correlation matrices may be decay parameters of an exponential decay model for the time, frequency and spatial correlation matrices respectively. Alternatively, or in addition, generating the channel estimate may involve: interpolation and/or extrapolation of time-related pilot signal data following by processing by said first neural network; interpolation and/or extrapolation of frequency-related pilot signal data following by processing by said second neural network; and interpolation and/or extrapolation of spatial-related pilot signal data following by processing by said third neural network. FIG.13 is a flow chart showing an algorithm, indicated generally by the reference numeral 130, in accordance with an example embodiment. The algorithm 130 is a training algorithm in which we denote by θ the set of trainable parameters, found in the channel estimator and optionally in described

above. The algorithm 130 starts at operation 131, where the trainable parameters θ are initialised, e.g., randomly. At operation 132, B bit vectors are generated, where B is the batch size.

At operation 133, a forward pass through the transmitter and receiver of the communication system is carried out. When applied to the system 110, the operation 133 may include: • Generating, at a channel estimation module 111a to 111n of the receiver, a channel estimate and optionally an estimate of a covariance matrix of channel estimate error based on the one or more received pilot signals, wherein the channel estimation module includes the first set of trainable parameters; • Generating a corrected channel error covariance matrix of the channel using a channel error covariance module (114a to 114n), wherein the channel error covariance module comprises the second set of trainable parameters; • Generating, using the equalization module 112 of the receiver, an estimate of the one or more transmission symbols based on the received data symbols, the channel estimate and the corrected channel error covariance matrix of the channel estimate; • Generating an equivalent channel estimated variance using a variance module 115a to 115n, wherein the variance module comprises the third set of trainable parameters; • Generating, using the demapping module 113a to 113n of the receiver, an estimate of the one or more transmission bits based on the estimated transmission symbols and the equivalent channel estimated variance. At operation 134, a loss function is evaluated based on the one or more transmission bits (which are known at the receiver/trainer) and the estimate of the one or more transmission bits. The loss function may be related to one or more of block error rate, bit error rate, mutual information and categorical cross-entropy. The loss function may be defined as follows:

where is the probability that the

user is set to one for the

^^IJ training example and is computed by taking the sigmoid of the LLRs. At operation 135, the parameters θ are updated by applying one step of stochastic gradient descent (SGD) (or a variant thereof) on the loss. At operation 136, the algorithm either returns to operation 131 or is stopped, depending on whether a predefined stop criterion has been reached. The stop criterion can take multiple forms, e.g., a defined performance level, a defined number of iterations and/or when the loss function has not decreased for a predefined number of iterations. In the algorithm 130, the learning rate, batch size, and possibly other parameters of the SGD variant (Adam, RMSProp…) are optimization parameters. Note that the loss function ^ corresponds, up to a constant, to the bit metric decoding rate, which is an achievable rate for practical systems that operates on bits (and not symbols). In some embodiments, a machine-learning based channel estimator is trained as part of an end-to-end communication system by directly comparing the send bits to the estimated received bits (or of their probabilities). Such a technique has the benefit of not requiring the true knowledge of the channel realizations for training the channel estimator, which are inaccessible in practice. Therefore, training of the channel estimator from actual measurements is possible. In some embodiments, a channel estimator is trained together with an equalizer and a demapper. Such channel estimators may therefore be optimized specifically for the receiver for which it is part of. FIG.14 shows an example neural network that may be used in one or more example embodiments. The neural network 140 comprises an input layer 141, one or more hidden layers 142, and an output layer 143. The hidden layers 142 may comprise a plurality of hidden nodes, where the processing may be performed based on the received user device positioning information. FIG.15 is a plot, indicated generally by the reference numeral 150, showing simulation results in accordance with example embodiments. Simulation were performed on a 3GPP compliant dataset, with 8 users, 32 receiving antennas, and using a 64 QAM modulation. Coded BER curves are shown in the plot 150. The two baselines are a receiver with perfect channel estimation (referred to as “Perf” in the legend), and a receiver that performs least-square channel estimation at pilot positions using (4) followed by linear interpolation (“LS” in the legend). Note that this last baseline is commonly used in practice. The proposed schemes are: • MMSE: Corresponds to LMMSE estimation with learned correlation matrices. • NN: Correspond to the channel estimation using neural networks, as discussed above. • LS_corr/MMSE_corr/NN_corr refers to MMSE/NN with additional use of neural networks SS_^ and SS_^ to correct error correlation matrices and post-equalization noise-power. For completeness, FIG.16 is a schematic diagram of components of one or more of the example embodiments described previously, which hereafter are referred to generically as a processing system 300. The processing system 300 may, for example, be the apparatus referred to in the claims below. The processing system 300 may have a processor 302, a memory 304 closely coupled to the processor and comprised of a RAM 314 and a ROM 312, and, optionally, a user input 310 and a display 318. The processing system 300 may comprise one or more network/apparatus interfaces 308 for connection to a network/apparatus, e.g. a modem which may be wired or wireless. The network/apparatus interface 308 may also operate as a connection to other apparatus such as device/apparatus which is not network side apparatus. Thus, direct connection between devices/apparatus without network participation is possible. The processor 302 is connected to each of the other components in order to control operation thereof. The memory 304 may comprise a non-volatile memory, such as a hard disk drive (HDD) or a solid state drive (SSD). The ROM 312 of the memory 304 stores, amongst other things, an operating system 315 and may store software applications 316. The RAM 314 of the memory 304 is used by the processor 302 for the temporary storage of data. The operating system 315 may contain code which, when executed by the processor implements aspects of the algorithms 40, 50, 120 and 130 described above. Note that in the case of small device/apparatus the memory can be most suitable for small size usage i.e. not always a hard disk drive (HDD) or a solid state drive (SSD) is used. The processor 302 may take any suitable form. For instance, it may be a microcontroller, a plurality of microcontrollers, a processor, or a plurality of processors. The processing system 300 may be a standalone computer, a server, a console, or a network thereof. The processing system 300 and needed structural parts may be all inside device/apparatus such as IoT device/apparatus i.e. embedded to very small size. In some example embodiments, the processing system 300 may also be associated with external software applications. These may be applications stored on a remote server device/apparatus and may run partly or exclusively on the remote server device/apparatus. These applications may be termed cloud-hosted applications. The processing system 300 may be in communication with the remote server device/apparatus in order to utilize the software application stored there. FIGS.17A and 17B show tangible media, respectively a removable memory unit 365 and a compact disc (CD) 368, storing computer-readable code which when run by a computer may perform methods according to example embodiments described above. The removable memory unit 365 may be a memory stick, e.g. a USB memory stick, having internal memory 366 storing the computer-readable code. The internal memory 366 may be accessed by a computer system via a connector 367. The CD 368 may be a CD-ROM or a DVD or similar. Other forms of tangible storage media may be used. Tangible media can be any device/apparatus capable of storing data/information which data/information can be exchanged between devices/apparatus/network. Embodiments of the present invention may be implemented in software, hardware, application logic or a combination of software, hardware and application logic. The software, application logic and/or hardware may reside on memory, or any computer media. In an example embodiment, the application logic, software or an instruction set is maintained on any one of various conventional computer-readable media. In the context of this document, a “memory” or “computer-readable medium” may be any non-transitory media or means that can contain, store, communicate, propagate or transport the instructions for use by or in connection with an instruction execution system, apparatus, or device, such as a computer. Reference to, where relevant, “computer-readable medium”, “computer program product”, “tangibly embodied computer program” etc., or a “processor” or “processing circuitry” etc. should be understood to encompass not only computers having differing architectures such as single/multi-processor architectures and sequencers/parallel architectures, but also specialised circuits such as field programmable gate arrays FPGA, application specify circuits ASIC, signal processing devices/apparatus and other devices/apparatus. References to computer program, instructions, code etc. should be understood to express software for a programmable processor firmware such as the programmable content of a hardware device/apparatus as instructions for a processor or configured or configuration settings for a fixed function device/apparatus, gate array, programmable logic device/apparatus, etc. If desired, the different functions discussed herein may be performed in a different order and/or concurrently with each other. Furthermore, if desired, one or more of the above-described functions may be optional or may be combined. Similarly, it will also be appreciated that the flow diagrams of Figures 4, 5, 12 and 13 are examples only and that various operations depicted therein may be omitted, reordered and/or combined. It will be appreciated that the above described example embodiments are purely illustrative and are not limiting on the scope of the invention. Other variations and modifications will be apparent to persons skilled in the art upon reading the present specification. Moreover, the disclosure of the present application should be understood to include any novel features or any novel combination of features either explicitly or implicitly disclosed herein or any generalization thereof and during the prosecution of the present application or of any application derived therefrom, new claims may be formulated to cover any such features and/or combination of such features. Although various aspects of the invention are set out in the independent claims, other aspects of the invention comprise other combinations of features from the described example embodiments and/or the dependent claims with the features of the independent claims, and not solely the combinations explicitly set out in the claims. It is also noted herein that while the above describes various examples, these descriptions should not be viewed in a limiting sense. Rather, there are several variations and modifications which may be made without departing from the scope of the present invention as defined in the appended claims.

Claims

Claims 1. An apparatus comprising means for performing: receiving, at a receiver of a transmission system, one or more pilot signals and one or more data symbols, wherein the transmission system comprises a transmitter, a channel and the receiver, wherein the transmitter includes a transmitter algorithm for converting one or more transmission bits into one or more transmission symbols for transmission over the channel; generating, at a channel estimation module of the receiver, a channel estimate based, at least in part, on the one or more received pilot signals, wherein the channel estimation module is trained using machine learning principles; generating, using a channel error covariance module, a corrected channel error covariance matrix of the channel, wherein the channel error covariance module is trained using machine learning principles; generating, using an equalisation module of the receiver an estimate of the one or more transmission symbols based on the received data symbols, the channel estimate and the corrected channel error covariance matrix of the channel; and generating, using a demapping module of the receiver, a soft estimate of the one or more transmission bits.

2. An apparatus as claimed in claim 1, wherein: the channel estimation module is configured to generate an estimate of a covariance matrix of channel estimate error based, at least in part, the one or more received pilot signals, wherein the channel estimation module is trained using machine learning principles; and the channel error covariance module is configured to generate the corrected channel error covariance matrix of the channel by refining the estimate of the covariance matrix of channel estimate error generated by the channel estimation module.

3. An apparatus as claimed in claim 1 or claim 2, further comprising means configured to perform: generating, using a variance module, an equivalent channel estimated variance, wherein the variance module is trained using machine learning principles, wherein the demapping module of the received generates the soft estimate of the one or more transmission bits based, at least in part, on the estimated transmission symbols and the equivalent channel estimated variance.

4. An apparatus as claimed in claim 3, further comprising means configured to perform: training the variance module.

5. An apparatus as claimed in any one of the preceding claims, further comprises means configured to perform: training said channel estimation module to generate said channel estimate.

6. An apparatus as claimed in any one of the preceding claims 5, wherein generating said channel estimate comprises generating independent time, frequency and spatial correlation matrices.

7. An apparatus as claimed in claim 6, wherein generating said channel estimate comprises determining decay parameters of an exponential decay model for the time, frequency and spatial correlation matrices respectively.

8. An apparatus as claimed in claims 5 to 7, wherein said means for training said channel estimation module is configured to perform: training a plurality of neural networks.

9. An apparatus as claimed in claim 8, wherein the plurality of neural networks comprises a first, second and third neural networks configured to process time, frequency and spatial relationships respectively.

10. An apparatus as claimed in claim 9, wherein the means configured to perform generating the channel estimate is further configured to perform: interpolation and/or extrapolation of time-related pilot signal data following by processing by said first neural network; interpolation and/or extrapolation of frequency-related pilot signal data following by processing by said second neural network; and interpolation and/or extrapolation of spatial-related pilot signal data following by processing by said third neural network.

11. An apparatus comprising means for performing: initialising a first set of trainable parameters of a channel estimation module of a receiver of a transmission system, wherein the transmission system comprises a transmitter, a channel and the receiver, wherein the transmitter includes a transmitter algorithm for converting one or more transmission bits into one or more transmission symbols and the receiver includes an equalisation module and a demapping module for converting one or more received symbols into one or more received bits; initialising a second set of trainable parameters; receiving, at the receiver, one or more pilot signals and one or more data symbols transmitted by the transmitter over the channel; generating, at a channel estimation module of the receiver, a channel estimate based, at least in part, on the one or more received pilot signals, wherein the channel estimation module includes the first set of trainable parameters; generating a corrected channel error covariance matrix of the channel using a channel error covariance module, wherein the channel error covariance module comprises the second set of trainable parameters; generating, using the equalisation module of the receiver, an estimate of the one or more transmission symbols based on the received data symbols, the channel estimate and the corrected channel error covariance matrix of the channel estimate; generating, using the demapping module of the receiver, a soft estimate of the one or more transmission bits based on the estimated transmission symbols and the equivalent channel estimated covariance; generating a loss, using a loss function, based on the one or more transmission bits and the estimate of the one or more transmission bits; updating the first and second sets of trainable parameters to minimise the loss function; and repeating the receiving, generating and updating until a first condition is reached.

12. An apparatus as claimed in claim 11, further comprising means configure to perform: generating an equivalent channel estimated variance using a variance module, wherein the covariance module comprises a third set of trainable parameters.

13. An apparatus as claimed in any one of the preceding claims, wherein the channel error covariance module and/or the covariance module are implemented using neural networks.

14. A method comprising: receiving, at a receiver of a transmission system, one or more pilot signals and one or more data symbols, wherein the transmission system comprises a transmitter, a channel and the receiver, wherein the transmitter includes a transmitter algorithm for converting one or more transmission bits into one or more transmission symbols for transmission over the channel; generating, at a channel estimation module of the receiver, a channel estimate based, at least in part, on the one or more received pilot signals, wherein the channel estimation module is trained using machine learning principles; generating, using a channel error covariance module, a corrected channel error covariance matrix of the channel, wherein the channel error covariance module is trained using machine learning principles; generating, using an equalisation module of the receiver an estimate of the one or more transmission symbols based on the received data symbols, the channel estimate and the corrected channel error covariance matrix of the channel; and generating, using a demapping module of the receiver, a soft estimate of the one or more transmission bits.

15. A computer program comprising instructions for causing an apparatus to perform at least the following: receiving, at a receiver of a transmission system, one or more pilot signals and one or more data symbols, wherein the transmission system comprises a transmitter, a channel and the receiver, wherein the transmitter includes a transmitter algorithm for converting one or more transmission bits into one or more transmission symbols for transmission over the channel; generating, at a channel estimation module of the receiver, a channel estimate based, at least in part, on the one or more received pilot signals, wherein the channel estimation module is trained using machine learning principles; generating, using a channel error covariance module, a corrected channel error covariance matrix of the channel, wherein the channel error covariance module is trained using machine learning principles; generating, using an equalisation module of the receiver an estimate of the one or more transmission symbols based on the received data symbols, the channel estimate and the corrected channel error covariance matrix of the channel; and generating, using a demapping module of the receiver, a soft estimate of the one or more transmission bits.