Reconstruction of clipped signals

Abstract

Using the information contained in clipped samples from analog-to-digital (ADC) conversions to improve receiver performance, for example by reducing clipping artifacts caused by the ADC due to its data resolution limitations. This offers advantages over existing solutions, which perform poorly because they discard information in the clipped samples.

Description

Technical Field

[0001] Embodiments related to reconstruction of clipped signals, including reconstruction of clipped signals in massive multiple-input multiple-output (MIMO) systems, are disclosed. Background Art

[0002] Clipping is a form of distortion that limits signal values ​​above or below a certain threshold. In practice, clipping may be necessary due to system limitations (for example, to avoid overmodulating an audio transmitter). In discrete systems, it can be caused unintentionally due to data resolution limitations (for example, when a sample exceeds the maximum value that can be represented), or intentionally when simulating a process where the signal values ​​are limited.

[0003] Clipping is a nonlinear operation that introduces frequency components that were not present in the original signal. In the digital domain, when the frequencies of these new components exceed the Nyquist limit, they are reflected back to the baseband, resulting in aliasing.

[0004] Massive MIMO systems are now a mature technology that forms the backbone of fifth-generation (5G) 3GPP mobile networks. With massive MIMO, the number of antennas at the base station (BS) is increased by several orders of magnitude compared to traditional multi-antenna systems, with the goal of achieving significant gains, such as higher capacity and energy efficiency.

[0005] In a traditional multi-antenna BS, each radio frequency (RF) port is connected to a pair of high-resolution analog-to-digital converters (ADCs) (typically, the in-phase and quadrature signal components are quantized with a resolution of more than 10 bits). Scaling this architecture to massive MIMO with hundreds or thousands of active antenna elements would result in prohibitively high power consumption and hardware costs. The hardware complexity and power consumption of the ADC grow roughly exponentially with the number of quantization bits. Therefore, an effective solution to keep power consumption and system cost within the desired limits is to reduce the accuracy of the ADC (e.g., up to 8 bits). An additional motivation for reducing the resolution of the employed ADC is to limit the amount of data that must be transmitted over the link connecting the RF components (also called the radio unit (RU)) and the baseband processing unit (BBU), which can be located remotely from the RU.

[0006] An ADC can be modeled as two processes: sampling and quantization. Sampling converts a continuous, time-varying voltage signal into a discrete-time signal—that is, a sequence of real numbers. Quantization replaces each real number with an approximation from a finite set of discrete values, and performs clipping when the input exceeds the supported range to limit the output to that range. The error introduced by this clipping is called overload distortion. The amount of spacing between the quantizer's selectable output values ​​within the limits of the supported range is called its granularity, and the error introduced by this spacing is called granularity distortion. The design of a quantizer typically involves determining the appropriate balance between granularity distortion and overload distortion. For a given supported number of possible output values, reducing the average granularity distortion may involve increasing the average overload distortion, and vice versa. Summary of the Invention

[0007] There are certain challenges. For example, overload distortion can severely impact the quality of digital signals by corrupting the data they represent. In fact, even a very low percentage of clipped samples can result in significant overload distortion. In multi-user MIMO (MU-MIMO) systems, overload distortion can lead to inaccurate channel state information (CSI), which degrades data estimates at the base station and / or at the user equipment (UE) to which the BS provides network access.

[0008] The present disclosure proposes utilizing the information contained in clipped samples from ADC conversions to improve receiver performance, for example by reducing clipping distortion caused by the ADC due to its data resolution limitations. This provides an advantage over existing solutions, which perform poorly because they discard information in the clipped samples.

[0009] Therefore, in one aspect, a method is provided for reconstructing clipped samples and thereby reducing overload distortion by exploiting the correlation between clipped samples and unclipped samples. In one embodiment, the method includes receiving a signal y and sampling y to produce a sample set. The method also includes quantizing each sample in the sample set to produce a quantized received signal r, wherein quantizing each sample in the sample set includes clipping at least M samples in the sample, wherein M>0, such that r includes M clipped samples. The method also includes obtaining information representing clipped samples and unclipped samples, and using the probability density function of y and the information representing clipped samples and unclipped samples to obtain a probability density function G(x) of an unknown sample in y, the unknown sample having been clipped conditioned on the quantized received vector r. The method also includes modifying r by replacing the clipped sample value with an expected value corresponding to the clipped sample value for each clipped sample value in r, wherein the expected value is based on G(x), thereby producing a reconstructed received signal.

[0010] In another aspect, a receiver apparatus is provided, configured to: sample a received signal y to produce a sample set; quantize each sample in the sample set to produce a quantized received signal r, wherein quantizing each sample in the sample set includes clipping at least M samples in the sample, wherein M>0, so that r includes M clipped samples; obtain information representing clipped samples and unclipped samples; use a probability density function of y and the information representing clipped samples and unclipped samples to obtain a probability density function G(x) of unknown samples in y, which have been clipped conditioned on the quantized received vector r; and modify r by replacing the clipped sample value with an expected value corresponding to the clipped sample value for each clipped sample value in r, wherein the expected value is based on G(x), thereby producing a reconstructed received signal. BRIEF DESCRIPTION OF THE DRAWINGS

[0011] The accompanying drawings, which are incorporated herein and form a part of the specification, illustrate various embodiments.

[0012] Figure 1 A clipping aware (CA) receiver according to an embodiment is shown.

[0013] Figure 2A is a graph illustrating the improvement provided by a CA-MMSE receiver according to an embodiment.

[0014] Figure 2B is a graph illustrating the improvement provided by a CA-MMSE receiver according to an embodiment.

[0015] Figure 3A is a graph illustrating the improvement provided by a CA-MMSE receiver according to an embodiment.

[0016] Figure 3B is a graph illustrating the improvement provided by a CA-MMSE receiver according to an embodiment.

[0017] Figure 4 is a flowchart illustrating a process according to an embodiment.

[0018] Figure 5 is a flowchart illustrating a process according to an embodiment.

[0019] Figure 6 is a block diagram of an apparatus according to an embodiment.

[0020] Figure 7 is a schematic block diagram of an apparatus according to an embodiment. DETAILED DESCRIPTION

[0021] Figure 1Components of a clipping-aware receiver 100 (CA-MMSE receiver 100) based on minimum mean square error (MMSE) are shown. CA-MMSE receiver 100 includes a quantizer 102 that receives an input signal y and produces a quantized output r. Quantizer 102 (e.g., an ADC) also outputs information (C) indicating which samples of y have been clipped. For example, set C represents the indices of clipped elements of a received vector r. Set G represents the indices of unclipped elements of a received vector r.

[0022] The CA-MMSE receiver 100 further comprises a reconstructor 104 which reconstructs the clipped samples conditioned on the quantized received vector r. For example, given the observed receive vector and In the case of , the reconstructor 104 replaces each clipped sample value in r with its corresponding expected value To reconstruct the clipped samples This approach is optimal because it minimizes the unknown received signal and its estimate The mean square error between . Mathematically speaking:

[0023]

[0024] The CA-MMSE receiver 100 further includes a decoder 106, which decodes the information output by the reconstructor 104 (ie, in, ) for decoding.

[0025] 1.1 System Model

[0026] Consider a single-cell MU-MIMO system consisting of a BS equipped with N antennas communicating with K single-antenna UEs, and assuming that the BS and UEs are fully synchronized and run a time division duplex (TDD) protocol with universal frequency reuse. The Nx1 receive vector at the BS is:

[0027]

[0028] Where H is the NxN small-scale channel coefficient matrix between the K UEs and the base station. Furthermore, x is a Kx1 vector of independent unit-power symbols transmitted simultaneously by the K UEs, with an average transmit power of ρ per UE. Finally, n is additive white Gaussian noise (AWGN).

[0029] 1.2 Quantization of complex-valued vectors

[0030] The in-phase and quadrature components of the received signal at each antenna are quantized separately by an ADC with b-bit resolution. More precisely, we model the ADC as a symmetric uniform quantizer with a step size of Δ, and each ADC is characterized by a set of L = 2b quantization levels in

[0031]

[0032] In addition, we define a set of L+1 quantization thresholds So that ∞=τ0<τ1<…<τ L-1 <τ L =∞, and

[0033]

[0034] A practical technique for controlling the signal amplitude (or equivalently, the quantization step size Δ) to achieve an appropriate balance between granularity distortion and overload distortion is to use automatic gain control (AGC). Next, we define the nonlinear quantizer mapping function that describes the joint operation of the 2N b-bit ADCs at the BS. For convenience, we first define the Cartesian product And let y n and r n are the nth element of the N×1 vectors y and r, respectively. Then, the quantizer mapping function can be represented by the function To describe, the function maps the received continuous-valued signal y to the quantized output r in the following way: if and Then r n =l k +jl m Therefore, the quantized received signal r can be written as

[0035]

[0036] Where A is the NxN diagonal matrix of the automatic gain control (AGC) that scales the received signal y. In addition, and represent the real and imaginary parts of the signal respectively.

[0037] 1.3 Augmented Real-Valued Representation

[0038] Since the ADC quantizes the real and imaginary parts of the signal separately (or quantizes the in-phase component and the orthogonal component separately), channel estimation and data detection should allow the real and imaginary parts of the received signal to be processed separately. Therefore, the complex-valued problem (1) can be conveniently converted into an equivalent augmented real-valued representation using the following definition:

[0039]

[0040] Then the quantized signal can be written as

[0041]

[0042] For the sake of notation, for the rest of the document, we use y, r, and x to denote y, r, and x, respectively. R 、r R and x R .

[0043] We only focus on the reconstruction of the clipped samples of the quantized received signal r, while the samples that are not clipped (i.e., within the quantizer's granularity) remain unchanged after quantization. For ease of notation, we first define the following two index sets of the quantized received vector r, namely, and

[0044]

[0045] That is, the set represents the index of the clipped element of the receiving vector r, while the set The indices in represent the elements of r that belong to the granularity region of the quantizer. We then define the vector and Please note that and denote the quantized received vector of the observed clipped samples and the vector lying within the quantizer’s granularity region, respectively. On the other hand, the vector and represents the unknown continuous signal, which is quantized to produce the vector and

[0046] Next, we introduce the proposed clipping recovery receiver, which reconstructs only the clipped samples conditioned on the quantized received vector r. Given the observed receive vector and In the case of , the proposed receiver replaces each clipping sample value in r with its corresponding The expected value of This approach is optimal because it minimizes the unknown received signal and its estimate Mathematically, we have the following receiver:

[0047]

[0048] Therefore, when determining the expectation in (2), we first need the posterior probability density function (exist Figure 5 It is also expressed as G(x), where x refers to The unknown continuous sample y is conditional C ), the posterior probability density function produces all feasible values Observe the probability of the quantized vector r. Next, we give a more detailed description of the proposed receiver in view of the fact that the received continuous signal follows a Gaussian distribution.

[0049] Assume that the mth element of vector r The elements have been clipped. That is, r m =l j , where j∈{0, L-1}. In addition, let Represents a vector The mth unknown continuous sample y m Therefore, according to the definition of expectation in (2), we have:

[0050]

[0051] Among them, the posterior is a one-sided truncated normal distribution in the interval [α′, β′]. Now, let α′ m and β′ m are the mth elements of vectors α′ and β′ respectively. Note that the interval [α′ m , β′ m ] represents the unknown continuous value y given its clipping observation sample m The feasible range of , and therefore calculate its expected value in this interval. is the condition, the unknown vector Each component of can be (a) left truncated, i.e., α′ m =τ0=-∞ and β′ m =τ1, or (b) right truncation, i.e., α′ m =τ L-1 And β′ m =τ L =∞.

[0052] It is worth mentioning that the elements of the received consecutive vectors are correlated, and therefore by observing It also gives us some information about unknown clipping samples. This correlation is captured by the covariance matrix of the posterior pdf in (3).

[0053] The integral in expression (3) cannot be evaluated in closed form, so we can use numerical integration to implement the estimator. However, for a large number of clipped samples, this becomes impractical. Therefore, we next provide a low-complexity iterative algorithm to approximate the mean E{y C |r C , r G}. This algorithm relies on the one-sided truncated normal distribution The expectation of a closed-form one-dimensional condition, where M represents the cardinality of the clipping samples.

[0054] Specifically, the jth iteration of this process returns the vector And it looks like this:

[0055]

[0056]

[0057]

[0058] Repeat this process iteratively until the number Until it falls below a constant δ or exceeds the maximum number of iterations J. Next, we generate a closed-form expression for the above expected value.

[0059] For the convenience of notation, we define the vector Among them, the vector The i-th element of has been removed. Then, each expectation of the iterative algorithm is given by the following closed-form expression, which is the mean of the univariate truncated normal distribution (see NL Johnson, S. Kotz, N. Balakrishnan, "Continuous Univariate Distributions", 2nd edition, volume 1, 1994, Wiley):

[0060]

[0061] in, and also, is the probability density function of the standard normal distribution, is its cumulative distribution function, and erfc(·) is the complementary error function. and Given by the following formula

[0062]

[0063]

[0064] Among them, the (N-1)×(N-1) matrix ∑ -i-i is formed by removing the i-th row and i-th column from ∑, and the (N-1)×1 vector σ -i is the i-th column of ∑ after removing the i-th element, and the (N-1)×1 mean vector μ -i is obtained by removing the i-th element from μ. In addition, the variance σ ii corresponds to the i-th diagonal element of the covariance matrix ∑. Finally, the parameters μ and ∑ are given by the following formulas respectively

[0065]

[0066]

[0067] in, and

[0068] Regarding computational complexity, the iterative algorithm requires evaluating simple closed-form formulas. The main computational burden is due to computing the M inverse matrices that appear in (5) and (6).

[0069] The proposed algorithm is a deterministic approximation of the Gibbs sampler, where the randomly generated samples of the Gibbs sampler are replaced by the mean of the corresponding conditional distribution. It is worth mentioning that the deterministic approximation of the Gibbs sampler has also been used for the semi-supervised hyperspectral unmixing problem (see, for example, KE Themelis and AARontogiannis and KD Koutroumbas, "A Novel Hierarchical Bayesian Approach for Sparse Semisupervised Hyperspectral Unmixing", IEEE Journal of Signal Processing, vol. 60, no. 2, pp. 585-599, February 2012).

[0070] If the matrix ∑- 1 If the norm of the i-th diagonal element of is greater than all the entries in its corresponding row, then the mapping in expression (3) is About The contraction of the norm (and hence its convergence to a unique fixed solution).

[0071] We now evaluate the mean square error (MSE) using the proposed receiver on a massive MU-MIMO uplink system, where each RF port at the BS is equipped with a resolution-limited ADC. The AGC is chosen to minimize the mean square error (MSE) between the unquantized received vector y and the quantized vector r.

[0072] On this basis, by assuming 3-bit, 4-bit and 6-bit resolution ADCs, on average 10%, 2% and 0.25% of the samples in the received vector r are clipped, respectively.

[0073] Finally, we assume that the entries of the channel matrix H are independent and distribution, and define the average signal-to-noise ratio as

[0074] First, in Figure 2A and Figure 2B In

[15] , we visualize the improvements of the Clipping Aware MMSE (CA-MMSE) receiver on the average granularity per sample and the overload distortion. Recall that CA-MMSE only reconstructs the clipped samples, while the granularity samples remain unchanged, and therefore does not improve the granularity distortion. However, CA-MMSE brings significant improvement on the overload distortion. For example, at b = 4 bits ( Figure 2A ) and high signal-to-noise ratio (SNR) (i.e., 20 dB), it reduces the overload distortion by 83% compared to the quantization-unaware case (QU) where the quantized samples are not reconstructed. Indeed, when the ADC resolution is increased to b = 6 bits ( Figure 2B ), then the gain of CA-MMSE on overload distortion increases to 95%. The reason is that in ADCs with higher resolution, the granularity distortion is lower, and therefore CA-MMSE can reconstruct the clipped samples with higher accuracy. The latter result is particularly important because it means that even a small proportion of clipped samples can cause significant overload distortion (recall that when b = 6 bits, the percentage of clipped samples is very low, i.e., 0.25%). In addition, note that the overload distortion of quantization-aware MMSE (QA-MMSE) (which reconstructs both clipped samples and samples lying within the granularity region) is similar to that of CA-MMSE, indicating how efficiently the latter reconstructs clipped samples.

[0075] Finally, note that with b = 4 bits and high SNR, QA-MMSE produces 37% lower granularity distortion than CA-MMSE, while when the ADC resolution is increased to b = 6 bits, its granularity gain is negligible, as expected. Although QA-MMSE outperforms CA-MMSE in granularity distortion, it does not produce a significant improvement in data estimation. The reason is that the overload distortion dominates the per-sample granularity distortion, and therefore compensating for the overload distortion is more important.

[0076] We now turn our attention to the convergence of the iterative algorithm and confirm that the CA-MMSE receiver 100 is adopted instead of the more computationally expensive QA-MMSE (i.e., is enough. Figure 3A and Figure 3B In the estimation symbol The MSE of is shown as a function of the average SNR and ADC resolution for different numbers of BS antennas. More precisely, we compare the CA-MMSE implemented by the iterative algorithm with clipping-aware receivers based on Gibbs sampling (GS-AC-MMSE) and QA-MMSE. We note that if b = 4 bits ( Figure 3A ), CA-MMSE converges to GS-AC-MMSE after only 15 iterations. However, note that CA-MMSE converges to GS-CA-MMSE faster when the number of BS antennas is relatively small. The reason is that in the case of N=16, there are much fewer clipping samples to estimate than in the case of N=64. Therefore, the smaller the number of BS antennas, the faster the iterative CA-MMSE receiver. Similarly, when the number of bits in the ADC is large, convergence is faster. Therefore, in the case of b=6 bits ( Figure 3B ), convergence occurs at 5 iterations.

[0077] It is also worth mentioning that after 15 iterations, CA-MMSE is almost identical to the best QA-MMSE receiver, confirming that it is sufficient to use CA-MMSE and reconstruct only the clipped samples, while the unclipped samples within the quantizer's granularity can remain unchanged. This result is particularly important because it means that we can achieve almost optimal performance with a lower-complexity receiver.

[0078] Figure 4 is a flow chart illustrating a process 400 according to an embodiment. The process 400 may start at step s402.

[0079] Step s402 includes: the receiver 100 receives a signal y.

[0080] Step s404 includes: the receiver 100 samples y to generate a sample set, and then quantizes each sample in the sample set to generate a quantized received signal r, wherein quantizing each sample in the sample set includes clipping at least M samples in the sample, wherein M>0, so that r includes M clipped samples.

[0081] Step s406 includes: the receiver 100 obtains information indicating clipped samples and unclipped samples, for example, obtaining a list of clipped samples and / or unclipped samples.

[0082] Step s412 includes the receiver 100 using the probability density function of y and the information representing the clipped samples and the unclipped samples to obtain the probability density function G(x) of the unknown samples in y, which have been clipped according to the received quantization vector r, where x=(x1, ..., x M ) represents the clipping to vector r c The previous unknown value.

[0083] Step s414 includes the receiver 100 modifying r by replacing each clipping sample value in r with an expected value corresponding to the clipping sample value, wherein the expected value is based on G(x), thereby generating a reconstructed received signal.

[0084] The process 400 may further include steps s408, s410, and s416. Steps s408 and s410 respectively include: receiving training data, and using the received training data to obtain the probability density function of y. Step s416 includes: to decode.

[0085] Figure 5 is a flow chart illustrating a process 500 for implementing step s414 according to an embodiment. The process 500 may begin at step s501.

[0086] Step s501 includes initializing the expected value of the clipped samples in y, i.e., μ (0) =(μ1 (0) , μ2 (0) ,...μ M (0) ). The initial values ​​here are equal to their corresponding clipping values.

[0087] That is, in step s501, the first vector μ is initialized (0) , where the first vector is represented by the value μ1 (0) , μ2 (0) , μ3 (0) ,...,μ M (0) composition.

[0088] Step s502 includes defining termination conditions δ and J.

[0089] Step s504 includes setting j=1.

[0090] Step s506 includes determining μ (j) , which is the vector of expected values. Determine μ (j) Include: For i = 1 to M, calculate each element μ of the vector i (j) , where M is equal to the total number of clipping samples. That is, determine μ(j) Includes calculations:

[0091] μ1 (j) =E(x1|μ2 (j-1) , μ3 (j-1) ,...,μ M (j-1) )

[0092] μ2 (j) =E(x2|μ1 (j) μ3 (j-1) ,...,μ M (j-1) )

[0093]

[0094] μ M (j) =E(x M |μ1 (j) , μ2 (j) ,…,μ M-1 (j) ).

[0095] E(x1|μ2 (j-1) , μ3 (j-1) ,...,μ M (j-1) ) is based on The expected value of the first clipping sample of (j) , μ3 (j-1) ,...,μ M (j-1) ) is based on The expected value of the second clipping sample of ; ...; and E(x M |μ1 (j) , μ3 (j) ,…,μ M (j) ) is based on The expected value of the Mth clipping sample.

[0096] Determine μ (j) Then, in step s508, it is determined whether:

[0097] ||μ (j) -μ (j-1) ||<δ or j=J.

[0098] If ||μ (j) -μ (j-1)If ||<δ or j=J is true, the process proceeds to step s510, otherwise the process proceeds to step s509, in which j is incremented by 1. After step s509, the process returns to step s506.

[0099] Step s510 includes setting For example, as a way to determine ||μ (1) -μ (0) || is less than δ, replace the first clipping sample value in r with μ1 (1) , replace the second clipping sample value in r with μ2 (1) , ..., and replace the Mth clipping sample value in r with μ M (1) .

[0100] Figure 6 is a block diagram of a receiver apparatus 600 according to some embodiments. The receiver apparatus 600 may be used to implement the receiver 100. Figure 6As shown, the receiver device 600 may include: a processing circuit (PC) 602, which may include one or more processors (P) 655 (e.g., a general-purpose microprocessor and / or one or more other processors, such as an application-specific integrated circuit (ASIC), a field-programmable gate array (FPGA), etc.), which may be co-located in a single housing or a single data center, or may be geographically distributed (i.e., the receiver device 600 may be a distributed computing device); a network interface 648, including a transmitter (Tx) 645 and a receiver (Rx) 647, for enabling the device 600 to send data to and receive data from other nodes connected to a network 110 (e.g., an Internet Protocol (IP) network), the network interface 648 being connected to the network 110; and a local storage unit (also referred to as a "data storage system") 608, which may include one or more non-volatile storage devices and / or one or more volatile storage devices. In embodiments where the PC 602 includes a programmable processor, a computer program product (CPP) 641 may be provided. CPP 641 includes a computer-readable medium (CRM) 642 storing a computer program (CP) 643 including computer-readable instructions (CRI) 644. CRM 642 can be a non-transitory computer-readable medium, such as a magnetic medium (e.g., a hard drive), an optical medium, a storage device (e.g., random access memory, flash memory), etc. In some embodiments, CRI 644 of computer program 643 is configured such that, when executed by PC 602, CRI causes apparatus 600 to perform the steps described herein (e.g., the steps described herein with reference to the flowcharts). In other embodiments, apparatus 600 can be configured to perform the steps described herein without requiring code. That is, for example, PC 602 can consist solely of one or more ASICs. Thus, the features of the embodiments described herein can be implemented in hardware and / or software.

[0101] Figure 7 7 is a schematic block diagram of a receiver apparatus 600 according to some other embodiments. The receiver apparatus 600 includes one or more modules 700, each of which is implemented in software. The modules 700 provide the functionality of the apparatus 600 described herein (e.g., the steps described above, e.g., regarding Figure 4 and / or Figure 5 ).

[0102] Although various embodiments of the present disclosure are described herein, it should be understood that they are only proposed in an illustrative and non-restrictive manner. Therefore, the width and scope of the present disclosure should not be subject to any one of the limitations of the above-mentioned exemplary embodiments. Generally, unless clearly given and / or different meanings are implied from the context in which they are located, all terms used in this article will be interpreted according to their common meaning in the relevant technical field. Unless otherwise clearly stated, all references to "one / an / described element, equipment, component, device, step etc." should be openly interpreted as referring to at least one instance in element, equipment, component, device, step etc. Unless otherwise indicated herein or otherwise clearly conflicting with context, any combination of the above-mentioned elements and all their possible variations is included in the present disclosure.

[0103] Additionally, although the processes described above and illustrated in the accompanying drawings are shown as a series of steps, this is for illustrative purposes only. Therefore, it is contemplated that some steps may be added, some steps may be omitted, the order of steps may be rearranged, and some steps may be performed in parallel. That is, the steps of any method disclosed herein do not necessarily need to be performed in the exact order disclosed, unless one step is explicitly described as preceding or following another step and / or a step is implicitly described as preceding or following another step.

Claims

1. A method (400) for reconstructing clipped samples implemented by a receiver (100) having N antennas, wherein N>1, the method comprising: receiving (s402) a signal vector y, wherein y is an N x 1 vector having N elements and each element of y is a received signal from a different one of the N antennas; Sampling each of the N elements of y (s404) to generate a sample set; quantizing (s404) each sample in the sample set to generate a quantized received vector r, wherein quantizing each sample in the sample set comprises clipping at least M samples of the samples, wherein M>0, such that r includes M clipped samples; obtaining (s406) information representing the clipped samples and the unclipped samples; The probability density function of y (s412) and the information representing the clipped samples and the unclipped samples are used to obtain the probability density function G(x), where x=(x1,...,x M ) represents the unknown value before being clipped to said M clipping samples of r; and r is modified (s414) by replacing each clipped sample value in r with an expected value corresponding to the clipped sample value, wherein the expected value is based on G(x), thereby producing a reconstructed received signal.

2. The method according to claim 1, further comprising: The reconstructed received signal is decoded.

3. The method according to claim 1 or 2, further comprising: Training data is received, and a probability density function of y is obtained using the received training data.

4. The method according to claim 1 or 2, further comprising: Determine the corresponding expected value.

5. The method according to claim 4, wherein Determining the corresponding expected value includes: Initialize (s501) the first vector μ as a vector of expected values (0) , where μ (0) is the expected value μ1 corresponding to the M clipping samples of r (0) , μ2 (0) , μ3 (0) ,...,μ M (0) The vector of components; Define (s502) a first termination condition δ and a second termination condition J; Based on μ (0) and G(x) to calculate (s506) the second vector μ (1) ;as well as Determine (s508) whether ||μ (1) -μ (0) || is less than δ or determines whether J is equal to 1.

6. The method according to claim 5, wherein: Based on μ (0) and G(x) to calculate μ (1) include: Calculate μ1 (1) = E(x1|μ2 (0) ,μ3 (0) ,...,μ M (0) ), where E(x1|μ2 (0) ,μ3 (0) ,...,μ M (0) ) is given μ2 (0) ,μ3 (0) ,...,μ M (0) and The expected value of the first clipping sample in the case of (1) = E(x2|μ1 (1) ,μ3 (0) ,...,μ M (0) ), where E(x2|μ1 (1) ,μ3 (0) ,...,μ M (0) ) is given μ1 (1) ,μ3 (0) ,...,μ M (0) and The expected value of the second clipped sample in this case.

7. The method according to claim 6, wherein: Modifying r by replacing each clipped sample in r with its corresponding desired value involves: If it is determined that ||μ (1) -μ (0) || is less than δ or J is equal to 1, then the first clipping sample value in r is replaced by μ1 (1) , and replace the second clipping sample value in r with μ2 (1) .

8. A receiver apparatus (100) for reconstructing clipped samples, the receiver apparatus (100) having N antennas, wherein N>1, and the receiver apparatus (100) being configured to: a received signal vector y, where y is an N x 1 vector having N elements and each element of y is a received signal from a different one of the N antennas; Sample each of the N elements of y to produce a sample set; quantizing each sample in the sample set to generate a quantized received vector r, wherein quantizing each sample in the sample set comprises clipping at least M samples of the samples, wherein M>0, such that r includes M clipped samples; obtaining information representing the clipped samples and the unclipped samples; The probability density function of y and the information representing the clipped samples and the unclipped samples are used to obtain the probability density function G(x), where x=(x1,...,x M ) represents the unknown value before being clipped to said M clipping samples of r; and Modifying r by replacing each clipped sample value in r with an expected value corresponding to the clipped sample value, wherein the expected value is based on G(x), thereby producing a reconstructed received signal.

9. The receiver apparatus according to claim 8, further comprising: The reconstructed received signal is decoded.

10. The receiver device according to claim 8 or 9, further comprising: Training data is received, and a probability density function of y is obtained using the received training data.

11. The receiver device according to claim 8 or 9, further comprising: Determine the corresponding expected value.

12. The receiver device according to claim 11, wherein Determining the corresponding expected value includes: Initialize the first vector μ as the vector of expected values (0) , where μ (0) is the expected value μ1 corresponding to the M clipping samples of r (0) , μ2 (0) , μ3 (0) ,...,μ M (0) The vector of components; Define the first termination condition δ and the second termination condition J; Based on μ (0) and G(x) to calculate the second vector μ (1) ;as well as Determine whether ||μ (1) -μ (0) || is less than δ or determines whether J is equal to 1.

13. The receiver device according to claim 12, wherein: Based on μ (0) and G(x) to calculate μ (1) include: Calculate μ1 (1) = E(x1|μ2 (0) ,μ3 (0) ,...,μ M (0) ), where E(x1|μ2 (0) ,μ3 (0) ,...,μ M (0) ) is given μ2 (0) ,μ3 (0) ,...,μ M (0) and The expected value of the first clipping sample in the case of (1) = E(x2|μ1 (1) ,μ3 (0) ,...,μ M (0) ), where E(x2|μ1 (1) ,μ3 (0) ,...,μ M (0) ) is given μ1 (1) ,μ3 (0) ,...,μ M (0) and The expected value of the second clipped sample in this case.

14. The receiver device according to claim 13, wherein: Modifying r by replacing each clipped sample in r with its corresponding desired value involves: If it is determined that ||μ (1) -μ (0) || is less than δ or J is equal to 1, then the first clipping sample value in r is replaced by μ1 (1) , and replace the second clipping sample value in r with μ2 (1) .

15. A computer-readable storage medium containing a computer program for reconstructing clipped samples and comprising instructions which, when executed by processing circuitry (602) of a receiver apparatus (600) having N antennas, cause the receiver apparatus to: a received signal vector y, where y is an N x 1 vector having N elements and each element of y is a received signal from a different one of the N antennas, where N>1; Sample each of the N elements of y to produce a sample set; quantizing each sample in the sample set to generate a quantized received vector r, wherein quantizing each sample in the sample set comprises clipping at least M samples of the samples, wherein M>0, such that r includes M clipped samples; obtaining information representing the clipped samples and the unclipped samples; The probability density function of y and the information representing the clipped samples and the unclipped samples are used to obtain the probability density function G(x), where x=(x1,...,x M ) represents the unknown value before being clipped to said M clipping samples of r; and Modifying r by replacing each clipped sample value in r with an expected value corresponding to the clipped sample value, wherein the expected value is based on G(x), thereby producing a reconstructed received signal.

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