In many channel measurement applications, one needs to estimate some characteristics of the channels based on a limited set of measurements. This is mainly due to the highly time varying characteristics of the channel. In this contribution, it will be shown how free probability can be used for channel capacity estimation in MIMO systems. Free probability has already been applied in various application fields such as digital communications, nuclear physics and mathematical finance, and has been shown to be an invaluable tool for describing the asymptotic behaviour of many large-dimensional systems. In particular, using the concept of free deconvolution, we provide an asymptotically (w.r.t. the number of observations) unbiased capacity estimator for MIMO channels impaired with noise called the free probability based estimator. Another estimator, called the Gaussian matrix mean based estimator, is also introduced by slightly modifying the free probability based estimator. This estimator is shown to give unbiased estimation of the moments of the channel matrix for any number of observations. Also, the estimator has this property when we extend to MIMO channels with phase off-set and frequency drift, for which no estimator has been provided so far in the literature. It is also shown that both the free probability based and the Gaussian matrix mean based estimator are asymptotically unbiased capacity estimators as the number of transmit antennas go to infinity, regardless of whether phase off-set and frequency drift are present. The limitations in the two estimators are also explained. Simulations are run to assess the performance of the estimators for a low number of antennas and samples to confirm the usefulness of the asymptotic results.
Deep Dive into Channel Capacity Estimation using Free Probability Theory.
In many channel measurement applications, one needs to estimate some characteristics of the channels based on a limited set of measurements. This is mainly due to the highly time varying characteristics of the channel. In this contribution, it will be shown how free probability can be used for channel capacity estimation in MIMO systems. Free probability has already been applied in various application fields such as digital communications, nuclear physics and mathematical finance, and has been shown to be an invaluable tool for describing the asymptotic behaviour of many large-dimensional systems. In particular, using the concept of free deconvolution, we provide an asymptotically (w.r.t. the number of observations) unbiased capacity estimator for MIMO channels impaired with noise called the free probability based estimator. Another estimator, called the Gaussian matrix mean based estimator, is also introduced by slightly modifying the free probability based estimator. This estimator i
Øyvind Ryan, Member, IEEE and Mérouane Debbah, Member, IEEE Abstract-In many channel measurement applications, one needs to estimate some characteristics of the channels based on a limited set of measurements. This is mainly due to the highly time varying characteristics of the channel. In this contribution, it will be shown how free probability can be used for channel capacity estimation in MIMO systems. Free probability has already been applied in various application fields such as digital communications, nuclear physics and mathematical finance, and has been shown to be an invaluable tool for describing the asymptotic behaviour of many large-dimensional systems. In particular, using the concept of free deconvolution, we provide an asymptotically (w.r.t. the number of observations) unbiased capacity estimator for MIMO channels impaired with noise called the free probability based estimator. Another estimator, called the Gaussian matrix mean based estimator, is also introduced by slightly modifying the free probability based estimator. This estimator is shown to give unbiased estimation of the moments of the channel matrix for any number of observations. Also, the estimator has this property when we extend to MIMO channels with phase off-set and frequency drift, for which no estimator has been provided so far in the literature. It is also shown that both the free probability based and the Gaussian matrix mean based estimator are asymptotically unbiased capacity estimators as the number of transmit antennas go to infinity, regardless of whether phase off-set and frequency drift are present. The limitations in the two estimators are also explained. Simulations are run to assess the performance of the estimators for a low number of antennas and samples to confirm the usefulness of the asymptotic results.
Index Terms-Free Probability Theory, Random Matrices, deconvolution, limiting eigenvalue distribution, MIMO.
Random matrices, and in particular limit distributions of sample covariance matrices, have proved to be a useful tool for modelling systems, for instance in digital communications [1], nuclear physics [2] and mathematical finance [3]. A typical random matrix model is the information-plus-noise model,
R n and X n are assumed independent random matrices of dimension n×N , where X n contains i.i.d. standard (i.e. mean 0, variance 1) complex Gaussian entries. (1) can be thought of as the sample covariance matrices of random vectors r n +σx n .
r n can be interpreted as a vector containing the system characteristics (direction of arrival for instance in radar applications or impulse response in channel estimation applications).
x n represents additive noise, with σ a measure of the strength of the noise. Classical signal processing estimation methods consider the case where the number of observations N is highly bigger than the dimensions of the system n, for which equation ( 1) can be shown to be approximately:
Here, Γ n is the true covariance of the signal. In this case, one can separate the signal eigenvalues from the noise ones and infer (based only on the statistics of the signal) on the characteristics of the input signal. However, in many situations, one can gather only a limited number of observations during which the characteristics of the signal does not change. In order to model this case, n and N will be increased so that lim n→∞ n N = c, i.e. the number of observations is increased at the same rate as the number of parameters of the system (note that equation (2) corresponds to the case c = 0).
Previous contributions have already dealt with this problem. In [4], Dozier and Silverstein explain how one can use the eigenvalue distribution of Γ n = 1 N R n R H n to estimate the eigenvalue distribution of W n by solving a given equation. In [5], [6], we provided an algorithm for passing between the two, using the concept of multiplicative free convolution, which admits a convenient implementation. The implementation performs free convolution exactly based solely on moments.
In this paper, channel capacity estimation in MIMO systems is used as a benchmark application by using the connection between free probability theory and systems of type (1). For MIMO channels with and without frequency off-sets, we derive explicit asymptotically unbiased estimators which perform much better than classical ones. We do not prove directly that the proposed estimators work better than the classical ones, but present simulations which indicate that they are superior. We remark that the proposed capacity estimators will not be unbiased, it is needed that either the number of transmit antennas or the number of observations be large to obtain precise estimation. This limitation is most severe for channels with frequency off-sets, where it is needed in any case that the number of transmit antennas is large to obtain precise estimation. A case of study where channel estimation using free deconvolution has been use
…(Full text truncated)…
This content is AI-processed based on ArXiv data.
