A simple importance sampling technique for orthogonal space-time block codes on Nakagami fading channels

In digital communications, the bit error rate (BER) serves as a fundamental performance measure. For complicated systems, analytical assessment of the BER becomes very difficult, if not impossible, and Monte Carlo (MC) simulations must be applied to estimate the BER. In the case of low BER, however, the computation time associated with MC simulations can be very long, since many bits must be sent to generate a sufficient number of bit errors. In order to speed up these simulations without losing precision, importance sampling (IS) [1] can be employed. The latter technique aims to reduce the variance of the BER estimator by using a biased distribution for the input variables that increases the bit error probability during simulation. The use of the biased distribution is corrected for by weighting the results with the ratios of the actual to the biased probability density functions (PDFs), yielding an unbiased BER estimate with lower variance.

Although the application of IS is well documented, the search for a convenient biased distribution remains a challenging task. In conventional importance sampling (CIS) [2], the variance of the additive Gaussian noise is increased, resulting in a scaled noise distribution and more errors. The improved importance sampling (IIS) technique proposed in [3], on the other hand, shifts the mean of the actual PDF of the input random variables (RVs) directly towards the error region. Both techniques are combined in [4] for efficient simulation in Rayleigh fading. In order to speed up the simulation of orthogonal space-time block codes (OSTBCs) on Rayleigh-faded multiple-input multiple-output (MIMO) channels, it is shown in [5] that scaling the channel distribution is more efficient than scaling the noise distribution. However, an optimal biased channel distribution was not derived. An IS methodology based on the stochastic gradient descent (SGD) algorithm is presented in [6] for simulation of adaptive systems in frequency-flat Rayleigh fading channels.

In spite of the numerous papers on IS, an easy-to-use biased distribution for simulation over non-Rayleigh fading channels is not yet available from the literature to the best of our knowledge. In this contribution, we propose an IS technique for simulation over a frequency-flat MIMO fading channel, where we maintain the actual distribution for the channel noise and the data symbols, but use a biased channel distribution. Furthermore, we illustrate how the presented technique enables to derive a practical biased channel distribution for the case of OSTBCs transmitted over a spatially correlated Nakagami-m fading channel. Although perfect channel knowledge (PCK) is assumed for the derivation of the biased distribution, this distribution allows accurate BER estimation in a wide range of scenarios, such as pilot-based [7] or blind channel estimation, and in the presence of residual frequency offset, IQ imbalance, and phase noise. Moreover, the technique can easily be extended to the case of multicarrier communication on dispersive MIMO channels, using for instance OSTBCs or orthogonal space-frequency block codes.

which yields σ^∗2=0. However, the biased distribution from (7) is impractical, as it depends on the unknown bit error rate P_b that is to be estimated by simulation. Nevertheless, (7) indicates that an efficient biased distribution should be proportional to an approximation of F(x)p(x).

where ∝ denotes proportionality. However, a closed-form expression for $\tilde{F}\left(\mathbf{h}\right)$ from (11) is usually not available or too complex to yield a practical biased distribution.

The exact expression of the conditional bit error probability P_b,n(h) depends on the observation model and the type of receiver considered and is often unknown. Hence, a suitable approximation of P_b,n(h) is needed to obtain a biased distribution (17) that adequately reduces the variance of the estimate ${\widehat{P}}_{\mathrm{b},n}^{\ast }$.

which is independent of the bit index n, so that the bit index can be dropped. Hence, the same biased distribution can be used to reduce the variance of the individual bit error probability estimates ${\widehat{P}}_{\mathrm{b},n}^{\ast }$ for n=1,…,N_b, and therefore, it is expected to efficiently reduce the variance of the BER estimate ${\widehat{P}}_{\mathrm{b}}^{\ast }$ as well.

where the L×L diagonal matrix Ω is given by Ω=diag{Ω₁,Ω₂,…,Ω _L } and ${\mathit{\Sigma }}_{\mathbf{G}}={\mathit{\Sigma }}^{\circ \frac{1}{2}}$, with ${\mathbf{X}}^{\circ \frac{1}{2}}$ denoting the element-wise square root of a matrix X.

It is important to note that, although (18) is obtained under the assumption of PCK, the resulting biased distribution is convenient also in the case of imperfect channel estimation (ICE), as demonstrated in Section 4, or in the presence of impairments, such as residual frequency offset, IQ imbalance, and phase noise, in which case $P_{\mathrm{b},n}^{\left(\text{ML}\right)}\left(\mathbf{h}\right)$ in (18) is regarded as an approximation of the actual P_b,n(h). In addition, the proposed IS technique can easily be extended to the case of multicarrier communication on dispersive MIMO channels, using for instance OSTBCs or orthogonal space-frequency block codes. In multicarrier systems, the relevant channel distribution is the joint distribution of the channel transfer function values at the subcarrier frequencies, which is to be derived from the joint distribution of the channel impulse response samples. In the important case of jointly Gaussian channel impulse response samples (i.e., Rayleigh or Rice fading), the channel transfer function values are also jointly Gaussian.

In the following, the lower bound (28) will be computed from the simulations by substituting P_b and var[F(x)] by the sample mean and variance resulting from {F(x _i )}, respectively.

The receiver has perfect channel knowledge and performs ML detection. In the simulations, for each BPSK symbol, a new realization of x is generated. For MC simulations with and without IS, Figure 1 shows the minimal number of realizations N required to ensure a given accuracy associated with ε²=10⁻⁴ as a function of E_b/N₀, where E_b=E_s/ log2(M) denotes the energy per information bit (for BPSK, M=2). Although in both cases the required simulation time increases with the signal-to-noise ratio (SNR), it is observed that huge run-time savings of several orders of magnitude can be achieved by using the proposed biased sampling distribution, especially for high SNR values where the BER is low.

In the simulations, a fixed number of N=10⁴ data frames is used, and for each data frame, a new realization of x is generated. Figure 2 displays the simulated BER, for both 4-QAM and 16-QAM transmissions. With IS, N=10⁴ frames yield accurate results for BER values down to 10⁻¹⁰. In particular, an accuracy of ε²=0.025 is obtained for E_b/N₀=20 dB and 4-QAM transmission. When no IS is used, a substantial deterioration of the accuracy can be observed for BER values below 10⁻⁴; for BER ∈(10⁻⁶,10⁻⁴), the deviation w.r.t. IS is up to 37.8%, and for BER values below 10⁻⁶, the results without IS are highly unreliable. The number of frames required to obtain an accuracy of ε²=0.025 with and without using IS is shown in Figure 3 for 4-QAM and 16-QAM transmissions. Note that for given E _b /N₀, 16-QAM requires fewer frames than does 4-QAM to achieve a given estimation accuracy, because the formed constellation yields a larger BER than the latter. Again, it is observed from the figure that substantial run-time savings can be achieved using the proposed IS technique.

where the variances var[F(x)]|_IS and var[ F(x)]|_MC for MC simulations with and without IS, respectively, are obtained from the MC simulation results {F(x _i )} conducted with and without using IS. In order to investigate the impact of spatial correlation and the number of receive antennas on the efficiency gain, we consider Alamouti’s code transmitted on a Nakagami-m MIMO channel with fading parameter m=2 and captured by an ML receiver equipped with 1, 2, or 3 antennas. Taking into account that the power correlation matrix of the MIMO channel is given by Σ=Σ_t⊗Σ_r, the diagonal elements of Σ_t and Σ_r are assumed to be given by 1, whereas the non-diagonal elements are given by ρ. Furthermore, we consider 4-QAM transmission, LMMSE channel estimation, and data frames consisting of 100 coded data symbols and 14 pilot symbols per transmit antenna. In Figure 4, the simulated BER is shown for a fixed number of N=10⁶ data frames transmitted. Obviously, straightforward MC simulation allows accurate BER estimation down to about 10⁻⁷, whereas employing IS enables accurate BER values down to 10⁻¹⁷ with the same number of data frames. In addition, it is observed from the figure that spatial correlation with ρ=0.85 results in a substantial degradation of the BER as compared to the case where all spatial channels are uncorrelated (ρ=0). In Figure 5, we display the efficiency gain γ_IS achieved by IS corresponding to the BER curves from Figure 4. For SNR values below 10 dB, efficiency gains between 2 and 50 can be observed. For SNR values above 10 dB, however, rapidly increasing efficiency gains up to 5∗10³ can be appreciated. Moreover, even higher efficiency gains are to be expected for the receiver with three antennas and, in the case of uncorrelated channels, the dual-antenna receiver. In general, it follows from the figure that for high SNR, the efficiency gain grows when the BER decreases, i.e., for increasing number of receive antennas and decreasing level of spatial correlation. As a result, the proposed IS sampling distribution allows practical BER simulation for OSTBC systems on spatially uncorrelated and correlated Nakagami fading channels, even when very low BERs are considered.

In this contribution, we presented a simple but efficient IS technique to speed up by orders of magnitude the BER simulations of OSTBC systems operating on spatially correlated frequency-flat Nakagami-m fading channels. While maintaining the actual distributions for the channel noise and the data symbols, we proposed a suitable biased distribution for the fading channel.

This IS technique can be extended to the case of multicarrier communication on dispersive MIMO channels, using for instance OSTBCs or orthogonal space-frequency block codes (OSFBCs). In multicarrier systems, the relevant channel distribution is the joint distribution of the channel transfer function values at the subcarrier frequencies, which is to be derived from the joint distribution of the channel impulse reponse samples. In the important case of jointly Gaussian channel impulse response samples, i.e., Rayleigh or Rice fading, the channel transfer function values are also jointly Gaussian.