WO2013034202A1

WO2013034202A1 - Technique for performing a transmission over a channel having a state history

Info

Publication number: WO2013034202A1
Application number: PCT/EP2012/000108
Authority: WO
Inventors: Eckhard Ohlmer; Udo Wachsmann
Original assignee: Telefonaktiebolaget LM Ericsson AB
Current assignee: Telefonaktiebolaget LM Ericsson AB
Priority date: 2011-09-06
Filing date: 2012-01-11
Publication date: 2013-03-14
Anticipated expiration: 2014-03-06
Also published as: EP2754256A1

Abstract

A technique for performing a transmission over a channel (300; 400) is provided. As to one method aspect of the technique, the method comprises the steps of obtaining a transmission measure (f, I; I _out; P _out; ϒp, Hp) and transmitting over the channel (300; 400). The transmission measure is based on a state history (Η_Δ) of a channel useable or the channel (300; 400) used for the transmission. The transmission uses a transmission rate (f; I'; I(ϒ,), based on the transmission measure.

Description

Technique for Performing a Transmission over a Channel

having a State History

Technical Field

The present disclosure relates to a transmission technique. More specifically, and not by way of limitation, the disclosure relates to a technique for performing a transmission over a channel having a state history.

Background

Modern societies increasingly rely on rapid and reliable exchange of information. Various mobile communication standards, including Universal Mobile Telecommunications System (UMTS) and Long Term Evolution (LTE), have been defined for wireless information exchange.

The supported, i.e., theoretically possible data throughput of a wireless communication often changes with time. One reason for the time-dependency may be that one or both of the communication partners are moving. Furthermore, objects acting as obstacles or reflections in the path of signal propagation may move.

These or other factors lead to uncertainty as to the actually supported transmission rate. As a consequence, some existing transmission techniques assign overly conservative transmission rates to a sender.

Summary

There is a need for a technique for performing a transmission at a transmission rate that is at least in some environments or under certain conditions more efficient.

According to one aspect, a method of performing a transmission over a channel is provided. The method comprises the steps of obtaining a transmission measure based on a state history of a channel useable or used for the transmission; and transmitting over the channel using a transmission rate based on the transmission measure. The method may allow obtaining the transmission measure derived from the state history of the same channel that is later used for the transmission or of another channel that is equivalent to the one that is used. In some embodiments, basing the transmission rate on the transmission measure may reduce a computational complexity of computing or obtaining or adjusting the transmission rate. The method may include receiving a feedback. Alternatively or in addition, basing the transmission rate on the transmission measure may reduce an overhead of the communication or the feedback.

In some embodiments, the method may allow that the channel can be used efficiently, i.e., the transmission rate that is supported by the channel is exploited by the transmission and hence less transmission resources remain unused or wasted. Basing the transmission rate on the transmission measure may, in some embodiments, avoid a complex dependency on the full channel state or on the full state history. In same or some other embodiments, the method may allow defining the transmission rate on a block-by-block basis. Defining the transmission rate may encompass setting, changing, updating and/or adjusting the transmission rate in a time-variant communication environment.

The transmission measure may relate to the time of the transmission. The transmission measure may indicate a quality of the channel. The transmission measure may indicate a channel quality for the time of the transmission.

The state history may also be referred to as Channel State Information or CSI. The state history of the channel may relate to a time prior to the transmission. The state history of the channel may indicate at least one of a past quality of the channel, a past change of the channel quality, a sequence of the channel quality in the past, and a temporal course of the channel quality in the past. The state history may include statistics of a time-dependent channel quality. "Past" may refer to any time prior to the transmission or a time prior to a delay (which is also denoted by Δ) prior to the transmission. "Past" may also include a zero delay. Alternatively or in addition, one block may be transmitted out of a series of blocks to be transmitted. The state history may relate to previous transmissions of blocks in the series.

The channel may comprise one or more of a sender of the transmission, a receiver of the transmission and a connection between sender and receiver. The state history may indicate an end-to-end quality of an "effective channel" usable or to be used for the transmission. The state history of the channel may relate to least one of a pre- coding or any other processing at the sender, the connection, and an equalizing or any other processing at the receiver. Alternatively or in addition, the state history may include at least one of characteristics of a signal processing at the sender (e.g., of a linear precoding), a condition or quality of the connection, and a signal processing at the receiver (e.g., of an equalizer). The connection may be partially or completely wireless. The wireless connection may be a radio connection. The channel may be a radio channel. At least one of the receiver and the sender may be collocated with a mobile communication device or with a base station of a mobile communication network, respectively or vice versa.

The transmission may be structured in blocks. The transmission rate may correspond to an amount of information, or any other measure of information content, that is output in one block. Alternatively or in addition, the channel may be structured in a plurality of sub-channels. The transmission rate (also denoted by f) may correspond to an amount of information (also denoted by I) that is input to one or all of the plurality of sub-channels. The information I may be any measure of information content. The transmission rate (which may be measured in bits per channel use, bits per block, per sub-channel and/or per time) may be commensurable with mutual information defined by the logarithm to base two or any other predefined base. The mutual information may be the mutual information between input and output of the channel. A definition of the mutual information can be found in T. M. Cover and 1 A. Thomas, Elements of Information Theory. Wiley-Interscience, 2006.

The step of transmitting may include applying or changing the transmission rate at the sender prior to the transmission. The transmission rate may be applied or changed by defining the amount of information transmitted per block and/or per subchannel. The transmission rate may be changed so as to adaptively adjust of the transmission rate in a time-variant and essentially block-static communication environment. A clock of a modulation of the transmission may be set so that the amount of information corresponds to the transmission rate. E.g., in the case of the transmission measure being a mutual information, the information transmitted per block and/or per sub-channel may be based on the mutual information.

A state of the channel (to which is also referred to as "the state" for brevity) may be represented by a matrix of state coefficients. A number of rows and/or a number of columns of the state matrix may correspond to a number of sub-channels of the channel.

The state history may include at least one past state of the channel. Each of the at least one past state may indicate a state of the channel at a different point in time. The points in time may be periodic. The state history may include a sequence of past states.

Alternatively or in addition, the state history may include statistics of the at least one past state of the channel. "Alternatively" may encompass that the at least one past state, which statistics is included or represented by the state history, need not be included in the state history. The statistics of the state history may include at least one statistical characteristics of the at least one past state. The statistical characteristics may include at least one of a mean value, a variance, a covariance and a correlation, or any other statistical parameter involving a first moment, a second moment or any higher moment of the at least one past state. The state history may further include joint statistics of the at least one past state and an actual channel state during the transmission. E.g., the state history may include a correlation between at least one past state (which may be known) and the actual channel state during the transmission (which may be unknown at the time of the obtaining). Alternatively or in addition, the state history may include joint first and second order statistics of a first function of the past channel states and a second function of the actual channel state during the transmission. The first function may represent a SINR corresponding to the past channel states. The second function may represent a SINR corresponding to the actual channel state during transmission.

The at least one past state may include one or more or all states in a time frame prior to the transmission. The time frame may be defined relative to the time of the transmission (which is also referred to as a running time frame). The time frame may have a predefined duration. The time frame may be separated from the time of the transmission by the delay. The time frame may lag the time of the transmission by the delay. Alternatively or in addition, the at least one past state may include a predefined number M of past states. For example, the at least one past state may include the M latest past states. "Latest past states" may refer to the latest states known at the sender (e.g. received by means of the feedback). The latest past state (e.g, a most up to date state that is known) may be delayed by Δ. Other past states may be subject to a higher delay. The number M may assume any one of the preset values 1, 2, 3, 4, 5, 6, 10, or 12, or any value larger than 2, or any other natural or integer number. A past state that is older than the M latest past states may be deleted or excluded from the state history or may not be considered in the state history.

A number P of past states, on which the statistics of the state history is based, may be multiple times larger than the number M of the at least one past state. The statistics may be based on a plurality of state histories. The statistics may be statistics of the M past states.

Obtaining the transmission measure may include at least one of deriving the transmission measure and receiving the transmission measure. The transmission measure may be derived at the sender of the transmission. In addition, the state history may be determined at the sender. Determining the state history may include maintaining the state history. Alternatively or in addition, determining the state history may include receiving a part of the state history ("partial state history") or receiving the state history in its entirety ("complete state history"). The at least one past channel state and/or the at least one statistical characteristic may be estimated or measured at the receiver, at the sender, or partly at both the sender and the receiver. The partial state history or the complete state history may be fed back from the receiver to the sender. Further, the transmission measure may be received at the sender of the transmission. The transmission measure may be received or processed by an obtaining unit of the sender. The transmission measure may be at least one of derived at a receiver of the transmission and fed back from the receiver of the transmission to the sender. For deriving the transmission measure at the receiver, the state history may be maintained at the receiver.

The transmission measure may be derived from the state history. The transmission measure may be derived solely from the statistics of the state history.

Maintaining the state history may include maintaining a pool of past states. The pool of past states may encompass the P past states on which the statistics is based. The at least one statistical characteristic may be derived from the pool of past states.

The method may further comprise receiving one or more of the at least one past state and the at least one statistical characteristic. The at least one past state and/or the at least one characteristic may be received from the receiver of the transmission. The state history may be maintained at the sender. The received partial state history may include at least one past state of the channel. Alternatively or in addition, the received partial state history may include at least one statistical characteristic of the at least one past state. Maintaining the state history may include updating the state history based on the received partial state history. The received partial state history may augment the maintained state history with a more recent past state. The more recent past state may be more recent than past states on which the state history had been based. The received complete state history may include a plurality of past states of the channel or a complete set of statistical characteristics. Maintaining the state history may include replacing the state history by the received complete state history. Updating or replacing the state history may be trigged by the reception.

The transmission measure may be a prediction (also referred to as "predicted transmission measure"). Deriving the transmission measure may include the prediction. The transmission measure may be predicted based on the state history. The prediction may use a maximum likelihood estimator. The predicted transmission measure may be predicted for the time of the transmission. The predicted transmission measure may predict the channel quality. The channel quality may relate to a quality of at least one of the sender, the receiver and the connection from the sender to the receiver. The prediction may account for at least one of noise and interference in at least one of the sender, the receiver and a transmission path therebetween. The noise may be externally induced. The interference may be induced between subchannels of the channel.

The transmission measure may have a lower complexity than the state history or the at least one past state. The transmission measure may have the lower complexity by having a reduced number of parameters compared to the state history or by having less than M parameters. The transmission measure may allow for a numerical representation that is shorter than a numerical representation of the state history or the at least one past state. The transmission measure may be a single real or complex value. The state history may be a record of values, a vectorial quantity or a matrix quantity. The transmission measure may be a function of the state history. The function may be non-invertible, non-injective or non-surjective.

The transmission measure may include at least one of a supported transmission rate, a Signal to Interference Ratio, a Signal to Noise Ratio, a Signal to Interference and Noise Ratio (SINR), a mutual information, at least one state of the channel, an out- age probability of the transmission, an effective mutual information and any post equalization signal, or a prediction thereof (i.e., the predicted transmission measure). The SINR may be a ratio of received signal strength of a desired signal to the received strength of undesired signals (e.g., noise and/or interference). Outage may encompass an unsuccessful transmission, e.g., an unsuccessful transmission of one block. The effective mutual information is also referred to as "mutual information with outage". The effective transmission rate may indicate or predict the mutual information or the transmission rate, respectively, reduced according to the outage probability of the transmission using the transmission rate for the transmission. The mutual information with outage may be computed based on the mutual information and the outage probability. The effective transmission rate may account for losses in the transmission due to outage of the transmission.

Outage may occur if a transmission rate applied at the transmission is higher than the supported transmission rate at the time of the transmission, or if the transmitted information content is higher than the mutual information at the time of the transmission. The mutual information may define the supported transmission rate. The SINR may be a function of the mutual information. The mutual information may be a function of the SINR. The function relating the SINR and the transmission rate may be tabulated. The function may depend on the modulation of the inputs. The modulation may be different for different inputs. The function may be a monotonic function.

The predicted transmission measure may include at least one of a prediction of a supported transmission rate (also referred to as "predicted transmission rate" and denoted by f_p), a predicted Signal to Interference Ratio (SIRp), a predicted Signal to Noise Ratio (SNRp, also denoted by γ_ρ ), a predicted Signal to Interference and Noise

Ratio (SINRp, also denoted by γ_ρ ), a predicted mutual information (I_p) of the channel or between the sender and the receiver, at least one predicted state (H_p) of the channel, a predicted outage probability (P₀ut)_/ and a predicted effective mutual information (lout). The predicted transmission rate may result from an optimization. The predicted outage probability may predict an outage of the transmission when using the transmission rate or assuming usage of the transmission rate. Alternatively or in addition, the predicted outage probability or the predicted effective transmission rate may be a function of the transmission rate, the mutual information or the SINR. The optimization may maximize the effective transmission rate. The optimization may determine the transmission rate so as to maximize the effective transmission rate. The effective mutual information may be the mutual information reduced by an expected loss (at a transmission rate set or to be set at the sender).

The method may further comprise inferring a probability distribution of the transmission measure. The probability distribution may be represented by a cumulative distribution function (also abbreviated by cdf) or a probability density function (also abbreviated by pdf). The probability distribution may be inferred at the sender. The probability distribution may be updated based on the obtained transmission measure

The probability distribution may be an approximation or estimation. The approximated probability distribution or estimated probability distribution may have one or more parameters. The probability distribution may be approximated or estimated by setting the parameters of the probability distribution. The parameters may be moments of the probability distribution, e.g. the first moment and the second moment of the transmission measure. The probability distribution may be recursively estimated. The parameters may be iteratively or recursively estimated or updated, e.g. block-by-block. The approximated probability distribution or estimated probability distribution may be a probability distribution of a logarithm of the transmission measure.

The probability distribution may be a joint probability distribution or a conditional probability distribution. The probability distribution may represent a correlation between an actual transmission measure and the predicted transmission measure. The actual transmission measure may be the transmission measure prevailing or occurring at the time of the transmission. More specifically, the actual transmission measure may be a random variable of the transmission measure at the time of the transmission. The joint probability distribution may be a (bivariate) probability distribution of the actual transmission measure and the predicted transmission measure. The joint probability distribution may be represented by a copula function, a first marginal distribution function of the actual transmission measure and a second marginal distribution function the predicted transmission measure. Alternatively or in addition, the joint probability distribution may be represented by a Bayesian network including the conditional probability distribution. The conditional probability distribution may be a probability distribution of the actual transmission measure conditioned to the predicted transmission measure. The transmission rate may be computed based on the joint probability distribution or the conditional probability distribution. The transmission rate may be computed by conditioning the probability distribution on the transmission measure, e.g. the predicted transmission measure. The transmission rate may be computed by optimizing the actual transmission rate and conditioning the probability distribution on the predicted transmission rate.

The method may further comprise predicting a channel state (also denoted by H_p) based on the state history. The channel state may be predicted for the time of the transmission. The channel state may be predicted according to a maximum likely- hood estimator or a minimum mean square error estimator. The statistics of the state history may include an auto covariance of an actual channel state at the time of the transmission and/or an auto covariance of the at least one past state (e.g., the M past states). The statistics may further include a cross covariance between the actual channel state and the at least one past state (e.g., the Λ/past states). The auto covariance and/or the cross covariance may be represented by matrices. The predicted channel state may be predicted by computing a matrix product including the cross covariance and an inverse of the auto covariance.

The transmission measure, e.g. the predicted transmission measure, may be derived from the predicted channel state. A relation defining the transmission measure as a function of the predicted channel state may depend on an equalizer (also referred to as linear receiver or being part of a linear receiver). The equalizer may be a Zero Forcing equalizer (ZF equalizer) or a Minimum Mean Square Error equalizer (MMSE equalizer). The equalizer may be non-linear (also referred to as successive cancellation receiver).

The channel may provide a plurality of sub-channels. The channel may include more than one input. The more than one input can be transmitted via a respective one of the sub-channels. The channel may comprise more than one output. The channel may be a Multiple Input Multiple Output channel, or MIMO channel. A number of outputs of the channel may be equal to or greater than a number of inputs. The transmission may be spatially multiplexed. The channel may comprise a plurality of sending antennas. Each input may be associated with a different one of the plurality of sending antennas. The channel may comprise a plurality of receiving antennas. Each output may be associated with a different one of the plurality of receiving antennas. At least one of the plurality of sending antennas and the plurality of receiving antennas may be spatially distributed. The spatially distributed antennas may be physically separated from one another.

The channel may allow individually receiving each of the one or more inputs. The inputs may be superimposed by the channel. The superposition may yield the outputs. A decoupled estimate of the inputs can be derived from the outputs. The inputs may be received by decoupling (yielding the post-equalization signal). The inputs may be received by multiplying the one or more outputs with an equalizer matrix. The equalizer matrix may be configured to suppress spatial interference (according to the ZF equalizer). Alternatively or in addition, the equalizing matrix may be configured to minimize a mean square error between the channel input and a result of the decoupling (according to the MSE equalizer). Alternatively or in addition, the decoupling may include a successive cancellation (also referred to as a non-linear receiver), sub-channel by sub-channel or in groups of sub-channels, of an interference. The interference may the interference of previously detected inputs. The successive cancelation may be implemented at the receiver. The successive cancelation may be based on knowledge of the channel state. The successive cancelation may assume that all (prior) cancelation steps were successful. In the case of a linear or non-linear receiver, the transmission measure (e.g., the SINR) may be a transmission measure per sub-channel.

The sub-channels of the channel may correspond to spatial layers of the channel. The sub-channels arise from a concatenation of a precoder (at the sender) and a receiver-side processing (e.g. the linear on non-linear receiver). Alternatively, the sub-channels may be input to a (linear) precoder at the sender. The sub-channels may be mapped to the spatial layers by the precoder at the sender. The precoder may be represented by a precoding matrix.

The transmission rate may fulfill a target. The target may be predefined. The target may be a target quantity or a target interval. Fulfilling the target may include optimizing or maximizing a target function. The transmission rate may be computed so as to optimizing or maximizing the target function. The target function may be a function of the transmission rate and the predicted transmission measure. The transmission rate may be chosen or computed such that the target fulfills or evaluates to the target quantity for the predicted transmission measure. The target function may depend on the transmission rate through the actual transmission measure at the time of the transmission, particularly through the actual SINR at the time of the trans- mission. The function relating the SINR and the transmission rate may be used to compute the transmission rate fulfilling the target.

The target may be the outage probability. The target interval may be a range of the outage probability. The target function may be the outage probability as a function of the transmission rate or of the SINR. The predicted transmission measure, e.g., the predicted SINR γ_ρ , may be a parameter of the outage probability. The outage probability may indicate or predict the outage of the transmission using the transmission rate for the transmission. The outage probability may be computed based on the probability distribution.

Alternatively or in addition, the target may be the mutual information with outage or the effective transmission rate in the presence of outage. The target function may be the mutual information with outage, or the effective transmission rate, as a function of the transmission rate or of the SINR.

The channel may include a plurality of sub-channels. The sub-channels may correspond to the multiple inputs of the MIMO channel. The channel may use multiplexing in at least one of space and frequency. The sub-channels may correspond to the spatial and/or frequency multiplexing.

The transmission measure may relate to the channel in its entire frequency spectrum, in its entire spatial distribution, or in both frequency and space. Alternatively, the transmission measure may relate to one or few sub-channel of the channel.

Alternatively or in addition to each of above relations, the transmission measure may include an individual transmission measure for each of the sub-channels. For example, the mutual information may relate to the entire channel or may individually relate to each of its sub-channels. As a further combinable example, the outage probability may relate to the entire channel or may relate to an individual outage of one or more of the sub-channels.

The transmission rate used for the transmission may be an individual transmission rate for each of the sub-channels. The individual transmission rates used for the transmission may be different or independently optimized for different sub-channels. In the case of successive interference cancelation, a joint optimization over the spatial sub-channels can be employed. The complexity of the transmission measure may be independent of the number of sub-channels or spatial layers.

According to another aspect, a method of performing a transmission over a channel is provided. The method comprises the steps of obtaining a transmission measure based on a state history of a channel useable or used for the transmission; and receiving over the channel using a transmission rate based on the transmission measure.

The state history may include at least one past state of the channel. The at least one state of the channel may be determined at the receiver of the transmission. The at least one state may be determined by estimating or measuring the state or the quality of the channel. The receiver of the transmission may repeatedly provide statistics of the at least one past state to the sender.

Any of the steps and features mentioned in the context of the one aspect can be implemented in the context of any one of the other aspects, and vice versa. E.g., the method may further comprise equalizing the channel output. Obtaining the transmission measure may include deriving or predicting the transmission measure from the state history at the receiver of the transmission. The method may further comprise providing the transmission measure to the sender of the transmission.

According to a further aspect, a computer program product is provided. The computer program product comprises program code for performing the steps of any one of aforementioned methods when the computer program product is executed on one or more computer devices, optionally connected in a data network. Advantageously, computational resources can be reallocated depending on a state (online/offline) of a receiver or a sender of the transmission. The computer program product may be stored on a computer-readable recording medium or provided for download on the computer-readable recording medium in the data network.

According to a still further aspect, a device for performing a transmission over a channel is provided. The device comprises an obtaining unit adapted to obtain a transmission measure based on a state history of a channel useable or used for the transmission; and a transmitting unit adapted to transmit over the channel using a transmission rate based on the transmission measure. According to another aspect, a device for performing a transmission over a channel is provided. The device comprises an obtaining unit adapted to obtain a transmission measure based on a state history of a channel useable or used for the transmission; and a receiver unit adapted to receive the transmission over the channel using a transmission rate based on the transmission measure.

Any of the devices presented herein may be realized as or comprised by a mobile or stationary terminal, a mobile telephone, smartphone, data or network card, network node (e.g., of an access network), and in other ways. A particular device may be figured to transmit wirelessly (e.g., based on UMTS or LTE) or via a wired (e.g., cable-base) connection.

Any one of above devices may further be adapted to perform the steps of any one of aforementioned methods and/or may comprise any one of the features mentioned in the context of aforementioned method aspects.

Brief Description of the Drawings

Further features, advantages and technical effects of the disclosure will become apparent in below detailed description of exemplary embodiments with reference to the accompanying drawings, in which:

Fig. 1 schematically illustrates an embodiment on a sender side of a device for performing a transmission over a channel;

Fig. 2 schematically illustrates an embodiment on a receiver side of a device for performing a transmission over a channel;

Fig. 3 schematically illustrates the channel of Figs. 1 and 2 on a spatial layer including a plurality of sending antennas on the sender side and a plurality of receiving antennas on the receiver side;

Fig. 4 schematically illustrates the channel of Figs. 1 and 2 on a logical layer including a plurality of sub-channels; Fig. 5 is a flow diagram illustrating an embodiment of a method of performing a transmission over the channel of Fig. 3 or 4 on a sender side performable by the device of Fig. 1;

Fig. 6 is a flow diagram illustrating an embodiment of a method of performing a transmission over the channel of Fig. 3 or 4 on a receiver side performable by the device of Fig. 2;

Fig. 7 is a flow diagram illustrating a more detailed embodiment of the method of Fig. 5;

Fig. 8 is a flow diagram illustrating a further more detailed embodiment of the method of Fig. 5;

Fig. 9 is a flow diagram illustrating a still further more detailed embodiment of the method of Fig. 5;

Fig. 10 shows two diagrams of a logarithmized post-equalizing Signal to Interference and Noise Ratio (SINR) for Zero Forcing (ZF) equalization and Minimum Mean Square Error (MMSE) equalization, respectively;

Fig. 11 shows for ZF equalization two diagrams of an outage probability and an effective mutual information, respectively; and

Fig. 12 shows for MMSE equalization two diagrams of an outage probability and an effective mutual information, respectively.

Detailed Description

In the following detailed description, for purposes of explanation and not limitation, numerous specific details are set forth in order to provide a thorough understanding of the invention. It will be apparent to one skilled in the art that the present invention may be practiced without some or all of these specific details, or in other embodiments that depart from these specific details. In other instances, well-known methods, procedures, components and circuits have not been described in detail so as not to obscure the present invention. Those skilled in the art will further appreciate that the functions explained herein may be implemented using individual hardware circuitry, using software functioning in conjunction with a programmed microprocessor or general purpose computer, using an Application Specific Integrated Circuit (ASIC) or using one or more DSPs. It will also be appreciated that the technique described herein could be embodied in a microprocessor and a memory coupled to the microprocessor, wherein the memory is encoded with one or more programs that perform the methods and method aspects disclosed herein when executed by the processor.

Fig. 1 schematically illustrates an embodiment of a device 100 for performing a transmission over a channel. The device 100 comprises an obtaining unit 110 and transmitting unit 120. The obtaining unit 110 obtains a transmission measure based on a state history Η_Δ of a channel. The transmission measure can be a predicted transmission rate f_P/ a predicted mutual information I_out, a predicted outage probability Pout, a predicted Signal to Interference and Noise Ratio γ_ρ or a state H_p of the channel predicted for the transmission period. Exemplary channels 300 and 400 are described below with reference to Figs. 3 and 4, respectively. The state history relates to the channel that is subsequently used for the transmission. The transmitting unit 120 transmits over the channel at a transmission rate computed using the transmission measure.

Fig. 2 schematically illustrates an embodiment of a device 200 for performing a transmission over a channel. The device 200 comprises an obtaining unit 210 and a receiving unit 220. The obtaining unit 210 obtains a transmission measure as described herein based on a state history Η_Δ of a channel. The state history relates to the channel that is subsequently used for the reception. The receiving unit 220 receives the transmission over the channel using a transmission rate computed using the transmission measure.

The transmission setup is described with reference to Figs. 3 and 4. Channels 300 and 400 are shown schematically on a spatial level and on a logical level in Figs. 3 and 4, respectively. The channel 300 is a Multiple Input Multiple Output (MIMO) channel. The transmission is spatially multiplexed over the slowly time-variant and essentially block-static fading channel 300. The channel 300 comprises the transmitting unit 120, a wireless connection 310 and at least one receiving unit 220 with a linear equalizer 222 (which is also referred to as linear receiver). The wireless connection 310 is a radio connection characterized by a channel state H. The channel state H is encoded in a numerical matrix representation in storage of the obtaining units 110 and 210.

Modern wireless communications systems are extensively relying on multi-antenna techniques and exploit an available channel quality by a fast adaptation of the transmission parameters to the (ideally instantaneous) channel state. This is enabled by fast feedback, from the device 200 to the device 100, of information about at least one of precoding parameters, the channel state and the supported transmission rate on the channel 300 or 400. Relying on feedback appears feasible as long as the channel changes slowly as compared to the duration of a transmitted block and the feedback delay. A variety of methods to determine the transmission parameters have been developed. Such methods include, e.g., mutual information based methods published by E. Ohlmer and G. Fettweis, "Linear and Non-Linear Detection for MIMO- OFDM Systems with Linear Precoding and Spatial Correlation," in IEEE Wireless Communications and Networking Conference, 2010. The known methods are mostly based on the assumption of instantaneous and hence accurate feedback. However, feedback delays on the order of milliseconds inevitably occur due to the time required for channel measurement at the device 200 on the receiver side, feedback encoding, feedback transmission and adaptation at the transmitting unit 120. From an adaptation perspective, this delay introduces uncertainty as to the channel quality and hence the supported transmission rate during a future transmission if the channel is time-variant. Even a small uncertainty may result in transmission errors and the transmission rate needs to be adapted such that a certain target error probability is achieved. In that regard, it is useful to exploit the correlation of a state history (i.e. channel state information, or CSI, that is available during adaptation) and a transmission measure (e.g., a CSI during the actual transmission) in order to not assign transmission rates that are unnecessary conservative.

Much progress has been made concerning rate adaption for SISO or multi-antenna combining systems according to G. E. 0ien, H. Holm, and K. J. Hole, "Adaptive Coded Modulation with Imperfect Channel State Information: System Design and Performance Analysis Aspects," in SWC-2002: Book of Extended Abstracts, 2002, to J. F. Paris and A. J. Goldsmith, "Adaptive Modulation for MIMO Beamforming under Average BER Constraints and Imperfect CSI," in IEEE International Conference on Communications, 2006, and to A. Maarev and S. A^'ssa, "Optimized Rate-Adaptive PSAM for MIMO MRC Systems with Transmit and Receive CSI Imperfections," IEEE Transactions on Communications, vol. 3, pp. 821-870, 2009. Those results rely on the accurate knowledge of the channel quality statistics, conditioned on outdated CSI. As an example of relevance also in the context of LTE communication, for Multiple Input Multiple Output (MIMO) spatial multiplexing systems with a linear receiver as the equalizer 222, no satisfactory technique as to the transmission rate is available. Particularly, channel quality statistics conditioned on outdated CSI had not been considered. Detailed embodiments based on a transmission measure that reduces computation complexity are presented. In the context of the equalizer 222 being a linear receiver (also abbreviated by LR), channel quality can be measured in terms of a post-equalization Signal to Interference and Noise Ratio (SINR). The embodiments described herein can perform two approaches to model the SINR statistics, conditioned on the outdated CSI which is available during adaptation in the obtaining unit 110 or 210. Both approaches rely on a log-transformed SINR and yield compact, closed-form approximate expressions for the outage probability per spatial layer. The expressions can be beneficially exploited in order to adjust the transmission rate such that it exceeds the mutual information of the actual transmission with a target outage probability, as described below in more detail.

The communication from the device 100 to the device 200 is a spatial multiplexing transmission over the narrow-band MIMO channel 300. The channel 300 is represented by a matrix H e C^RxT using τ transmit antennas 320 and ^R - ^T receive antennas 330. In a real-world implementation, the elements of the channel input

2 T vector x e C^{Tx l} (indicated by reference sign 340 in Fig. 3) have a covariance ^σχίτ . Their statistics correspond to an independently encoded symbol sequences. More specifically, each channel input carries a separate code word. The rate per channel input can be adjusted individually. The channel output vector ^{e cKx l} (indicated by reference sign 350 in Fig. 3) reads y = ⁱ/v Hx + v, (!) wherein ^v ~ Ncto^i^) denotes additive white Gaussian receiver noise. The normalization models a constant transmit power, regardless of the number of antennas. The signal-to-noise-ratio (SNR) is defined as τ ^{= σχ}/^Τσ . Block static fading often present. In other words, the channel matrix H can be regarded as remaining essentially static during the transmission of a complete block, but may change in between blocks. The change is tracked by the state history Η_Δ/ based on which a future channel state H_p is predicted for the time of the transmission, as is described below.

The rate at which blocks are transmitted on the f-th sub-channel is denoted by r_t. An ideal transmission rate on the f-th channel would be ?^deal = I_t ^■ r_t, wherein I_t s the current mutual information of the f-th channel. It is to be noted that, at the time of transmission or at the time of configurating the transmitting unit 120 of the transmission, the mutual information at the transmission is not known. The ideal transmission rate f^deal is a maximum transmission rate in the sense that an attempt to transmit on the t-t channel at a transmission rate that is higher than the ideal transmission rate,

is unsuccessful. The unsuccessful transmission may imply a complete loss of the block that was attempted to be transmitted. The loss is also referred to as an outage.

In an environment of changing channel quality, and thus changing mutual information, it may be beneficial to choose a transmission rate that entails a significant chance of outage. For example, in a spatio-temporal pattern of channel quality exhibiting short negative peaks of reduced channel quality, the outage may be limited to the short negative peaks. As a result, an effective transmission rate may still be higher than choosing a lower transmission rate that avoids outage with high certainty or high probability.

Throughout, symbols with superscript tilde ( ·~ ) denote quantities on the receiver side after equalization. ΛΓ( , σ²) and -^(μ, σ·²) denote the normal and Complex normal distribution with mean ⁴, variance σ ). italic letters (such as lower-case a and upper-case A bold face letters denote scalars, vectors and matrices.

Pr (·), p( ), F(-) _ancj Ε_α[·] denote a probability, a probability density function, a cumulative distribution function and the expectation with regard to a random vector a. IT. ( )^r and (·)^Η denote the identity matrix of dimension T x T, the matrix transpose and the Hermitian operator, respectively.

The equalizer 222 decouples the channel inputs, superimposed by the MIMO channel 300. Interference and noise are in particular present along the wireless connection 310. The equalizer 222 decouples the channel inputs into a vector ^χ e C^{Tx l} (indicated by reference sign 360 in Figs. 3 and 4) by multiplying the received signal vector Υ with an equalizer matrix. In one variant, the equalizer 222 is configured to completely suppress spatial interference (also referred to as Zero Forcing or ZF). In another variant, the equalizer 222 is configured to minimize the mean square error between channel input and equalizer output (also referred to as Minimum Mean Square Error or MMSE). The receiver 220 measures the MIMO channel matrix H based on received reference symbols. The receiver 220 estimates coefficient values of H for each of the subchannels.

The channel 400 in Fig. 4 schematically illustrates a resulting transmission setup on a logical level. The plurality 410 of sub-channels 420 is subject to noise 430, 432 and mutual interference 434 in one or more of the transmitting unit 120, the wireless connection 310 and the receiving unit 220.

The resulting τ sub-channels ^xt ^→ %t are treated as independent during detection of the transmitted data. Each of the sub-channels t = 1, ... T is characterized by a post-equalization SINR

in the case of a ZF equalizer 222, or by a post-equalization SINR t^MMSE = 7 - 1

^'(Η^ίίΗ + ₇-¹Ι_τ)-¹ J1 £, (3) in the case of a MMSE equalizer 222. The mutual information resulting using a linear receiver (cf. T. M. Cover and J. A. Thomas, Elements of Information Theory. Wiley- Interscience, 2006)

LR ^{T T}

I(x; x| H) = ^ I(i_t; _t | H) =

t.=i 5^ I(7t)

t=i ₍₄₎ is given by the sum over the (individual) mutual information of each of the subchannels. The mutual information ^τ(¾ is a monotonically increasing function of the post-equalization SINR It on each sub-channel for a particular input distribution p(^xt). It is noted that in realistic implementations, any residual post-equalization interference is often Gaussian distributed. Further, the Gaussian distribution is an approximation if X is drawn from non-Gaussian or discrete alphabets. Each of the T parallel sub channels ^xt ^→ t is hence an additive white Gaussian noise channel with SINR 7t. For a detailed discussion is provided in E. Ohlmer, U. Wachsmann, and G. Fettweis, "Mutual Information of MIMO Transmission over Correlated Channels with Finite Symbol Alphabets and Link Adaption," in IEEE Globecom, 2010.

Figs. 5 and 6 show high-level flow charts of methods 500 and 600 of performing a transmission over a channel. The channel can be the channel 300 or 400. The methods 500 and 600 can be performed by the devices 100 and 200, respectively. The method 500 comprises a step 510 of obtaining a transmission measure. The transmission measure is computed using a state history Η_Δ of the channel by one of the device 100 or 200. In the first alternative, the transmission measure, or updating information thereof, is feedback via a dedicated feedback channel or using the channel 300, 400 (in reversed transmission direction, from the device 200 to the device 100) that is also used later for the transmission. The method 500 further comprises a step 520 of deriving a transmission rate from the transmission measure. In some embodiments detailed below and compatible with the method 500, the transmission rate is computed directly from the transmission measure. In further embodiments also detailed below and compatible with the method 500, the transmission rate is computed using the transmission measure as a statistical condition, i.e., as a partial knowledge that reduces uncertainty as to an optimal transmission rate for the later transmission.

The method 600 comprises a step 610 of obtaining a transmission measure. The transmission measure of the method 500 may essentially be identical to the transmission measure in the sense of the method 500 (e.g., the predicted SINR), in which case it is feedback from the device 200 to the device 100. The transmission measure may also be any other quantity used for computing the transmission measure in the sense of the method 500 at the device 100, such as a recently measured state of the channel 300 or 400, or a prediction of the channel state. The channel state is feed^¬ back from the device 200 to the device 100.

A more detailed embodiment of a method 700 is described with reference to Fig. 7. The method 700 includes the steps of the method 500 indicated by like reference signs. For obtaining the transmission measure according to the step 510, statistics of the state history Η_Δ are maintained at the device 100 in a step 710. In an implementation of the step 710, H e C^{TRx l} denotes an actual MIMO channel vector during the transmission of a particular block of interest. The channel vector has a numerical representation encoded in storage of the device 100. The encoding is related to the matrix H of the state by stacking the columns of H. At the time of obtaining the transmission measure in the step 510, the channel vector H \s not known. In other words, H is a random variable. Furthermore, Η_Δ denotes the channel state history, i.e. the CSI available for adaptation Δ^* T_b seconds before the transmission of this frame, wherein Δ-denotes the adaptation delay in number of blocks and 7b is the block duration in seconds, during which the channel remains static. More specifically, ^ηΔ is composed of the M MIMO channel vectors

[Hj_{il 5} . . . , ¾ _M]^r, wherein ^HA_;m with m = 1, M is delayed by Δ+ 77 -Ι blocks as compared to the actual block of the transmission. In the following it is assumed that the unknown future channel state H and the state history Η_{Δ are} jointly complex Gaussian distributed with zero mean, auto covariance matrices ¾ and ^ςΗ_ΔΗ_Δ/ and cross covariance matrix^∑ΗΗ_λ- Based on the statistics of the state history Η_Δ the transmission measure is predicted for the time of the transmission in step 720 as part of the step 510.

As part of transmitting using a certain transmission rate according to the step 520, in a step 730 of the method 700, a probability distribution for the transmission measure is determined. In one variant, the probability distribution is determined by a variational principle. To this end, the obtaining unit 110 maximizes an entropy of the probability distribution under constrains defined by the predicted transmission measure. In another variant, the probability distribution is determined by fixing parameters of a family of distributions. The family of distributions is predefined. The constraints defining the variation or the parameters include a predicted mean value and a predicted variance of the transmission measure.

The probability distribution is a joint or conditional probability distribution that relates the obtained or predicted transmission measure to the transmission rate in a step 740. It is sufficient to relate the obtained or predicted transmission measure to the "actual" transmission measure, i.e. to the random variable representing the transmission measure at the time when the block of interest is transmitted. The transmission rate is then readily derived from computation result for the "actual" transmission measure by means of a tabulated function. The step 520 concludes with the transmission over the channel using the computed transmission rate in a step 750. Fig. 8 shows a flow diagram of a further more detailed method embodiment 800 compatible with each of the methods 500 and 700. The step 720 of predicting the transmission measure includes a step 810 of predicting a state **_P of the channel 300 or 400 for the time of the transmission.

A general result for multivariate Gaussian random variables (cf. , e.g., Chapt. 7.5 in L. L. Scharf, Statistical Signal Processing. Addison-Wesley, 1991), a Minimum Mean Square Error (MMSE) channel predictor, which exploits spatial and temporal correlation, is symbolically representable by

The parameter M of the technique determines a prediction filter length. The parameter M is chosen so as to determine a prediction accuracy and/or a computational complexity of the technique. Throughout, the numerical examples assume that each channel coefficient has unit variance.

For the advantage of a small parameter space, spatio-temporal correlation coefficients between each two channel coefficients at time instants T_x and T∑ and receive- transmit antenna pairs riti and r₂t₂ are defined as

E [H ^tlH ^t2*] (6)

A numerical implementation in accordance with Eq. (6) assumes or represents

1) separability of transmitter and receiver spatial correlation with identical receiver and transmitter correlation coefficients Ps (cf. hereto: A. van Zelst and J. S. Hammerschmidt, "A Single Coefficient Spatial Correlation Model for Multi^¬ ple-Input Multiple-Output (MIMO) Radio Channels," in Gen. Ass. of the Intern. Union of Radio Science, 2002); and/or

2) separability of spatial and temporal correlation.

The temporal correlation coefficient depends on the time difference and the maximum Doppler frequency ^^d-^max as Ρί (^τ2 - Ά) = ίο(2π/_ύΛαχ(τ₂ - Ti))_r wherein ^Jo(-) denotes the zeroth order Bessel function of the first kind, which is a representtation of the so-called Jakes doppler spectrum used in the numerical examples (cf. Chapt. 4.1 in W. C. Jakes, Ed., Microwave Mobile Communications, Wiley-IEEE Press, 1994).

The step 720 further includes a step 820 of computing the predicted transmission measure using the predicted channel state ^hP. In the case of the post-equalization SINR It being used for the transmission measure, the computation of step 820 uses Eq. (2) or (3), or a similar relation between channel state and SINR depending on the employed equalizer 222.

Fig. 9 illustrates a still further detailed embodiment of a method 900 of performing a transmission over the channel 300 or 400. The method 900 is compatible with each of the method embodiments 500, 700 and 800 described above. Corresponding reference signs indicate related or interchangeable steps.

The method 900 allows to predefine a target for the computation of the transmission rate f, or the corresponding mutual information ^' , per sub-channel, given the CSI

ΗΔ, which is outdated from the perspective of the actual transmission. Assuming with a reasoning in L. H. Ozarow, S. Shamai, and A. D. Wyner, "Information Theoretic Considerations for Cellular Mobile Radio," IEEE Transactions on Vehicular Technology, vol. 43, pp. 359-376, 1994, that the transmitted blocks are long enough such that the mutual information ^!(7t) can be achieved. However, the transmitted blocks are short as compared to the channel coherence time, such that the block- static fading assumption holds. The only error event is hence the event of a mutual information outage, which occurs if the selected transmission rate exceeds the mutual information: ⁷ί > ^*). In that regard we pursue two tasks: either to maximize the mutual information with outage or to ensure that a certain target outage proba- bility is achieved. Both tasks can be treated independently for each sub-channel.

The probability of outage is

Eq. (8) states that an outage occurs equivalents to Eq. (7) if the post-equalization SINR It during the future transmission is smaller than some predicted threshold value . and the rate has been chosen as ^Jt' = This holds since the mutual information is monotonically increasing in 7t . The mutual information with outage is thus out,_t = (l - iW¾)) I(¾) - (9)

A notable difficulty in computing the outage probability in Eq. (8), particularly under limited resources of mobile devices, is the lack of closed-form expressions for the conditional probability density function (pdf) Ρ(7^. | Η_Δ)_# for arbitrary spatially correlated channels and arbitrary linear equalizers. In fact, not even the marginal pdf p( t ) is known in the arbitrary spatially correlated channel case or for MMSE based equalization. Further approximations based on the Gamma distribution in the case of MMSE equalization are provided by P. Li, D. Paul, R. Narasimhan, and J. Cioffi, "On the Distribution of SINR for the MMSE MIMO Receiver and Performance Analysis," IEEE Transactions on Information Theory, vol. 52, pp. 271-286, 2006. It is noted in passing that for the problem at hand, the actual channel state at the future transmission, conditioned on the outdated channel Η_{Δ r} has non-zero mean, which further complicates deriving the probability distribution p( t l ^HA ).

To address the problem, the transmission measure, which may be any function of the state history Η_δ (j._e. the outdated CSI), is used to decrease uncertainty to some extent about the future transmission measure, e.g. the future SINR. Such a function would contain the same information about 7t as compared to Η_{Δ #} jf it was a sufficient statistics with regard to 7t . A possible option is to use the post-equalization SINR i_:p, which is computed from the predicted channel ^hP (see the MMSE predictor Eq. (5)), which is highly correlated to the actual SINR it if the outdated CSI Η_Δ was. Again, the joint (bivariate) pdf of actual and predicted SINR ^p rr _P) is hard to obtain in closed form, since even its marginal distributions are not known in general. Therefore, we attempt to construct it from approximations to its marginal distributions and p(7-,p). More specifically, we consider approximations of the marginal distributions of the log-transformed SINR = logfit) an ^log(7t,_P) by the Gum- bel and the Gaussian distribution, in order to construct

₀r tft Wp).

Considerations leading to the construction of the log-distribution are provided below. Similar to Eq. (8), however based on the transmission measure rather than the full state history, the outage probability is given by

if the transmission rate has been chosen as I_t' = I(exp( i')) . Hereto, it is reminded that the mutual information I is a monotonically increasing function in ϊ\ ^{= lo}s(7i). An outage equivalents occurs in case < it but also if ^< % .

The two low-complexity approximations are described below. The approximations are based on a log-transformed post-equalization SINR. Due to compact and closed-form expressions, the technique can be beneficially implemented in mobile devices.

Starting point for the approximation of the marginal distribution Ρ( Ϊ) is the observation that the post-equalization SINR with /?=rantennas, ZF-based equalization and without spatial correlation yields an exponential distribution (cf., e.g., D. Gore, R. Heath Jr., and A. Paulraj, "On Performance of the Zero Forcing Receiver in Presence of Transmit Correlation," in IEEE ISIT, 2002): Pirit) = ^λΗ ^ -^/_Ί \ Consequently, the random variable can be shown to follow an extreme value distribution (cf., e.g., J. F. Lawless, Statistical Models and Methods for Lifetime Data, Wiley-Inter- science, 2003), which is also referred to as Gumbel distribution: ρ(ϊΙ) = i/b_t exp(_¾ - exp(z_¾)) ^_Q^ with ¾ = (^ -"')/6,. The shift and shape parameters a_t and b_t are related to mean and variance ^σ of ΊΪ by b_t = σ,ι _π ^, at = μ^ι -btu, u Euler's constant (11)

(cf. P. F. Rasmussen and N. Gautam, "Alternative PWM-Estimators of the Gumbel Distribution," Journal of Hydrology, vol. 280, 2003). The parameters a_t and b_t are estimate using the relation (11). For the zero forcing equalizer it is found that a_t relates to the SNR as ^a* ^{= lo}§( ) and = l. in contrast, the present embodiment keeps those parameters variable in order to adjust the distribution in case of spatially correlated channels or MMSE based equalization. The respective cumulative distribution function (cdf) is ( i) = l - exp(-exp(z)) (12)

Fig. 10 shows two diagrams of the cumulative probability distribution of the post- equalization log-transformed SINR for ZF-based and MMSE-based linear equalization, respectively. The cumulative probability distributions of the Gumbel approximation and the Gaussian approximation are shown in comparison to a distribution resulting from numerical simulation of a 2x2-MIMO channel. In the left-hand side and right- hand side diagrams, the numerical comparison of the log-transformed post-equalization SINR distribution is obtained from simulations with uncorrelated and severely spatially correlated channels, respectively.

The Gumbel distribution matches the simulated SINR distribution perfectly in case of spatially uncorrelated channels and ZF equalization, as expected. In contrast, the skewness of the SINR distribution is reduced in case of MMSE-based equalization and/or severely correlated channels. In latter case, a representation by the Gumbel distribution becomes inaccurate. The observed reduction in skewness is accounted for by a second approximation of the log-transformed post-equalization SINR, namely an approximation using the Gaussian distribution. The Gaussian distribution is matched to the mean and the variance of the log-transformed post-equalization SINR ¾ for the t-th sub-channel, which result is also shown in each of the diagrams of

Fig. 10.

In the step 730 (compatible with each of the method embodiments), a bivariate joint distribution of ¾_P and ¾ is constructed from its marginal distributions. It is noted that in principle there exists infinitely many solutions for a joint distribution with given marginal distributions (cf. E. J. Gumbel, "Bivariate exponential distributions," Journal of the American Statistical Association, vol. 55, pp. 698-707, 1960 and earlier work). An exact constructing the bivariate distribution requires knowledge of its marginal distributions and, in addition, a dependence function describing the coupling between both random variables (cf. Chapt. 19 in S. Kotz, N. Balakrishnan, and N. L. Johnson, Continuous Multivariate Distributions Volume 1: Model and Applications. Wiley-Intersc, 2000).

In a first variant, the step 730 includes constructing a bivariate Gumbel distribution. E. J. Gumbel, "Bivariate Logistic Distributions studied in "Journal of the American Statistical Association, vol. 56, pp. 335-349, 1961, several ways on how to construct a bivariate distribution from given marginals. The following form appears particularly well suited for the present technique. m I "

-. llr = ^e* - [^~ ^)Γ + (-log ( ¾_p)))

(13) with marginal cdfs according to Eq. (12) with respect to both the actual transmission measure and the predicted transmission measure. One criterion for suitability is the temporal correlation coefficient pt between predicted and actual SINR, which has to satisfy ⁰≤ Pt≤ 1 For other options having a range of the correlation coefficient that is restricted to be smaller are provided, loco citato, by E. J. Gumbel. The parameter™> > l in Eq. (13) characterizes the statistical dependency between _t ^l and i,_p. The parameter m is related to the temporal correlation coefficient Pt through m = (l - pt)-¹'² (cf. chapt. 53 in S. Kotz, N. Balakrishnan, and N. L. Johnson, Continuous Multivariate Distributions Volume 1: Model and Applications. Wiley-Inter- science, 2000). For m = 1, corresponding to Pt = 0, Eq. (13) decomposes into a product of the marginal cumulative distribution functions.

The outage probability can be obtained from the joint distribution in Eq. (13) by using general results for distribution functions (available, e.g., from Chapt. 2, (2.1- 32) in J. Proakis, Digital Communications. McGraw-Hill, 2001):

By calculating the derivative in Eq. (14), the outage probability Ρ_οχΛ is determined in a step 910 according to

P- m = exp (- («& + ¾,) *) + C ,) *³" ^C¾-¹ ₁ - exp(-Lp(_¾J) (15) with

¾_P

c i' = - ¹°g (¹ - ^exp(-^exP(¾'))) > ¹¹ b t Eq. (15) is thus fully specified by the temporal correlation coefficient pt and the shift and shape parameters a_tl b_t and _t,_p, b _v, which are derived from the (joint) first and second order moments of η[ and ¾_p using Eq. (11).

In a second variant, the step 730 includes constructing a bivariate Gaussian probability distribution. Based on the observation that the log-transformed post-equalization SINR approximately follows a Gaussian distribution if MMSE-based equalization is employed or the channel becomes spatially correlated, the second variant constructs for t and 7t,_p a joint or conditional Gaussian probability distribution. Basic results for Gaussian random variables are provided, e.g., in Chapt. 7.5 of L. L. Scharf, Statistical Signal Processing, Addison-Wesley, 1991). Mean μί and variance

of the conditional pdf Ρ {ΊΪ \ Ί[,_Ρ) are derivable from the joint first and second order moments of 7t and 7i,_P as follows:

2 _ 2 4 -2

it ~t~t , p 7t ,_{p <} (17)

The outage probability can then be obtained from ρ(7ί Ι τ.,_ρ) as solution to the integral

wherein erfc(u) = ~

denotes the complementary error function.

Eq. (18) provides a remarkably efficient means to evaluate the outage probability in a numerical implementation of the step 910. Eq. (18) thus provides numerically very efficient means to adjust, based on 7t,_p obtained directly from the predicted channel

(using Eq. (3), (4), or a similar relation), as

In Eq. (19), the term b) is the expectation value ^Ejri solely depending on the predicted log-transformed SINR through Eq. (16). The term a) is a constant correction term, which does only depend on the variance σΐ of the conditional pdf that characterizes the uncertainty contained in ϊ[ about _t ^l and the desired target outage probability P₀'_ut. Increasing the uncertainty

e.g. through higher velocities, or decreasing P₀'_ut results in choosing ϊ[' (and thus the transmission rate) more conservatively. Eq. (19) is preferably implemented by means of a lookup table of

An implementation of the step 730 using a recursive parameter estimation is described. Both variants (i.e., approximating the probability distribution by a Gumbel probability distribution or by a Gaussian probability distribution) require knowledge of the joint first and second order moments of η[ and ϊ[,_ρ, which are not known a priory, but are estimated recursively during the transmission of multiple frames (or blocks) for any of above embodiment as described below.

Let °W and \ be the -th realization of a random process fulfilling (essentially) wide- sense stationarity. Estimates of mean and covariance (likewise for the variance) are obtained from the recursive mean estimator i— 1 1

α[ϊ] =——μα[ϊ - i] + -a[i]

*M = i— L - ΐ] + %— 1 - - /Mⁱl) · (2°)

In one implementation, _t ^l is available Δ blocks after t,_p such that the moments cannot be estimated during the first frames (or block) of a transmission. In another implementation this is circumvented by starting the estimation a number of frames (or blocks) before the actual transmission (i.e., before the time of the transmission), which also improves the estimation accuracy.

Numerical examples are described with reference to the Figs. 11 and 12, each of which shows a diagram of the outage probability per sub-channel and a diagram of the mutual information with outage. More specifically, the mutual information is an average sum. A 2x2 MIMO channel is simulated with 7 = 10dB _ancj Gaussian input signals. The MIMO system operates at 2.5 GHz. The state history is represented by MIMO channel vectors ί¾_Δ. The state history comprises M=2 blocks with a delay of

Δ=8 blocks. 7[_> denotes the block duration. If 7^=1 ms holds, the delay is 7£,·Δ=8 ms. The M=2 blocks are available to the obtaining unit 110 or 210 during adaptation. Relative velocity is on the abscissa. Results at different velocities and for ZF-based and MMSE-based equalization, with and without spatial correlation, are shown in Figs. 11 and 12, respectively. and hence the rate which achieves a

from the Eq. (15) or (18) for the outage probability using a bisection algorithm. In the Figs. 11 and 12, the respective results are denoted by the word "Gumbel" and "Gauss" depending on the used approximate probability distribution.

In a second substep of the step 910, the mutual information with outage (i.e. the effective mutual information) according to Eq. (9) is maximized using a combination of bisection and direct search algorithm (cf. for further details D. P. Bertsekas, Nonlinear Programming. Athena Scientific, 1999). The respective results are denoted "Gumbel max" and "Gauss max" in the diagrams of Figs. 11 and 12.

While the substep 910 yields the SINR, the step 740 provides the transmission rate using in step 920 a monotonic function, which is preferably tabulated in storage of the device 100.

For a comparison, we consider a benchmark embodiment of the transmission technique. The post-equalization SINR 7^t,p is computed from the MMSE channel prediction according to Eq. (5), and the post-equalization SINR 7t,p is directly used to select the transmission rate as I_t' = _P). (This MMSE-adaptation is not to be confused with an MMSE-based equalization.) Results for the benchmark embodiment are indicated by "MMSE" in the diagrams of Figs. 11 and 12.

The MMSE-adaptation of the benchmark embodiment causes the outage probability and the mutual information with outage to decrease with increasing velocity. This can be explained from the fact that the channel prediction becomes increasingly biased with increasing velocity. E.g., in a further embodiment for a SISO transmission (not shown in Figs. 11 and 12), a single channel coefficient is predicted from a single outdated channel coefficient according to ^HP ⁼ PtH . The predicted channel thus approaches zero as the correlation approaches zero. The outage probability Ρ_{0 Λ} achieved with the benchmark embodiment is about 0.5 in the low velocity regime, which highlights the advantages of the more reliable rate adaptation of embodiments based on the target function (such as _ut and particularly ^out)- The non-monotonic shape of the curves is due to the temporal correlation function of the channel which follows a Bessel function. It can be seen that the Gumbel and Gauss approximations (i.e., wherein the outage probability Ρ_οιΛ is the target function) can guarantee not to exceed the predefined outage probability target.

Further, the transmission rate is still "conservative" in the low velocity regime, which indicates that the construction of the bivariate distribution Ρ^[ [_Ρ) not completely captures the high statistical dependency between [ and

in the low velocity regime. In case of MMSE-based equalization, the Gaussian approximation turns out to be more accurate due to the reduced skewness of the log-SINR probability distribution (as pointed out above with reference to Fig. 10). As an additional advantage, the statistical dependency between Η_Δ and H is at least partially exploited. This advantage is shown in Figs. 12 as the mutual information with outage significantly increases towards the low velocity regime (cf. the diagram on the right-hand side) at a constant outage probability (cf. the diagram on the left-hand side). It almost approaches the mutual information with perfect adaptation, i.e. no outages, indicated by "perf. CSI" in each of Fig. 11 and 12.

Higher velocities and, hence, an increased uncertainty about the channel during the actual transmission cause a large mutual information loss compared to the case of perfect CSI, i.e. perfect knowledge of the channel state at the time of transmission. The uncertainty is numerically represented by the conditional pdf p(7t l¾_p) becoming increasingly spread with increasing velocities, since the uncertainty about the future channel quality cannot be reduced by the outdated CSI, i.e. by the state his^¬ tory, which is available during adaptation. The time of adaptation includes the obtaining and the application of the transmission rate. In a more advanced variant (which results are not included in the Figs. 11 and 12), a higher prediction filter length M would increase the velocity range, in which the velocity-caused loss remains small.

In what follows, the results indicated "Gumbel max" and "Gauss max" for using the outage probability -£,_ut as the target function are described. Regarding the maximal mutual information with outage, it can be seen that small outage probabilities are optimal in the low velocity regime. An outage probability of about 0.4 (i.e., a less conservative transmission) becomes optimal at high velocities. As a result, the mutual information of the MMSE-based adaptation of the benchmark embodiment is exceeded over the complete velocity range. Again, the Gaussian approximation turns out to be more accurate for an MMSE-based equalizer 222, or in any other case of spatial correlation, as compared to the Gumbel distribution, and hence achieves a higher mutual information with outage.

Advanced variants compatible with each of above embodiments incorporate the channel estimation noise, allow for a transmission over multiple independently fading frequency bands in broadband multi-carrier systems, apply successive interference cancelation receivers, or combine the presented technique with higher layer error correction mechanisms such as hybrid ARQ.

As has become apparent from the above description of advantageous implementations, some embodiments of the technique disclosed herein allow a transmission at a transmission rate that is more frequently, or with less computational complexity, adjusted to a changing channel state. In some embodiments, uncertainty about a current channel quality and/or a currently supported transmission does not necessarily lead to assigning overly conservative (e.g., lower than optimal) transmission rates. A certain target error probability may be predefined and/or achieved. Computational requirements can be reduced where applicable. Some battery driven devices may achieve a longer time of operation due to the lower computational complexity. The embodiments can adapt the transmission rate per spatial layer based on outdated channel state information.

Two approximate low-complexity approaches, based on the log-transformed post- equalization SINR, have been presented to adjust the transmission rate adaptively on a per-block basis. Both approaches can be applied in order to maintain a certain target outage probability or to maximize the mutual information with outage. The methodology is suited for arbitrary input alphabets. Some of above embodiments allow a rate adaptation for MIMO spatial multiplexing receivers based on a state history representing outdated CSI. The log-transformed post-equalization SINR distribution can be approximated or predicted by a Gumbel probability distribution and, more accurately, by a Gaussian probability distribution, in some embodiments. These approximations serve as a basis or as auxiliary quantities to express the outage probability in closed form. The provided closed-form expression may allow for an efficient numerical implementtation. Certain expressions are suited to adjust the transmission rate adaptively on a per-block basis in order to achieve the target outage probability or to maximize the mutual information with outage. While the technique presented herein has been described in relation to its preferred embodiments, it is to be understood that this description is for illustrative purposes only. Accordingly, it is intended that the invention be limited only by the scope of the claims appended hereto.

Claims

1. A method (500) of performing a transmission over a channel (300; 400), the method comprising:

- obtaining (510) a transmission measure (f_r I; j^; Out; γ_ρ ) γ_ρ ,' H_p) based on a state history (Η_Δ) of a channel (300; 400) useable for the transmission; and

- transmitting (520) over the channel (300; 400) using a transmission rate (f; Γ; /(/,)) based on the transmission measure.

2. The method of claim 1, wherein the state history (Η_Δ) includes at least one of at least one past state of the channel (300; 400) and at least one statistical characteristic of the at least one past state of the channel (300; 400).

3. The method of claim 1 or 2, wherein obtaining the transmission measure (f, Γ, -&_ut; /'out; Y_p ; r_P ) H_p) includes at least one of deriving the transmission measure and receiving the transmission measure.

4. The method of any one of claims 1 to 3, wherein the transmission measure (/; /; out; Out; Y_P ) Y_P ) Hp) is predicted based on the state history (Η_Δ).

5. The method of any one of claims 1 to 4, wherein a complexity of the transmission measure (f, I; 7₀ut; /out; Y_P ; Y_P ; H_p) is lower than a complexity of the state history

(ΗΔ).

6. The method of any one of claims 1 to 5, wherein the transmission measure (γ_ρ ) is a Signal to Interference and Noise Ratio (SINR) of the channel (300; 400).

7. The method of any one of claims 1 to 6, further comprising inferring a probability distribution ( ρ{γ, γ_ρ ) ) p(Y I Y_p) ) of the transmission measure ( I; J ; ^out; Y_P ; Y_P ; Hp), optionally including recursive estimates of the probability distribution.

8. The method of claim 7, wherein the transmission rate (f; Γ; ΐ(γ,)) is computed based on the probability distribution ( ρ(γ , γ_ρ ) ; ρ(γ \ γ_ρ) ).

9. The method of claim 7 or 8, wherein the transmission rate {f, I ΐ{γ,)) is computed by conditioning the probability distribution ( ρ{γ \ γ_ρ) ) on the transmission measure ( I; 7₀ut; Pout; r_P ; r_p ; H_p).

10. The method of any one of claims 1 to 9, further comprising predicting a channel state (Hp) based on the state history (Η_Δ).

11. The method of claim 10, wherein the transmission measure is derived from the predicted channel state (H_p).

12. The method of any one of claims 1 to 11, wherein the transmission rate (f; Γ; l(y,)) fulfills a predefined target (P_out; W.

13. The method of claim 12, wherein the predefined target (P_out) includes an outage probability.

14. The method of claim 12 or 13, wherein the predefined target (I_out) includes a mutual information with outage.

15. The method of any one of claims 1 to 14, wherein the channel (300; 400) includes a plurality of sub-channels (410), and wherein at least one of the transmission measure and the transmission rate (f; Γ; ΐ{γ,)) includes, for each of the sub-chan^¬ nels (410), an individual transmission measure ( ) or an individual transmission rate U{y,)).

16. A method (600) of performing a transmission over a channel (300; 400), the method comprising:

- obtaining (610) a transmission measure (f, , jo_Ut; /Out; r_P ϊ_Ρ Ί ^HP) based on a state history (Η_Δ) of a channel (300; 400) useable for the transmission; and

- receiving (620) over the channel (300; 400) using a transmission rate (f; Γ; ΐ{γ,)) based on the transmission measure.

17. A computer program product comprising program code for performing the steps of any one of claims 1 to 16 when the computer program product is executed on a computer system.

18. The computer program product of claim 17 stored on a computer-readable recording medium.

19. A device (100) for performing a transmission over a channel (300; 400), the device comprising:

- an obtaining unit (110) adapted to obtain a transmission measure (ή I; ^_ut; /Out; χ_ρ ; γ_ρ ) Hp) based on a state history (Η_Δ) of a channel (300; 400) useable for the transmission; and

- a transmitting unit (120) adapted to transmit over the channel (300; 400) using a transmission rate (f; Γ; l(y ) based on the transmission measure.

20. A device (200) for performing a transmission over a channel (300; 400), the device comprising:

- an obtaining unit (210) adapted to obtain a transmission measure ( , 7₀ut; out; Y_{P <} ' Y_{P <}' ^HP) based on a state history (Η_Δ) of a channel (300; 400) useable for the transmission; and

- a receiving unit (220) adapted to receive the transmission over the channel (300; 400) using a transmission rate (f; Γ; l{y,)) based on the transmission measure.

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