Performance Analysis of Adaptive Modulation for Precoded MIMO Systems with a GMD Zero-Forcing Transceiver

Abstract

Geometric mean decomposition (GMD) has emerged as an alternative method to design multiple-input multiple-output (MIMO) transceivers. The MIMO-GMD scheme decouples the MIMO channel into multiple independent links with identical gains. The GMD-based system with zero-forcing decision feedback equalizer (ZF-DFE) is known to minimize the bit error rate (BER) for high signal-to-noise ratios (SNRs). In addition, adaptive modulation has been widely used to enhance the average spectral efficiency (ASE) while maintaining a target BER and transmit power. In this paper, we present an analytic study of the adaptive modulation for GMD-ZF-DFE systems under Rayleigh flat fading correlated channels. In order to adjust the constellation size, the SNR at the equalizer output is sent back to the transmitter. The SNR at the DFE output is a function of the determinant of a Wishart complex matrix. The complementary cumulative distribution function (CCDF) is then an important key to our analysis. To evaluate the performance of the considered system, we use some bounds on the CCDF of the determinant and the trace of a Wishart matrix. Closed-form expressions of the BER, the ASE and the outage probability are derived and compared to Monte Carlo simulation results. Furthermore, we analyze the effect of the channel spatial correlation.

Appendix: Average BER for Constellation \(M_i\)

The average BER for constellation $M_i$ is

$$\begin{aligned} \text {BER}(M_i) =\int \limits _{v_i}^{v_{i+1}} h_{i}(v) f_{V}(v)dv \end{aligned}$$

(52)

Using integration by parts, we have

$$\begin{aligned} \text {BER}(M_i)&= \left[ h_{i}(v) F_V(v) \right] _{v_i}^{v_{i+1}} - \int \limits _{v_i}^{v_{i+1}} h'_{i}(v)F_V(v) d v \nonumber \\&= \left[ h_{i}(v) F_V(v) \right] _{v_i}^{v_{i+1}} - \int \limits _{v_i}^{v_{i+1}} h'_{i}(v) \left[ 1 - {\bar{F}}_V(v) \right] d v \nonumber \\&= \left[ h_{i}(v) F_V(v) -h_{i}(v) \right] _{v_i}^{v_{i+1}} + \int \limits _{v_i}^{v_{i+1}} h'_{i}(v) {\bar{F}}_V(v) d v \nonumber \\&= A\left( M_i, {v_i},v_{i+1} \right) + B\left( M_i, {v_i},v_{i+1} \right) \end{aligned}$$

(53)

where

$$\begin{aligned} A\left( M_i, {v_i},v_{i+1} \right) = \left[ - h_{i}(v) {\bar{F}}_V(v) \right] _{v_i}^{v_{i+1}} \end{aligned}$$

(54)

and

$$\begin{aligned} B\left( M_i, {v_i},v_{i+1} \right) = \int \limits _{v_i}^{v_{i+1}} h'_{i}(v) {\bar{F}}_V(v) d v \end{aligned}$$

(55)

To evaluate $B\left( M_i, {v_i},v_{i+1} \right)$, we use the polynomial approximation of the Gaussian Q-function given in [38, eq. (4)]

$$\begin{aligned} Q(t) = 1- \sum _{p=0}^l \sum _{q=0}^l \delta _{p,q}^l \; t^q \end{aligned}$$

(56)

The first derivative of $Q(t)$ is

$$\begin{aligned} Q'(t) = - \sum _{p=0}^l \sum _{q=0}^l q \delta _{p,q}^l \; t^{q-1} \end{aligned}$$

(57)

Hence, we have

$$\begin{aligned} h'_{i}(v) = - \frac{a_i }{2M} \sum _{p=0}^l \sum _{q=0}^l {q \delta _{p,q}^l \; d_i^{q}} v^{(q/2M-1)} \end{aligned}$$

(58)

To evaluate $B\left( M_i, {v_i},v_{i+1} \right)$, we consider the CCDF bound of (18). Therefore, we have

$$\begin{aligned} h'_{i}(v) {\bar{F}}_V(v) = - \sum _{k=0}^{N-M} \sum _{p=0}^l \sum _{q=0}^l \varDelta (i,p,q,l,k) v^{\alpha _{kq}} e^{-\beta _k v} \end{aligned}$$

(59)

where

$$\begin{aligned}&\varDelta (i,p,q,l,k) = \frac{a_i}{(2M) k!} \left( \prod _{\,m=1}^{M} \frac{(N-m-k)!}{(N-m)!} \right) q \delta _{p,q}^l \; d_i^{q } \nonumber \\&\beta _k = \frac{(N-M-k)!}{(N-1-k)!} \text { and } \alpha _{kq}= \frac{q-2M}{2M}+k \end{aligned}$$

(60)

Using the upper incomplete Gamma function, we can evaluate $B\left( M_i, {v_i},v_{i+1} \right)$ as

$$\begin{aligned} B\left( M_i,{v_i}, v_{i+1}\right)&= -\sum _{k=0}^{N-M} \sum _{p=0}^l \sum _{q=0}^l \frac{\varDelta (i,p,q,l,k)}{\beta _k^{\alpha _{kq}+1}} \cdot \nonumber \\&\quad \times \left[ \varGamma (\alpha _{kq}+1,\beta _k v_i) - \varGamma (\alpha _{kq}+1,\beta _k v_{i+1}) \right] \end{aligned}$$

(61)