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In
wireless communication Wireless communication (or just wireless, when the context allows) is the transfer of information between two or more points without the use of an electrical conductor, optical fiber or other continuous guided medium for the transfer. The most ...
s, channel state information (CSI) is the known channel properties of a communication link. This information describes how a signal propagates from the transmitter to the receiver and represents the combined effect of, for example, scattering, fading, and power decay with distance. The method is called Channel estimation. The CSI makes it possible to adapt transmissions to current channel conditions, which is crucial for achieving reliable communication with high
data rates In the pursuit of knowledge, data (; ) is a collection of discrete values that convey information, describing quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpreted. ...
in multiantenna systems. CSI needs to be estimated at the receiver and usually quantized and feedback to the transmitter (although reverse-link estimation is possible in TDD systems). Therefore, the transmitter and receiver can have different CSI. The CSI at the transmitter and the CSI at the receiver are sometimes referred to as CSIT and CSIR, respectively.


Different kinds of channel state information

There are basically two levels of CSI, namely instantaneous CSI and statistical CSI. Instantaneous CSI (or short-term CSI) means that the current channel conditions are known, which can be viewed as knowing the impulse response of a
digital filter In signal processing, a digital filter is a system that performs mathematical operations on a sampled, discrete-time signal to reduce or enhance certain aspects of that signal. This is in contrast to the other major type of electronic filter, t ...
. This gives an opportunity to adapt the transmitted signal to the impulse response and thereby optimize the received signal for
spatial multiplexing Spatial multiplexing or space-division multiplexing (often abbreviated SM, SDM or SMX) is a multiplexing technique in MIMO wireless communication, fibre-optic communication and other communications technologies used to transmit independent chan ...
or to achieve low
bit error rate In digital transmission, the number of bit errors is the number of received bits of a data stream over a communication channel that have been altered due to noise, interference, distortion or bit synchronization errors. The bit error rate (BER) ...
s. Statistical CSI (or long-term CSI) means that a statistical characterization of the channel is known. This description can include, for example, the type of fading distribution, the average channel gain, the line-of-sight component, and the
spatial correlation In wireless communication, spatial correlation is the correlation between a signal's spatial direction and the average received signal gain. Theoretically, the performance of wireless communication systems can be improved by having multiple anten ...
. As with instantaneous CSI, this information can be used for transmission optimization. The CSI acquisition is practically limited by how fast the channel conditions are changing. In fast fading systems where channel conditions vary rapidly under the transmission of a single information symbol, only statistical CSI is reasonable. On the other hand, in slow fading systems instantaneous CSI can be estimated with reasonable accuracy and used for transmission adaptation for some time before being outdated. In practical systems, the available CSI often lies in between these two levels; instantaneous CSI with some estimation/quantization error is combined with statistical information.


Mathematical description

In a
narrowband Narrowband signals are signals that occupy a narrow range of frequencies or that have a small fractional bandwidth. In the audio spectrum, narrowband sounds are sounds that occupy a narrow range of frequencies. In telephony, narrowband is usua ...
flat-fading channel with multiple transmit and receive antennas (
MIMO In radio, multiple-input and multiple-output, or MIMO (), is a method for multiplying the capacity of a radio link using multiple transmission and receiving antennas to exploit multipath propagation. MIMO has become an essential element of wi ...
), the system is modeled as :\mathbf = \mathbf\mathbf + \mathbf where \scriptstyle\mathbf and \scriptstyle\mathbf are the receive and transmit vectors, respectively, and \scriptstyle\mathbf and \scriptstyle\mathbf are the channel matrix and the noise vector, respectively. The noise is often modeled as circular symmetric complex normal with :\mathbf \sim \mathcal(\mathbf,\,\mathbf) where the mean value is zero and the noise covariance matrix \scriptstyle\mathbf is known.


Instantaneous CSI

Ideally, the channel matrix \scriptstyle\mathbf is known perfectly. Due to channel estimation errors, the channel information can be represented as :\mbox (\mathbf_) \sim \mathcal(\mbox(\mathbf),\,\mathbf_) where \scriptstyle\mathbf_ is the channel estimate and \scriptstyle\mathbf_ is the estimation error covariance matrix. The
vectorization Vectorization may refer to: Computing * Array programming, a style of computer programming where operations are applied to whole arrays instead of individual elements * Automatic vectorization, a compiler optimization that transforms loops to vec ...
\mbox() was used to achieve the column stacking of \scriptstyle\mathbf, as
multivariate random variable In probability, and statistics, a multivariate random variable or random vector is a list of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its value. ...
s are usually defined as vectors.


Statistical CSI

In this case, the statistics of \scriptstyle\mathbf are known. In a
Rayleigh fading Rayleigh fading is a statistical model for the effect of a propagation environment on a radio signal, such as that used by wireless devices. Rayleigh fading models assume that the magnitude of a signal that has passed through such a transmission m ...
channel, this corresponds to knowing that :\mbox (\mathbf) \sim \mathcal(\mathbf,\,\mathbf) for some known channel covariance matrix \scriptstyle\mathbf.


Estimation of CSI

Since the channel conditions vary, instantaneous CSI needs to be
estimated Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is der ...
on a short-term basis. A popular approach is so-called training sequence (or pilot sequence), where a known signal is transmitted and the channel matrix \scriptstyle\mathbf is estimated using the combined knowledge of the transmitted and received signal. Let the training sequence be denoted \mathbf_1,\ldots,\mathbf_N, where the vector \mathbf_i is transmitted over the channel as :\mathbf_i = \mathbf\mathbf_i + \mathbf_i. By combining the received training signals \mathbf_i for i=1,\ldots,N, the total training signalling becomes :\mathbf= mathbf_1,\ldots,\mathbf_N= \mathbf\mathbf + \mathbf with the training matrix \scriptstyle \mathbf= mathbf_1,\ldots,\mathbf_N/math> and the noise matrix \scriptstyle \mathbf= mathbf_1,\ldots,\mathbf_N/math>. With this notation, channel estimation means that \scriptstyle \mathbf should be recovered from the knowledge of \scriptstyle \mathbf and \scriptstyle \mathbf.


Least-square estimation

If the channel and noise distributions are unknown, then the least-square estimator (also known as the
minimum-variance unbiased estimator In statistics a minimum-variance unbiased estimator (MVUE) or uniformly minimum-variance unbiased estimator (UMVUE) is an unbiased estimator that has lower variance than any other unbiased estimator for all possible values of the parameter. For pra ...
) is :\mathbf_ = \mathbf \mathbf^H(\mathbf \mathbf^H)^ where ()^H denotes the
conjugate transpose In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex con ...
. The estimation Mean Square Error (MSE) is proportional to :\mathrm (\mathbf \mathbf^H)^ where \mathrm denotes the
trace Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * ''The Trace'' (album) Other uses in arts and entertainment * ''Trace'' ...
. The error is minimized when \mathbf \mathbf^H is a scaled
identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...
. This can only be achieved when N is equal to (or larger than) the number of transmit antennas. The simplest example of an optimal training matrix is to select \scriptstyle\mathbf as a (scaled) identity matrix of the same size that the number of transmit antennas.


MMSE estimation

If the channel and noise distributions are known, then this
a priori ("from the earlier") and ("from the later") are Latin phrases used in philosophy to distinguish types of knowledge, justification, or argument by their reliance on empirical evidence or experience. knowledge is independent from current ...
information can be exploited to decrease the estimation error. This approach is known as
Bayesian estimation In estimation theory and decision theory, a Bayes estimator or a Bayes action is an estimator or decision rule that minimizes the posterior expected value of a loss function (i.e., the posterior expected loss). Equivalently, it maximizes the pos ...
and for Rayleigh fading channels it exploits that :\mbox (\mathbf) \sim \mathcal(0,\,\mathbf), \quad \mbox(\mathbf) \sim \mathcal(0,\,\mathbf). The
MMSE estimator In statistics and signal processing, a minimum mean square error (MMSE) estimator is an estimation method which minimizes the mean square error (MSE), which is a common measure of estimator quality, of the fitted values of a dependent variable. In ...
is the Bayesian counterpart to the least-square estimator and becomes :\mbox(\mathbf_) = \left(\mathbf^ + (\mathbf^T \, \otimes\, \mathbf)^H \mathbf^ (\mathbf^T \, \otimes\, \mathbf) \right)^ (\mathbf^T \, \otimes\, \mathbf)^H \mathbf^ \mbox(\mathbf) where \otimes denotes the
Kronecker product In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer product (which is denoted by the same symbol) from vectors ...
and the identity matrix \scriptstyle \mathbf has the dimension of the number of receive antennas. The estimation Mean Square Error (MSE) is : \mathrm \left(\mathbf^ + (\mathbf^T \, \otimes\, \mathbf)^H \mathbf^ (\mathbf^T \, \otimes\, \mathbf) \right)^ and is minimized by a training matrix \scriptstyle \mathbf that in general can only be derived through numerical optimization. But there exist heuristic solutions with good performance based on waterfilling. As opposed to least-square estimation, the estimation error for spatially correlated channels can be minimized even if N is smaller than the number of transmit antennas. Thus, MMSE estimation can both decrease the estimation error and shorten the required training sequence. It needs however additionally the knowledge of the channel correlation matrix \scriptstyle\mathbf and noise correlation matrix \scriptstyle\mathbf. In absence of an accurate knowledge of these correlation matrices, robust choices need to be made to avoid MSE degradation.


Data-aided versus blind estimation

In a data-aided approach, the channel estimation is based on some known data, which is known both at the
transmitter In electronics and telecommunications, a radio transmitter or just transmitter is an electronic device which produces radio waves with an antenna (radio), antenna. The transmitter itself generates a radio frequency alternating current, which i ...
and at the receiver, such as training sequences or pilot data. In a blind approach, the estimation is based only on the received data, without any known transmitted sequence. The
tradeoff A trade-off (or tradeoff) is a situational decision that involves diminishing or losing one quality, quantity, or property of a set or design in return for gains in other aspects. In simple terms, a tradeoff is where one thing increases, and anot ...
is the accuracy versus the overhead. A data-aided approach requires more
bandwidth Bandwidth commonly refers to: * Bandwidth (signal processing) or ''analog bandwidth'', ''frequency bandwidth'', or ''radio bandwidth'', a measure of the width of a frequency range * Bandwidth (computing), the rate of data transfer, bit rate or thr ...
or it has a higher overhead than a blind approach, but it can achieve a better channel estimation
accuracy Accuracy and precision are two measures of ''observational error''. ''Accuracy'' is how close a given set of measurements ( observations or readings) are to their ''true value'', while ''precision'' is how close the measurements are to each oth ...
than a blind estimator.


See also

* Channel sounding


Weblinks


Atheros CSI Tool

Linux 802.11n CSI Tool


References

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Impact of antenna correlation on the capacity of multiantenna channels
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E. Björnson, B. Ottersten
A Framework for Training-Based Estimation in Arbitrarily Correlated Rician MIMO Channels with Rician Disturbance
IEEE Transactions on Signal Processing, vol 58, pp. 1807-1820, 2010.
J. Kermoal, L. Schumacher, K.I. Pedersen, P. Mogensen, F. Frederiksen
A Stochastic MIMO Radio Channel Model With Experimental Validation
{{Webarchive, url=https://web.archive.org/web/20091229154616/http://www.its.caltech.edu/~taocui/page/tutorial/mimo_channel.pdf# , date=2009-12-29 , IEEE Journal on Selected Areas Communications, vol 20, pp. 1211-1226, 2002.
M. Biguesh and A. Gershman
Training-based MIMO channel estimation: a study of estimator tradeoffs and optimal training signals
{{webarchive , url=https://web.archive.org/web/20090306172253/http://www.comm.ccu.edu.tw/~comtsliu/CourseInformation/DetectionEstimation07Fall/DetectionEstimation07FallFinalPaper.pdf , date=March 6, 2009 , IEEE Transactions on Signal Processing, vol 54, pp. 884-893, 2006.
Y. Li, L.J. Cimini, and N.R. Sollenberger
Robust channel estimation for OFDM systems with rapid dispersive fading channels
IEEE Transactions on Communications, vol 46, pp. 902-915, July 1998.
M. D. Nisar, W. Utschick and T. Hindelang
Maximally Robust 2-D Channel Estimation for OFDM Systems
IEEE Transactions on Signal Processing, vol 58, pp. 3163-3172, June 2010.
A. Zhuang, E.S. Lohan, and M. Renfors, "Comparison of decision-directed and pilot-aided algorithms for complex channel tap estimation in downlink WCDMA systems", in Proc. of 11th IEEE Personal and Indoor Mobile Radio Communications (PIMRC), vol. 2, Sept. 2000, p. 1121-1125. Wireless Information theory Radio resource management Telecommunication theory