Low SNR Capacity of Keyhole MIMO Channel
in Nakagami-$m$ Fading With Full CSI

Continuing from the capacity expression in (8), we recall that the optimal power distribution over a scalar fading channel $\bm{\lambda}$ is the closed-form ‘waterfilling formula’ expressed as $P(\lambda)=\left(\tfrac{1}{\lambda_{0}}-\tfrac{1}{\lambda}\right)^{+}$ [23]. Here, $z^{+}$ denotes $\mathrm{max}\{0,z\}$. The parameter $\lambda_{0}$ is chosen to meet the power constraint (9). Substituting $P(\lambda)$ in (8), we get

where, for convenience, we have defined a scaled random variable $\bm{\mu}:=\dfrac{\bm{\lambda}}{b_{r}b_{t}}$ with the distribution as follows:

Accordingly, the power constraint (9) in terms of $\bm{\mu}$ becomes

Note that $\mu_{0}:={\lambda}_{0}/\left(b_{t}b_{r}\right)$. It is easy to verify from (15) that as $\textrm{SNR}\to 0$, the threshold $\mu_{0}\to\infty$. The low-SNR asymptotic capacity formula is stated next.

To illustrate the accuracy of the asymptotic low-SNR capacity formulae proposed in Theorem 2, we now present some numerical results. The exact (i.e., non-asymptotic) capacity curves with and without channel knowledge at the transmitter (for reference/comparison purpose) are computed by standard numerical integration methods; the threshold $\lambda_{0}$ is also computed numerically from the average power constraint. For simplicity, we have normalized all the fading gains to unity, i.e., $\Omega_{r}=\Omega_{t}=1$. The channel settings in Figures 2(a) and 2(b) corresponds to $n>0$ case, and in Figures 2(c) and 2(d) corresponds to $n<0$ case. From these Figures, it is clear that the curves of the asymptotic capacity expressions in Theorem 2 follow the same shape as of the exact capacity curves in the displayed SNR range. In both cases, we have verified that by further reducing the $\mathrm{SNR}$ considerably, the gap to the exact capacity reduces significantly. Notice from the Figures 2(a) and 2(b) that at sufficiently low SNR values, the Log function based characterization of the asymptotic capacity in (17) is always an upper bound on the Lambert $W$ function based characterization in (16) for $n>0$; likewise, from the Figures 2(c) and 2(d), we note that (17) is always a lower bound on (16) at low SNRs for $n<0$ (see Appendix A for the detailed proofs). We have also observed that both of these asymptotic capacity characterizations are better for $n$ values close to zero.

The ergodic rate performance of an on-off power scheme has recently been explored for a few important fading channels and is found to be asymptotically optimal in the low-SNR regime [28]-[31]. The attractive feature of this transmission scheme is that it requires feedback of only one-bit channel information (i.e., whether current channel state is in good or bad condition) at the transmitter for power adaptation. We will explore the ergodic rate performance of the on-off transmission scheme over the keyhole MIMO channel in the low-SNR conditions and show that the on-off transmission rates follows the keyhole MIMO channel capacity closely at low-SNRs. The mathematical details are relegated to Appendix B.

Under no path loss considerations and only normalized small-scale Rayleigh fadings present, the low-SNR capacity of the MIMO fading channel (no keyhole) follows asymptotically as [28, Eq. (6)]

When subjected to keyhole condition, Theorem 2 suggests that the low-SNR capacity of this MIMO fading channel scales asymptotically as

It is now straightforward to conclude that (40) exceeds (39) in the low-SNR regime, see Fig. 3 for numerical illustration. Precisely, (39) is improved by a factor of

when subjected to keyhole condition which improves unboundedly as $\mathrm{SNR}\to 0$; this is depicted in Fig. 3 by black vertical double arrows for the $2\,\times\,2$ MIMO IID Rayleigh channel in the low-SNR regime. Notice that the exact capacity of the MIMO Rayleigh fading channel under keyhole effect begins to exceed the capacity pure MIMO Rayleigh fading channel starting at $\mathrm{SNR}$ close to $-2$ $\mathrm{dB}$. We define this event as ‘crossover phenomenon’ and the respective $\mathrm{SNR}$ is termed as ‘crossover point’. To justify this improvement, note that the ‘spatial degrees of freedom gain’ is no longer a dominant factor for the MIMO fading channel (with and without keyhole) in the low-SNR regime. On the other hand, a keyhole MIMO channel is double-faded and thus more severe than the MIMO fading channel without keyhole effect. With this increased fading severity, the knowledge of the frequent channel fluctuations (e.g., peaks and nulls) at the transmitter helps to improve the performance in the low-SNR regime by dynamically allocating power and transmitting opportunistically only when the channel state is good. The low-SNR capacity formulas derived recently for a few important scalar fading channels such as Nakagami-$m$ fading [29], Gamma-Gamma atmospheric turbulence fading [30], and Rayleigh-product fading [31] also clearly allude to our observation of enhanced low-SNR capacity with higher fading severity levels. It is expected that with the Nakagami-$m$ fading shape parameter $m$ increasing (i.e., fading severity level is decreasing), the capacity enhancement of the keyhole MIMO channel in the low-SNR regime narrows down (when compared to MIMO channel without keyhole) as there are lesser opportunistic communications possible due to relatively mild channel fluctuations.

The low-SNR capacity improvement of MIMO fading channel in the presence of keyhole (as indicated in Fig. 3) is over-simplified due to the lack of suitable large-scale attenuation (or path loss) laws. It is important to note that path loss in wireless propagation conditions diminishes the received signal power very rapidly. As we will discuss later that large-scale attenuation (path-loss) laws are significantly different for propagation conditions with and without keyhole effect. More precisely, the large-scale signal attenuation (path loss) increases significantly when signal propagation is subjected to keyhole condition. Therefore, the performance improvement suggested in Fig. 3 is overly optimistic and unrealistic. Nevertheless, this simplistic analysis provides a critical insight that the improvement factor (see (41)) increases unboundedly as $\mathrm{SNR}$ gets smaller which, in turn, suggests that at sufficiently low-SNR, the keyhole MIMO channel capacity should surpass the pure MIMO channel capacity irrespective of the huge difference in path losses. Motivated by this insight, we will now perform a detailed low-SNR capacity analysis of keyhole MIMO channels under more realistic propagation conditions. We will incorporate suitable large-scale attenuation (path loss) laws along with the small-scale IID Rayleigh fadings in the MIMO channel with and without keyhole condition. To account for the large-scale attenuation, we employ a widely used distance-dependent path loss model where the average received signal power decays logarithmically with distance. This observation is theoretically-consistent and also supported quite well by emperical measurements for both indoor and outdoor wireless channels [27]. Assuming that the transmit power is $P_{T}$, the average received signal power is as follows:

which decays with the $\gamma$-th power of the distance $d_{TR}$ between the transmitter and receiver sides of the MIMO channel, and the proportionality constant $K_{mimo}$ is determined from measurements at some close-in free space reference distance. For example, with $\lambda$ as the operating wavelength, $K_{mimo}=(\lambda/4\pi)^{2}$ is the free space path loss at reference distance of $1\,\textrm{m}$. The value of $\gamma$, also known as path loss exponent, is dependent on the specific signal propagation conditions; e.g., the value $\gamma$ equals $2$ for line of sight propagation (or free space propagation) and larger value for more difficult propagation conditions [27, Chapter 3].

The channel matrix for the MIMO IID Rayleigh channel with large-scale fading incorporated and without keyhole condition can be expressed as

where the constant $\varrho$ is the large-scale attenuation corresponding to the total distance $d_{TR}$ between the multi-antenna transmitter and receiver, i.e., $\varrho:={P_{R}/P_{T}}={K_{mimo}/d_{TR}^{\gamma}}\,$. The $r\,\times\,t$ Rayleigh channel matrix $\textbf{H}_{Rayleigh}$ has normalized IID Rayleigh fading (small-scale) entries, i.e., each entry in $\textbf{H}_{Rayleigh}\sim\mathcal{CN}(0,1)$. We will borrow the optimal power and capacity laws for the MIMO channel $\textbf{H}_{Rayleigh}$ from [25] and scale these laws appropriately for $\textbf{H}_{mimo}$. The ergodic capacity integral formula for $\textbf{H}_{Rayleigh}$ is available in [25, Eq. (2)] and can be appropriately altered for $\textbf{H}_{mimo}$ as follows:

where $m:=\mathrm{min}\{r,t\}$ and the second equality follows from the substitution of the optimal waterfilling scheme ${P}^{\,*}(\zeta)=\left(\dfrac{1}{\zeta_{0}}-\dfrac{1}{\varrho\zeta}\right)^{+}$. The eigenmode distribution $f_{{\bm{\zeta}}}(\zeta)$ is given by

where, in turn, $n:=\mathrm{max}\{r,t\}$ and $L_{k}^{n-m}(\zeta)$, the associated Laguerre polynomial of order $k$, has a closed-form expression given as

The corresponding average power constraint is obtained by modifying from [25, Eq. (4)] as follows:

As mentioned earlier, the low-SNR capacity of the MIMO IID Rayleigh channel $\textbf{H}_{Rayleigh}$ has been investigated in [28] and is shown to follow (asymptotically) as

Keeping (46) and (49) in mind for the channel $\textbf{H}_{mimo}$ and following similar steps as in the proof of the main asymptotic capacity result in Section III in [28], it can be easily shown that the low-SNR capacity of the channel $\textbf{H}_{mimo}$ scales asymptotically as

For the MIMO fading channel under the keyhole condition, the distance-dependent path loss model results in the average received signal power as follows:

where $d_{TK}$ and $d_{KR}$ are the distance from the keyhole to the transmitter and receiver, respectively. The proportionality constant $K_{keyhole}:=\lambda^{2}A/(4\pi)^{3}$ where $A$ is the area of the keyhole. The equation (52) with $K_{keyhole}$ defined earlier is well known in the literature as “radar equation” and, roughly speaking, it is valid when A is ‘small’ (i.e., $A/\lambda<<\mathrm{min}(d_{TK},d_{KR})$). It is important to note that the keyhole size plays a critical role in the amount of scattering possible; a small keyhole on the order of the wavelength provides good scattering (coverage) in all directions with rank reduction while a large keyhole does not result in a rank reduction as anticipated in a keyhole effect [10]-[11]. It is clear from (42) and (52) that, similar to the fact that small-scale fading characteristics vary significantly when keyhole condition is introduced, the large-scale attenuation law is also modified significantly with wireless signal propagation subjected to keyhole effect. To give a simple example, assuming that $d_{TK}=d_{KR}=d_{TR}/2$, i.e., the keyhole is located in the middle of the channel between the transmitter and receiver sides, we obtain $P_{R}\propto P_{T}/d_{TR}^{2\gamma}$ and thus, the path loss exponent is twice larger (i.e., more severe) than that without keyhole condition. This explains why it is imperative to incorporate large-scale attenuation laws (path losses) along with the small-scale fadings for an accurate comparison study of the MIMO fading channel capacity with and without keyhole effect. Now, we will incorporate the modified large-scale attenuation law (fixed path loss) into the optimal waterfilling power and capacity laws for the MIMO fading channel with keyhole degeneracy.

Applying the path loss scaling law in (52) with transmitter-to-keyhole distance as $d_{TK}$ and keyhole-to-receiver distance as $d_{KR}$, the keyhole MIMO channel in (2) is modified as

where the constant $\xi$ is the large-scale attenuation corresponding to the distances $d_{TK}$ and $d_{KR}$ respectively as $\xi:=P_{R}/P_{T}=K_{keyhole}/\left(d_{TK}^{\gamma}d_{KR}^{\gamma}\right)$. Note that $\textbf{h}_{r}$ and $\textbf{h}_{t}$ are small-scale fadings similar to as defined in the system model in Section II. Accordingly, the ergodic capacity in (8) becomes

with the optimal waterfilling power adaptation $P(\lambda)=\left(\dfrac{1}{\lambda_{0}}-\dfrac{1}{\xi\lambda}\right)^{+}$. Substituting this waterfilling power scheme in (54) gives

Correspondingly, the average power constraint becomes

with $\mathrm{SNR}$, $\bm{\lambda}$ and its PDF $f_{\bm{\lambda}}(\cdot)$ are exactly the same as defined in Section II. Now keeping (IV-B) and (56) in mind and following steps similar to as in the proof of Theorem 2, it is a straightforward exercise to show that the low-SNR capacity of this keyhole MIMO Rayleigh channel $\textbf{H}_{keyhole}$ scales asymptotically as

With (51) as the low-SNR asymptotic capacity for the pure MIMO fading channel $\textbf{H}_{mimo}$ and (57) as the low-SNR asymptotic capacity for the keyhole MIMO fading channel $\textbf{H}_{keyhole}$ under IID Rayleigh fadings condition, we now state the main result in the following theorem.

For numerical demonstration, we assume the path loss related parameters as follows: wavelength of operation $\lambda=3/50$ m (equivalently RF frequency of $5$ GHz), $K_{mimo}=10^{-4.642}$ (or $-46.42$ dB) and $\gamma=2$. The choice $K_{mimo}=10^{-4.642}$ or $\,46.42$ dB attenuation can be shown to be accurate at a reference distance of $1$ meter with frequency of operation $5$ GHz and unity gain antennas. Also, the selection of the path loss exponent $\gamma=2$ is reasonable for microcellular radio operation [27]. The keyhole area is taken as $A=\lambda^{2}$ which can be considered a relatively larger hole (see [10]-[11] for keyhole’s size and measurements related experimental work). For these settings, the keyhole channel’s path loss constant $K_{keyhole}=-81.85$ dB. Also, we take distances $d_{TK}=10\lambda$, $d_{KR}=4\lambda$ and $d_{TK}+d_{KR}\approx d_{TR}$ (thus, $d_{TR}=14\lambda$). At this point, it can be argued that some of the selected channel parameters are very optimistic such as the $d_{TK}$ and $d_{KR}$ distances are chosen small, path loss exponent $\gamma=2$ is low and a relatively larger keyhole area. This is done only for the sake of computational convenience as we have already provided a theoretical guarantee of the ‘crossover phenomenon’ in the low-SNR regime in Theorem 3 irrespective of how large the difference is between the path losses $\xi$ and $\varrho$. For these path loss settings, the low-SNR capacity results for the MIMO Rayleigh channel with and without keyhole effect are shown in the Figures 4(a), 4(b) and 4(c).

A few remarks are now in order:

• •

  Comparing the lowerbound in (58) with (41), it is clear that the overall effect of incorporating suitable path loss models for the MIMO fading channel with and without keyhole effect, precisely with more severe path loss under keyhole condition than without it, is a delayed crossover point.
  

• •

  Allowing path losses to increase in both MIMO fading channels with and without keyhole effect, possibly may be due to increasing distances and/or larger path loss exponents and/or keyhole related impairments such as lower keyhole size, signal attenuation at the keyhole etc., it can be shown that the crossover point will further slide down in the low-SNR regime.
  

• •

  The numerical results in the Figures 4(a), 4(b) and 4(c) for the very optimistic large-scale fading conditions assumed there indicates that the crossover point will generally lie in the extremely low-SNR regime. This also suggests that it will be extremely difficult if not impossible to validate this crossover phenomenon via experimental measurements in realistic practical conditions.
  

• •

  Though we have utilized distance-dependent path loss models for $\xi$ and $\varrho$ in the build-up to Theorem 3, any other suitable large-scale attenuation models (e.g., emperical path loss models) can be applied with no change in the asymptotic low-SNR capacity behaviour suggested by Theorem 3.
  

The low-SNR asymptotic capacity improvement of MIMO fading channel under keyhole effect in Theorem 3 is proved under the assumption of IID Rayleigh fading conditions. Similar low-SNR capacity comparison for the general $r\times t$ MIMO fading channel with and without keyhole effect in IID Nakagami-$m$ fading conditions is slightly constrained due to the lack of eigenmode distributions (joint and marginal) except for some specific cases [32], [33]. For the $2\times 2$ MIMO Nakagami-$m$ fading channel, the joint and marginal eigenmode distribution is derived in a closed form for integer values of $m$ [32, Theorem 1], while the joint eigenmode distribution is available for the general $r\times t$ MIMO Nakagami-$m$ fading channel for $m=0.5$ only [33], [34, p. 107]. For these special MIMO Nakagami-$m$ fading channels with path loss conditions kept the same as mentioned earlier (except for $m=2$ case where we take $d_{TK}=6\lambda$, $d_{KR}=4\lambda$ and thus $d_{TR}=10\lambda$), we have numerically computed the exact MIMO channel capacity with and without keyhole effect in the low-SNR regime, and plotted in Figures 5(a)-5(c) and Figures 6(a)-6(c). These numerical evidences and the fact that the rank of MIMO channel has no effect on the capacity at low-SNRs strongly suggests that for the general $r\times t$ MIMO Nakagami-$m$ fading channel case, the double-faded nature under keyhole effect provides more opportunistic communications at low-SNRs (compared to no keyhole condition), and thus resulting in improved capacity.