TECHNICAL FIELD

The present invention relates to a precoder for the downlink of a multiuser commu- nication system, arranged to precode a first set of sample values to a second set of complex sample values, wherein the first set of sample values comprises a first predetermined number (K) of values destined for a corresponding number of receivers and wherein the second set of complex sample values comprises a second predetermined number ( ) of complex sample values destined for M available emitter units.

BACKGROUND ART

This invention is concerned with multiuser multiple-input multiple-output (MU- MIMO) systems, herein defined as wireless communication systems comprising a emitter unit with at least one antenna communicating with at least one receiver, each receiver having at least one antenna. In a MU-MIMO system, by virtue of the multiple transmitter antennas, the base station can communicate with more than one receiver in the same frequency band simultaneously in time. In a typical implementation of MU-MIMO, the base station has multiple antennas arranged in the form of an array, but the multiple antennas can also be distributed at different geographical locations. Of particular interest herein is coding and transmission schemes for the downlink of a MU-MIMO system, defined as the set of transmission links from the base station to each of the receivers. A practical implementation of MU-MIMO requires low-cost transmitter equipment and transmitter precoding techniques that can be performed at low computational cost. A major cost of deploying multiple antennas at the base station is that of the power amplifiers, both in terms of initial expenditures and in terms of power con- sumption during operation. Generally, higher linearity requirements on the power amplifier design translate into a lower efficiency in terms of the ratio between the electrical power spent and the radio frequency power radiated by each antenna. Generally, the peak-to-average variations of the input signal envelope determine the linearity requirements that the power amplifier must meet. Specifically, for signals with a large peak-to-average variation, highly linear, and therefore power- inefficient, power amplifiers are needed. By contrast, for signals with small peak-to- average variations, the linearity requirements are weaker and therefore more power- efficient power amplifiers can be used. From this perspective, the type of signal that facilitates the use of most power-efficient power amplifiers is a signal with unity peak-to-average ratio. Such a signal has a constant envelope, and information is contained only in its phase.

A practical implementation of MU-MIMO requires a precoder that maps the set of information symbols to be transmitted to the receivers in a given time-frequency resourceonto a set of signals that are transmitted by the antennas. When reviewing the prior art on precoding techniques for the MU-MIMO downlink, the prior art falls into the following two main categories.

In the first category, the base station obtains the signals transmitted by the antennas from the information symbols by a linear mapping. The linear mapping can be chosen to maximize the received signal-to-noise-ratio at each receiver, subject to a given total transmit power constraint. This results in so-called maximum-ratio transmission, where the transmitted vector is obtained by multiplying the vector of information symbols by a scaled version of the conjugate transpose of the channel matrix. As another example of a linear mapping, called zero-forcing, the transmitter can choose the mapping to maximize the received signal-to-noise ratio at each receiver subject to a total power constraint and subject to the constraint that no receiver should overhear information intended for another transmitter.

Linear precoders are simple to implement, but the rates achieved by each receiver are generally smaller than what could theoretically be achieved. Moreover, the transmitted signals have a high peak-to-average ratio. This is a problem since expensive and power-inefficient power amplifiers must then be used.

In the second category, the base station obtains the transmitted symbols from the information symbols by a non-linear mapping. One known non-linear mapping is based on a technique called dirty-paper coding. It is known that if the receiver noise is Gaussian and the base station perfectly knows the channels between the base station and all receivers then this strategy can be used to achieve any point on the rate region of the MU-MIMO channel. However, this technique has very high computational complexity and moreover, the transmitted signals generally have a high peak-to-average ratio.

Another nonlinear mapping known in the literature is vector perturbation. The idea there is to apply a linear mapping to a perturbed version of the vector of information symbols. The perturbation consists of adding a vector whose elements are obtained by multiplying a complex-valued constant by an integer, where the complex-valued constant is known to the receiver whereas the integer is unknown. The effect of this is that the received signal at a particular receiver is equal to the information symbol intended for that user modulo a constant value. Vector perturbation has been shown to perform closer to optimal as compared to linear precoding techniques in terms of sum-rate performance. However, it is computationally more complex than linear precoders. Moreover, the transmitted vector generally has a high peak-to-average ratio. SUMMARY OF THE INVENTION

One object of the present invention is to provide an improved precoder.

This has been achieved by means of a precoder for a multiuser communication system arranged to precode a first set of sample values to a second set of complex sam- pie values, wherein the first set of sample values comprises a first predetermined number (K) of values destined for a corresponding number of receivers, wherein the second set of complex sample values comprises a second predetermined number ( ) of complex sample values for transmission destined to a number of available emitter units for transmission. The precoder further has channel information cor- responding to a channel to each receiver from each emitter unit, and is arranged to form the second set of complex sample values based on the first set of sample values and the channel information, wherein the complex sample values of the second set have a constant envelope.

One advantage of the present invention as defined above is that the complex sample values in the said second set have constant envelope, which results in the peak- to-average ratio of the transmitted signals being unity. Thereby amplifiers of the emitter units can be non-linear without adversely affecting performance.

In one option, the envelope is constant over all complex sample values in the second set and for consecutive second sets over a predetermined time interval.

In one option, the channel information comprises a channel gain matrix (H) having elements related to the channel to the kt receiver from the mth emitter unit, wherein k E {1, K} and wherein m E {1, M}.

In one option, the precoder is arranged to perform processing so as to minimize the Euclidean distance between the channel as defined by the channel gain matrix (H) multiplied with the second set of complex sample values and the first set of sample values destined for the K receivers.

In one option, the precoder can be arranged to perform processing so as to minimize the Euclidean distance between the channel as defined by the channel gain matrix (H) multiplied by the second set of complex sample values and any third set of complex sample values that falls into an equivalence class associated with the first set of sample values. For example, the equivalence class can comprise a group of third sets of sample values that yield the desired received sample values when performing a modulo-operation.

In one option, the difference between the phase of a signal transmitted by any emitter unit at a given point in time and the phase of the signal transmitted by said emitter unit at an immediately preceding point in time can be less than a predefined threshold.

In one option, the precoder is arranged to minimize the sum of said Euclidean distance and the sum of a set of penalty terms, where each penalty term is associated with an element of the second set of complex sample values, and where each penalty term grows monotonically with the difference between the phase of the correspond- ing element of the second set of complex sample values at a given point in time and the phase of said element at an immediately preceding point in time.

In one option, the precoder is arranged to perform a gradient search so as to minimize the Euclidean distance. In another option, the precoder is arranged to perform an alternating optimization procedure consisting of a gradient search and a poten- tially suboptimal solution of an integer-constrained least-squares problem, the latter performed by methods known in the art.

The invention also relates to a transmitter system comprising a precoder according to the above and at least the second predetermined number of emitter units connected to the precoder, wherein the precoder is arranged to feed the complex sample values of the second set to the corresponding emitter units.

The emitter units can each comprise at least one non-linear power amplifier.

A base station for a cellular communication system comprises in one embodiment a transmitter system according to the above. Alternatively, an acoustic system comprises a transmitter system according to the above, wherein the emitter units each comprise at least one loudspeaker. Further, the acoustic system may comprise the first predetermined number of receivers, wherein the receivers are microphones.

In one example, the present invention also relates to a method for precoding in a transmitter system, comprising

• obtaining a first set of sample values comprising a first predetermined number (K) of values destined for a corresponding number of available receivers, and

• forming a second set of complex sample values for transmission having a constant envelope based on channel information corresponding to a channel to each receiver from each emitter unit and based on the first set of sample values, wherein the number of complex sample values of the second set corresponds to the number of available emitter units for transmission and wherein the second predetermined number is larger than or equal to the first predetermined number.

BRIEF DESCRIPTION OF THE DRAWINGS

• Fig. 1 shows an example of a wireless communication system.

• Fig. 2 shows an example of a transmitter system of the wireless communication system of Fig. 1.

• Fig. 3 shows an example of a precoder in the transmitter system of Fig. 2 in accordance with a first and a second example of the invention.

• Fig. 4 shows an example of a precoder in the transmitter system of Fig. 2 in accordance with a further example of the invention.

• Fig. 5 shows an example of a receiver in the wireless communication system of Fig. 1. • Fig. 6 is a flowchart over a precoding method according to a first example of the invention.

• Fig. 7 is a flowchart over a precoding method according to a second example of the invention. · Fig. 8 is a flowchart illustrating the operation of the precoder. DETAILED DESCRIPTION

In Fig. 1 , a wireless communication system 100 comprises a transmitter system 1 10, a channel 120 and a plurality of receivers 130χ , ...,130^ scheduled to receive data from the transmitter system. The system is in one example a multiuser MIMO system. The transmitter system comprises a plurality of emitter units 140 , 140^. The channel 120 represents the transmission paths from the emitter units 140χ , 140^f to the receivers 130χ , ...,130^. The number of emitter units, in this example denoted M, is characteristically larger than or equal to the number of receivers, herein denoted K. In one example the number of emitter units is substantially larger than the number of receivers. The emitter units 140 , 140^ each comprise a non-linear amplifier 241 , 241^ and an antenna or the like 242 _b ..., 242 _M (see Fig. 2).

In an example wherein the communication system 100 is a cellular radio communication system, the transmitter system 1 10 is a part of a base station comprising one or a plurality of transmitter systems each covering a cell. As stated above, the emitter units 140 , 140^ each comprise a non-linear power amplifier and an antenna. The channel 120 is a radio downlink. The receivers 130 , ...,130_¾- in accordance with this example are user terminals each provided with at least one antenna for reception of signals transmitted from the transmitter system 1 10 over the channel 120.

In an alternative example, wherein the system is an acoustic system, the channel 120 is an acoustic channel. Each of the emitter units 140 , 140^ comprise in accordance with this example a non-linear power amplifier and a loudspeaker.

In Fig. 2, a transmitter system 210 comprises a precoder 211 connected to M emitter units 240χ, 240 _j ^f . The precoder 211 is arranged to precode a first set of sample values «]_ , ..., comprising K samples to a second set of complex sample values xi , ..., XM comprising M samples. Each sample value of the first set comprises information destined for a corresponding receiver. The precoded complex sample values in the second set correspond to the available emitter units 240χ, 240^- . The precoder is arranged to supply the complex sample values of the second set to the corresponding emitter units for transmission over the channel.

The precoder 211 comprises at least a means for digital signal processing 212 and a memory 213. The precoder has access to channel information H corresponding to a channel to each receiver from each emitter unit. The channel information is in one example obtained from received training data. The channel information is in another example obtained from channel state information feedback from the re- ceivers. Methods for estimating the characteristics of communication channels and for feedback of channel state information from the receivers are known in the art and do not constitute part of this disclosure. It is just assumed that the channel characteristics are known and that information related to the channel is present in the precoder. The precoder is arranged to form the second set of sample values based on the first set of sample values and the channel information in such a manner that the complex sample values of the second set have a constant envelope. The precoder is in one example arranged to form the second set of sample values so that the envelope is constant over all complex sample values in the second set and for consecutive second sets over a predetermined time interval.

In accordance with one example, the first set of sample values is formed as an information symbol vector (u). Further, the second set of complex sample values can be formed as a symbol vector (x). The channel information can comprise a channel gain matrix (H) of dimension K x M whose (k, m)th element contains a channel gain between the mth emitter unit at the transmitter system and the kth receiver.

Below, the operation of the precoder will be described by way of examples. The following notation will be used. A /^-dimensional vector y is defined comprising signals received at the K receivers, with its kth component being the complex sample value received at the kth receiver. The vector of K received signals is given by

y = H x + n. (1) u = [u U2 · · · u _j ^] ^T is a /^-dimensional information symbol vector, with be- ing the information symbol destined for the kth receiver. The goal of the transmitter system is to communicate the information symbol to the kth receiver in a reliable manner. The information symbols are mapped to a transmit vector x = ^{a wa ma} t the mth element x _m of x has constant envel m = 1, 2, . . . , M where P represents the total trans- mitted power. This mapping is henceforth referredto as the "matching" process. This matching is done in such a way that each receiver can accurately recover its information symbols from its received signals.

In a first example of the invention, the precoder is arranged to perform processing so as to minimize the Euclidean distance between the channel as defined by the channel gain matrix (H) multiplied with the transmit vector x and a vector desired to be received at the receivers. In the example described herein, the vector desired to be received at the receivers is the information symbol vector u.

The minimization of the Euclidean distance involves in one example finding a transmit vector x that minimizes a first least-squares criterion. In more detail, x = (2)

where D is a diagonal weighing matrix that can be chosen in order to maximize performance. The matrix D may be chosen to be the identity matrix. In another option, it be chosen to be diagonal matrix whose (k, k)th diagonal entry is

D k,k (3)

P \\h.

where is the kth row of H.

The optimization problem in (2) may be solved in different ways. Since the signals transmitted from all emitter units have constant envelope, we can express them as xm l _ e., m 1, 2, . . . , (4) where j is the imaginary unit and 9 _m E -7Γ , 7Γ) is the phase of the signal transmitted by the mth emitter unit.

The optimization problem in (2) can be described equivalently as follows. Let Θ = [0]_ #2 " " " denote the vector of phases of the complex sample values transmitted by the M emitter units. The optimally precoded transmit vector is given

,opt _^ opt ,opt T

where the optimal phase angles Θ°Ρ* 1 °2 M are given by

©O * = (6)

where

Δ

H D H (V) and where h^, denotes the fcth column of H. In (6), we can consider to be an approximation of the information symbol vector u. For the sake of brevity, define the Euclidean norm of the approximation error (u—

(8) The function e(©) is continuous and differentiable with respect to all the M phase variables. However, it is generally neither convex nor concave with respect to Θ, which makes it difficult to use standard convex optimization techniques for minimizing it with respect to Θ. Nevertheless, it has been observed that for a given K, as the number of emitter units M becomes large (i.e., M K), most of the local minima of the function e(©) have a small magnitude which means that they constitute a good approximation. Therefore, in one example, a gradient search algorithm is used which is guaranteed to converge at least to a local minimum. The gradient search algorithm is iterative in nature, and therefore we denote the value of the phase variables after the pth iteration by Θ^) = [θ^ ^■■■ Θ^] ^Τ '. The phase variables can be arbitrarily initialized. In one option, the phase variables are initialized to take on the value of zero. In the (p+ l)th iteration, the algorithm steers the current phase vector ©(P) in the direction opposite to the gradient of e(©) at © = ©(^) . Therefore, the hase vector after the (p + l)th iteration is given by

where λ > 0 is a positive constant that may be chosen to maximize performance and convergence speed, and Ve(©) is the gradient vector of the function e(©) which is given by

T

de{&) de(&) de{&)

Ve(©) (10)

8Θ ₁ 3Θ ₂ 8Θ M

The partial derivatives in (10) are given by

The algorithm terminates either after a fixed number of iterations or when the difference between the objective function at two consecutive iterations e(©(^) ) — e(©(P ^— 1) ) is below a predetermined threshold value. If the algorithm terminates after the mth iteration, then the transmit vector is given by

(12) In a second example of the invention, the precoder is arranged to perform processing so as to minimize the Euclidean distance between the channel as defined by the channel gain matrix (H) multiplied with the transmit vector x and any vector that falls into an equivalence class associated with the vector that is desired to be received at the receivers. In the example described herein, the vector desired to be received at the receivers is the information symbol vector u. In one example, the equivalence class comprises a set of vectors that yield the desired received vector when performing an element-wisemodulo-operation.

The minimization of the Euclidean distance involves in this example finding a trans- mit vector x that minimizes a second least-squares criterion. In more detail, argmm lu + rl - DHxll ² (13) where r may be chosen to maximize performance and where stands for the set of all /^-dimensional vectors with integer elements.

The optimization problem in (13) may in one example be solved in the following way. For a given information s mbol vector u, the transmitted vector is given by

where

M

(1*, Θ*) u + rl 0r. (15)

^{■ Θ} Μ ^Τ - For the sake of clarity the norm of the approximation error is here denoted by the function

M

Δ

β(θ, 1) |u + rl - ∑ h _m e ^J (16)

m=l

The optimization problem in (6) is a special case of the modified precoding scheme in (14) and (15), since the optimal Θ°Ρ* in (6) is the same as the optimal solution to (15) with the integer vector 1 fixed to 0. This therefore implies that, the modified precoding scheme in (14)— (15) results in a better constant envelope approximation to the information symbol vector u, i.e. However, the non-convexity of the objective function e(©, 1) with respect to Θ and the discreteness due to 1 makes it difficult to use standard optimization tools to solve (15). The function e(©, 1) may in one option be minimized by using a cyclic minimization algorithm. The algorithm starts with an initial estimate of Θ and 1, and modifies them iteratively so as to reduce the norm of the approximation error in each iteration. Let the values of Θ and 1 after the pth iteration be denoted by ©(p) and respectively.

In one option, the starting value for Θ may be taken to be the output of the gradient search algorithm for solving the optimization problem in (6). In one option, the starting value of 1 may be taken to be l^) = 0. Each iteration of the cyclic algorithm consists of two phases, phase I and phase II. In phase I of the (p + l)th iteration, the value of Θ is fixed to and the following optimization problem is solved to give lCP+!) ': i(p+i) argmm u + rl (18)

For small values of K this minimization may be performed by using standard methods for finding the exact or approximate solution to integer-constrained least- squares problems. For large values of K, local search techniques known in the art can be used. In phase II of the (p + l)th iteration, 1 is fixed to l(P+l) , and the algorithm tries to further reduce the approximation error by updating & p) to ©^ ⁺¹ ) , where ©(P+l) is given by

©(P+l) argmm u (19)

0,6» _m 6 [-7T ,7r) This optimization problem is effectively the same as the problem in (6) and can therefore be solved using the gradient search algorithm proposed earlier.

After phase II of the {jp + l)th iteration, the algorithm moves to phase I of the (p + 2)th iteration. Since the algorithm minimizes the approximation error norm in a cyclic manner alternating between optimization with respect to Θ and 1, it follows that

e(©fr ⁺¹ ) , lfr ⁺¹ ) ) < e(©fr) , lfr)). (20)

Therefore, the cyclic algorithm described above guarantees a monotonic descent in the value of the approximation error norm from one iteration to the next. Since the approximation error norm is non-negative, the monotonically decreasing sequence {β(θ( ¹ ) , lW ), β(θ( ² ) , l( ² ) ), β(θ( ³ ) , l( ³ ) ), · · · } is bounded from below and therefore the limit

exists. Hence, the cyclic algorithm is guaranteed to converge at least to a local optimum.

In a variation of this example, rl is replaced by [τχΐχ · · · τχ\χ] ^Τ where one parameter 7¾ is chosen individually for each receiver, instead of using the same r for all receivers. As a skilled person in the art will recognize, similar optimization methods as described above can be used to solve the resulting optimization prob- lems.

In an extended example of the invention, the precoder is arranged to perform precod- ing so that the difference between the phase of a signal transmitted by any emitter unit at a given point in time and the phase of the signal transmitted by said emitter unit at an immediately preceding point in time is limited. In one option, said phase differences may be constrained to fall below a predefmedthreshold. In another option, a least-squares criterion such as the first least-squares criterion in (2) or the second least-squares criterion in (13) may be augmented by a term that penalizes said phase differences. In accordance with this example, the transmitter system col- lectively maps the information symbols to be transmitted over several time instances onto sets of complex samples supplied to the emitter units.

In the description of this extended example, time is considered to be discrete and t is the time index. For example, x(i) = [#]_ (£) a¾ W ' ' ' # (^)] ^{r re} P ^resents th ^{e vector} of transmitted complex symbols at time t, with x _m {t) being the signal transmitted from the mth emitter unit at time t. Similarly, u _k (t) denotes the information symbol destined for the kth receiver at time t. Time interval of interest is considered to be from t = 0 to t = T— 1. It is assumed that the channel H is essentially constant over the time interval of interest. It is further assumed that the transmitter system may obtain knowledge of H at least to within a prescribeddegree of accuracy.

The information symbols destined for each receiver may in one option be scrambled across time. This feature can help in averaging out the approximation error of the matching process over T consecutive information symbols. The following is a description of the scrambling procedure. For the kth receiver, the vector of T infor- mation symbols u _k = [u _k (0) u _k {l) · · · u _k (T— is scrambled using a T x T unitary matrix W¾. The scrambled vector for the kth receiver is given by s _k = [¾(0) _¾ (1) · · · s _k (T - l)] ^T W _k u _k , k = l, ..., K. (22)

At time t, the transmitter system communicates the scrambled symbol vector s(t) = [si (t) s ₂ (t) ^■■■ s _K (t)] ^T , i = 0, ..., T - l (23) with s _k {t) being the scrambled symbol intended for the kth user. In the option with scrambling, the transmitter system first maps the vectors of information symbols {u(0), u(T— 1)} onto the vectors of scrambled symbols {s(0), s(T— 1)}. The scrambled symbols are then mapped onto a set of transmit vectors {x(0), x(T— 1)} in such a way that the elements x _m (f) of x(i) have constant envelope, i.e., m = 1, 2, . . . , M, t = 0, T - 1 where P represents the total transmit power. The scrambling matrices may be taken to be equal to the identity matrix, = I, which then means that no scrambling is effectively taking place. In the extended example, the precoder may involve finding {x(0) , x(T— 1) } in a manner that minimizes the first least-squares criterion (2), with u(t) being replaced by its possibly scrambled version s(t), subject to constraints on the time variation of the transmitted phase from each emitter unit. More precisely,

{Θ°^(0) , ..., Θ°^(Τ - 1) } (24)

T-l 2 argmin s(t) - ∑ h _m eJ ^' t) {Θ(0),...,Θ(Γ-1)} m=l

-7Γ,7Γ) , |0 _TO (i+l)-0 _TO (i) | <?7 _TO 7r

where 0 _m (t) is the phase of the signal x _m (t) transmitted by the mth emitter unit at time t, and ^r q _m is a factor which constrains the time variation of the phase transmitted from the mth emitter unitln another option, the transmitter system may alternatively find {x(0) , x(T— 1) } by minimizing the second least-squares criterion (13), with (t) being replaced by its possibly scrambled version s(t), subject to said constraints on the time variation of the transmitted phase from each emitter unit. More precisely,

{θ*(0) , ..., θ* (T - l) }

argmm

{Θ(0),...,Θ(Γ-1)} , {1(0),...,1(Γ-1)} \ <J _im Ti , l(t) eZ ^K

where r may be chosen to maximize performance and where stands for the set of all /^-dimensional vectors with integer elements. The optimization problem in (24) can be solved efficiently using interior-point methods or barrier methods known in the art. For solving (25), a cyclic algorithm similar to the one proposed for solving (13) can be used, the only difference being that in phase-IIof each iteration an instance of (24) would be solved using an interior-point method.

In another option, the precoder would find (x(0) , ... , x(T— 1 ) } in a manner that minimizes the first least-squares criterion (2), with (t) being replaced by its possibly scrambled version s(i), augmented with a term that penalizes large time variations of the transmitted phase from each emitter unit. In more detail, the phases of the transmitted signals may be found via °Ρ ¹

where g(- ) is a function chosen such that g(0) = 0 and such that g{x) is monotoni- cally increasing with \x \ . The function g(- ) may be chosen as g (-) = (·)^ . Alternatively, {x(0) , x(T— 1) } may be found by minimizing the second least-squares criterion (13), with u(i) being replaced by its possibly scrambled version s(t) , and augmented with a term that penalizes time variation of the transmitted phase from each emitter unit. More precisely,

{Θ*(0) , ..., Θ*(Τ - 1) } = argmin

As a skilled person in the art will recognize, criteria (26) and (27) may be minimized by using gradient searches or alternating optimization schemes as discussed above.

Fig. 3 illustrates the function implemented in the precoder in the first and second examples. In Fig. 3, a precoder 31 1 comprises a matching module 314. The matching module 314 is arranged to take as input the information symbol vector u and to output the transmit vector x.

Fig. 4 illustrates the functions implemented in the precoder in the extended example. In Fig. 4, a precoder 41 1 comprises a scrambler 415 and a matching module 414. The scrambler 415 is arranged to take as input information symbol vectors

{u(i) } _i= Q and to output scrambled symbol vectors {s(i) } _i=0 . The scrambled symbol vectors {s(i) } ₊₌ n are fed to the matching module 414 in order to process τ-ι

t=0 and output the resulting transmit vectors

In Fig. 5, a receiver 530 comprises a receiving device 531 in the form of an antenna, a microphone or the like. The received signals is fed to a detector 532 that performs signal processing optionally comprising descrambling, and outputs the restored signal. The modulo-constant used for each receiver and the scrambling matrix are known beforehand to each receiver.

The receivers 530 served by the transmitter system are in one example selected using an optimization criterion that maximizes the weighted sum of the transmis- sion rates offered to the respective receivers. The weights used are in one example selected based on a fairness criterion.

In Fig. 6, a method 600 for precoding in a transmitter system according to the first example comprises the following steps. In a first step 670, a first set of samples destined for the receivers along with information related to a communication channel H between the transmitter system and a plurality of receivers is obtained. In a second step 675, a second set of complex sample values is initialized. In a third step 680, a gradient vector of the approximation error is computed. In a fourth step 685, the second vector of complex sample values is updated based on said gradient vector. In a fifth step 690, it is evaluated whether the gradient search has converged to within satisfactory accuracy. In one option, the search is considered to have converged if a predetermined number of iterations has been carried out. In another option, the search is considered to have converged if the decrease in approximation error from one iteration to the next is less than a predetermined threshold. The process is iteratively repeated from the step 680 until convergence has been determined. In Fig. 7, a method 700 for precoding in a transmitter system according to the second example comprises the following steps. In a first step 770, a first set of samples destined for the receivers along with information related to a communication channel H between the transmitter system and a plurality of receivers is obtained. In a second step 775, an iterative algorithm is initiated by initalizing a perturbation vector. In a third step 780, a second set of complex sample values is computed keeping the perturbation vector fixed. In one option, this is accomplished by means of a gradient search 600. In a fourth step 785, the perturbation vector is updated by solving an integer-constrained least-squares problem, keeping fixed said second set of complex sample values. In a fifth step 790, it is evaluated whether the iterative search has converged to within satisfactory accuracy. In one option, the search is considered to have converged if a predetermined number of iterations has been carried out. In another option, the search is considered to have converged if the decrease in approximation error from one iteration to the next is less than a predetermined threshold. In a further option, the search is considered to have converged if the perturbation vector has not changed from one iteration to the next. The process is iteratively repeated from the step 780 until convergence has been determined.

In Fig. 8, a method 800 for precoding comprises the steps of obtaining channel in- formation 870, obtaining the first set of sample values 875, and forming the second set of complex sample values 880 based on the first set of sample values and the channel information, wherein the complex sample values of the second set have a constant envelope.