A computing device and a method thereof

Abstract

Embodiments of the invention relate to a computing device (100) and a communication device comprising such a computing device. The computing device (100) is configured to obtain an observation vector y comprising m real values obtained from m measurements; obtain a complex measurement matrix A; and recover x based on y and A according to the equation: y = |Ax|2, where x is an unknown sparse complex vector and | | is the modulus operator, and wherein each element in A is a rational function of a set of measurement variables ϕ comprising at least one pole. Furthermore, the invention also relates to a corresponding method.

Description

[0001]A COMPUTING DEVICE AND A METHOD THEREOF TECHNICAL FIELD Embodiments of the invention relate to a computing device. Furthermore, embodiments of the invention also relate to a corresponding method and a communication device comprising such a computing device. BACKGROUND A classical problem is to recover signals from their power spectral density. This problem is often referred to in literature as the phase retrieval problem and is of principal importance inmany fields of applied sciences such as in physics, astronomy, mathematics, andcommunications. More specifically, the measurements of the signal are obtained through application of a known linear measurement operator (a matrix) followed by outputting thesquared magnitude of each measurement.Throughout the years, dating as way back as 1950s, many algorithms for reconstruction have been proposed such as GAMP-like algorithms, lifting procedure algorithms, and many more.Beside proposal of recovery algorithms, another area of interest is the design of linearoperators (matrices) that are used for measurement. One result in this direction is the classicmatrix construction, from which one can provably recover any unknown ^ dimensionalcomplex signal (up to a phase) from 4^ − 4 measurements. Another result shows that generic(4^ − 4) × ^ matrices also provide unique recovery up to a phase.SUMMARY An objective of embodiments of the invention is to provide a solution which mitigates or solves the drawbacks and problems of conventional solutions. Another objective of embodiments of the invention is to provide a solution having lower computational complexity compared to conventional solutions. The above and further objectives are solved by the subject matter of the independent claims. Further embodiments of the invention can be found in the dependent claims. According to a first aspect of the invention, the above mentioned and other objectives are achieved with a computing device configured to: obtain an observation vector ^ comprising ^ real values obtained from ^measurements;obtain a complex measurement matrix ^; andrecover ^ based on ^ and ^ according to the equation:^ = |^^|^where ^ is an unknown sparse complex vector and | | is the modulus operator, and whereineach element in ^ is a rational function of a set of measurement variables ^ comprising atleast one pole. An advantage of the computing device according to the first aspect is that it enables recoveryof ^ with as few as ^(^) measurements and with ^(^) recovery complexity where ^(. ) standsfor Big ^ notation.In an implementation form of a computing device according to the first aspect, ^ is an ^-dimensional unknown sparse complex vector with ^ non-zero elements, where ^ ≥ 4^ − 1.An advantage with this implementation form is that it enables recovery of ^ with as few as ^(^)measurements and with ^(^) recovery complexity.In an implementation form of a computing device according to the first aspect, ^ ≥ 4^ − 1 or^ ≥ 8^ − 3.An advantage with this implementation form is that it enables recovery of ^ with ^measurements and with ^(^) recovery complexity if ^ ≥ 8^ − 3. Moreover, if 4^ − 1 ≤ ^ <8^ − 3, the recovery complexity is at most ^(^^2^) for some finite integer ^ > 0.In an implementation form of a computing device according to the first aspect, the at least one pole has a multiplicity of at least 1.An advantage with this implementation form is that it enables recovery of ^ with as few as ^(^)measurements and with ^(^) recovery complexity.In an implementation form of a computing device according to the first aspect, the at least one pole has a multiplicity 1.An advantage with this implementation form is that it enables recovery of ^ with as few as ^(^)measurements and with ^(^) recovery complexity. In an implementation form of a computing device according to the first aspect, each pole is associated with a non-zero element in ^. An advantage with this implementation form is that it is possible to retrieve the support of ^, i.e., positions of non-zero elements, by recovering the poles. In an implementation form of a computing device according to the first aspect, the computing device is configured to compute: to recover ^, where ^^^^ is at least one additional measurement, is a vector comprisingat least one non-zero element, and ^ is the transpose operator.An advantage with this implementation form is that, combined with the previous ^measurements stated above, it enables recovery of ^ up to a phase.In an implementation form of a computing device according to the first aspect, random vector. An advantage with this implementation form is that the correct signal will, with probability 1, produce unique measurement value(s) from which it can be recovered. In an implementation form of a computing device according to the first aspect, the real parts and the imaginary parts of the elements of the random vector are derived from a real valued continuous distribution. An advantage with this implementation form is that the resulting random vector will be able toprovide a unique measurement value for the correct signal.In an implementation form of a computing device according to the first aspect, the real valued continuous distribution is any of: a Gaussian distribution, a uniform distribution, and an exponential distribution. In an implementation form of a computing device according to the first aspect, at least onemeasurement variable in the set of measurements variables ^ takes on different measurementvalues for different measurements. An advantage with this implementation form is to make sure that different measurements correspond to changing values of variables in ^. In an implementation form of a computing device according to the first aspect, elements in the same column in ^ have the same set of poles and their associatedmultiplicities, and elements in the same row in ^ have the same set of measurement values.An advantage with this implementation form is that it enables recovery of ^ with as few as ^(^)measurements and with ^(^) recovery complexity.In an implementation form of a computing device according to the first aspect, ^ is expressedas a product of two Fourier transform matrices.An advantage with this implementation form is that this is a practical construction of ^ thatenables recovery of ^ with as few as ^(^) measurements and with ^(^) recovery complexity.In an implementation form of a computing device according to the first aspect, ^ is expressedas: ^(^, ^) = ^(^)^^(^)where ^ is a Fourier transform matrix, ^ is a first set of real numbers associated with the setof measurement variables ^, and ^ is a second set of real numbers associated with the atleast one pole. An advantage with this implementation form is that this is an explicit, practical construction of^ that enables recovery of ^ with as few as ^(^) measurements and with ^(^) recoverycomplexity. In an implementation form of a computing device according to the first aspect, the Fouriertransform matrix ^ > 0, is defined as: where ^ is the dimension of ^ with ^ non-zero elements, and ^ ≥ 4^ − 1. According to a second aspect of the invention, the above mentioned and other objectives are achieved with a communication device comprising a computing device according to embodiments of the invention. In an implementation form of a communication device according to the second aspect, the communication device is configured to: measure ^ to obtain the observation vector ^.According to a third aspect of the invention, the above mentioned and other objectives are achieved with a method for a computing device, the method comprises: obtaining an observation vector ^ comprising ^ real values obtained from ^measurements; obtaining a complex measurement matrix ^; andrecovering ^ based on ^ and ^ according to the equation:^ = |^^|^where ^ is an unknown sparse complex vector and | | is the modulus operator, and whereineach element in ^ is a rational function of a set of measurement variables ^ comprising atleast one pole. The method according to the third aspect can be extended into implementation forms corresponding to the implementation forms of the computing device according to the first aspect. Hence, an implementation form of the method comprises the feature(s) of the corresponding implementation form of the computing device. The advantages of the methods according to the third aspect are the same as those for the corresponding implementation forms of the computing device according to the first aspect. Embodiments of the invention also relate to a computer program, characterized in program code, which when run by at least one processor causes the at least one processor to execute any method according to embodiments of the invention. Further, embodiments of the invention also relate to a computer program product comprising a computer readable medium and the mentioned computer program, wherein the computer program is included in the computer readable medium, and may comprises one or more from the group of: read-only memory (ROM), programmable ROM (PROM), erasable PROM (EPROM), flash memory, electrically erasable PROM (EEPROM), hard disk drive, etc. Further applications and advantages of embodiments of the invention will be apparent fromthe following detailed description.BRIEF DESCRIPTION OF THE DRAWINGS The appended drawings are intended to clarify and explain different embodiments of the invention, in which: ^Fig.1 shows a computing device according to an embodiment of the invention;^ Fig. 2 shows a flow chart of a method for a computing device according to anembodiment of the invention;^ Fig.3 shows a communication device according to an embodiment of the invention;^ Fig.4 shows a communication system according to an embodiment of the invention;^ Fig.5 illustrates measurements on beam projections in a communication system; and^ Fig.6 and 7 shows a flow chart of Algorithm 1 and 2, respectively.DETAILED DESCRIPTION In the present disclosure the so-called sparse phase retrieval problem is solved, i.e., the equation: ^= |^^|^ (1)where ^ is a complex ^ × ^ measurement matrix that we have the freedom to construct, ^ isa ^ × 1 unknown ^-sparse complex vector meaning that it only has ^ non-zero elements. Atask is to recover ^ up to a global phase ^^^, i.e., estimate ^^^^ for any ^, which is what wefrom now on define as recovery, from the knowledge of ^ and ^. The recovery should be donewith as few measurements ^ as possible.As aforementioned, this problem arises in many scientific fields. As in many applications, thesignal dimension ^ is very large, the required number of measurements will thus also beprohibitively large. Instead, if some prior knowledge exists about the signal, there is a possibilityto reduce the number of measurements. One assumption is so called sparsity – assuming thatthe signal has only ^ non-zero elements, with ^ much smaller than ^. In this case, we aredealing with the sparse phase retrieval. Despite the vast progress in phase retrieval and sparse phase retrieval, there are several openproblems left. In this disclosure, we focus on an open issue in sparse phase retrieval.Specifically, we look at the open issue of constructing a measurement matrix which enablesexact recovery with at most ^ ^^log ^^ ^^^ measurements and at most ^(^) recoverycomplexity. The disclosure thus proposes a measurement matrix construction and an associatedalgorithm to exactly recover ^ in absence of noise with ^(^) measurements and a polynomialrecovery complexity in ^. To our knowledge, this is the first algorithm achieving exact recoverywith ^(^) measurements and a polynomial recovery complexity in ^ . Current recoveryalgorithms only achieve probabilistic recovery with a probability strictly less than 1 for a finite number of measurements. Fig. 1 shows a computing device 100 according to an embodiment of the invention. In the embodiment shown in Fig.1, the first communication device 100 comprises a processor 102, and a memory 104. The processor 102 is coupled to the memory 104 by communication means 108 known in the art. The processor 102 may be referred to as one or more general-purpose central processing units (CPUs), one or more digital signal processors (DSPs), one or more application-specific integrated circuits (ASICs), one or more field programmable gate arrays (FPGAs), one or more programmable logic devices, one or more discrete gates, one or more transistor logic devices, one or more discrete hardware components, or one or more chipsets. The memory 104 may be a read-only memory, a random access memory (RAM), or a non-volatile RAM (NVRAM). The memory 104 and / or processor 102 may be implemented in separate chipsets or may be implemented in a common chipset. According to embodiments of the invention the computing device 100 is configured to obtainan observation vector ^ comprising ^ real values obtained from ^ measurements; obtain acomplex measurement matrix ^; and recover ^ based on ^ and ^ according to the equation:^ = |^^|^where ^ is an unknown sparse complex vector and | | is the modulus operator, and whereineach element in ^ is a rational function of a set of measurement variables ^ comprising atleast one pole.The complex measurement matrix ^ is known to the computing device 100 which may beunderstood as that ^ is programmed / implemented into the computing device 100 so that thecomputing device 100 applies ^ onto ^ to obtain the (magnitude only)observation / measurement vector ^ as above.Furthermore, in an embodiment of the invention, the computing device 100 comprises aprocessor and a memory having computer readable instructions stored thereon which, whenexecuted by the processor, cause the processor to: obtain an observation vector ^ comprising^ real values obtained from ^ measurements; obtain a complex measurement matrix ^; andrecover ^ based on ^ and ^ according to the equation:^ = |^^|^where ^ is an unknown sparse complex vector and | | is the modulus operator, and whereineach element in ^ is a rational function of a set of measurement variables ^ comprising atleast one pole. The computing device 100 may be a standalone device or integrated in another device such as in a communication device as will be described in the following disclosure. Thus, thecomputing device 100 may obtain the observation vector ^ and / or the complex measurementmatrix ^ from a memory, indirectly from measurements and observations performed by otherdevices or directly by own measurements. Fig.2 shows a flow chart of a corresponding method 200 which may be executed in computing device 100, such as the one shown in Fig.1. The method 200 comprises obtaining 202 anobservation vector ^ comprising ^ real values obtained from ^ measurements; obtaining 204a complex measurement matrix ^; and recovering 206 ^ based on ^ and ^ according to theequation: ^= |^^|^where ^ is an unknown sparse complex vector and | | is the modulus operator, and whereineach element in ^ is a rational function of a set of measurement variables ^ comprising atleast one pole. Fig.3 shows a communication device 300 according to an embodiment of the invention. In the embodiment shown in Fig.3, the second communication device 300 comprises a processor 302, a transceiver 304 and a memory 306. The processor 302 is coupled to the transceiver 304 and the memory 306 by communication means 308 known in the art. The wireless communication capability may be provided with an antenna or antenna array 110 coupled to the transceiver 304, while the wired communication capability may be provided with a wired communication interface 312 e.g., coupled to the transceiver 304. The processor 302 may be referred to as one or more general-purpose CPUs, one or more DSPs, one or more ASICs, one or more FPGAs, one or more programmable logic devices, one or more discrete gates, one or more transistor logic devices, one or more discrete hardware components, one or more chipsets. The memory 306 may be a read-only memory, a RAM, or a NVRAM. The transceiver 304 may be a transceiver circuit, a power controller, or an interface providing capability to communicate with other communication modules orcommunication devices. The transceiver 304, the memory 306 and / or the processor 302 maybe implemented in separate chipsets or may be implemented in a common chipset. That the communication device 300 is configured to perform certain actions can in this disclosure be understood to mean that the second communication device 300 comprises suitable means,such as e.g., the processor 302 and the transceiver 304, configured to perform the actions.Fig.4 shows a communication system 500 according to an embodiment of the invention. The communication device 300 may e.g., be a network access node of a RAN or a network node of a core network (CN) of a 5G new radio (NR) communication system. The network access node, such as a gNB, may be configured to communicate in the downlink (DL) and the uplink (UL) with a client device 300´ such as a UE. One instance of solving Eq. 1 that appears in telecommunications is inferring a signal from power measurements of different beam projections as illustrated in Fig.5. A gNB is transmitting a signal which is received in multiple beams and the UE measures the signal with different beam projections. If the received signal is a column vector, each beam becomes a row in the measurement matrix, and the squared magnitude of the resulting matrix vector product are the power measurements of the different beam projections. The reason for power measurements only and no phase information is because beam changes introduce unknown phase rotations which destroy any phase information about the signal.Thus, the communication device 300 may be configured to measure ^ to obtain theobservation vector ^, and an instance of solving Eq. (1) appears in telecommunications wheninferring a received signal ^ from power measurements of different beam projections, whichbecome the elements of ^. Each beam becomes a row in the measurement matrix ^. In embodiments the computing device 100 may be employed in other applications. For example, the computing device 100 may be used in physics, chemistry and astronomy for retrieving signals. The computing device 100 may also be used for solving mathematical problems.In this disclosure, the elements of the complex measurement matrix ^ may be chosen to berational functions of a variable that is set to different values for each measurement. Eachrational function may have at least one pole that is associated with a position in ^. Hence, torecover the unknown ^ positions in ^ that contain the non-zero elements, we need to recoverthe poles of these rational functions. We show that pole recovery of a summation of rational functions with single poles, that passesthrough the non-linear magnitude operator, can be accomplished in an exact andcomputationally simple way. Moreover, it turns out that valuable information about theelements of ^ is embedded in the zeroes of the rational function that is the summation of theabove said rational functions. As a result, we are able to propose a solution that performs exactrecovery of a ^-sparse ^. Moreover, we can achieve a polynomial recovery complexity in ^ withas low as 8^ − 2 measurements. If an exponential average recovery complexity in ^ is allowed,then we can lower the number of measurements to 4^. Another possibility is to apply 1-norm minimization methods for recovery. These are all significant improvements to conventional solutions for solving Eq. (1).Hence, it is proposed a novel complex measurement matrix ^ that allows exact recovery of ^in a relatively simple way. Each pole is associated with a non-zero element in ^ which is an ^-dimensional unknown sparse complex vector with ^ non-zero elements, where ^ ≥ 4^ − 1. Inembodiments, ^ ≥ 4^ − 1 or ^ ≥ 8^ − 3.The at least one pole has a multiplicity of at least 1, and in embodiments the at least one polehas a multiplicity 1. In its most general formulation, an element ^^,^ of ^ on row ^ and column^, where 0 ≤ ^ ≤ ^ − 1, 0 ≤ ^ ≤ ^ − 1, equals where ^^,^(^) is a rational function in ^with a set / vector of poles ^^, where in each pole in ^^ occurs with at least a multiplicity of 1.Note that from this construction, the elements ^:,^ on the same column ^ of ^ have the sameset of poles (and their multiplicities) while the elements ^^,: on the same row of ^ have thesame values ^^ of the measurement variables ^.When ^ is expressed as a product of two Fourier transform matrices, see below, then ^becomes a single variable ^ that takes on value ^^ for measurement ^ and ^^,^(^) has a singlepole ^^ with multiplicity 1. The pole ^^ is an encoding of position ^ in ^, i.e., recovering ^ = ^^tells us that there is a non-zero element in ^ on position ^ . The poles ^^ and values ofmeasurements variables ^^are chosen by the designer.The at least one measurement variable in the set of measurements variables ^ may take ondifferent measurement values for different measurements. Furthermore, in embodiments of the invention, one or more additional measurements may beneeded to recover ^ with higher precision. Thus, the computing device 100 may be configuredto compute:to recover ^, where ^ is a least one additional measurement, is a vectorat least one non-zero element, and ^ is the transpose operator. The number of additionalmeasurements ^ is at least 1, i.e., ^ ≥ 1. With these additional measurements, and themeasurements ^, it is possible to solve for a unique ^ (up to a global phase ^^^). The exactsolution steps are given by two exemplary algorithms below. Vector ^^^^^has some interesting properties which may be noted. Firstly, ^^^^^may be a random vector. Secondly, the real parts and the imaginary parts of the elements of the random vector ^^^^^are derived from a real valued continuous distribution. Thirdly, the real valued continuous distribution is any of: a Gaussian distribution, a uniform distribution, and an exponential distribution. These are exemplary distributions for which the resulting ^ ^ ^^^produces, with probability 1, distinct measurement values for each of the candidate solutionsfor ^ arising from just the measurements ^. This enables to identify the unique solution (up toa global phase from these candidates that produces the observed ^^^^.Moreover, in embodiments of the invention, ^ may be expressed as a product of two Fouriertransform matrices. Thus, ^ may in examples be expressed as:^(^, ^) = ^(^)^^(^)where ^ is a Fourier transform matrix, ^ is a first set of real numbers associated with the setof measurement variables ^, and ^ is a second set of real numbers associated with the atleast one pole.An example of ^ as described above is ^(^, ^) = ^(^)^^(^), with ^ = [^^, … , ^^^^] and ^ =[^ , … , where ^(^) is a Fourier matrix with frequencies ^ = [^^, … , ^^^^], for some ^0, i.e., ^^,^ = ^^^^^^ , 0 ≤ ^ ≤ ^ − 1, 0 ≤ ^ ≤ ^ − 1.For this construction, each ^^ in ^ is the pole of all the elements on column ^ of ^(^, ^) while^^ is the value of the (single) measurement variable ^ at measurement ^.Let ^ denote a random measurement vector with respect to ^. Hence, we take the following^ + 1 measurements ^^, … , ^^ to obtain where ^ = [^^, … , ^^^^]^. For the measurements in Eq. (2), the following two recovery algorithms may be proposed denoted Algorithm 1 in Fig.6, and Algorithm 2 in Fig.7. The mentioned algorithms can solve the measurement problem in Eq. (2). Since the measurement in Eq. (2) can be easily implementable in practice, this invention finds practical use in allscenarios where phase-less measurements with a specifiable measurement matrix occur. Algorithm 1 The complexity of Algorithm 1 is exponential in ^. This is due to operations in steps 6-10. Themain benefit of Algorithm 1 is that it solves Eq. (2) in just 4^ measurements.The input to Algorithm 1 is: ^Vector ^ of values for the measurement variable ^, in which 4^ − 1 elements in ^are of the form , 0 ≤ ^ ≤ 4^ − 2, for some ^ and distinct integers ^^ suchthat 0 ≤ ^^ ≤ ^ − 1, and the rest of the elements are arbitrary distinct numberswithin the interval [-1,1] which reflects the phases. ^Vector ^ of ^ distinct values within [-1,1] corresponding to the poles.The output of Algorithm 1 is: ^^^^^ for some unknown real number ^, where ^^^ is an unknown phase rotation.Algorithm 1 comprises: 1. Let ^^ = ^^^^^ and construct the matrix ^^ = [^ ^ ^^ − ^^ −^ −^^^] whereAny operations on of its smallestsingular value are also allowed. 2. Keep reducing the value of ^ (and thus the dimension of ^^) by one until ^^ has aunique smallest singular value (assume ^ = ^ when this happens).3. Denote by ^ the eigenvector of ^^ corresponding to its smallest singular value.4. Construct the Laurent polynomials ^^(^) = ∑^^ ^^^ ^^^ ^^^ , ^^(^) = 5. The 2^ roots of ^^(^) occur with multiplicity of 2. Let the ^ distinct roots of ^^(^) be^ , … , . These roots can be obtained by any other polynomial construction,long as it produces the exact same roots b^, … , b^^^. The ^ non-zero positions^^, … , ^^^^ in ^ can be obtained as^^ = arg 0 ≤ ^ ≤ ^ − 1 or by any other measures of distance that produce the same outputs ^^ , 0 ≤ ^ ≤^ − 1.6. The 2^ − 2 roots of ^(^) can be paired as . From eachchoose one element. This choice can be done in 2^^^ ways and let ^^ , 0 ≤ ^ ≤2^^^ − 1, denote the ^:th choice.7. For each ^^ , 0 ≤ ^ ≤ 2^^^ − 1 , solve the ^ − 1 linear equations equationsproduce the same ^^up to a scalar. 8. Define ^^ such that ^ ^ ^ ^ ^^^ = ^^ , 0 ≤ ^ ≤ ^ − 1, and ^^ = 0 for other elements in ^ .9. Scale ^ so that holds. 10. LetFind the minimum value ^ let ^ be theachieving it. Any other distance metrics are allowed as long as they produce the same solution ^^. 11. Output ^ = ^^.Algorithm 2Instead, Algorithm 2 solves Eq. (2) with a polynomial complexity in ^ since alloperations in Algorithm 2 are of polynomial complexity. The input to Algorithm 2 is: ^Vector ^ of values for the measurement variable ^, in which all elements of ^ arearbitrary distinct numbers within the interval [-1,1]. ^Vector ^ of ^ distinct values within [-1,1] corresponding to the poles. The output of Algorithm 2 is: ^^^^^ for some unknown real number ^, where ^^^ is an unknown phase rotation.Algorithm 2 comprises: 1. Let ^^ = ^^^^^and construct the matrix ^= [^ ^ ^^ − ^^ −^ −^^^]where Any operations on ^ that preserve the one-dimensional space of its smallestsingular value are also allowed. 2. Keep reducing the value of ^ (and thus the dimension of ^) by one until ^ has aunique smallest singular value (assume ^ = ^ when this happens).3. Denote by ^ the eigenvector of ^ corresponding to its smallest singular value.4. Construct the Laurent polynomials ^^(^) = ∑^^^^^^^^ ^^^ , ^^(^) =∑^^^^ ^^^^^ ^^^^ ^^^ ^^^^^^^^ , ^(^) = ∑ ^^^ ^^^^^^^^^^^. 5. The 2^ roots of ^^(^) occur with multiplicity of 2. Let ^^, … , ^^^^ denote the ^ distinctroots of ^^(^). These roots can be obtained by any other polynomial construction, as long as it produces the exact same roots ^^, … , ^^^^. The ^ non-zero positions^^, … , ^^^^ in ^ are obtained as^^ = arg m^in1 ^^^− ^^ , ^or by any other measures of distance that produce the same outputs ^^ , 0 ≤ j ≤^ − 1.6. Let ^(^) = ^^^(^)^^ and (^( ) ^ ^) + ^^ (^) − 4^(^)^ ^ =2 denote by ^^, … , ^^^^ the ^ − 1 roots of ^(^). These roots can be obtained by anyother polynomial construction, as long as it produces the exact same roots7. Solve the ^ − 1 linear equations for ^ = [^^ , … , ^^^^] up to a scalar; or any other set of linear equations thatproduce the same ^ up to a scalar.8. Let 9. Let ^ be an all zero vector of ^ elements and set ^^^^ = ^^ , 0 ≤ ^ ≤ ^ − 1. Similarly,let ^ be an all zero vector of ^ elements and set ^^^ = ^^^, 0 ≤ ^ ≤ ^ − 1.10. Scale ^ and ^ so that ^ = |^(^, ^)^ |^ and ^ = |^(^, ^)^ |^ hold.11. Let Find ^ = arg^m∈{i^n,^}|^^ − ^^|. Any other distance metrics are allowed as long asthey produce the same solution ^. 12. Output ^.Beside these two algorithms, one can instead apply well known algorithms (such as GAMP or the LASSO algorithm with the magnitude only constraint) to solve the following optimization problem: m | ^in ^^|^ ^ subject to ^= |^^|^where The exact solution to this optimization problem is (with probability 1) ^^^^ for some unknownreal number ^ (where ^^^ is an unknown phase rotation). The complexity of solving thisoptimization problem is at least ^(^). Furthermore, any method according to embodiments of the invention may be implemented in a computer program, having code means, which when run by processing means causes the processing means to execute the steps of the method. The computer program is included in a computer readable medium of a computer program product. The computer readable medium may comprise essentially any memory, such as previously mentioned a ROM, a PROM, an EPROM, a flash memory, an EEPROM, or a hard disk drive. Moreover, it should be realized that the communication device 300 comprises the necessary communication capabilities in the form of e.g., functions, means, units, elements, etc., for performing or implementing embodiments of the invention. Examples of other such means, units, elements and functions are: processors, memory, buffers, control logic, encoders, decoders, rate matchers, de-rate matchers, mapping units, multipliers, decision units, selecting units, switches, interleavers, de-interleavers, modulators, demodulators, inputs, outputs, antennas, amplifiers, receiver units, transmitter units, DSPs, TCM encoder, TCM decoder, power supply units, power feeders, communication interfaces, communication protocols, etc. which are suitably arranged together for performing the solution. Therefore, the processor(s) of the communication device 300 may comprise, e.g., one or more instances of a CPU, a processing unit, a processing circuit, a processor, an ASIC, a microprocessor, or other processing logic that may interpret and execute instructions. The expression “processor” may thus represent a processing circuitry comprising a plurality of processing circuits, such as e.g., any, some or all of the ones mentioned above. The processing circuitry may further perform data processing functions for inputting, outputting, and processing of data comprising data buffering and device control functions, such as call processing control, user interface control, or the like. Finally, it should be understood that the invention is not limited to the embodiments described above, but also relates to and incorporates all embodiments within the scope of the appended independent claims.

Claims

CLAIMS 1. A computing device (100) configured to: obtain an observation vector ^ comprising ^ real values obtained from ^measurements; obtain a complex measurement matrix ^; andrecover ^ based on ^ and ^ according to the equation:^ = |^^|^where ^ is an unknown sparse complex vector and | | is the modulus operator, and whereineach element in ^ is a rational function of a set of measurement variables ^ comprising atleast one pole.

2. The computing device (100) according to claim 1, wherein ^ is an ^-dimensional unknownsparse complex vector with ^ non-zero elements, where ^ ≥ 4^ − 1.

3. The computing device (100) according to claim 2, wherein ^ ≥ 4^ − 1 or ^ ≥ 8^ − 3.

4. The computing device (100) according to any one of the preceding claims, wherein the at least one pole has a multiplicity of at least 1.

5. The computing device (100) according to claim 4, wherein the at least one pole has amultiplicity 1.

6. The computing device (100) according to any one of the preceding claims, wherein each pole is associated with a non-zero element in ^.

7. The computing device (100) according to any one of the preceding claims, further configured to compute:to recover ^, where ^^^^ is at least one additional measurement,is a vector comprisingat least one non-zero element, and ^ is the transpose operator.

8. The computing device (100) according to claim 7, whereinis a random vector.

9. The computing device (100) according to claim 8, wherein the real parts and the imaginary parts of the elements of the random vector ^^^^^are derived from a real valued continuous distribution.

10. The computing device (100) according to claim 9, wherein the real valued continuous distribution is any of: a Gaussian distribution, a uniform distribution, and an exponential distribution.

11. The computing device (100) according to any one of the preceding claims, wherein at leastone measurement variable in the set of measurements variables ^ takes on differentmeasurement values for different measurements.

12. The computing device (100) according to any one of the preceding claims, wherein elements in the same column in ^ have the same set of poles and their associatedmultiplicities, and elements in the same row in ^ have the same set of measurement values.

13. The computing device (100) according to any one of the preceding claims, wherein ^ isexpressed as a product of two Fourier transform matrices.

14. The computing device (100) according to claim 13, wherein ^ is expressed as:where ^ is a Fourier transform matrix, ^ is a first set of real numbers associated with the setof measurement variables ^, and ^ is a second set of real numbers associated with the atleast one pole.

15. The computing device (100) according to claim 14, wherein the Fourier transform matrix^(^) with a frequency as:where ^ is the dimension of ^ with ^ non-zero elements, and ^ ≥ 4^ − 1.

16. A communication device (300) comprising a computing device (100) according to any one of the preceding claims.

17. The communication device (300) according to claim 16, configured to: measure ^ to obtain the observation vector ^.

18. A method (200) for a computing device (100), the method (200) comprising:obtaining (202) an observation vector ^ comprising ^ real values obtained from ^measurements; obtaining (204) a complex measurement matrix ^; andrecovering (206) ^ based on ^ and ^ according to the equation:^ = |^^|^where ^ is an unknown sparse complex vector and | | is the modulus operator, and whereineach element in ^ is a rational function of a set of measurement variables ^ comprising atleast one pole.

19. A computer program with a program code for performing a method according to claim 18 when the computer program runs on a computer.

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