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Achievable rate (AR) is significant to communications. As to multi-input multi-output (MIMO) digital transmissions with finite alphabets inputs, which greatly improve the performance of communications, it seems rather difficult to calculate accurate AR. Here we propose an estimation of con-siderable accuracy and low complexity, based on Euclidean measure matrix for given channel states and constellations. The main contribution is explicit expression, non-constraints to MIMO schemes and channel states and constellations, and controllable estimating gap. Numerical results show that the proposition is able to achieve enough accurate AR computation. In addition the estimating gap given by theoretical deduction is well agreed.

Achievable rate (AR), defined as information entropy collected from receiving signals―mutual information, is a fundamental means to evaluate and optimize communications. It is demonstrated that AR is inputs related, and achieves its maximum―channel capacity with Gaussian inputs [

Reconsider definition of mutual information in [

Comparing to current proposals, the main contribution of this letter is to propose AR estimation of low complexity, analytically explicit expression and controllable gap, without constraints to inputs and MIMO channels. This work is organized as follows. Section 2 formulates the problem and premiss; Section 3 describes details of proposed solution, and analyzes estimating gap; Section 4 gives numerical results, and further discuss on computational complexity and estimating gap; finally conclusions are drawn in Section 5.

This work uses the following notations. Italic character in lower and upper case denotes variable. Bold italic character in lower and upper case denotes vector and matrix respectively. The superscript

Consider digital transmissions over

•

•

Define

•

•

Note that, Equation (4) is AR value for the whole transmitting and receiving vector. Considering spatial multiplexing mode of MIMO, AR value for each element in

Firstly, Equation (4) is rewritten as

Given

Given

Definition: Euclidean measure matrix

Recall Equation (3), and then Equation (6) is rewritten as

where

Then normalize

So we rewrite Equation (8) as

Theorem 1. AR is estimated by exponentially weighted average of Euclidean measure matrix as

Proof. see Appendix 1.

Theorem 2. AR is decomposed as

where

Proof. see Appendix 2.

With Theorem 1 and 2, the formulated problem in Section 2 is solved.

Theorem 3. Gap between true and estimated AR given in proposition 1 is bounded by exponentially weighted average of minimum Euclidean measure as

where

Proof. see Appendix 3.

Recalling Theorem 3, the gap between true and estimated AR for

To verify Theorem 1, 2 and 3, numerical results are provided. For generality,

Also high and low correlated scenarios with correlation coefficient of 0.1 and 0.9 are considered respectively,

where correlation matrices

The 3 transmitted symbols in

Numerical results in

Despite slightness, numerical results show that the gap remains. However, such estimating gap can be quantized by Equation (14) and (15), and numerical results are shown in

A low-complexity AR estimation is presented in this work. Numerical results show that it is accurate enough, and the deductive theoretic bound of estimating gap is well matched. Moreover, the most encouraging thing is that, the proposed estimation is of no constraints to finite-alphabet constellations and MIMO channels. Besides, as shown in Equation (12), this proposition deduces integral of AR calculation into an weighted average of Euclidean measure matrix for given channel states and constellations, which is explicit enough for analytical applications.

We thank the Editor and the referee for their comments. This work is funded by the NSFC program (61172021 and 61471030), the Fundamental Research Funds for the Central Universities (2014JBZ023), Beijing city science and technology special program (Z141101004414091), and Research on the development of science and technology plan Chinese Railway Corporation (2014X012-B, Z2014-X002). This support is greatly appreciated.

Proof. Following approximations are easily achieved with numerical methods,

Although gap still remains, following deduction will show that such gap makes no difference on AR computations. Define

To prove of Equation (12) equals to prove

where

Use inductive reasoning, define

For

Then assume

This implies that, within the operative domain of

following approximation is valid.

Recall Equation (18), for

where

And the right side of Equation (22) is

So that

Using monotonic property of exponentiation and logarithm, it is demonstrated that,

Recalling Equation (23), (24) and (30), assigning

Proof. Use

Then designate

Proof. Recall Equation (12) and (30), the maximum gap between true and estimated AR value for

Since that the sequence of

Theorem 3 is proved.