FREQUENCY OFFSETS IN OFDM SYSTEMS

Technical Field

The present invention relates to an estimator. Furthermore, the present invention also relates to a communication device comprising such an estimator, a corresponding method and a computer program.

Background

In Orthogonal Frequency-Division Multiplexing (OFDM) systems it is of great importance to estimate the Frequency Offset (FO) between the frequencies used by a transmitter and the frequencies used by a receiver. In a situation when K multiple autonomous transmitters transmit to a single receiver, a problem is the joint estimation of the K FOs. The estimation is a complex task for the receiver.

This situation occurs in at least two scenarios within cellular wireless communication systems, such as in Long Term Evolution (LTE). Scenario 1 is in the downlink where the multiple autonomous transmitters are different E-UTRAN Node Bs (eNodeBs). One of the eNodeB may be serving/transmitting to a User Equipment (UE) being the single receiver. The remaining eNodeB:s may be acting as transmitting interferers. Scenario 2 is in the uplink where the multiple autonomous transmitters are different UEs and the single receiver is the serving eNodeB.

Conventional solutions normalize one or more FOs to a subcarrier spacing, such that a dimensionless quantity is obtained. Thereafter, each FO or each normalized FO is divided into two parts, i.e. an Integer Frequency Offset (IFO) and a Fractional Frequency Offset (FFO). The IFO is typically always an integer, and the FFO is typically a number limited to 0.5 in magnitude.

Conventional solutions may further estimate the IFO during the initial acquisition process. As the FFO may drift slightly over time, conventional solutions may therefore continuously estimate the FFO. Conventional solutions for dealing with the estimation of multiple FFOs may further make the assumption that the remaining signals, e.g. from K— 1 eNodeBs out of total K eNodeBs not serving the UE, act as additive noise. This assumption means reduced estimation performance as the assumption ignores the fact that there is a joint optimization problem to be solved. The problem is joint over the different FOs since the received signal is affected by K unknowns, i.e. the K different FOs. The assumption that the remaining signals can be seen as additive noise greatly simplifies the estimation procedure, since it renders K one-dimensional problems to solve. This solution reduces computational complexity but also results in poor estimation performance.

Another conventional approach is to assume that pilot signals have been designed in a way such that estimation by the relaxation to K one-dimensional problems performs as well is possible. However, this conventional solution has the disadvantage that the solution is restricted to specialized pilot designs and will not work for an arbitrary set of pilot signals.

Yet another conventional solution is to make use of Zadoff-Chu sequences. The solution based on Zadoff-Chu sequences is only aiming at the relaxation into K one-dimensional problems but does not address the joint estimation problem. However, in systems where Zadoff-Chu sequences have not been implemented this conventional solution is by definition not available. Summary

An objective of embodiments of the present invention is to provide a solution which mitigates or solves the drawbacks and problems of conventional solutions.

Another objective of embodiments of the present invention is to provide improved fractional frequency offset estimations compared to conventional solutions.

An "or" in this description and the corresponding claims is to be understood as a mathematical OR which covers "and" and "or", and is not to be understand as an XOR (exclusive OR).

The above and further objectives are solved by the subject matter of the independent claims. Further advantageous implementation forms of the present 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 an estimator for determining fractional frequency offsets, FFOs, in a communication signal, the estimator comprising a processor configured to

obtain a communication signal s comprising at least K FFOs, where K is an integer larger than 1 ;

determine Log Likelihood Function, LLF, λ for the K FFOs; approximate the determined LLF X as a series expansion comprising M basis functions φ _τη and associated expansion coefficients a _m , where m = 1, 2, ... , ;

determine the K FFOs by maximizing the series expansion. Determining the LLF may further comprise determining a mathematical form of the LLF.

The communication signal s may be a superimposed signal comprising signals from K number of autonomous transmitters, where K is an integer larger than 1. The communication signal s may be received at one single receiver.

The communication signal s may be received in the uplink direction, e.g. from multiple UEs to a single eNodeB. The communication signal s may also be received in the downlink direction, e.g. from multiple NodeBs to a single UE.

An advantage of an estimator according to the first aspect is that computational complexity can be reduced as multiple FFOs can be estimated with reduced number of log-likelihood computations due to the present approximation. A further advantage of an estimator according to the first aspect is that up to a certain SNR value, there is no loss in performance compared with a full ML estimator while having reduced computational complexity.

A further advantage of an estimator according to the first aspect is that the SNR value may be adapted by controlling the number of log-likelihoods that are computed.

In a first possible implementation form of an estimator according to the first aspect, the processor further is configured to evaluate the determined LLF X for M sets of predetermined FFOs so as to obtain corresponding log likelihood values X _m , where m = 1, 2, ... , M; and to determine each expansion coefficient a _m based on the obtained log likelihood values X _m .

The first possible implementation form means that by having evaluated the determined LLFs the solution can easily be implemented in hardware. In a second possible implementation form of an estimator according to the first implementation form of the first aspect, the processor is configured to determine the expansion coefficients a _m according to

where A is an MxM matrix.

This implementation form means that the matrix A can be precomputed rendering low computational complexity.

In a third possible implementation form of an estimator according to the second implementation form of the first aspect, the processor is configured to determine matrix A so as to minimize the total approximation error between the determined LLF λ and the series expansion.

This implementation form means that the approximation loss between the series expansion and the true LLF is minimized.

In a fourth possible implementation form of an estimator according to any of the first implementation form to the third implementation form of the first aspect or the first aspect as such, the M basis functions φ _τη are one of pulse functions: φ _τη = sincQ , φ _τη = exp (), and <p _m = rootrcQ . The sine stands for the sine function. The exp stands for the exponential function. The rootrc stands for the root raised cosine filter function.

These basis functions are easy to evaluate therefore rendering low computational complexity.

In a fifth possible implementation form of an estimator according to the fourth implementation form of the first aspect, the pulse functions can be expressed as

1 ' ·ε _κ ) = rootrc. r

where γ and μ are design parameters, and c™ are localization parameters for the basis functions φ _τη , where k = 1, 2, ... , K. The s _k is the frequency offset for the kt receive-transmit pair.

These specific basis functions can be tuned for improved estimations. By tuning the design and localization parameters the estimations can be adapted to different radio conditions.

In a sixth possible implementation form of an estimator according to the fifth implementation form of the first aspect, the design parameter γ has a value in the interval 0.5 - 3.0 and the design parameter μ has a value in the interval 0.3 - 1 .0.

This implementation form provides the values for a specific implementation. In a seventh possible implementation form of an estimator according to the fifth implementation form or the sixth implementation form of the first aspect, further comprising a memory, and

wherein the processor is configured to retrieve the design parameters γ and μ and the localization parameters c™ from a memory.

This implementation form means that no online tuning is needed. Only a small database comprising the mentioned parameters is needed.

In an eighth possible implementation form of an estimator according to any of the fifth implementation form to the seventh implementation form of the first aspect, the processor is configured to determine the localization parameters c™ calculable by

This implementation form provides geometrical simplicity since the localization parameters are located on a square grid. In a ninth possible implementation form of an estimator according to the eighth implementation form of the first aspect, the processor is configured to determine τ so as to minimize the total approximation error between the determined LLF λ and the series expansion.

This implementation form means minimized approximation loss rendering improved estimation performance.

In a tenth possible implementation form of an estimator according to the eighth implementation form or the ninth implementation form of the first aspect, the processor is configured to determine the number of basis functions M as M = P ^K .

This implementation form provides the relation between the number of basis functions M and the positive integer P.

In an eleventh possible implementation form of an estimator according to any of the eighth implementation form to the tenth implementation form of the first aspect, where P = 3, 4 or 5.

This implementation form implies a low value for P which means low complexity since the complexity is proportional to the term P ^K .

According to a second aspect of the invention, the above mentioned and other objectives are achieved with a communication device for a wireless communication system. The communication device comprises an estimator according to the first aspect. The communication device further comprises a receiver configured to receive the communication signal s.

According to a third aspect of the invention, the above mentioned and other objectives are achieved with a method comprising

obtaining a communication signal s comprising at least K FFOs, where K is an integer larger than 1 ;

determining (a form for) a Log Likelihood Function, LLF, λ for the K FFOs;

approximating the determined LLF 1 as a series expansion comprising M basis functions <p _m and associated expansion coefficients a _m , where m = 1, 2, ... , M

determining the K FFOs by maximizing the series expansion. In a first possible implementation form of a method according to the third aspect, the method further comprising evaluating the determined LLF A for sets of predetermined FFOs so as to obtain corresponding log likelihood values _m , where m = 1, 2, ... , M and determining each expansion coefficient a _m based on the obtained log likelihood values _m .

In a second possible implementation form of a method according to the first implementation form of the third aspect, the method further comprising determining the expansion coefficients a _m according to:

where A is an MxM matrix.

In a third possible implementation form of a method according to the second implementation form of the third aspect, the method further comprising determining matrix A so as to minimize the total approximation error between the determined LLF λ and the series expansion.

In a fourth possible implementation form of a method according to any of the first implementation form to the third implementation form of the third aspect or the third aspect as such, the M basis functions φ _τη are one of pulse functions: φ _τη = sincQ , φ _τη = exp (), and φ _τη = rootrcQ.

In a fifth possible implementation form of method according to the fourth implementation form of the third aspect, the method further comprising expressing the pulse functions as

^(^•••½) = ^rooirc i r where γ and μ are design parameters, and are localization parameters for the basis functions φ _τη , where k = 1, 2, ... , K. In a sixth possible implementation form of a method according to the fifth implementation form of the third aspect, the design parameter γ has a value in the interval 0.5 - 3.0 and the design parameter μ has a value in the interval 0.3 - 1 .0. In a seventh possible implementation form of a method according to the fifth implementation form or the sixth implementation form of the third aspect, the method further comprising retrieving the design parameters γ and μ and the localization parameters c™ from a memory.

In an eighth possible implementation form of a method according to any of the fifth implementation form to the seventh implementation form of the third aspect, the method further comprising determining the localization parameters c™ by calculating

{c™, -, c _K ^m }=^T, -,q _K ^m r}

where τ is a real positive number, and q™<≡{-P/2, ^■■■ ,Ρ/ΐ] for even P and ™e {-( -l)/2, - - -,( -l)/2} for odd P, where P is a positive integer.

In a ninth possible implementation form of a method according to the eighth implementation form of the third aspect, the method further comprising determining τ so as to minimize the total approximation error between the determined LLF λ and the series expansion. In a tenth possible implementation form of a method according to the eighth implementation form or the ninth implementation form of the third aspect, the method further comprising determining the number of basis functions M as M = P ^K .

In an eleventh possible implementation form of a method according to any of the eighth implementation form to the tenth implementation form of the third aspect, where P = 3, 4 or 5.

The advantages of any method according to the third aspect are the same as those for the corresponding estimator according to the first aspect. According to a fourth aspect of the invention, the above mentioned and other objectives are achieved with a computer program with a program code for performing a method according to the third aspect when the computer program runs on a computer.

The present invention also relates to a computer program, characterized in code means, which when run by processing means causes said processing means to execute any method according to the present invention. Further, the invention also relates to a computer program product comprising a computer readable medium and said mentioned computer program, wherein said computer program is included in the computer readable medium, and comprises of one or more from the group: ROM (Read-Only Memory), PROM (Programmable ROM), EPROM (Erasable PROM), Flash memory, EEPROM (Electrically EPROM) and hard disk drive.

Further applications and advantages of the present invention will be apparent from the following detailed description.

Brief Description of the Drawings

The appended drawings are intended to clarify and explain different embodiments of the present invention, in which:

Fig. 1 shows an estimator according to an embodiment of the present invention.

Fig. 2 shows a communication device according to an embodiment of the present invention. Fig. 3 shows a method according to an embodiment of the present invention.

Fig. 4a shows a first scenario in a wireless communication system according to an embodiment of the present invention.

Fig. 4b shows a second scenario in a wireless communication system according to an embodiment of the present invention.

Fig. 5 illustrates selection of localization parameters for two different setups according to an embodiment of the present invention.

Fig. 6 shows a diagram of RMS error performance for an estimator according to an embodiment of the present invention.

Detailed Description

In the following detailed description system assumptions, mathematical models and approximations are used, explained and described so as to provide a deeper understanding of embodiments of the invention. In this respect sometimes 3GPP Long Term Evolution (LTE) and LTE Advanced system context and thereto terminology is used, such as OFDM, UE (corresponding to a user device) and NodeB (corresponding to a base station or an access point). It is however noted that embodiments of the invention are not limited to these assumptions, mathematical models, approximations, and communication systems. A general mathematical model is first described for a deeper understanding of embodiments of the invention. The received communication signal s, after perfect removal of the Cyclic Prefix (CP) is assumed to have the form

_S =∑D( _£k )H _k Qp _k + n

k=l

where

• s is the received communication signal represented as a Nxl column vector, and k is the kth transmitter among a total of K transmitters.

• D(S _f e) is a diagonal matrix representing the kth FFO, wherein its nth diagonal element is exp(2njs _k (n— 1)/N).

· H _k is a circular convolution matrix representing the radio channel from the k th transmitter, out of the total K transmitters, to the estimator/receiver. The nth column of the convolution matrix is a downward shift of the (n - l)th column. Moreover, the first column of the convolution matrix H _k comprises independent complex Gaussian zero mean variables whose variances are given by the elements of the kth Power Delay Profile (PDP). All matrices H _k are unknown to the estimator but their PDPs are assumed to be known.

• Q is an NxN Inverse Fast Fourier Transform (IFFT) matrix.

• p _k is an Nxl vector containing pilot/reference signals/symbols which are known to the estimator. We also define the NxN diagonal matrix P _k as a matrix containing the elements of p _k along its main diagonal.

• n is an Nxl noise vector and each element of the noise vector is a zero mean, complex Gaussian random variable with variance N ₀ . The value N ₀ is assumed to be known or at least an estimate which is considered as a perfect estimate. Further possible assumptions in an OFDM system may according to embodiments of the invention include:

• Each of the K receive-transmit pairs may have an unknown FFO limited to 0.5 in magnitude.

• Each of the K transmitters may be configured to send a single OFDM symbol comprising N sub-carriers.

• The K OFDM symbols may be assumed to be perfectly overlapping in time.

• The time-synchronization among the K transmitters may be assumed to be ideal, such that the cyclic prefix (CP) can be perfectly removed. • Each of the K transmitters may be assumed to send a pilot vector in the frequency domain. The K vectors may be assumed to be known to the estimator.

• The current channel parameters, such as the noise density or an estimate thereof, at the receiver and the Power Delay Profiles (PDPs), or estimates thereof, for each pair of the receiver and each of the K transmitters may be assumed to be known to the estimator.

In an embodiment it is assumed that all PDPs are known to the estimator. This assumption may be considered restrictive but is in fact not. First of all, the PDPs may be used in many other parts of the estimator/receiver, such as for channel estimation purposes, so they may typically be at hand. Secondly, the estimator may assume a robust choice for the PDPs and may use a uniform PDP with a length equal to the CP, without any noticeable impact on FFO estimation performance. A LLF removing all additive and multiplicative constants, of the K FFOs, given that communication signal s has been received can be expressed as

where

∑ = E{SS ^H }=∑D(s _K )QP _K Q ^H diag{PDP _K )QP _K ^H Q ^H D(S + 1N ₀ .

k=l

where I is the NxN identity matrix. It is understood that the term diag(PDP _k ) means an NxN diagonal matrix with the kt PDP on its main diagonal.

It has been realized by the inventors that the estimation problem according to the Maximum Likelihood (ML) criterion can be formulated as

{έ - ; έ _κ } = argmax^ _{... ¾} l(s; s ,- - -s _K ).

where we maximize the LLF over the set of the K FOs. The FO combination that maximizes the LLF is taken as the estimator output.

Conventional solutions to the ML problem in this context involve significant computational complexity. In particular, the problem of the ML approach is that each function evaluation of

A(s;s ₁ ,- - -s _K ) is computationally complex. This is mainly due to the term∑ ¹ above which requires taking the inverse of a very large matrix. Thus, embodiments of the present invention relate to an estimator 100, a communication device 300 and a method 200 to approximate the ML problem using series expansions which significantly reduce the computational complexity.

Fig. 1 shows an estimator 100 according to an embodiment of the present invention. The estimator 100 is configured to determine FFOs in a communication signal s. The estimator 100 comprises a processor 102 configured to obtain a communication signal s comprising K number of FFOs, where K is an integer larger than 1. The processor 102 is further configured to determine a LLF λ for the K FFOs. The processor 102 is further configured to approximate the determined LLF λ as a series expansion comprising M basis functions φ _τη and associated expansion coefficients a _m , where m = 1, 2, ... , M. The processor 102 is further configured to determine the K FFOs by maximizing the series expansion.

The processor 102 may further be configured to determine a so called mathematical form for the LLF λ as described above, i.e.

The mathematical form is used for approximating the LLF λ.

The estimator 100 may be a standalone device or be part of or integrated in another device, such as a communication device. Fig. 2 shows a communication device 300 according to an embodiment of the present invention. The present communication device 300 comprises an estimator 100 according to embodiments of the invention, such as the one shown in Fig. 1. Further, the communication device 300 comprises a receiver 104 configured to receive the communication signal s. The receiver 104 is communicably coupled to the estimator 100 and/or directly to the processor 102 of the estimator 100. The receiver 104 is further configured to send or provide the communication signal s to the processor 102 for processing. The communication device 300 may also include an optional memory 106 coupled to the processor 102. The optional memory 106 is described in the following disclosure. The (wireless) communication signal s may be received in the downlink, e.g. as in the previously described scenario 1 , or received in the uplink, e.g. as in the previously described scenario 2.

Fig. 3 shows a corresponding method 200 according to an embodiment of the present invention which may be executed in an estimator 100 or in a communication device 300, such as the one shown in Fig. 1 and Fig. 2. The method 200 comprises obtaining 202 a communication signal s comprising at least K number of FFOs, where K is an integer larger than 1. The method 200 further comprises determining 204 a LLF λ for the K FFOs. The method 200 further comprises approximating 206 the determined LLF λ as a series expansion comprising M basis functions φ _τη and associated expansion coefficients a _m , where m = 1,2, ...,M. The method 200 further comprises determining 208 the K FFOs by maximizing the series expansion. Determining the LLF may further comprise determining a mathematical form for the LLF as described above.

In another embodiment, the processor 102 is configured to evaluate the determined LLF λ for M sets of predetermined FFOs so as to obtain corresponding log likelihood values _m , where m = 1,2, ...,M. The processor 102 may further be configured to determine each expansion coefficient a _m based on the obtained log likelihood values _m .

In an example, the true M log-likelihoods may be evaluated, where the M true log-likelihoods are defined by the relation:

l _m =l(s;s ,--,s ), l≤k≤K.

The true log-likelihoods values are not computed from the series expansion. The series expansion approximate the true values.

Each expansion coefficient a _m may according to an embodiment be determined by resolving the equation:

where A is an MxM matrix, e.g.

A

φ _χ (ε _χ ,— ,ε _κ ) ··· φ _Μ (ε _ι ,···,ε _κ )

By selecting matrix A according to this expression the series expansion will be located exactly at the positions for which the LLF was computed.

As aforementioned, it has also been realized by the inventors that the estimation problem may be defined by the expression:

{^,---,^} = argmax _{ff] ...¾} X(s;e _l ,— £ _K ).

In an embodiment, the above estimation problem is solved using an optimization technique, e.g. a gradient based algorithm known in the art. In an example, it can be assumed that the true log-likelihood function has been evaluated M times at the values

{ { ^£ i > ^, "'¾}> { ⁱ i '" ''¾}" ^,) | ^e i > ^{' "} > { ^£ I > ^{' "} > ^£ K } }-

This generates M true values of the log-likelihood function denoted as λ _ί ,..., λ _Μ . By approximating the present approximation can be forced to

be accurate on the M evaluated points such that

In other words, expressed for all M values λ _χ to λ _Μ we have

Thus, the expansion coefficients a ₁ ,..., a _M may be recovered by determining the inverse matrix of the matrix above which is expressed as:

0¾(*ί>···> ) ··· ί¾(4···> ) φ _λ (ε?,-,ε ) - φ _Μ (ε? ,-,ε _κ ^Μ )

By denoting the center matrix as A it may also be expressed in compact form as:

as previously described.

The matrix A may be designed according an optimization criterion. Therefore, in an embodiment, the processor 102 is configured to determine the matrix A so as to minimize the total approximation error between the determined LLF λ and the approximation as the series expansion. In an example, the processor 102 is configured to minimize the total approximation error defined by the expression:

^ ^■■■ $\A(s;e ₁ ,---£ _K )-A(s;e ₁ ,---£ _K ] άε _λ ^■■■ άε _κ .

This ensures that the series expansion is close as possible to the LLF, not only at the positions where the LLF is evaluated but also on other positions. In an embodiment, the M number of basis functions φ _τη are any of the pulse functions: φ _τη = sincQ, φ _τη = exp (), and φ _ιη = rootrcQ.

In a particular case of this embodiment, the pulse functions φ _ηι = sincQ, φ _ιη = exp (), and (p _m = rootrcQ are expressed as: r ∑(¾ - ²

k=l φ ^ε _{χ >} · · ·ε _κ ) = rootrc . r ∑(£ _k - cD ² where γ and μ are design parameters, and c™ are localization parameters for the basis functions φ _τη , where k = 1, 2, ... , K. The mentioned parameters can be seen as specifying the centres of the basis functions in a coordinate system where the FOs are the coordinates. Especially, the design parameters control the shapes of the basis functions. Thereby, the estimations can be adapted to different radio channel conditions for improved performance.

The pulse functions may also be expressed by any other pulse shape that is well localized, which implies that most of the mass of the pulse shape is close to its mass of gravity. The parameters γ and μ are design parameters that can be adjusted. These parameters may in turn depend on other parameters, such as the above described N ₀ , N, K, PDP. The series expansion is approximating the likelihood function. However, the statistical behavior of the likelihood depends on these mentioned parameters. Therefore, the basis functions that is trying to approximate the LLF should preferably depend on the same parameters as well.

The expression "rootrc" denotes the standard root-raised-cosine pulse with excess bandwidth β. The excess bandwidth β may be set to zero, i.e. such that a sine pulse is obtained. Thus, the first basis/pulse function φ _τη = sincQ is a subset of the third basis/pulse function φ _τη = rootrcQ given above. The first pulse function φ _τη = sincQ may be of particular interest, due to its advantage of reduced computational complexity compared to the third pulse function.

In an embodiment, the design parameter γ has a value in the interval 0.5 - 3.0 and the design parameter μ has a value in the interval 0.3 - 1.0. In an example, the expansion coefficients £½ ,..., _M may be approximately 1 , i.e. = l , and the design parameter μ may be approximately 0.5, i.e. μ = 1/2. These values have proved good estimation performance.

In an embodiment, the estimator 100 further comprises, or have access to a memory 106 as was shown in Fig. 2. The processor 102 is in this embodiment configured to retrieve the design parameters γ and μ and the localization parameters c™ from the memory 106. The parameters γ and μ may be predetermined and stored in a database in the memory 106, e.g. as a Look-Up Table (LUT) accessible by the processor 102. The database may comprise sets of values for γ and μ together with corresponding different system parameters, e.g. mentioned N ₀ , N, K, PDP configured to act or be used as a Look-Up index in the database. The corresponding system parameters may be given as different values differing with a predetermined step size. The parameters c™ may be defining a location in a K dimensional space of the mth basis function, thereby forming a Look-Up index for the sets of values. The parameters c™ may be further comprised in the LUT as sets of values, together with the previously mentioned sets of values for γ and μ, and indexed by the same corresponding system parameters.

The estimator 100 or the communication device 300 may therefore comprise the memory 106. The memory 106 is configured to store and retrieve parameters or data values, e.g. design parameters such as γ and μ and the localization parameters c™. The memory 106 is further coupled to the processor 102 and configured to send the retrieved parameters or data values to the processor 102, e.g. upon receiving a request from the processor 102. The memory 106 is further configured to store parameters or data values received from the processor 102, e.g. upon receiving a request from the processor 102.

In an embodiment, the processor 102 is configured to determine the localization parameters c™ calculable by

{ m m \ J m _ m \

ci ,— , c _K )= T,— , q _K T)

where τ is a real ositive number, and q™≡{-Ρ/2, · · ·,Ρ/2] for even P and q™≡{-(P-X)l2, - - for odd P, where P is a positive integer.

The dimension of the present series expansion equals the number of basis functions M spanning the series expansion. In an embodiment, the processor 102 is further configured to determine the number of basis functions M as M = P ^K . In an embodiment, the number P may be selected in the range [3 5], i.e. P = 3, 4 or 5 which means low complexity.

Fig. 4a shows a first scenario in a wireless communication system 400 comprising K transmitters Tx-i , TX ₂ , TX _K each transmitting a communication signal s _{k=1 2i K} . The superimposed communication signal s includes the communication signals s _k=li2i ... _iK . The communication signal s is received by a single receiver RX. The receiver RX may be comprised in a communication device 300 as shown in Fig. 4a as a UE. The multiple autonomous transmitters Tx-i , TX ₂ , TX _K may be different eNodeBs, where one of the eNodeBs, e.g. TX^ is a serving eNodeB for a UE being the single receiver. Therefore, the remaining K— 1 eNodeB:s TX ₂ , TX _K are acting as interferers.

Fig. 4b shows a second scenario in a wireless communication system 400 of K transmitters Tx-i , TX ₂ , TX _K each transmitting a communication signal s _k=li2i ... _iK - The superimposed communication signal s includes the communication signals s _{k=1 2i K} . The receiver RX may be comprised in a communication device 300 as shown in Fig. 4b as a eNodeB. The second scenario differs from the one shown in Fig. 4a in that the multiple autonomous transmitters Tx-i , TX ₂ , TX _K may be different UEs whereof one is transmitting to a single serving eNodeB, and the remaining K— 1 UEs are acting as interferers.

Fig. 5 illustrates a selection of the localization parameters for two different setups according to an embodiment of the present invention. As shown in Fig. 5, the basis pulses are located uniformly in the (ε ₁₍ ε ₂ ). This choice renders the series approximation to have approximately the same quality at every point. The horizontal localization is shown on the X-axis as ε ₁ and the vertical localization is shown on the Y-axis as ε ₂ . The X-axis is scaled from 0 to 0.5 and the Y-axis is also scaled from 0 to 0.5.

The first setup, with localization shown as black squares, uses the values P = 3 and K = 2. For the first setup, the resulting M = 9 and the localization parameters (horizontal X, vertical Y):

{(— r^ Ti) , (0, ^), (r^ Ti) , (— r ₁₍ 0), (0,0), (τ ₁₍ 0), (— r ₁₍ -τ^), (Ο, -τ^), τ^ -τ^)}.

The second setup, with localization shown as hollow circles, uses the values P = 2 and K = 2. For the second setup, the resulting M = 4 and the localization parameters (horizontal X, vertical Y):

{(-T ₂ , T ₂ ) , (T ₂ , T ₂ ) , (-τ ₂ , -τ ₂ ), (T ₂ , -T ₂ )}. The values for τ _ί and τ ₂ , which represent the spacing between two adjacent basis functions, is different for the two cases in Fig. 5 and may be individually optimized.

5 The processor 102 may further be configured to evaluate the LLF by evaluating the true LLF M number of times. The processor 102 may further be configured to select positions for g-likelihoods, i.e. to select

1100 TThhee ppoossiittiioonnss ffoorr wwhhiicchh ttoo eevvaalluuaattee tthhee MM ttrruuee lloogg--lliikkeelliihhooooddss mmaayy bbee sseelleecctteedd iinn aa ssiimmiillaarr mmaannnneerr aass tthhee llooccaalliizzaattiioonn ppaarraammeetteerrss,, ii..ee..,, [[εε"" ,, ·· ·· ··,, wwhheerree ©© iiss aa rreeaall positive number, and for even P we have {-Ρ/2, · · ;Ρ/2} while for odd P we have

-l)/2, -,( -l)/2}

15 The number © may be determined as close to variable τ as possible. The number Θ may further be optimized. The optimization may in an example be performed by an off-line check of the resulting estimator performance in which different values for Θ is evaluated.

In an embodiment, the processor 102 is configured to determine τ so as to minimize the total 20 approximation error between the determined LLF λ and the series expansion.

In an example, the number τ may be optimized based on an error function. In yet an example, the number τ may be approximately determined as τ = 1/(2(P + 1)). The number τ may be approximately determined such that the approximation error in the below expression is 25 minimized:

Fig. 6 shows a Root Mean Square (RMS) error performance diagram for an estimator 100 according to an embodiment of the present invention. The number of transmitters K in this example is 2, i.e. K = 2. The basis function for this approximation is selected to be a sine pulse φ _ηι = sincQ . The results shown in Fig. 6 was generated through simulation of a multiuser (K = 2) OFDM system. A uniform PDP, of a duration of 14 symbols, with an IFFT of size 128 subcarriers, and a frequency interleaved pilot symbol distribution among both users. The diagram shows Signal to Noise Ratio (SNR) of the communication signal s on the X-axis with a linear scale and the RMS estimation error on the Y-axis in a logarithmical scale.

The diagram in Fig. 6 shows, as a reference, the RMS estimation error of a true Maximum- Likelihood Estimation (MLE) as the lower curve comprising a solid line and square shaped markers. The remaining curves shows the RMS estimation error using approximations to the ML function according to the present invention with P = 3, 4 and 5. The 2 ^nd curve from the bottom shows the RMS estimation error for the approximation using P = 5 ( = 25 ) comprising a dashed line and diamond shaped markers. The 2 ^nd curve from the top shows the RMS estimation error for the approximation using P = 4 ( = 16) comprising a dotted line and "x" shaped markers. The curve at the top shows the RMS estimation error of the approximation using P = 3 ( = 9 ) comprising a dash-dotted line and triangular shaped markers. For each of the approximations to the ML function, P = 3, 4, 5, the same number of samples as the number of functions in the basis were required.

In the following description two different exemplary implementations of an estimator 100 according to embodiments of the invention are envisioned.

In the first exemplary implementation, the first step is to establish parameter values for the design parameters of the basis pulses of the series expansion. We have found that suitable values, for all three families, are anywhere in the vicinity of a = 1 and μ = 0.5.

Let us first express M as P ^K . For the localization parameters we propose in this exemplary implementation to use the following construction

[c, , - - -, c _K }= ^ ₁ T, - - -, q _K T]

where τ is a real positive number, and for even P we have q™ <≡{- P/2, ^■■■ ,Ρ/ΐ] while for odd P we have e {-(/ ^> - 1)/ 2, ^■■■ , (P- The number τ needs to be optimized as well, but as a general guideline, τ = 1 /(2(P+1 )). In general, τ may be optimized such that the approximation error is minimized, i.e. minimize the expression ^ ^■■■ ^A(s; ₁ , - - - £ _K ) - (s; ₁ , - - - £ _K άε _χ · · · άε _κ .

The above description is illustrated in Fig. 5 for K = 2.

At this point, we have fully characterized the basis functions. We now evaluate the true log- likelihood function M times. In order to do so, we must first decide for which values to evaluate the M true log-likelihoods, i.e., we must decide upon { ^■■ ·, ε _κ } }.

We select these positions in exactly the same way as the localization parameters were selected, i.e., where Θ is a real positive number, and for even P we have ε{- /2,···, /2} while for odd P we havew™e{-(/ ^J -l)/2,---,(/ ^J -l)/2}. In general, the number Θ needs to be optimized, but as a general rule of thumb, we may take Θ to be equal to τ.

When ating the M log-likelihoods

We then recover the expansion coefficients through the described relation

where, e.g.

A ^:

φ _λ (ε _λ ,···,ε _κ ) ··· φ _Μ (ε _λ ,···,ε _κ )

Hence, at this point we can address the estimation problem itself, i.e., we can try to solve

{^,---,^} = argmax _{ff] ...¾} X(s;e _l ,— £ _K ).

Many techniques to solve the above estimation problem can be envisioned. It has been realized by the inventors that a simple gradient based algorithm known in the art works very well. Such a gradient based algorithm converges to the global optimum in only a few iterations.

In a second exemplary implementation, we define "region of uncertainty" as all possible values that the K number of FOs can take. Values outside the "region of uncertainty" will not be searched by the algorithm. With "full region", we mean the region of uncertainty at the estimator start, i.e. - 0.5≤s _k ≤ 0.5, V& .

The second exemplary implementation comprises: 1 . At the start of estimating K number of FFOs, the region of uncertainty is the full region ( - 0.5 < ^ < 0.5, Vfc ).

2. 2 ^K true log-likelihoods are evaluated in a square grid around the origin.

3. We construct an approximate LLF based on the 2 ^K log-likelihoods obtained in step 2.

4. We Taylor expand the approximate LLF to second order, i.e., we approximate the approximation with a quadratic function.

5. For the quadratic function, we maximize it in closed form, and denote the argument for which the function is maximized as .

6. Around the optimum, we put a new region of uncertainty of size ξ (which can be designed), so that the new region of uncertainty is given by ε ^ορ '—ξ < s _k < ε° ^ρ ' + ξ, Vk . If the boundary of the new region of uncertainty falls outside of [-0.5,0.5] the new region is truncated.

7. Within this much smaller new region of uncertainty we evaluate P ^K true log- likelihoods.

8. With these P ^K true log-likelihoods at hand, a P ^K dimensional LLF approximation can be constructed by finding the coefficients for the basis functions according to the relation

9. Based on these coefficients, the approximate LLF can be optimized by any standard optimization method, such as a gradient based method.

At the end of step 3, we have a 2 ^K dimensional approximation at hand, i.e. α _πι φ _πι (ε _λ , · · · ε _κ ) . Around the origin, we can Taylor expand the

approximation as

i{s; s _l , - - - s _K

οε _ι οε _{ι m=} i δε _ι δε _{κ m=1}

οε _κ οε _{ι m=} i δε K^δ ^U ε ^C " K m=\ where it should be understood that all derivatives are evaluated at the origin. Since the basis functions are taken as standard functions, the derivatives are straightforward. Further, since the basis functions are always evaluated in the origin, the derivatives are always constant.

Thus, we can rewrite the Taylor series as

where g represents the gradient, and H the Hessian, respectively. Both the gradient g and the Hessian H are linear combinations, where a _n are the expansion coefficients of predefined vectors/matrices. These vectors/matrices are obtained by taking the first and second order differential of the basis functions. Since the basis functions are determined in advance in this particular case, mentioned vectors/matrices can be precomputed offline and stored, e.g. in a memory. Maximizing the above expression results in the closed form

_^ opt

lg .

_^ opt

A remark is that in step 7, the region of uncertainty is rather small and therefore P can also be small, e.g. 2 or 3.

A generalization of the above description is to repeat steps 1 to 6 in an iterative manner. For example, in the reduced region of uncertainty in step 6, we can go back to step 1 and evaluate 2 ^K true log-likelihoods, Taylor expand the 2 ^K true log-likelihoods and solve for the optimum, and thereby obtain an even reduced region of uncertainty. However, it has been found that also one iteration gives excellent performance.

For the iterative approach the iterative loop can stop when the region of uncertainty is smaller than the precision (e.g. in the form of a threshold value acting as a stop criterion) required for the system, such as an OFDM system. For example, if the FO is smaller than 0.01 the OFDM system will work well. If the region of uncertainty is smaller than this, it doesn't matter if the iterations are continued or not since the OFDM system will work well anyway. Note that other communication systems may have other FO requirements for functioning well. Regarding steps 6 and 7, we have selected, with excellent outcome ξ = 0.2. Then, it may suffice to choose P = 2 or 3 in step 7. With the mentioned parameters and operations this particular estimator 100 can perform as well as the ML estimator up to around 12 dB for P = 2 or 20 dB for P = 3 , after which the performance saturates. Thus, for K = 2 , it is sufficient to evaluate only 2 ² +3 ² = 13 true likelihoods.

The present communication device 300 may be an eNodeB, which in some networks may be referred to as (radio) network node, access node, access point, base station, Radio Base Station (RBS), transmitter, eNB, eNodeB, NodeB or B node, depending on the technology and terminology used. The radio network nodes may be of different classes such as, e.g., macro eNodeB, home eNodeB or pico base station, based on transmission power and thereby also cell size. The radio network node can be a Station (STA), which is any device that contains an IEEE 802.1 1 -conformant Media Access Control (MAC) and Physical Layer (PHY) interface to the Wireless Medium (WM). The present communication device 300 may further be a User Equipment (UE), mobile station (MS), wireless terminal or mobile terminal which is enabled to communicate wirelessly in a wireless communication system, sometimes also referred to as a cellular radio system. The UE may further be referred to as mobile telephones, cellular telephones, computer tablets or laptops with wireless capability. The UEs in the present context may be, for example, portable, pocket-storable, hand-held, computer-comprised, or vehicle-mounted mobile devices, enabled to communicate voice or data, via the radio access network, with another entity, such as another receiver or a server. The UE can be a Station (STA), which is any device that contains an I EEE 802.1 1 -conformant Media Access Control (MAC) and Physical Layer (PHY) interface to the Wireless Medium (WM). Furthermore, any method according to the present 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 comprises of essentially any memory, such as a ROM (Read-Only Memory), a PROM (Programmable Read-Only Memory), an EPROM (Erasable PROM), a Flash memory, an EEPROM (Electrically Erasable PROM), or a hard disk drive.

Moreover, it is realized by the skilled person that the present communication device 300 and estimator 100 comprises the necessary communication capabilities in the form of e.g., functions, means, units, elements, etc., for performing the present solution. 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, MSDs, TCM encoder, TCM decoder, power supply units, power feeders, communication interfaces, communication protocols, etc. which are suitably arranged together for performing the present solution. Especially, the processors of the present devices may comprise, e.g., one or more instances of a Central Processing Unit (CPU), a processing unit, a processing circuit, a processor, an Application Specific Integrated Circuit (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 present 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.