Abstract
Information rate for discrete signaling constellations is significant. However, the computational complexity makes information rate rather difficult to analyze for arbitrary fading multiple-input multiple-output (MIMO) channels. An analytical method is proposed to compute information rate, which is characterized by considerable accuracy, reasonable complexity, and concise representation. These features will improve accuracy for performance analysis with criterion of information rate.
1. Introduction
Information rate plays an important role in performance analysis for discrete signaling constellations (m-PSK, m-QAM, etc.) [1–4]. Currently, there have been three metrics to evaluate information rate. They are accurate values, lower or upper bounds, and intermediate variables.
According to definition of information rate, direct computation is rather hard for arbitrary fading MIMO channels [5, 6]. Therefore, Monte Carlo (MC) trials turn out to be a direct and accurate computation [5]. To reduce computational complexity, an improved particle method is proposed in [6]. However, it is iterative and implicit, which makes it ambiguous to analyze. Recently a bitwise computation with concise analytical expression is proposed [4]. Unfortunately, further studies have shown that it is limited to some scenarios, single-input single-output (SISO) and MIMO channels with constellation of BPSK, QPSK, 16QAM, and 64QAM. On the other hand, to the issue of complexity, lower or upper bounds are used to profile information rate instead [7–9]. However, differences between bounds and accurate information rate are still notable. Meanwhile there are also researches which suggest intermediate variables to implement qualitative analysis [10–13]. However, they are handling some special MIMO channels, such as diagonal MIMO channel.
In this work, we are focusing on analytical computation of information rate for arbitrary fading MIMO channels and proposing a symbol-wise algorithm. It is characterized by considerable accuracy, reasonable complexity, and analytical expression, which enable IR to be applicable for analysis. This work is organized as follows. In Section 2, a basic review on analytical computation is presented. Then demonstration of the proposed symbol-wise computation is detailed in Section 3. In Section 4, comparison on accuracy and computational complexity between MC simulation and symbol-wise analytical computation is presented. Finally, conclusions are drawn in Section 5.
2. Basic Review of Analytical Computation
Consider the problem of computing information ratebetween input vector and output vector over MIMO channels with additive white Gaussian noise (AWGN). Use dimensional matrix – H to denote coefficient for arbitrary fading MIMO channels. and are numbers of transmitting and receiving antennas, respectively. Then we have [10, 13]where w is AWGN vector. Using Complex- denotes complex Gaussian with zero mean and variance of and submits to Every element () in s is selected from the kth finite subconstellation uniformly and independently. Therefore, is uniformly distributed over finite discrete signaling constellations – , and is the Cartesian product of all subconstellations. Assuming that size of is , probability density function (PDF) for s isProvided channel states – H, definition of information rate gives in [10] aswhere W is domain of AWGN vector – w and represents 2-norm for vector . Generally speaking, (5) requires at least 2 dimensional integral. Therefore, it is difficult to implement directly. And then MC method is used to compute accurate information rate in [8] asNeither MC computation is simple, nor it can reveal explicit relation between information rate and channel states. Consequently, bitwise computation is developed, using sum of several adjusted Gaussians to approximate PDF of logwise likelihood ratio (LLR), and then information rate is computed bywhere means the number of adjusted Gaussians which is determined by preliminary simulations. Adjustment of is also determined by simulations. denotes variance of the lth Gaussian, defined in [4]. And isThis bitwise computation achieves acceptable accuracy for single-input single-output (SISO) and MIMO channels with BPSK, QPSK, 16QAM, and 64QAM [4].
3. Proposed Analytical Computation
Since that bitwise computation of information rate is limited to some scenarios, we propose a symbol-wise algorithm. In this section, we present strict demonstration and extend this computation to general MIMO scenarios with the help of mutual distance vector asInformation rate (5) is rewritten aswhereBecause is AWGN vector, the PDF of is Gaussian,And denotes Gaussian with zero mean and variance of . Normalize Gaussian variance asWe haveConsequently, it is pivotal to compute numerical integral. Use Taylor expansion asThe denotation is defined as Equation (16) points out that is arithmetic mean of progression to the power of . For the sake that it is difficult to compute (15) with directly, a suboptimal computation is proposed. Regarding as a progression of elements, we are trying to find another geometric progression . This geometric progression is characterized by minimum mean square error to the original progression . And then, this characteristic guarantees the minimum mean square error between computations of (15) with and . Consequently, we accomplish this geometric progression with least-squares fitting,Numerical approximation givesThus, integral is approximated asRecall (14), we get analytical computation of information rate as
4. Numerical Results
This section presents numerical results for validation. Accurate information rate is computed with MC method (6) as basic reference for comparison. It is clear that information rate is determined by digital signaling constellation , channel states , and AWGN variance . To assure that the numerical results are self-contained, we will classify [, and ] into several orthogonal spaces.
4.1. Numerical Results for Arbitrary Fading SIMO Channels
We analyze single-input multiple-output (SIMO) channels first. The simplest scenario, single-input single-output (SISO) channels, can be seen as a subset of SIMO. Considerwith maximum ration combination (MRC); this transmission is effective toFor generality, constellations BPSK, 8PSK, 64QAM, and 256QAM are assigned to , respectively. Numerical results on computation of information rate are illustrated in Figure 1. It is clear that the proposed method achieves considerable accuracy and tells the accurate information rate for different digital signaling constellations.
4.2. Numerical Results for Arbitrary Fading MISO Channels
Then we consider multiple-input single-output (MISO) channels. Consider example as follows:Since is complex, MISO channel is of at least degrees freedom, so it is impossible to profile full classification. Therefore, we have to make the following yields.(1) is selected as 2, 3, and 4, for example.(2)For each value of , is randomly chosen with complex Gaussian.(3)Assuring generality, constellations BPSK, QPSK, 8PSK, and 16QAM are assigned to each symbol in vector s, respectively.Numerical results are illustrated in Figure 2. It is also clear that symbol-wise computation achieves considerable accuracy for SIMO channels.
4.3. Numerical Results for Arbitrary Fading MIMO Channels
As to MIMO channel, H is consisted of complex coefficients, so we make similar yields.(1) is 3, and is selected as 2, 3, and 4, for example.(2)All elements of H are randomly chosen with complex Gaussian.(3)Assuring generality, constellations BPSK, QPSK, 8PSK, and 16QAM are assigned to each symbol in vector s, respectively.Numerical results illustrated in Figure 3 show that symbol-wise computation achieves considerable accuracy also.
4.4. Numerical Results for Resolution of Information Rate
Consider another problem as computing information rate of any component within input vector accurately. Firstly, it is proven that information rate can be resolved as follows [11]:where is residual subvector by excluding from s. This tells that we can compute arbitrary information rate provided computation of (5). Consequently, this part of results validates (24) with symbol-wise computation of information rate. For example, that H is and randomly chosen similarly as previous sections. Used constellations are BPSK, QPSK, 8PSK, and 16QAM for each symbol in vector s, respectively. Interesting components are individual symbol in vector s. Information rate of every transceiver is illustrated in Figure 4. It is the same as before that symbol-wise computation achieves considerable accuracy.
4.5. Numerical Results for Erasure Channel
Besides MIMO channels mentioned before, there is a very special kind of channel as follows:where s1 and s2 are both BPSK modulated. This kind of transmission is a typical erasure channel, where is 1 and is 2. Numerical results are illustrated in Figure 5. It also validates the proposed symbol-wise computation.
5. Discussion
We have presented an analytical computation of information rate for arbitrary fading MIMO channels. Based on simulation, we have further discussion as follows.
For the “Generality” of the proposal, similarly as presented in [5, 6, 9, 10], we demonstrate computation of information rate for MIMO channel, without supplemental conditions except the knowledge of constellations – , power of AWGN, and channel status – to the receiver. And then, we carry out validation with numerical results on selected MIMO cases. To ensure that the selected MIMO cases are general, numbers of transmitting and receiving antenna (, ) vary from 1 to 4, as presented in Section 4; the adopted constellations vary from QPSK to 256QAM; and simulated channel status – H are randomly generated. In addition, (24) (in Section 4.4) can be used to compute information rate for each MIMO stream, which also improves generality, whereas bitwise computation is quite limited by tuning factors related to selected MIMO scenarios [4].
As to “Accuracy” of the proposal, validations in Section 4 show that the maximum gap between information rate computed by proposed and MC methods is lower than 0.063 bits/symbol. Reference information rate is computed by MC method [5], and particle method achieves exactly accurate numerical results [6]. With SNR based intermediate variables, estimation [4] and upper/lower bounds [9, 10] are proposed. The gap between reference information rate by MC and upper/lower bounds is about 0.15 bits/symbol [9, 10], which is a little worse than computation proposed in this work. The accuracy of estimation in [4] is not compared for the sake that there are quite a lot MIMO and constellations cases when information rate is unavailable.
Then we turn to “Complexity.” Computation shown as (20) will need exponential processes and N logarithms, which is approximately equivalent to those presented in [9, 10]. This is much simpler than MC method [5]. The comparison on complexity between the proposed and particle methods is dependent to scale of MIMO and constellation, because it tells in [6] that particle methods need a sequence length of 104 to obtain convergent calculation, while the suggested method in this work is of complexity varying with N.
To sum up, the proposed computation makes sense that it is interpreted in a general and concise analytical expression, so that it facilitates further studies on performance and optimization of wireless MIMO transmissions with information rate criterion.
Disclosure
This work was presented in part at 2010 International Conference on Communications.
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is financially supported by National Natural Science Foundation of China (NSFC) 61471030 and 61631013 and Research Project of Railway Corporation (2016J011-H).