The case for SIMO random access in multi-antenna multi-hop wireless networks

Abstract

In this paper, we demonstrate that multiple concurrent asynchronous and uncoordinated Single-Input Multiple-Output (SIMO) transmissions can successfully take place even though the respective receivers do not explicitly null out interfering signals. Hence, we propose simple modifications to the widely deployed IEEE 802.11 Medium Access Control (MAC) to enable multiple non-spatially-isolated SIMO sender-receiver pairs to share the medium. Namely, we propose to increase the physical carrier sense threshold, disable virtual carrier sensing, and enable message-in-message packet detection. We use experiments to show that while increasing the peak transmission rate, spatial multiplexing schemes such as those employed by the IEEE 802.11n are highly non-robust to asynchronous and uncoordinated interferers. In contrast, we show that the proposed multi-flow SIMO MAC scheme alleviates the severe unfairness resulting from uncoordinated transmissions in 802.11 multi-hop networks. We analytically compute the optimal carrier sense threshold based on different network performance objectives for a given node density and number of receive antennas.

Appendices

Appendix 1: Analytical proof of SIMO robustness to unknown interference

Here, we analytically prove SIMO robustness properties using outage analysis. Time is slotted and in each slot the Rayleigh channel matrices between different sender-receiver pairs are unchanged. Given our interference-limited network model, consider a tagged sender that transmits n _s independent data streams from n _s antennas each with rate r and power P. Due to the independence of the data streams transmitted by a n _s  × N flow, the achievable flow rate can be calculated as

$$ R_s = n_s r (1-p_{out})^{n_s} $$

(24)

where, p _out is the outage probability per stream (or antenna). Without loss of generality, we consider a single interferer that uses n _i antennas. The maximal ratio combiner uses the channel gain of the tagged sender as the weights of the combiner [1, 10]. The received SINR of the kth stream at the output of the maximal ratio combiner is given by

$$ SINR = { \frac{ \tilde{P}_s {\bf h}_{k,s}^{\dag} {\bf h}_{k,s}} {\sum_{l=1}^{n_i} \tilde{P}_i \frac{\mid {\bf h}_{k,s}^{\dag} {\bf h}_{l,i} \mid^2}{{\bf h}_{k,s}^{\dag} {\bf h}_{k,s}} + \sigma^2}} $$

(25)

where $\tilde{P}_s$ and $\tilde{P}_i$ are the path-loss components of received signal power from the tagged and interfering senders per antenna, respectively, and ${}^{\dag}$ is the complex conjugate transpose. The vector h _k,s and h _l,i represent the channel gain vectors of the kth transmit antenna of the tagged sender s, and the lth interfering antenna of the interferer i, respectively, and the receiver. We define the following terms to describe the output of the maximal ratio combiner: $\gamma = {\bf h}_{k,s}^{\dag} {\bf h}_{k,s} = \sum_{m=1}^N \mid h_{km,s} \mid^2$ as the effective SIMO channel of the kth transmit antenna at the output of the combiner, where h _km,s is the channel fading coefficient between the kth transmit antenna and the mth receive antenna; and $\tilde{\gamma}= \sum_{l=1}^{n_i} \frac{ \mid {\bf h}_{k,s}^{\dag} {\bf h}_{l,i} \mid^2}{{\bf h}_{k,s}^{\dag} {\bf h}_{k,s}}$ as the effective interference at the combiner output. Hence, (25) can be rewritten as

$$ SINR = { \frac{SIR \gamma}{\tilde{\gamma} + \frac{\sigma^2}{\tilde{P}_i}}} $$

(26)

where $SIR=\tilde{P}_s/\tilde{P}_i$ is the signal to interference ratio per antenna. Substituting in (18), the outage probability can be expressed as

$$ p_{out}= Prob \left[\frac{\gamma}{\tilde{\gamma} + \frac{\sigma^2}{\tilde{P}_i}} < \frac{(2^{r}-1)} {SIR}\right] $$

(27)

For Raleigh fading channel coefficients, $\mid h_{km,s} \mid^2$ and $\mid {\bf h}_{k,s}^{\dag} {\bf h}_{l,i} \mid^2 {\bf h}_{k,s}^{\dag} {\bf h}_{k,s}$ are exponentially distributed [1, 10]. The Chi-square $(\chi_m^2)$ distribution with m degrees of freedom nominally applies to the sum of m i.i.d. exponential random variables. Since the channel fading coefficients are i.i.d., γ and $\tilde{\gamma}$ are Chi-square distributed with 2N and 2n _i degrees of freedom, respectively. Thus, the outage probability in (27) is calculated as

$$ p_{out} = \int\limits_0^\infty f_{\tilde{\gamma}} (\tilde{\gamma}) \int\limits^{A (\tilde{\gamma}+ \frac{\sigma^2}{\tilde{P}_i} )}_0 f_\gamma (\gamma) d\gamma d\tilde{\gamma} $$

(28)

$$ = 1 \! - \! \frac{e^{\frac{-A\sigma^2}{\tilde{P}_i}}}{\Upgamma(\!n_i \!)} \!\!\sum_{s=0}^{N\!-\!1} \! \frac{A^s}{s!} \!\! \int\limits_0^\infty \!\!\!\! \tilde{\gamma}^{n_i\!-\!1} \left(\tilde{\gamma} \!\!+\!\! \frac{\sigma^2}{\tilde{P}_i}\right)^s e^{-\tilde{\gamma}(1\!+\!A)} d\tilde{\gamma} $$

(29)

where $A = \frac{(2^{r}-1)}{SIR}$. For interference-limited networks, the signal and interference powers are much higher than the noise power (i.e., $\tilde{P}_s \gg \sigma^2$ and $\tilde{P}_i \gg \sigma^2$), the intractable integral in (29) is reduced to a tractable one that equals $(s+n_i-1)!/(1+A)^{s+n_i},$ and hence (29) is equal to

$$ p_{out} \cong 1 \! - \! \frac{1}{\Upgamma(n_i)(1\!+\!A)^{n_i}} \!\! \sum_{s=0}^{N-1} \!\! \left(\frac{A}{1\!+\!A}\right)^s \frac{(s\!+\!n_i\!-\!1)!}{s!} $$

(30)

Thus, the outage probability, and hence, the achievable flow rate R _s is a function n _s , n _i , the SIR, and the stream rate r.

To illustrate the increase in the required SIR to obtain spatial multiplexing gain, we plot the achievable rate R _s normalized to the stream rate r for different number of transmit antennas. We consider two symmetric uncoordinated flows (i.e., n _s  = n _i ) with 4-antenna receivers. As depicted in Fig. 10(a), SIMO robustness yields almost unity normalized flow rate even at low SIR. As the number of antennas per flow increases, the SIR required to obtain the promised gain increases, as experimentally demonstrated in Sect. 3.2. This trend is independent of the stream rate r. As r increases, the SIR required for higher spatial multiplexing degrees further increases.

Fig. 10

Theoretical validation of SIMO properties. a Single symmetric interferer (n _s  = n _i ). b Multiple interferers with fixed power P _tot

Similar analysis can be performed considering L SIMO interferers each with power $\frac{P_{tot}}{L}.$ We reevaluate the outage probability in such a multi-interferer scenario. Figure 10(b) depicts the achievable SIMO rate for different values of L with the total power fixed. Similar increase in the SIMO flow rate given more interferers was experimentally shown in Sect. 3.2. It is worth mentioning that for a given P _tot , the variance of the cumulative interference term in SINR is inversely proportional to L.

Appendix 2: Characteristic function of received interference

Conditioning on a certain channel instance γ _k  = γ, the path-loss component of the received power per interferer $\tilde{P} = P_0 \left(\frac{d_0}{d}\right)^{\alpha}$ has the following distribution

$$ f_{\tilde{P}}(x)=\frac{2P_0^{2/\alpha} d_0^2}{\alpha(D^2-\varepsilon^2)x^{(\alpha+2)/\alpha}} $$

(31)

defined in the interval $\left[\left(\frac{d_0}{D}\right)^{\alpha}P_0, \left(\frac{d_0}{\varepsilon}\right)^{\alpha}P_0\right]$.

By definition, the characteristic function of the random variable P _k\γ is given by

$$ \phi_{P_{k \backslash \gamma}}(\omega) = {\mathbb{E}}\left[e^{(j\omega P_k)} \backslash \gamma_k = \gamma\right] $$

(32)

$$ = \int\limits_{\left(\frac{d_0}{D}\right)^\alpha P_0}^{\left(\frac{d_0}{\epsilon}\right)^\alpha P_0} e^{(j\omega x \gamma)} f_{\tilde{P}}(x) dx $$

(33)

$$ = \frac{2P_0^{2/\alpha} d_0^2}{\alpha(D^2-\varepsilon^2)} \int\limits_{\left(\frac{d_0}{D}\right)^\alpha P_0}^{\left(\frac{d_0}{\epsilon}\right)^\alpha P_0} \frac{e^{(j\omega x \gamma)}}{x^{(\alpha+2)/\alpha}} dx $$

(34)

We use the distribution of the effective channel fading process given by (3) to remove the conditioning in (32) as follows,

$$ \phi_{P_{k}}(\omega) = \int\limits_{0}^{\infty} \phi_{k \backslash \gamma}(\omega) f_{\gamma}(\gamma) d\gamma $$

(35)

$$ = \frac{2P_0^{2/\alpha} d_0^2}{\alpha(R^2-\varepsilon^2)} \int\limits_{0}^{\infty} \int\limits_{\left(\frac{d_0}{D}\right)^\alpha P_0}^{\left(\frac{d_0}{\epsilon}\right)^\alpha P_0} \frac{e^{(j\omega x \gamma)} \gamma^{N-1} e^{- \gamma}}{\Upgamma(N) x^{(\alpha+2)/\alpha}} dx d\gamma $$

(36)

$$ = \frac{2P_0^{2/\alpha} d_0^2}{\alpha(D^2-\varepsilon^2)} \int\limits_{\left(\frac{d_0}{D}\right)^\alpha P_0}^{\left(\frac{d_0}{\epsilon}\right)^\alpha P_0} \frac{1}{x^{(\alpha+2)/\alpha}(1-j\omega x)^N} dx $$

(37)

which is what is given by (9).