FIELD

The present invention relates to the field of digital communications in overloaded channels.

BACKGROUND

It is estimated that by 2030, over 100 billion wireless devices will be interconnected through emerging networks and paradigms such as the Internet of Things (loT), fifth generation (5G) cellular radio, and its successors. This future panorama implies a remarkable increase in device density, with a consequent surge in competition for resources. Therefore, unlike the preceding third generation (3G) and fourth generation (4G) systems, in which spreading code overloading and carrier aggregation (CA) were add-on features aiming at moderately increasing user or channel capacity, future wireless systems will be characterized by nonorthogonal access with significant resource overloading.

The expressions “resource overloading” or “overloaded communication channel” typically refers to a communication channel that is concurrently used by a number of users, or transmitters, T whose number N _T is larger than the number NR of resources R. At a receiver the multiplicity of transmitted signals will appear as one superimposed signal. The channel may also be overloaded by a single transmitter that transmits a superposition of symbols and thereby goes beyond the available channel resources in a “traditional” orthogonal transmission scheme. The “overloading” thus occurs in comparison to schemes, in which a single transmitter has exclusive access to the channel, e.g., during a time slot or the like, as found in orthogonal transmission schemes. Overloaded channels may be found, e.g., in wireless communication systems using Non-Orthogonal Multiple Access (NOMA) and underdetermined Multiple-Input Multiple-Output (MIMO) channels. Figures 1 and 2 illustrate basic properties of orthogonal multiple access and non-orthogonal multiple access, respectively. Figure 1 shows one exemplary embodiment of the ordered access of transmit resources to channels of a shared transmission medium, e.g., in a wireless communication system. The available frequency band is split into a number of channels. A single channel or a combination of contiguous or non-contiguous channels may be used by any one transmitter at a time. Different transmitters, indicated by the different hashing patterns, may transmit in discrete time slots or in a number of subsequent timeslots and may change the channels or combination of channels in which they transmit for each transmission. Note that, as shown in figure 1 , any transmitter may use one channel resource over a longer period of time, while another transmitter may use two or more channel resources simultaneously, and yet another transmitter may to both, using two or more channel resources over a longer period of time. In any case, only one transmitter uses any channel resource or combination thereof at a time, and it is relatively easy to detect and decode signals from each transmitter.

Figure 2a shows the same frequency band as shown in figure 1 , but there may not always be a temporary exclusive assignment of one or more individual channels to a transmitter. Rather, at least a portion of the frequency band may concurrently be used by a plurality of transmitters, and it is much more difficult to detect and decode signals from individual transmitters. Again, different hashing patterns indicate different transmitters, and the circled portions indicate where wo or more transmitters concurrently use a resource. While, beginning from the left, at first three transmitters use temporary exclusive channel resources in an orthogonal manner, in the next moment two transmitters transmit in channels that partially overlap. The transmitter represented by the horizontal hashing pattern has exclusive access to the channel shown at the bottom of the figure, while the next three channels used by this transmitter are also used by another transmitter, represented by diagonal hashing pattern in the dashed-line oval. The superposition is indicated by the diagonally crossed hashing pattern. A similar situation occurs in the following moment, where each of two transmitters exclusively uses two channel resources, while both share a third one. It is to be noted that more than two transmitters may at least temporarily share some or all of the channel resources each of them uses. These situations may be called partial-overloading, or partial-NOMA.

In a different representation, figure 2b shows the same frequency band as figure 2a. Since there is no clear temporary exclusive assignment of one or more individual channels to a transmitter, and at least a portion of the frequency band is at least temporarily concurrently used by a plurality of transmitters, the difficulty to detect and decode signals from individual transmitters is indicated by the grey filling pattern that does not allow for identifying any single transmitter. In other words, all transmitters use all channels.

Signals from some transmitters may be transmitted using higher power than others and may consequently be received with a higher signal amplitude, but this may depend on the distance between transmitter and receiver. Figures 2a and 2b may help understanding the situation found in non-orthogonal multiple access environments.

One of the main challenges of such overloaded systems is detection at the receiver, since the bit error rate (BER) performances of well-known linear detection methods, such as zero-forcing (ZF) and minimum mean square error (MMSE), are far below that of maximum likelihood (ML) detection, which is a preferred choice for detecting signals in overloaded communication channels. ML detection methods determine the Euclidian distances, for each transmitter, between the received signal vector and signal vectors corresponding to each of the symbols from a predetermined set of symbols that might have been transmitted, and thus allow for estimating transmitted symbols under such challenging conditions. The symbol whose vector has the smallest distance to the received signal’s vector is selected as estimated transmitted symbol. It is obvious, however, that ML detection does not scale very well with larger sets of symbols and larger numbers of transmitters, since the number of calculations that need to be performed for large sets in a discrete domain increases exponentially. In order to circumvent this issue, several signal detection methods based on sphere decoding have been proposed in the past, e.g. by C. Qian, J. Wu, Y. R. Zheng, and Z. Wang in “Two-stage list sphere decoding for under-determined multiple-input multiple-output systems," IEEE Transactions on Wireless Communication, vol. 12, no. 12, pp. 6476-6487, 2013 and by R. Hayakawa, K. Hayashi, and M. Kaneko in “An overloaded MIMO signal detection scheme with slab decoding and lattice reduction," Proceedings APCC, Kyoto, Japan, Oct. 2015, pp. 1-5, which illustrate its capability of asymptotically reaching the performance of ML detection at lower complexity. However, the complexity of the known methods grows exponentially with the size of transmit signal dimensions, i.e., the number of users, thus preventing application to large-scale systems.

R. Hayakawa and K. Hayashi, in “Convex optimization-based signal detection for massive overloaded MIMO systems," IEEE Transactions on Wireless Communication, vol. 16, no. 11 , pp. 7080-7091 , Nov. 2017, propose a low-complexity signal detector for large overloaded MIMO systems for addressing the scalability issue found in the previous solutions. This low complexity signal detector is referred to as sum-of-absolute-value (SOAV) receiver, which relies on a combination of two different approaches: a) the regularization-based method proposed by A. Aïssa-EI-Bey, D. Pastor, S. M. A. Sbaï and Y. Fadlallah in “Sparsity-based recovery of finite alphabet solutions of underdetermined linear system," IEEE Transactions on Information Theory, vol. 61 , no. 4, pp. 2008-2018, 2015, and b) the proximal splitting method described by P. L. Combettes and J.-C. Pesquet in “Proximal splitting methods in signal processing," Fixed-point algorithms for inverse problems in science and engineering, pp. 185-212, 2011.

While the SOAV decoder was found to outperform other state-of-the-art schemes, in terms of superior BER performance with significantly lower complexity, a shortfall of SOAV is that the ℓo-norm regularization function employed to capture the discreteness of input signals is replaced by an ℓi-norm approximation, leaving potential for further improvement. It is, thus, an object of the present invention to provide an improved method for estimating transmit symbol vectors, particularly in overloaded communication channels.

SUMMARY

The inventors recognize that, since the symbols used in digital communications are ultimately transmitted as analogue signals in the analogue, i.e., continuous domain, and attenuation, intermodulation, distortion and all kinds of errors are unavoidably modifying the signals on their way from the transmitter through the analogue communication channel to the receiver, the “detection” of the transmitted symbol in the receiver remains foremost an “estimation” of the transmitted signal, irrespective of the method used and, as the signals are in most if not all cases represented by signal amplitude and signal phase, in particular to the estimation of the transmitted signal’s vector. However, in the context of the present specification the terms “detecting” and “estimating” are used interchangeably, unless a distinction therebetween is indicated by the respective context. Once an estimated transmitted signal’s vector is determined it is translated into an estimated transmitted symbol, and ultimately provided to a decoder that maps the estimated transmitted symbol to transmitted data.

In the context of the present specification and claims, a communication channel is characterized by a set or matrix of complex coefficients. The channel matrix may also be referred to by the capital letter H. The communication channel may be established in any suitable medium, e.g., a medium that carries electromagnetic, acoustic and/or light waves. It is assumed that the channel properties are perfectly known and constant during each transmission of a symbol, i.e., while the channel properties may vary over time, each symbol’s transmission experiences a constant channel.

The expression “symbol” refers to a member of a set of discrete symbols c _i , which form a constellation C of symbols or, more profane, an alphabet that is used for composing a transmission. A symbol represents one or more bits of data and represents the minimum amount of information that can be transmitted at a time in the system using constellation C. In the transmission channel a symbol may be represented by a combination of analogue states, e.g., an amplitude and a phase of a carrier wave. Amplitude and phase may, e.g., be referred to as a complex number or as ordinate values over an abscissa in the cartesian space, and may be treated as a vector. A vector whose elements are symbols taken from C is referred herein by the small letter s. Each transmitter may use the same constellation C for transmitting data. However, it is likewise possible that the transmitters use different constellations. It is assumed that the receiver has knowledge about the constellations used in the respective transmitters.

A convex domain is a domain in which any two points can be connected by a straight line that entirely stays within the domain, i.e., any point on the straight line is a point in the convex domain. The convex domain may have any dimensionality, and the inventors recognize that the idea of a straight line in a 4-or-more-dimensional domain may be difficult to visualise.

The terms “component” or “element” may be used synonymously throughout the following specification, notably when referring to vectors.

As was mentioned before, in typical ML detection schemes one constraint is the strong focus on the discrete signal vectors for symbols c _i of the constellation C, which prevents using, e.g., known-effective fractional programming (FP) algorithms for finding the signal vector and thus the symbol having the minimum distance to the received signal’s vector. The strong focus is often expressed through performing individual calculations for symbols of the constellation C in equations describing the detection. Some schemes try to enable the use of FP algorithms for estimating the most likely transmitted symbol and replace the individual calculations for symbols by describing the discreteness of the constellation C through a ℓi-norm that is continuous and can thus be subjected to FP algorithms for finding minima. However, using the ℓi-norm introduces a fair amount of estimation errors, which is generally undesired. The detection scheme for overloaded systems of the method presented herein does not rely on the loose relaxation of the ℓo-norm by resorting to a ℓi-norm. Rather, in the inventive method a function that is a tight ℓo-norm approximation is employed, which allows utilizing an efficient and robust FP framework for the optimization of non-convex fractional objectives, which is less computationally demanding, and shown via simulations to outperform SOAV.

In the following, the theoretical base of the inventive detection scheme will be explained with reference to an exemplary underdetermined wireless system with N _T transmitters and N _R < N _T receive resources, such that the overloading ratio of the system is given and the received signal, after well-known signal realization, can be modelled as y = Hs + n,

(1 ) is a transmit symbol vector with each element sampled from a constellation set C of cardinality 2 ^b , with b denoting the number of bits per symbol, is a circular symmetric complex additive white Gaussian noise (AWGN) vector with zero mean and covariance matrix and describes a flat fading channel matrix between the transmitter and receiver sides.

In conventional detectors a maximum likelihood (ML) detection may be used for estimating a transmit signal vector s ^ML for a received signal y. The ML detection requires determining the distances between the received signal vector y and each of the symbol vectors s of the symbols c _i of the constellation C. The number of calculations exponentially increases with the number N _T of transmitters T.

The discreteness of the target set to the ML function prevents using effective FP algorithms, which are known to be effective for finding minima in functions having continuous input, for estimating the transmit signal vector ŝ for a received signal y. In accordance with the present invention the discrete target set for the ML function is first transformed into a sufficiently similar continuous function, which is open to solving through FP algorithms.

To this end, the alternative representation of the discrete ML function

(2a) s.t.

(2b) is first transformed into a penalized mixed ℓo- ℓ2 minimization problem that retains ML-like performance for the approximation of the constraint: where w and l are weighting parameters. The notation indicates that the approximation still has the potential to achieve near-ML performance, as long as the weights w and l are properly optimized. T is the number of transmitters and may also be referred to by N _T .

In order to address the intractable non-convexity of the ℓo-norm without resorting to the ℓi-norm, the ℓo-norm is replaced with the asymptotically tight expression:

(4) where x is an arbitrary sparse vector of length T.

The tight approximation of the ℓo-norm is then used as a substitute of the ℓo-norm in the penalized mixed ℓo- ℓ2 minimization problem, and a slack variable t _ij with the constraint is introduced, yielding

(5a)

(5b) with now a « 1 .

Since the ratios in equation (5a) possess a concave-over-convex structure due to the convex non-negative nominator and concave (linear) positive denominator, the required condition for convergence of the quadratic transform (QT) is satisfied, as has been shown by K. Shen and W. Yu in “Fractional programming for communication systems - Part I: Power control and beamforming,” IEEE Trans. Signal Process., vol. 66, no. 10, pp. 2616-2630, May 2018, such that equation (5a) can be reformulated into the following convex problem:

(6a)

(6b) ^where Thanks to the convergence of β _ij the equation can be solved through FP by iteratively updating β _ij and solving the equation for a given β _ij . The equation obtained by transforming the initial non-convex optimization problem into a convex optimization problem can be efficiently solved using known algorithms, such as augmented Lagrangian methods.

Thus, a computer-implemented method in accordance with the present invention of estimating transmit symbol vectors ŝ transmitted in an overloaded communication channel that is characterized by a channel matrix H of complex coefficients includes receiving, in a receiver R, a signal represented by a received signal vector y. The received signal vector y corresponds to a superposition of signals representing transmitted symbol vectors s selected from a constellation C of symbols c/that are transmitted from one or more transmitters T, plus any distortion and noise added by the channel.

In case of more than one transmitter the transmitters T are temporally synchronized, i.e., a common time base is assumed between the transmitters T and the receiver R, such that the receiver R receives transmissions of symbols from different transmitters T substantially simultaneously, e.g., within a predetermined time window. The symbols being received simultaneously or within a predetermined time window means that all temporally synchronized transmitted symbols are received at the receiver R before subsequent symbols are received, assuming that a transmitter T transmits a sequence of symbols one by one. This may include settings in which transmitters T adjust the start time of their transmission such that a propagation delay, which depends on the distance between transmitter T and receiver R, is compensated for. This may also include that a time gap is provided between transmitting subsequent symbols.

The method further comprises defining a convex search space including at least the components of the received signal vector y and of the transmit symbol vectors s for all symbols c _i of the constellation C. Further, continuous first and second functions f ₁ and f ₂ are defined in the search space. In this context, defining may include selecting factors or ranges of variables or the like for or in an otherwise predetermined function.

The continuous first function T is a function of the received signal vector y and the channel characteristics H and has a global minimum where the product of an input vector s from the search space and the channel matrix H equals the received signal vector y.

The continuous second function f ₂ is a function of input vectors s from the search space and has a significant low value for each of the transmit symbol vectors s of the symbols c _i of the constellation C.

In accordance with the invention the first function T and the second function h are combined into a third function f ₃ by weighted adding, and a fractional programming algorithm FP is applied to the third function f ₃ , targeted to finding an input vector s that minimizes the third function f ₃ . In other words, s is the optimal solution or outcome of applying the FP algorithm to the third function f ₃ for which the third function f ₃ has a minimum.

Once an input vector ŝ that minimizes the third function f ₃ is found, a mapping rule is applied thereto that translates the input vector ŝ into an estimated transmit vector ŝ _c , in which the index “C” indicates that every single component belongs to the constellation C. In other words, if the vector has two components, A and B, each of the components A and B of the input vector ŝ that minimizes the third function f ₃ can have any value in the search space. These values are translated into values A’ and B’ of the estimated transmit vector ŝ _c , each of which can only have a value that occurs in any one of the transmit symbol vectors s for the symbols a of the constellation C. The components may be mapped separately, e.g., by selecting the closest value of a corresponding component of any of transmit symbol vectors s of the symbols c _i of the constellation C.

After the mapping the estimated transmit symbol vector ŝ _c is output to a decoder to obtain the data bits of the transmitted message. In one or more embodiments the second function h has a tuneable factor that determines the gradient of the function in the vicinity of the significant low value at each of the vectors of the symbols of the constellation. The tuneable factor may help the FP algorithm to converge faster and/or to skip local minima that may be farther away from an optimal or at least better solution.

In some embodiments the tuneable factor may be different for different symbols of the constellation. For example, the gradient in the vicinity of a vector for a symbol that is farther away from the global minimum of the first function T may be very steep, but may be so only very close to the significant low value. Depending on the FP algorithm and the start value used this may help skipping local minima located at a greater distance from the global minimum of the first function T . On the other hand, the gradient in the vicinity of a vector for a symbol that is located close to the global minimum of the first function T may be rather shallow at a certain distance to the significant low value and growing steeper as the distance shrinks. Depending on the FP algorithm used this may help the function to quickly converge to a significant low value.

In some embodiments the first function T is monotonously increasing from the global minimum. The first function may be considered a coarse guidance function for the FP algorithm, which helps the FP algorithm to converge. It is, thus, advantageous if the first function itself does not have any local minima.

A receiver of a communication system has a processor, volatile and/or non-volatile memory and at least one interface adapted to receive a signal in a communication channel. The non-volatile memory may store computer program instructions which, when executed by the microprocessor, configure the receiver to implement one or more embodiments of the method in accordance with the invention. The volatile memory may store parameters and other data during operation. The processor may be called one of a controller, a microcontroller, a microprocessor, a microcomputer and the like. And, the processor may be implemented using hardware, firmware, software and/or any combinations thereof. In the implementation by hardware, the processor may be provided with such a device configured to implement the present invention as ASICs (application specific integrated circuits), DSPs (digital signal processors), DSPDs (digital signal processing devices), PLDs (programmable logic devices), FPGAs (field programmable gate arrays), and the like.

Meanwhile, in case of implementing the embodiments of the present invention using firmware or software, the firmware or software may be configured to include modules, procedures, and/or functions for performing the above-explained functions or operations of the present invention. And, the firmware or software configured to implement the present invention is loaded in the processor or saved in the memory to be driven by the processor.

The present method addresses difficulties in applying effective FP algorithms for estimating candidates of transmitted symbol vectors arising from the discrete nature of the constellation by transforming the discrete constraint present in the known ML method for determining the Euclidian distance between the received signal’s vector and the vectors of symbols of the constellation into a first function in a convex domain that presents significant low values for the vectors of symbols of the constellation. A minimum of the function in the convex domain can be found by applying known FP methods or algorithms that are more effective for finding a good estimate of a transmitted signal’s vector than brute-force calculations. A second continuous function in the convex domain is added to the first function that penalizes estimation results with increasing distance from the received signal’s vector.

While the invention has been described hereinbefore for detecting superimposed signals from transmitters that are all using the same constellation C it is also applicable to situations in which different transmitters use different constellations CT, i.e., if the symbols of a constellation C are considered letters of an alphabet, each transmitter may use a different alphabet.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention will be further explained with reference to the drawings in which Fig. 1 shows a simplified schematic representation of orthogonal multiple access to a shared medium,

Fig. 2 shows a simplified schematic representation of non-orthogonal access to a shared medium,

Fig. 3 shows an exemplary generalized block diagram of a transmitter and a receiver that communicate over a communication channel,

Fig. 4 shows an exemplary flow diagram of method steps implementing embodiments of the present invention,

Fig. 5 shows details of method steps of the present invention,

Fig. 6 shows exemplary and basic examples of a constellation, a transmitted and a received signal, and

Fig. 7 shows a simplified exemplary graphical representation of the third function determined in accordance with the present invention, that can be effectively solved using fractional programming.

In the drawings identical or similar elements may be referenced by the same reference designators.

DETAILED DESCRIPTION OF EMBODIMENTS

Figure 1 and 2 have been discussed further above and are not revisited here.

Figure 3 shows an exemplary generalized block diagram of a transmitter T and a receiver R that communicate over a communication channel 208. Transmitter T may include, inter alia, a source 202 of digital data that is to be transmitted. Source 202 provides the bits of the digital data to an encoder 204, which forwards the data bits encoded into symbols to a modulator 206. Modulator 206 transmits the modulated data into the communication channel 208, e.g. via one or more antennas or any other kind of signal emitter (not shown). The modulation may for example be a Quadrature Amplitude Modulation (QAM), in which symbols to be transmitted are represented by an amplitude and a phase of a transmitted signal.

Channel 208 may be a wireless channel. However, the generalized block diagram is valid for any type of channel, wired or wireless. In the context of the present invention the medium is a shared medium, i.e., multiple transmitters and receivers access the same medium and, more particularly, the channel is shared by multiple transmitters and receivers.

Receiver R receives the signal through communication channel 208, e.g. via one or more antennas or any other kind of signal receiver (not shown). Communication channel 208 may have introduced noise to the transmitted signal, and amplitude and phase of the signal may have been distorted by the channel. The distortion may be compensated for by an equalizer provided in the receiver (not shown) that is controlled based upon channel characteristics that may be obtained, e.g., through analysing pilot symbols with known properties transmitted over the communication channel. Likewise, noise may be reduced or removed by a filter in the receiver (not shown). A signal detector 210 receives the signal from the channel and tries to estimate, from the received signal, which signal had been transmitted into the channel. Signal detector 210 forwards the estimated signal to a decoder 212 that decodes the estimated signal into an estimated symbol. If the decoding produces a symbol that could probably have been transmitted it is forwarded to a de-mapper 214, which outputs the bit estimates corresponding to the estimated transmit signal and the corresponding estimated symbol, e.g., to a microprocessor 216 for further processing. Otherwise, if the decoding does not produce a symbol that is likely to have been transmitted, the unsuccessful attempt to decode the estimated signal into a probable symbol is fed back to the signal detector for repeating the signal estimation with different parameters. The processing of the data in the modulator of the transmitter and of the demodulator in the receiver are complementary to each other.

While the transmitter T and receiver R of figure 3 appear generally known, the receiver R, and more particularly the signal detector 210 and decoder 212 of the receiver in accordance with the invention are adapted to execute the inventive method described hereinafter with reference to figure 4 and thus operate different than known signal detectors.

Figure 4 shows an exemplary flow diagram of method steps implementing embodiments of the present invention. In step 102 a signal is received in an overloaded communication channel. The signal corresponds to a superposition of signals representing transmitted symbols selected from a constellation C of symbols c/ and transmitted from one or more transmitters T. In step 104 a search space is defined in a convex domain including at least the components of the received signal vector y and of transmit symbol vectors s for all symbols a of the constellation C. In step 106 a continuous first function fi is defined, which is a function of the received signal vector y and the channel characteristics H. The first function T has a global minimum where the product of an input vector s from the search space and the channel matrix H equals the received signal vector y. Further, in step 108 a continuous second function h is defined in the search space, which is a function of input vectors s from the search space. The second function h has a significant low value for each of the transmit symbol vectors s of the symbols a of the constellation C. It is to be noted that steps 104, 106 and 108 need not be executed in the sequence shown in the figure, but may also be executed more or less simultaneously, or in a different sequence. The first and second functions T, h are combined to a third continuous function h in step 110 through weighted adding. Once the third function f ₃ is determined a fractional programming algorithm is applied thereto in step 112 that is targeted to finding an input vector s that minimizes the third function f ₃ . The input vector s that is the result output from the fractional programming algorithm is translated, in step 114, into an estimated transmit vector Sc, in which every single component has a value from the list of possible values of corresponding components of transmit symbol vectors s of the symbols a of the constellation C. The translation may include selecting the value from the list that is nearest to the estimated value. The estimated transmit vector Sc is then output in step 116 to a decoder for decoding into an estimated transmitted symbol cfrom the constellation C. The transmitted symbol c may be further processed into one or more bits of the data that was transmitted, step 118.

Figure 5 shows details of the method steps of the present invention executed for finding an input vector s that minimizes the third function f3, in particular the function according to equation 6 described further above. In step 112-1 the fractional programming is initialised with a start value for the estimated transmit signal’s vector s _start and /? _£ is determined in step 112-2 for the start value of the estimated transmit vector s start- Then, a new candidate for s is derived in step 112-3 by solving the equation for the value determined in step 112-2. If the solution does not converge, “no”-branch of step 112-4, the value is determined based on the new candidate s derived in step 112-3 and the equation-solving process is repeated. If the solution converges, “yes”-branch of step 112-4, s is forwarded to step 114 of figure 4, for mapping the estimated transmit vector Sc whose components assume values from vectors s of symbols C from the constellation C.

Figure 6 a) shows exemplary and very basic examples of symbols d , c2, c3 and c4 from a constellation C. The symbols d , c2, c3 and c4 may represent symbols of a QAM-modulation. Figure 6 b) shows a symbol that was actually transmitted over a channel, in this case symbol c2. Figure 6 c) shows the signal that was actually received at a receiver. Due to some distortion and noise in the channel the received signal does not lie exactly at the amplitude and phase of symbol c2 that was sent. A maximum likelihood detector determines the distances between the received signal and each of the symbols from the constellation and would select that one as estimated symbol that is closest to the received signal. In the very simple example, this would be symbol c2. This process requires performing calculations for all discrete pairs of received signal and symbols from the constellation, and may result in a number of calculations that exponentially increases with the number of symbols in the constellation and the number of transmitters that possibly transmitted the signal.

Figure 7 shows a simplified exemplary graphical representation of the third function determined in accordance with the present invention that can be effectively solved using fractional programming. The graphical representation is based on the same constellation as presented in figure 6 a), and it is assumed that the same signal c2 was transmitted. The bottom surface of the three-dimensional space represents the convex search space for amplitudes and phases of signal vectors. The vertical dimension represents the values for the third function. Since the search space is convex, the third function has values for any combination of amplitude and phase, even though only 4 discrete symbols d , c2, c3 and c4 are actually in the constellation. The surface having a shape of an inverted cone represents the results of the continuous first function over the convex search space, and has a global minimum at the location of the received signal. The 4 spikes protruding downwards from the cone-shaped surface represent the continuous second function that has significant low values at the phases and amplitudes of the symbols from the constellation. The first and second function have been combined into the third function, which is still continuous and which can now be subjected to a fractional programming algorithm for finding the amplitude and phase that minimizes the third function. It is to be borne in mind that this representation is extremely simplified, but it is believed to help understanding the invention.