# 1-Bit MIMO for Terahertz Channels ††thanks: A. Lozano is with Univ. Pompeu
Fabra, 08018 Barcelona (e-mail: angel.lozano@upf.edu). His work is supported
by the European Research Council under the H2020 Framework Programme/ERC grant
agreement 694974, by MINECO’s Projects RTI2018-102112 and RTI2018-101040, and
by ICREA. Parts of this paper were presented at the 2021 Int’l ITG Workshop on
Smart Antennas [1].

Angel Lozano, Fellow, IEEE

###### Abstract

This paper tackles the problem of single-user multiple-input multiple-output
communication with 1-bit digital-to-analog and analog-to-digital converters.
With the information-theoretic capacity as benchmark, the complementary
strategies of beamforming and equiprobable signaling are contrasted in the
regimes of operational interest, and the ensuing spectral efficiencies are
characterized. Various canonical channel types are considered, with emphasis
on line-of-sight settings under both spherical and planar wavefronts,
respectively representative of short and long transmission ranges at mmWave
and terahertz frequencies. In all cases, a judicious combination of
beamforming and equiprobable signaling is shown to operate within a modest gap
from capacity.

##  I Introduction

As they evolve, wireless systems seek to provide ever faster bit rates and
lower latencies, and a key enabler for these advances in the increase in
bandwidth. From 1G to 5G, the spectrum devoted to wireless communication has
surged from a handful of MHz to multiple GHz, roughly three orders of
magnitude, and this growth is bound to continue as new mmWave bands open up
and inroads are made into the terahertz realm [2, 3, 4, 5, 6].

Besides bandwidth, another key resource is power. Leaving aside the power
spent in duties unrelated to communication, the power consumed by a device can
be partitioned as ${P_{\mathrm{t}}}/\eta+P_{\sf ADC}+P_{\sf other}$ where
${P_{\mathrm{t}}}$ is the power radiated by the transmitter, $\eta$ is the
efficiency of the corresponding power amplifiers, $P_{\sf ADC}$ is the power
required by the receiver’s analog-to-digital (ADC) converters, and $P_{\sf
other}$ subsumes everything else (including oscillators, filters, the
transmitter’s digital-to-analog (DAC) converters, and the receiver’s low-noise
amplifier). With $B$ denoting the bandwidth and $b$ the resolution in bits,
each ADC satisfies

$P_{\sf ADC}=\mathsf{FoM}\,B\,\kappa^{b}$ (1)

where $\mathsf{FoM}$ is a figure of merit and $\kappa$ ranges between two and
four [7, 8].

Power consumption has traditionally been dominated by ${P_{\mathrm{t}}}/\eta$
and thus high resolutions ($b=8$–$12$ bits) could be employed. In 1G and 2G, a
higher $\eta$ was facilitated by the adoption of (respectively analog and
digital) signaling formats tolerant of nonlinear amplification, but after 2G
this took a backseat to spectral efficiency. Linearity has since reigned,
despite the lower $\eta$, as ${P_{\mathrm{t}}}/\eta$ was well within the power
budget of devices for the desirable ${P_{\mathrm{t}}}$.

The advent of 5G, with the move up to mmWave frequencies and the enormous
bandwidths therein, is a turning point in the sense of $P_{\sf ADC}$ ceasing
to be secondary, and this can only accelerate moving forward [9]. Consider
this progression: with $b=10$ at a typical 4G bandwidth of $B=20$ MHz, $P_{\sf
ADC}$ is only a few milliwatts; for $B=2$ GHz, it is already on the order of a
watt; and for $B=20$ GHz, it would reach roughly 10 watts. Indeed, as $B$
continues to grow, $P_{\sf ADC}$ is bound to swallow up the entire power
budget of portable devices unless $\mathsf{FoM}$ or $b$ change. But
$\mathsf{FoM}$ is approaching a fundamental limit [8]. Moreover, while holding
steady up to about $B=100$ MHz, $\mathsf{FoM}$ drops sustainedly after that
mark, which is coincidentally the largest 4G bandwidth. Inevitably then, $b$
has to decrease and, ultimately, it should reach $b=1$, to drastically curb
the power consumption and to further enable dispensing with automatic gain
control at the receiver while simplifying the data pipeline between the ADCs
and the baseband processing [10].

While 1-bit ADCs curb the spectral efficiency at 1 bit per dimension, the vast
bandwidths thereby rendered possible make it exceedingly beneficial. Going
from $b=10$ down to $b=1$ cuts the spectral efficiency by a factor of 2–3, but
in exchange $B$ can grow by as much as 1000 under the same $P_{\sf ADC}$; the
net benefits in bit rate and latency are stupendous. Spectral efficiency is
then best recovered by expanding the number of antennas, which $P_{\sf ADC}$
is only linear in. This naturally leads to multiple-input multiple-output
(MIMO) arrangements with 1-bit ADCs.

Although 1-bit ADCs at the receiver do not necessarily entail 1-bit DACs at
the transmitter, and in some cases the spectral efficiency could improve
somewhat with richer DACs, it is inviting to take the opportunity and adopt
1-bit transmit signals. This not only minimizes the DAC power
consumption—somewhat lower than its ADC’s counterpart, yet also considerable
[11, 12, 13]—but it enables the power amplifiers to operate in nonlinear
regimes where $\eta$ is higher.

Altogether, 1-bit MIMO architectures might feature prominently in future
wireless systems, and not only for mmWave or terahertz operation: these
architectures are also a sensible way forward for lower-frequency extreme
massive MIMO, with antenna counts in the hundreds or even thousands [14]. All
this interest is evidenced by the extensive literature on transmission
strategies and the ensuing performance with 1-bit converters at the
transmitter or receiver only (see [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,
26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44,
45] and references therein), and by the smaller but growing body of work that
considers 1-bit converters at both ends [46, 47, 48, 49, 50, 51, 52, 53, 54,
55, 56, 57, 58, 59]. Chief among the difficulties in this most stringent case
stand (_i_) computing the information-theoretic performance limits for
moderate and large antenna counts, and (_ii_) precoding to generate signals
that can approach those limits.

On these fronts, and concentrating on single-user MIMO, this paper has a two-
fold objective:

  * •

To provide analytical characterizations of the performance of beamforming and
equiprobable signaling, two transmission strategies that are information-
theoretically motivated and complementary.

  * •

To show that a judicious combination of these strategies suffices to operate
within a modest gap from the 1-bit capacity in various classes of channels of
high relevance, foregoing general precoding solutions.

##  II Signal and Channel Models

###  II-A Signal Model

Consider a transmitter equipped with ${N_{\mathrm{t}}}$ antennas and 1-bit
DACs per complex dimension. The receiver, which features ${N_{\mathrm{r}}}$
antennas and a 1-bit ADC per complex dimension, observes

${\bm{y}}=\text{sgn}\\!\left(\sqrt{\frac{{\mathsf{SNR}}}{2{N_{\mathrm{t}}}}}\,{\bm{H}}{\bm{x}}+{\bm{z}}\right)$
(2)

where the sign function is applied separately to the real and imaginary parts
of each entry, such that $y_{n}\in\\{\pm 1\pm\mathrm{j}\\}$, while ${\bm{H}}$
is the ${N_{\mathrm{r}}}\times{N_{\mathrm{t}}}$ channel matrix,
${\bm{z}}\sim\mathcal{N}_{\mathbb{C}}({\bf 0},{\bm{I}})$ is the noise, and
${\mathsf{SNR}}$ is the signal-to-noise ratio per receive antenna. Each entry
of the transmit vector ${\bm{x}}$ also takes the values $\pm 1\pm\mathrm{j}$.

Each antenna in the foregoing formulation could actually correspond to a
compact subarray, in which case the model subsumes array-of-subarrays
structures for the transmitter and/or receiver [60, 61, 62, 63, 64] provided
${\mathsf{SNR}}$ is appropriately scaled.

For each given ${\bm{H}}$, the relationship in (2) embodies a discrete
memoryless channel with $4^{{N_{\mathrm{t}}}}\times 4^{{N_{\mathrm{r}}}}$
transition probabilities determined by

$\displaystyle p_{{\bm{y}}|{\bm{x}}}$
$\displaystyle=\prod_{n=0}^{{N_{\mathrm{r}}}-1}p_{\Re\\{y_{n}\\}|{\bm{x}}}\,p_{\Im\\{y_{n}\\}|{\bm{x}}},$
(3)

where the factorization follows from the noise independence per receive
antenna and complex dimension. Each such noise component has variance $1/2$,
hence

$\displaystyle p_{\Re\\{y_{n}\\}|{\bm{x}}}(1|{\bm{\mathsf{x}}})$
$\displaystyle=\text{Pr}\\!\left[\sqrt{\frac{{\mathsf{SNR}}}{2{N_{\mathrm{t}}}}}\Re\\{{\bm{h}}_{n}{\bm{\mathsf{x}}}+z_{n}\\}>0\right]$
(4)
$\displaystyle=\text{Pr}\\!\left[\Re\\{z_{n}\\}>-\sqrt{\frac{{\mathsf{SNR}}}{2{N_{\mathrm{t}}}}}\Re\\{{\bm{h}}_{n}{\bm{\mathsf{x}}}\\}\right]$
(5)
$\displaystyle=Q\\!\left(\\!-\sqrt{\frac{{\mathsf{SNR}}}{{N_{\mathrm{t}}}}}\Re\\{{\bm{h}}_{n}{\bm{\mathsf{x}}}\\}\\!\right)$
(6)

where ${\bm{h}}_{n}$ is the $n$th row of ${\bm{H}}$ (for
$n=0,\ldots,{N_{\mathrm{r}}}-1$) and $Q(\cdot)$ is the Gaussian Q-function.
Similarly,

$p_{\Re\\{y_{n}\\}|{\bm{x}}}(-1|{\bm{\mathsf{x}}})=Q\\!\left(\\!\sqrt{\frac{{\mathsf{SNR}}}{{N_{\mathrm{t}}}}}\Re\\{{\bm{h}}_{n}{\bm{\mathsf{x}}}\\}\\!\right).$
(7)

From (6) and (7),

$p_{\Re\\{y_{n}\\}|{\bm{x}}}(\Re\\{{\mathsf{y}}_{n}\\}|{\bm{\mathsf{x}}})=Q\\!\left(\\!-\Re\\{{\mathsf{y}}_{n}\\}\sqrt{\frac{{\mathsf{SNR}}}{{N_{\mathrm{t}}}}}\Re\\{{\bm{h}}_{n}{\bm{\mathsf{x}}}\\}\\!\right)$
(8)

and, mirroring it, finally

$\displaystyle p_{{\bm{y}}|{\bm{x}}}({\bm{\mathsf{y}}}|{\bm{\mathsf{x}}})$
$\displaystyle=\prod_{n=0}^{{N_{\mathrm{r}}}-1}Q\\!\left(\\!-\Re\\{{\mathsf{y}}_{n}\\}\sqrt{\frac{{\mathsf{SNR}}}{{N_{\mathrm{t}}}}}\Re\\{{\bm{h}}_{n}{\bm{\mathsf{x}}}\\}\\!\right)$
$\displaystyle\quad\cdot
Q\\!\left(\\!-\Im\\{{\mathsf{y}}_{n}\\}\sqrt{\frac{{\mathsf{SNR}}}{{N_{\mathrm{t}}}}}\Im\\{{\bm{h}}_{n}{\bm{\mathsf{x}}}\\}\\!\right).$
(9)

The transition probabilities correspond to (9) evaluated for the
$4^{N_{\mathrm{r}}}$ possible values of ${\bm{y}}$ and the
$4^{N_{\mathrm{t}}}$ values of ${\bm{x}}$. If ${\bm{H}}$ is known, these
transition probabilities can be readily computed. Conversely, if the
transition probabilities are known, ${\bm{H}}$ can be deduced.

The $4^{{N_{\mathrm{t}}}}$ transmit vectors ${\bm{x}}$ can be partitioned into
$4^{{N_{\mathrm{t}}}-1}$ quartets, each containing four vectors and being
invariant under a $90^{\circ}$ phase rotation of all the entries: from any
vector in the quartet, the other three are obtained by repeatedly multiplying
by $\mathrm{j}$. Since a $90^{\circ}$ phase rotation of ${\bm{x}}$ propagates
as a $90^{\circ}$ phase rotation of ${\bm{H}}{\bm{x}}$, and the added noise is
circularly symmetric, the four vectors making up each transmit quartet are
statistically equivalent and they should thus have the same transmission
probability so as to convey the maximum amount of information.

Likewise, the set of $4^{N_{\mathrm{r}}}$ possible vectors ${\bm{y}}$ can be
partitioned into $4^{{N_{\mathrm{r}}}-1}$ quartets, and the four vectors
${\bm{y}}$ within each received quartet are equiprobable.

###  II-B Channel Model

If the channel is stable over each codeword, then every realization of
${\bm{H}}$ has operational significance and ${\mathsf{SNR}}$ is well defined
under the normalization ${\rm
tr}({\bm{H}}{\bm{H}}^{*})={N_{\mathrm{t}}}{N_{\mathrm{r}}}$. Conversely, if
the coding takes place over a sufficiently broad range of channel
fluctuations, that significance is acquired in an ergodic sense with
$\mathbb{E}\big{[}{\rm
tr}({\bm{H}}{\bm{H}}^{*})\big{]}={N_{\mathrm{t}}}{N_{\mathrm{r}}}$ [65]. The
following classes of channels are specifically considered.

#### Line-of-Sight (LOS) with Spherical Wavefronts

LOS is the chief propagation mechanism at mmWave and terahertz frequencies,
and the spherical nature of the wavefronts is relevant for large arrays and
short transmission ranges. For uniform linear arrays (ULAs) [66],

${\bm{H}}={\bm{D}}_{\rm rx}\tilde{{\bm{H}}}{\bm{D}}_{\rm tx}$ (10)

where ${\bm{D}}_{\rm rx}$ and ${\bm{D}}_{\rm rx}$ are diagonal matrices with
entries

$\displaystyle[{\bm{D}}_{\rm rx}]_{n,n}$
$\displaystyle=e^{-j\pi\left[\frac{2n}{\uplambda}{d_{\rm r}}\sin\\!\theta_{\rm
r}\cos\\!\phi+\frac{n^{2}}{\uplambda D}d^{2}_{\rm
r}\,(1-\sin^{2}\\!\theta_{\rm r}\cos^{2}\\!\phi)\right]}$
$\displaystyle[{\bm{D}}_{\rm tx}]_{m,m}$
$\displaystyle=e^{-j\pi\left[\frac{2m}{\uplambda}{d_{\rm t}}\sin\\!\theta_{\rm
t}+\frac{m^{2}}{\uplambda D}d^{2}_{\rm t}\right]}$ (11)

and with $D$ the range, $\uplambda$ the wavelength, $d_{\rm t}$ and $d_{\rm
r}$ the antenna spacings at transmitter and receiver, $\theta_{\rm t}$ and
$\theta_{\rm r}$ the transmitter and receiver elevations, and $\phi$ their
relative azimuth angle. In turn, $\tilde{{\bm{H}}}$ is the Vandermonde matrix

$\displaystyle\tilde{{\bm{H}}}=\begin{bmatrix}e^{j2\pi\eta\frac{0\times
0}{{N_{\mathsf{max}}}}}&\cdots&e^{j2\pi\eta\frac{(N_{\rm t}-1)\times
0}{{N_{\mathsf{max}}}}}\\\ \vdots&\ddots&\vdots\\\
e^{j2\pi\eta\frac{0\times(N_{\rm
r}-1)}{{N_{\mathsf{max}}}}}&\cdots&e^{j2\pi\eta\frac{(N_{\rm
t}-1)\times(N_{\rm r}-1)}{{N_{\mathsf{max}}}}}\end{bmatrix}$ (12)

where ${N_{\mathsf{max}}}=\max({N_{\mathrm{t}}},{N_{\mathrm{r}}})$ while

$\eta=\frac{({d_{\rm r}}\cos\theta_{\rm r})({d_{\rm t}}\cos\theta_{\rm
t}){N_{\mathsf{max}}}}{\uplambda D}$ (13)

is a parameter that concisely describes any LOS setting with ULAs.

Uniform rectangular arrays can be expressed as the Kronecker product of ULAs,
and expressions deriving from (10) emerge [67]. For more complex topologies,
the entries of ${\bm{H}}$ continue to be of unit magnitude, but the pattern of
phase variations becomes more cumbersome.

#### LOS with Planar Wavefronts

For long enough transmission ranges, the planar wavefront counterpart to (10)
is obtained by letting $D\to\infty$, whereby the channel becomes rank-1 with

$\displaystyle h_{n,m}=e^{-\mathrm{j}\frac{2\pi}{\lambda}(nd_{\rm
r}\sin\theta_{\rm r}\cos\phi+md_{\rm t}\sin\theta_{\rm t})}.$ (14)

#### IID Rayleigh Fading

In this model, representing situations of rich multipath propagation, the
entries of ${\bm{H}}$ are IID and $h_{n,m}\sim\mathcal{N}_{\mathbb{C}}(0,1)$.

We note that the frequency-flat representation embodied by ${\bm{H}}$ is
congruous for the two LOS channel models, but less so for the IID model, where
the scattering would go hand in hand with frequency selectivity over the
envisioned bandwidths. The analysis presented for this model intends to set
the stage for more refined characterizations that account for the inevitable
intersymbol interference. In fact, even for the LOS channels, over a
sufficiently broad bandwidth there is bound to be intersymbol interference
because of spatial widening, i.e., because of the distinct propagation delays
between the various transmit and receive antennas [68, 69].

##  III 1-Bit Capacity

Denote by $p_{1},\ldots,p_{4^{{N_{\mathrm{t}}}-1}}$ the activation
probabilities of the transmit quartets, such that $\sum_{k}p_{k}=1$ and
$p_{\bm{x}}({\bm{\mathsf{x}}}_{k})=p_{k}/4$ with ${\bm{\mathsf{x}}}_{k}$ any
of the vectors in the $k$th quartet. Letting $\mathcal{H}(\cdot)$ indicate
entropy, and with all the probabilities conditioned on ${\bm{H}}$, the mutual
information is

$\displaystyle\\!\\!\mathcal{I}({\mathsf{SNR}},{\bm{H}})$
$\displaystyle=\mathcal{H}({\bm{y}})-\mathcal{H}({\bm{y}}|{\bm{x}})$ (15)
$\displaystyle=\sum_{\ell=1}^{4^{{N_{\mathrm{r}}}}}p_{{\bm{y}}}({\bm{\mathsf{y}}}_{\ell})\log_{2}\frac{1}{p_{{\bm{y}}}({\bm{\mathsf{y}}}_{\ell})}-\mathcal{H}({\bm{y}}|{\bm{x}})$
(16)
$\displaystyle=4\\!\sum_{\ell=1}^{4^{{N_{\mathrm{r}}}-1}}\\!p_{{\bm{y}}}({\bm{\mathsf{y}}}_{\ell})\log_{2}\frac{1}{p_{{\bm{y}}}({\bm{\mathsf{y}}}_{\ell})}-\mathcal{H}({\bm{y}}|{\bm{x}})$
(17)

where (17) follows from the equiprobability of the vectors in each received
quartet and ${\bm{\mathsf{y}}}_{\ell}$ is any of the vectors in the $\ell$th
such quartet while

$\displaystyle p_{{\bm{y}}}({\bm{\mathsf{y}}})$
$\displaystyle=\sum_{k=1}^{4^{{N_{\mathrm{t}}}-1}}\\!\frac{p_{k}}{4}\sum_{i=0}^{3}p_{{\bm{y}}|{\bm{x}}}({\bm{\mathsf{y}}}|\mathrm{j}^{i}{\bm{\mathsf{x}}}_{k})$
(18)

with $p_{{\bm{y}}|{\bm{x}}}$ depending on ${\mathsf{SNR}}$ and ${\bm{H}}$ as
per (9). Elaborating on (18),

$\displaystyle p_{{\bm{y}}}({\bm{\mathsf{y}}})$
$\displaystyle=\sum_{k=1}^{4^{{N_{\mathrm{t}}}-1}}\\!\frac{p_{k}}{4}\left[\prod_{n=0}^{{N_{\mathrm{r}}}-1}Q\\!\left(\\!-\Re\\{{\mathsf{y}}_{n}\\}\sqrt{\frac{{\mathsf{SNR}}}{{N_{\mathrm{t}}}}}\Re\\{{\bm{h}}_{n}{\bm{\mathsf{x}}}_{k}\\}\\!\right)\right.$
$\displaystyle\quad\cdot
Q\\!\left(\\!-\Im\\{{\mathsf{y}}_{n}\\}\sqrt{\frac{{\mathsf{SNR}}}{{N_{\mathrm{t}}}}}\Im\\{{\bm{h}}_{n}{\bm{\mathsf{x}}}_{k}\\}\\!\right)$
$\displaystyle\quad+\prod_{n=0}^{{N_{\mathrm{r}}}-1}Q\\!\left(\\!\Re\\{{\mathsf{y}}_{n}\\}\sqrt{\frac{{\mathsf{SNR}}}{{N_{\mathrm{t}}}}}\Im\\{{\bm{h}}_{n}{\bm{\mathsf{x}}}_{k}\\}\\!\right)$
$\displaystyle\quad\cdot
Q\\!\left(\\!-\Im\\{{\mathsf{y}}_{n}\\}\sqrt{\frac{{\mathsf{SNR}}}{{N_{\mathrm{t}}}}}\Re\\{{\bm{h}}_{n}{\bm{\mathsf{x}}}_{k}\\}\\!\right)$
$\displaystyle\quad+\prod_{n=0}^{{N_{\mathrm{r}}}-1}Q\\!\left(\\!\Re\\{{\mathsf{y}}_{n}\\}\sqrt{\frac{{\mathsf{SNR}}}{{N_{\mathrm{t}}}}}\Re\\{{\bm{h}}_{n}{\bm{\mathsf{x}}}_{k}\\}\\!\right)$
$\displaystyle\quad\cdot
Q\\!\left(\\!\Im\\{{\mathsf{y}}_{n}\\}\sqrt{\frac{{\mathsf{SNR}}}{{N_{\mathrm{t}}}}}\Im\\{{\bm{h}}_{n}{\bm{\mathsf{x}}}_{k}\\}\\!\right)$
$\displaystyle\quad+\prod_{n=0}^{{N_{\mathrm{r}}}-1}Q\\!\left(\\!-\Re\\{{\mathsf{y}}_{n}\\}\sqrt{\frac{{\mathsf{SNR}}}{{N_{\mathrm{t}}}}}\Im\\{{\bm{h}}_{n}{\bm{\mathsf{x}}}_{k}\\}\\!\right)$
$\displaystyle\quad\left.\cdot
Q\\!\left(\\!\Im\\{{\mathsf{y}}_{n}\\}\sqrt{\frac{{\mathsf{SNR}}}{{N_{\mathrm{t}}}}}\Re\\{{\bm{h}}_{n}{\bm{\mathsf{x}}}_{k}\\}\\!\right)\right].$
(19)

In turn, because of the factorization of $p_{{\bm{y}}|{\bm{x}}}$ in (9),

$\displaystyle\mathcal{H}({\bm{y}}|{\bm{x}})$
$\displaystyle=\\!\sum_{n=0}^{{N_{\mathrm{r}}}-1}\big{(}\mathcal{H}(\Re\\{y_{n}\\}|{\bm{x}})+\mathcal{H}(\Im\\{y_{n}\\}|{\bm{x}})\big{)}$
(20)
$\displaystyle=\\!\\!\sum_{k=1}^{4^{{N_{\mathrm{t}}}-1}}\\!\frac{p_{k}}{4}\sum_{i=0}^{3}\\!\sum_{n=0}^{{N_{\mathrm{r}}}-1}\big{(}\mathcal{H}(\Re\\{y_{n}\\}|{\bm{x}}=\mathrm{j}^{i}{\bm{\mathsf{x}}}_{k})$
$\displaystyle\quad+\mathcal{H}(\Im\\{y_{n}\\}|{\bm{x}}=\mathrm{j}^{i}{\bm{\mathsf{x}}}_{k})\big{)}$
(21)
$\displaystyle=\\!\\!\sum_{k=1}^{4^{{N_{\mathrm{t}}}-1}}\\!\frac{p_{k}}{4}\sum_{i=0}^{3}\\!\sum_{n=0}^{{N_{\mathrm{r}}}-1}\\!\left[\mathcal{H}_{\rm
b}\\!\left(Q\\!\left(\\!-{\textstyle\sqrt{\frac{{\mathsf{SNR}}}{{N_{\mathrm{t}}}}}}\Re\\{{\bm{h}}_{n}\mathrm{j}^{i}{\bm{\mathsf{x}}}_{k}\\}\\!\right)\\!\right)\right.$
$\displaystyle\quad\left.+\mathcal{H}_{\rm
b}\\!\left(Q\\!\left(\\!-{\textstyle\sqrt{\frac{{\mathsf{SNR}}}{{N_{\mathrm{t}}}}}}\Im\\{{\bm{h}}_{n}\mathrm{j}^{i}{\bm{\mathsf{x}}}_{k}\\}\\!\right)\\!\right)\right]$
(22)

where $\mathcal{H}_{\rm b}(p)=-p\log_{2}p-(1-p)\log_{2}(1-p)$ is the binary
entropy function. Since changing $i$ merely flips the sign of some of the
Q-funcion arguments, and $Q(-\xi)=1-Q(\xi)$ such that $\mathcal{H}_{\rm
b}(Q(-\xi))=\mathcal{H}_{\rm b}(Q(\xi))$, it follows that

$\displaystyle\mathcal{H}({\bm{y}}|{\bm{x}})$
$\displaystyle=\\!\sum_{k=1}^{4^{{N_{\mathrm{t}}}-1}}\\!p_{k}\sum_{n=0}^{{N_{\mathrm{r}}}-1}\\!\left[\mathcal{H}_{\rm
b}\\!\left(Q\\!\left(\\!-\sqrt{\frac{{\mathsf{SNR}}}{{N_{\mathrm{t}}}}}\Re\\{{\bm{h}}_{n}{\bm{\mathsf{x}}}_{k}\\}\\!\right)\\!\right)\right.$
$\displaystyle\quad\left.+\mathcal{H}_{\rm
b}\\!\left(Q\\!\left(\\!-\sqrt{\frac{{\mathsf{SNR}}}{{N_{\mathrm{t}}}}}\Im\\{{\bm{h}}_{n}{\bm{\mathsf{x}}}_{k}\\}\\!\right)\\!\right)\right].$
(23)

The combination of (17), (19), and (23) gives
$\mathcal{I}({\mathsf{SNR}},{\bm{H}})$, whose evaluation involves
$\mathcal{O}(4^{{N_{\mathrm{t}}}-1}4^{{N_{\mathrm{r}}}-1})$ terms. This
becomes prohibitive even for modest ${N_{\mathrm{t}}}$ and ${N_{\mathrm{r}}}$,
hence the interest in analytical characterizations. From
$\mathcal{I}({\mathsf{SNR}},{\bm{H}})$, the 1-bit capacity is

$C({\mathsf{SNR}},{\bm{H}})=\max_{\\{p_{k}\\}:\sum_{k}p_{k}=1}\mathcal{I}({\mathsf{SNR}},{\bm{H}})$
(24)

with maximization over $p_{1},\ldots,p_{4^{{N_{\mathrm{t}}}-1}}$. Since
$\mathcal{I}({\mathsf{SNR}},{\bm{H}})$ is concave in
$p_{1},\ldots,p_{4^{{N_{\mathrm{t}}}-1}}$ and these probabilities define a
convex set, (24) can be solved with off-the-shelf convex optimization tools.
Or, the Blahut-Arimoto algorithm that alternatively maximizes $p_{\bm{x}}$ and
$p_{{\bm{x}}|{\bm{y}}}$ can be applied, with converge guarantees to any
desired accuracy [70, 71].

In ergodic settings, what applies is the ergodic spectral efficiency

$\mathcal{I}({\mathsf{SNR}})=\mathbb{E}_{\bm{H}}\big{[}\mathcal{I}({\mathsf{SNR}},{\bm{H}})\big{]}$
(25)

and likewise for the ergodic capacity. Alternatively, if the channel is stable
over each codeword, then $\mathcal{I}({\mathsf{SNR}},{\bm{H}})$ and
$C({\mathsf{SNR}},{\bm{H}})$ themselves are meaningful for each ${\bm{H}}$.

The 1-bit capacity cannot exceed $2\min({N_{\mathrm{t}}},{N_{\mathrm{r}}})$
b/s/Hz, with three distinct regimes:

  * •

Low SNR. This is a key regime at mmWave and terahertz frequencies, given the
difficulty in producing strong signals, the high propagation losses, and the
noise bandwidth.

  * •

Intermediate SNR. Here, the spectral efficiency improves sustainedly with the
SNR.

  * •

High SNR. This is a regime of diminishing returns, once the capacity nears
$2\min({N_{\mathrm{t}}},{N_{\mathrm{r}}})$.

###  III-A Low SNR

The low-SNR behavior is most conveniently examined with the mutual information
expressed as function of the normalized energy per bit at the receiver,

$\frac{E_{\rm b}}{N_{0}}=\frac{{\mathsf{SNR}}}{\mathcal{I}({\mathsf{SNR}})}.$
(26)

Beyond the minimum required value of

$\displaystyle\frac{E_{\rm b}}{N_{0}}_{\rm\min}=\lim_{{\mathsf{SNR}}\to
0}\,\frac{{\mathsf{SNR}}}{\mathcal{I}({\mathsf{SNR}})},$ (27)

the mutual information behaves as [72, sec. 4.2]

$S_{0}\,\frac{\left.\frac{E_{\rm b}}{N_{0}}\right|_{\rm\scriptscriptstyle
dB}-\left.\frac{E_{\rm b}}{N_{0}}_{\rm\min}\right|_{\rm\scriptscriptstyle
dB}}{3\,{\rm\scriptstyle dB}}+\varepsilon,$ (28)

where $\varepsilon$ is a lower-order term, $S_{0}$ is the slope at
$\frac{E_{\rm b}}{N_{0}}_{\rm min}$ in b/s/Hz/($3$ dB), and
$z{|_{\mathrm{\scriptscriptstyle dB}}}=10\log_{10}z$.

$\displaystyle
S_{0}=\frac{2\,[\dot{\mathcal{I}}(0)]^{2}}{-\ddot{\mathcal{I}}(0)\log_{2}e}=\frac{\mathbb{E}\big{[}{\rm
tr}({\bm{H}}\bm{\Sigma}_{\bm{x}}{\bm{H}}^{*})\big{]}^{2}}{\frac{1}{2}\,\mathbb{E}\big{[}{\rm
tr}\big{(}(\text{nondiag}({\bm{H}}\bm{\Sigma}_{\bm{x}}{\bm{H}}^{*}))^{2}\big{)}\big{]}+\frac{\pi-1}{3}\,\mathbb{E}\big{[}\|\Re\\{{\bm{H}}{\bm{x}}\\}\|^{4}_{4}+\|\Im\\{{\bm{H}}{\bm{x}}\\}\|^{4}_{4}\big{]}}$
(29)

$\frac{E_{\rm b}}{N_{0}}_{\rm\min}$ and $S_{0}$ descend from the first and
second derivatives of $\mathcal{I}({\mathsf{SNR}})$ at ${\mathsf{SNR}}=0$,
which themselves emerge from (17), (19), and (23) after a tedious derivation
[31, 29]. Plugging these two derivatives into the definitions of $\frac{E_{\rm
b}}{N_{0}}_{\rm\min}$ and $S_{0}$ [72, sec. 4.2],

$\displaystyle\frac{E_{\rm b}}{N_{0}}_{\rm\min}$
$\displaystyle=\frac{1}{\dot{\mathcal{I}}(0)}=\frac{\pi{N_{\mathrm{t}}}}{\mathbb{E}\big{[}{\rm
tr}({\bm{H}}\bm{\Sigma}_{\bm{x}}{\bm{H}}^{*})\big{]}\log_{2}e}$ (30)

with
$\bm{\Sigma}_{\bm{x}}=\mathbb{E}\big{[}{\bm{x}}{\bm{x}}^{*}\big{]}=\sum_{k}p_{k}\,{\bm{\mathsf{x}}}_{k}{\bm{\mathsf{x}}}^{*}_{k}$,
while $S_{0}$ equals (29) where $\text{nondiag}(\cdot)$ returns a matrix with
its diagonal entries set to zero and $\|\cdot\|_{4}$ denotes L4 norm.

The expectations in (30) and (29) are conditioned on ${\bm{H}}$ when there is
operational significance attached to a specific such value, and unconditioned
in the ergodic case.

A worthwhile exercise is to appraise the expansion in (28) against its exact
counterpart, a contrast that Fig. 1 presents for
${N_{\mathrm{t}}}={N_{\mathrm{r}}}=1$ in Rayleigh fading. The characterization
provided by (28) is indeed precise, a fact that extends to all other channels
considered in the paper.

Figure 1: Capacity as a function of $E_{\rm b}/N_{0}$ for
${N_{\mathrm{t}}}={N_{\mathrm{r}}}=1$ and Rayleigh fading. The solid lines are
the exact capacities (1-bit and full resolution) while the dotted lines
represent their respective expansions as per (28).

###  III-B Intermediate SNR

In 1-bit communication, the intermediate-SNR regime steals relevance from its
high-SNR counterpart, which becomes unappealing. In order to delineate the
reach of this intermediate-SNR regime, it is of interest to establish the
limiting capacity for ${\mathsf{SNR}}\to\infty$. Let us define

$C_{\infty}({\bm{H}})=\lim_{{\mathsf{SNR}}\to\infty}C({\mathsf{SNR}},{\bm{H}})$
(31)

and consider channels satisfying ${\bm{h}}_{n}{\bm{x}}\neq 0$ with probability
1 for $n=0,\ldots,{N_{\mathrm{r}}}-1$, such that the transition probabilities
have a positive mass only at $0$ and $1$, meaning that ${\bm{y}}$ is fully
determined by ${\bm{x}}$. The vast majority of channels abide by the
condition, and in particular the ones set forth in Sec. II-B.

For ${N_{\mathrm{t}}}=1$, a single quartet is available for transmission and,
by virtue of its four equiprobable constituent vectors,
$C_{\infty}({\bm{H}})=2$. For ${N_{\mathrm{t}}}>1$ and ${N_{\mathrm{r}}}=1$,
it can be verified that (24) is maximized when a single quartet is activated,
depending on ${\bm{H}}$ [57]. Again, $C_{\infty}({\bm{H}})=2$.

For ${N_{\mathrm{t}}}>1$ and ${N_{\mathrm{r}}}>1$, it must hold that
$C_{\infty}\leq 2{N_{\mathrm{t}}}$, but this bound is generally not achievable
because some vectors ${\bm{x}}$ map to the same receive vector ${\bm{y}}$
[46]. As the transition probabilities are either $0$ or $1$, every binary
entropy function in (23) vanishes and $\mathcal{H}({\bm{y}}|{\bm{x}})\to 0$,
hence the mutual information comes to equal $\mathcal{H}({\bm{y}})$. Letting

$\mathcal{Y}({\bm{H}})=\Big{\\{}{\bm{y}}\,|\,{\bm{y}}=\text{sgn}({\bm{H}}{\bm{x}})\,\forall{\bm{x}}\in\\{\pm
1\pm\mathrm{j}\\}^{{N_{\mathrm{r}}}}\Big{\\}}$ (32)

denote the set of vectors ${\bm{y}}$ that can be elicited for channel
${\bm{H}}$, the maximization of $\mathcal{H}({\bm{y}})$ occurs when this set
is equiprobable. Then,

$C_{\infty}({\bm{H}})=\log_{2}|\mathcal{Y}({\bm{H}})|$ (33)

with $\mathbb{E}[C_{\infty}({\bm{H}})]$ being the limiting ergodic capacity.

The evaluation of (33) is far simpler than that of $C({\mathsf{SNR}})$ in its
full generality.

##  IV 1-Bit vs Full-Resolution Capacity

A naïve comparison of the 1-bit and full-resolution capacities would indicate
that the former always trails the latter. In terms of power, their gap in dB
is at least the difference between their $\frac{E_{\rm
b}}{N_{0}}_{\rm\min}{|_{\mathrm{\scriptscriptstyle dB}}}$ values; in a scalar
Rayleigh-faded channel, for instance, what separates the full-resolution mark
of $-1.59$ dB [72, sec. 4.2] from its 1-bit brethren of $0.37$ dB is $1.96$ dB
as noted in Fig. 1. As shall be seen in the sequel, this gap remains rather
steady with MIMO and over a variety of channels.

Such naïve comparison, however, only accounts for radiated power, disregarding
any other differences in power consumption between the full-resolution and
1-bit alternatives. While appropriate when the radiated power dominates, this
neglect becomes misleading when the digitalization consumes sizeable power
and, since this is the chief motivation for 1-bit communication, by definition
the comparison is somewhat deceptive.

Indeed, whenever the excess power of a full-resolution architecture, relative
to 1-bit, exceeds a $1.96$-dB backoff in ${P_{\mathrm{t}}}/\eta$, there is
going to be a range of SNRs over which, under a holistic accounting of power,
the 1-bit capacity is actually higher. For a very conservative assessment of
this phenomenon, let us assume that $\kappa=2$ in (1) and that $\eta$,
$\mathsf{FoM}$, and $P_{\sf other}$, are not affected by the resolution—in
actuality all of these quantities shall be markedly better in the 1-bit
case—to obtain the condition

$\frac{1}{10^{1.96/10}}\frac{{P_{\mathrm{t}}}}{\eta}+\mathsf{FoM}\,B\,2^{11}\geq\frac{{P_{\mathrm{t}}}}{\eta}+\mathsf{FoM}\,B\,4$
(34)

where we considered two ADCs (${N_{\mathrm{r}}}=1$) and $b=10$ bits for full
resolution. The above yields

$B\geq\frac{0.36\,{P_{\mathrm{t}}}/\eta}{\mathsf{FoM}\,(2^{11}-4)},$ (35)

which, for the sensible values ${P_{\mathrm{t}}}=23$ dBm and $\eta=0.4$, and
with a state-of-the-art $\mathsf{FoM}=10$ pJ/conversion [8], evaluates to
$B\geq 8.8$ GHz. This highly conservative threshold drops rapidly as the
number of digitally processed antennas grows large and thus, for bandwidths
well within the scope of upcoming wireless systems, 1-bit MIMO can be viewed
as information-theoretically optimum for at least some range of SNRs.

##  V Transmit Beamforming

Transmit beamforming corresponds to $\bm{\Sigma}_{\bm{x}}$ being rank-1, i.e.,
to ${\bm{x}}$ being drawn from a single quartet, with such quartet generally
dependent on ${\bm{H}}$. We examine this strategy with an ergodic perspective;
for nonergodic channels, the formulation stands without the expectations over
${\bm{H}}$.

###  V-A Low SNR

For vanishing SNR, transmit beamforming is not only conceptually appealing,
but information-theoretically optimum. Indeed, (30) can be rewritten as

$\displaystyle\frac{E_{\rm
b}}{N_{0}}_{\rm\min}=\frac{\pi{N_{\mathrm{t}}}}{\mathbb{E}\big{[}\sum_{k}p_{k}\,\|{\bm{H}}{\bm{\mathsf{x}}}_{k}\|^{2}\big{]}\log_{2}e},$
(36)

which is maximized by assigning probability $1$ to the quartet
$k^{\star}=\arg\max\|{\bm{H}}{\bm{\mathsf{x}}}_{k}\|^{2}$ for each realization
of ${\bm{H}}$. Therefore, it is optimum to beamform, and the optimum
beamforming quartet is the one maximizing the received power. The task is then
to determine $k^{\star}$ from within the $4^{{N_{\mathrm{t}}}-1}$ possible
quartets.

For ${N_{\mathrm{t}}}=1$, there is no need to optimize over $k$—only one
quartet can be transmitted—and thus

$\displaystyle\frac{E_{\rm b}}{N_{0}}_{\rm\min}$
$\displaystyle=\frac{\pi}{2{N_{\mathrm{r}}}\log_{2}e},$ (37)

which amounts to $0.37$ dB for ${N_{\mathrm{r}}}=1$ [73] and improves by $3$
dB with every doubling of ${N_{\mathrm{r}}}$ thereafter.

For ${N_{\mathrm{t}}}>1$, it is useful to recognize that the choices for
${\bm{x}}$ that are bound to yield high values for $\|{\bm{H}}{\bm{x}}\|^{2}$
are those that project maximally on the dimension of ${\bm{H}}$ that offers
the largest gain, namely the maximum-eigenvalue eigenvector of
${\bm{H}}^{*}{\bm{H}}$. This, in turn, requires that ${\bm{x}}$ mimic, as best
as possible, the structure of that eigenvector; since the magnitude of the
entries of ${\bm{x}}$ is fixed, this mimicking ought to be in terms of phases
only. Formalizing this intuition, it is possible to circumvent the need to
exhaustively search the entire field of $4^{{N_{\mathrm{t}}}-1}$ possibilities
and conveniently identify a subset of only ${N_{\mathrm{t}}}$ quartet
candidates that is sure to contain the one best aligning with the maximum-
eigenvalue eigenvector of ${\bm{H}}^{*}{\bm{H}}$, denoted henceforth by
${\bm{v}}_{0}$. Precisely, as detailed in Appendix A, if we let
$\varphi_{m}=\angle(v_{0,m})+\epsilon$ for $m=0,\ldots,{N_{\mathrm{t}}}-1$,
the ${N_{\mathrm{t}}}$ quartets in the subset can be determined as

${\bm{x}}_{k}=\text{sgn}\big{(}e^{\mathrm{j}\varphi_{k-1}}{\bm{v}}_{0}\big{)}\qquad\quad
k=1,\ldots,{N_{\mathrm{t}}}$ (38)

where $\epsilon$ is a small quantity, positive or negative. If the channel is
rank-1, then this subset is sure to contain the optimum
${\bm{x}}_{k^{\star}}$; if the rank is higher, then optimality is not
guaranteed, but the best value in the above subset is bound to yield excellent
performance.

Turning to the $\frac{E_{\rm b}}{N_{0}}_{\rm min}$ achieved by
${\bm{x}}_{k^{\star}}$, its explicit evaluation is complicated, yet its value
can be shown (see Appendix A again) to satisfy

$\displaystyle\frac{\pi}{2\,\mathbb{E}\big{[}\lambda_{0}\big{]}\log_{2}e}\leq\frac{E_{\rm
b}}{N_{0}}_{\rm\min}\leq\frac{\pi^{3}{N_{\mathrm{t}}}}{16\,\mathbb{E}\big{[}\lambda_{0}\|{\bm{v}}_{0}\|^{2}_{1}\big{]}\log_{2}e}$
(39)

where $\lambda_{0}$ is the maximum eigenvalue of ${\bm{H}}^{*}{\bm{H}}$ while
$\|\cdot\|_{1}$ denotes L1 norm. For ${N_{\mathrm{r}}}=1$, (39) specializes to

$\displaystyle\frac{\pi}{2{N_{\mathrm{t}}}\log_{2}e}\leq\frac{E_{\rm
b}}{N_{0}}_{\rm\min}\leq\frac{\pi^{3}}{16\left(1+({N_{\mathrm{t}}}-1)\frac{\pi}{4}\right)\log_{2}e}.$

Finally, $S_{0}$ can be obtained by plugging ${\bm{x}}={\bm{x}}_{k^{\star}}$
and $\bm{\Sigma}_{\bm{x}}={\bm{x}}_{k^{\star}}{\bm{x}}^{*}_{k^{\star}}$ into
(29).

###  V-B Intermediate SNR

The low-SNR linearity of the mutual information in the received power is the
root cause of the optimality of power-based beamforming in that regime. The
orientation on the complex plane of the received signals is immaterial—a
rotation shifts power from the real to the imaginary part, or vice versa, but
the total power is preserved. Likewise, the power split among receive antennas
is immaterial to the low-SNR mutual information.

At higher SNRs, the linearity breaks down and the mutual information becomes a
more intricate function of ${\bm{H}}{\bm{x}}$, such that proper signal
orientations and power balances become important, to keep
${\bm{h}}_{n}{\bm{x}}$ away from the ADC quantization boundaries for
$n=0,\ldots,{N_{\mathrm{r}}}-1$. This has a dual consequence:

  * •

Transmit beamforming ceases to be generally optimum, even if the channel is
rank-1.

  * •

Even within the confines of beamforming, solutions not based on maximizing
power are more satisfying.

As exemplified in Fig. 2 for ${N_{\mathrm{r}}}=1$, a beamforming quartet with
a better complex-plane disposition at the receiver may be preferable to one
yielding a larger magnitude. This is because, after a 1-bit ADC, only
$90^{\circ}$ rotations and no scalings are possible (in contrast with full-
resolution receivers, where ${\bm{h}}{\bm{x}}$ can subsequently be rotated and
scaled). The best beamforming quartet is the one that simultaneously ensures
large real and imaginary parts for ${\bm{h}}_{n}{\bm{x}}$ in a balanced
fashion for $n=0,\ldots,{N_{\mathrm{r}}}-1$, and the task of identifying this
quartet is a fitting one for learning algorithms [74, 58].

Figure 2: Complex plane representation of the four values of
${\bm{h}}{\bm{x}}$ for a given ${\bm{h}}$ and a given quartet, with the ADC
quantization boundaries indicated by dashed lines. Left-hand side, for
${\bm{\mathsf{x}}}_{k}$, which has a larger magnitude but worse orientation.
Right-hand side, for ${\bm{\mathsf{x}}}_{\ell}$, which has a smaller magnitude
but better orientation. On this channel, quartet $k$ yields a higher mutual at
low SNR while quartet $\ell$ yields a higher mutual information beyond the
low-SNR regime.

We note that, with full-resolution converters, multiple receive antennas play
a role dual to that of transmit beamforming [72, sec. 5.3], and the spectral
efficiency with $N$ transmit and one receive antenna equals its brethren with
one transit and $N$ receive antennas. With 1-bit converters, in contrast,
transmit beamforming optimizes ${\bm{h}}_{n}{\bm{x}}$ for
$n=0,\ldots,{N_{\mathrm{r}}}-1$, to mitigate the addition of noise prior to
quantization, while multiple receive antennas yield a diversity of quantized
observations from which better decisions can be made on which of the possible
vectors was transmitted. This includes majority decisions and erasure
declarations in the case of split observations.

##  VI Equiprobable Signaling

The complementary strategy to beamforming is to activate multiple quartets,
increasing the rank of $\bm{\Sigma}_{\bm{x}}$. Ultimately, all quartets can be
activated with equal probability, such that $\bm{\Sigma}_{\bm{x}}=2{\bm{I}}$.
This renders the signals IID across the transmit antennas, i.e., pure spatial
multiplexing. We examine this strategy with an ergodic perspective.

###  VI-A Low SNR

With equiprobable signaling, (30) gives

$\frac{E_{\rm b}}{N_{0}}_{\rm\min}=\frac{\pi}{2{N_{\mathrm{r}}}\log_{2}e}.$
(40)

In addition [31],

$\displaystyle\mathbb{E}_{\bm{x}}\big{[}\|$
$\displaystyle\Re\\{{\bm{H}}{\bm{x}}\\}\|^{4}_{4}+\|\Im\\{{\bm{H}}{\bm{x}}\\}\|^{4}_{4}\big{]}=6\,{\rm
tr}\big{(}(\text{diag}({\bm{H}}{\bm{H}}^{*}))^{2}\big{)}$
$\displaystyle-4\sum_{n=0}^{{N_{\mathrm{r}}}-1}\sum_{m=0}^{{N_{\mathrm{t}}}-1}\left(\Re\\{h_{n,m}\\}^{4}+\Im\\{h_{n,m}\\}^{4}\right),$
(41)

based on which $S_{0}$ in (29) simplifies considerably.

Combining (39) and (40), the low-SNR advantage of optimum beamforming over
equiprobable signaling, denoted by $\Delta_{\sf BF}$, is tightly bounded as

$\frac{8\,\mathbb{E}\big{[}\lambda_{0}\|{\bm{v}}_{0}\|^{2}_{1}\big{]}}{\pi^{2}{N_{\mathrm{t}}}{N_{\mathrm{r}}}}\leq\Delta_{\sf
BF}\leq\frac{\mathbb{E}\big{[}\lambda_{0}\big{]}}{{N_{\mathrm{r}}}}.$ (42)

This enables some general considerations:

  * •

The low-SNR advantage of beamforming is essentially determined by the maximum
eigenvalue of ${\bm{H}}^{*}{\bm{H}}$. The advantage is largest in rank-1
channels, and minimal if all eigenvalues are equal (on average or
instantaneously, as pertains to ergodic and nonergodic settings).

  * •

If all eigenvalues are equal, beamforming may still yield a lingering
advantage for ${N_{\mathrm{t}}}>{N_{\mathrm{r}}}$, but not otherwise. Indeed,
for ${N_{\mathrm{t}}}\leq{N_{\mathrm{r}}}$, if all eigenvalues all equal then
$\mathbb{E}\big{[}\lambda_{0}\big{]}={N_{\mathrm{r}}}$ and thus $\Delta_{\sf
BF}\leq 1$.

###  VI-B Intermediate SNR

While beamforming is optimum at low SNR, it is decidedly suboptimum beyond,
and activating multiple quartets becomes instrumental to surpass the 2-b/s/Hz
mark. This is the case even in rank-1 channels, where the activation of
multiple quartets allows producing richer signals; this can be seen as the
1-bit counterpart to higher-order constellations. And, given how the curse of
dimensionality afflicts the computation of the optimum quartet probabilities,
equiprobable signaling is a very enticing way of going about this. As will be
seen, not only is it implementationally convenient, but highly effective.

##  VII Channels of Interest

Capitalizing on the analytical tools set forth hitherto, let us now examine
the performance of transmit beamforming and equiprobable signaling in various
classes of channels, starting with the nonergodic LOS settings and progressing
on to the ergodic IID Rayleigh-faded channel.

###  VII-A LOS with Planar Wavefronts

This channel is rank-1, hence the optimum $\frac{E_{\rm b}}{N_{0}}_{\rm min}$
can be achieved with equality by the best beamforming quartet in subset (38).
More conveniently for our purposes here, we can rewrite (10) as
${\bm{H}}=\sqrt{{N_{\mathrm{t}}}{N_{\mathrm{r}}}}{\bm{u}}\,{\bm{v}}^{*}$ where

$\displaystyle u_{n}$
$\displaystyle=\frac{1}{\sqrt{{N_{\mathrm{r}}}}}\,e^{-j\pi\frac{2n}{\lambda}{d_{\rm
r}}\sin\\!\theta_{\rm r}\cos\\!\phi}$ (43) $\displaystyle v_{m}$
$\displaystyle=\frac{1}{\sqrt{{N_{\mathrm{t}}}}}\,e^{-j\pi\frac{2m}{\lambda}{d_{\rm
t}}\sin\\!\theta_{\rm t}}.$ (44)

Irrespective of the array orientations,
$\lambda_{0}={N_{\mathrm{t}}}{N_{\mathrm{r}}}$ and
$\|{\bm{v}}_{0}\|^{2}_{1}={N_{\mathrm{t}}}$ such that (39) reverts to

$\displaystyle\frac{\pi}{2\,{N_{\mathrm{t}}}{N_{\mathrm{r}}}\log_{2}e}\leq\frac{E_{\rm
b}}{N_{0}}_{\rm\min}\leq\frac{\pi^{3}}{16\,{N_{\mathrm{t}}}{N_{\mathrm{r}}}\log_{2}e},$
(45)

which depends symmetrically on ${N_{\mathrm{t}}}$ and ${N_{\mathrm{r}}}$. The
significance of $\frac{E_{\rm b}}{N_{0}}_{\rm\min}$ as the key measure of low-
SNR performance can be appreciated in Fig. 3, which depicts the low-SNR
capacity as a function of $\frac{E_{\rm b}}{N_{0}}$ for
${N_{\mathrm{t}}}={N_{\mathrm{r}}}=1$, $2$, and $4$ in an exemplary LOS
setting. Adding antennas essentially displaces the capacity by the amount by
which $\frac{E_{\rm b}}{N_{0}}_{\rm\min}$ changes.

Shown in Fig. 4 is how $\frac{E_{\rm b}}{N_{0}}_{\rm min}$ improves with the
number of antennas (${N_{\mathrm{t}}}={N_{\mathrm{r}}}$) for the same setting.
Also shown are the values for equiprobable signaling, undesirable in this case
as per (42). The low-SNR advantage of beamforming accrues steadily with the
numbers of antennas and the bounds in (39) tightly bracket the optimum
$\frac{E_{\rm b}}{N_{0}}_{\rm min}$. As anticipated, the gap of 1-bit
beamforming to full-resolution beamforming (included in the figure) remains
small.

Figure 3: Capacity as a function of $E_{\rm b}/N_{0}$ for
${N_{\mathrm{t}}}={N_{\mathrm{r}}}=1$, ${N_{\mathrm{t}}}={N_{\mathrm{r}}}=2$,
and ${N_{\mathrm{t}}}={N_{\mathrm{r}}}=4$, in a planar-wavefront LOS channel
with half-wavelength antenna spacings, $\theta_{\rm t}=0$, $\theta_{\rm
r}=\pi/6$, and $\phi=\pi/4$.  Figure 4: Minimum $E_{\rm b}/N_{0}$ as a
function of ${N_{\mathrm{t}}}={N_{\mathrm{r}}}$ for a planar-wavefront LOS
channel with half-wavelength antenna spacings, $\theta_{\rm t}=0$,
$\theta_{\rm r}=\pi/6$, and $\phi=\pi/4$: 1-bit beamforming (exact values in
solid, interval spanned by the bounds in shaded) vs equiprobable signaling.
Also shown is the performance with full resolution.

Moving up to intermediate SNRs, the beamforming and equiprobable-signaling
performance on another setting is presented in Fig. 5. Also shown is the
actual capacity with $p_{1},\ldots,p_{4^{{N_{\mathrm{t}}}-1}}$ optimized via
Blahut-Arimoto. Up to when the $2$-b/s/Hz ceiling is approached, beamforming
performs splendidly. Past that level, and no matter the rank-1 nature of the
channel, equiprobable signaling is highly superior, tracking the capacity to
within a roughly constant shortfall. This example represents well the
intermediate-SNR performance in planar-wavefront LOS channels, a point that
has be verified by contrasting the asymptotic performance of equiprobable
signaling in a variety of such channels against the respective $C_{\infty}$.

Figure 5: Spectral efficiency as a function of ${\mathsf{SNR}}$ for
${N_{\mathrm{t}}}={N_{\mathrm{r}}}=2$ and
${N_{\mathrm{t}}}={N_{\mathrm{r}}}=4$, in a planar-wavefront LOS channel with
half-wavelength antenna spacings, $\theta_{\rm t}=\pi/4$, $\theta_{\rm
r}=\pi/6$, and $\phi=\pi/4$. In solid, capacity and beamforming; in dashed,
equiprobable signaling.

###  VII-B LOS with Spherical Wavefronts

The scope of channels in this class is very large, depending on the array
topologies and relative orientations; for the sake of specificity, we
concentrate on ULAs, and draw insights whose generalization would be welcome
follow-up work. A key property of ULA-spawned channels within this class is
that [66]

${\bm{H}}^{*}{\bm{H}}\approx\frac{{N_{\mathsf{max}}}}{\eta}{\bm{D}}^{*}_{\rm
tx}{\bm{F}}\text{diag}(\underbrace{1,\ldots,1}_{\eta{N_{\mathsf{min}}}},0,\ldots,0){\bm{F}}^{*}{\bm{D}}_{\rm
tx}$

where the approximation sharpens with the numbers of antennas while ${\bm{F}}$
is a unitary Fourier matrix, ${\bm{D}}_{\rm tx}$ and $\eta$ are as introduced
in Sec. II-B, and
${N_{\mathsf{min}}}=\min({N_{\mathrm{t}}},{N_{\mathrm{r}}})$. Therefore,
$\lambda_{0}\approx{N_{\mathsf{max}}}/\eta$ and
$\|{\bm{v}}_{0}\|^{2}_{1}\approx{N_{\mathrm{t}}}$.

By specializing (39), the optimum $\frac{E_{\rm b}}{N_{0}}_{\rm min}$ attained
by beamforming is seen to satisfy

$\displaystyle\frac{\pi\eta}{2{N_{\mathsf{max}}}\log_{2}e}\lesssim\frac{E_{\rm
b}}{N_{0}}_{\rm\min}\lesssim\frac{\pi^{3}\eta}{16{N_{\mathsf{max}}}\log_{2}e},$
(46)

which indicates that a smaller $\eta$ is preferable at low SNR, meaning
antennas as tightly spaced as possible—this renders the wavefronts maximally
planar—and array orientations as endfire as possible—this shrinks their
effective widths. Indeed, wavefront curvatures trim the beamforming gains, and
reducing $\eta$ mitigates the extent of such curvatures.

With growing $\eta$, the low-SNR performance does degrade, but beamforming
retains an edge over equiprobable signaling for $\eta<1$ or
${N_{\mathrm{t}}}>{N_{\mathrm{r}}}$. Alternatively, for $\eta=1$ and
${N_{\mathrm{t}}}={N_{\mathrm{r}}}$, (46) is no better than the equiprobable-
signaling $\frac{E_{\rm b}}{N_{0}}_{\rm min}$ in (40). In fact, for this all-
important configuration whose eigenvalues are equal [75, 76, 67, 77, 78, 79],
_any_ transmission strategy achieves this same $\frac{E_{\rm b}}{N_{0}}_{\rm
min}$; this can be verified by using
${\bm{H}}^{*}{\bm{H}}={N_{\mathrm{r}}}{\bm{I}}$ and
$\|{\bm{\mathsf{x}}}_{k}\|^{2}=2{N_{\mathrm{t}}}$ in (36), whereby (40)
emerges irrespective of $p_{1},\ldots,p_{4^{N_{\mathrm{t}}}-1}$. This
coincidence of $\frac{E_{\rm b}}{N_{0}}_{\rm min}$ for all transmission
strategies when $\eta=1$ and ${N_{\mathrm{t}}}={N_{\mathrm{r}}}$ does not
translate to $S_{0}$, which is decidedly larger for equiprobable signaling,
indicating that this is the optimum low-SNR technique for this configuration
as illustrated in Fig. 6. Precisely, applying (29) and (41), a channel with
$\eta=1$ and ${N_{\mathrm{t}}}={N_{\mathrm{r}}}=N$ is seen to exhibit

$S_{0}=\frac{\frac{2}{\pi-1}N^{4}}{N^{3}-\frac{2}{3}{\displaystyle\sum_{n=0}^{N-1}\sum_{m=0}^{N-1}}\left[\cos^{4}\\!\big{(}2\pi\frac{nm}{N}\big{)}+\sin^{4}\\!\big{(}2\pi\frac{nm}{N}\big{)}\right]}.$
(47)

Figure 6: Spectral efficiency as a function of $E_{\rm b}/N_{0}$ for
${N_{\mathrm{t}}}={N_{\mathrm{r}}}=4$ and
${N_{\mathrm{t}}}={N_{\mathrm{r}}}=6$ in an LOS channel with $\eta=1$. In
solid, beamforming performance; in dashed, capacity with equiprobable
signaling.

Let us now turn to intermediate SNRs, where the full-resolution wisdom is that
the performance depends only on $\eta$ and it improves monotomically with
$\eta$ up to $\eta=1$, where capacity is achieved by IID signaling. All these
insights, underpinned by the approximate equality of the
$\eta{N_{\mathsf{min}}}$ nonzero eigenvalues of ${\bm{H}}^{*}{\bm{H}}$, cease
to hold in the 1-bit realm due to the transmitter’s inability of accessing
those singular values directly via precoding. Indeed, when the only ability is
to manipulate the quartet probabilities (see Fig. 7 for an example):

  * •

The performance does not depend only on $\eta$, but further on $\theta_{\rm
t}$, $\theta_{\rm r}$, $\phi$, $D$, and $d_{\rm t}$ and $d_{\rm r}$.

  * •

The optimum configuration need not correspond to $\eta=1$.

The main takeaway for our purpose, though, is that at intermediate SNRs
equiprobable signaling closely tracks the capacity.

Figure 7: Main plot: spectral efficiency as a function of ${\mathsf{SNR}}$ for
${N_{\mathrm{t}}}={N_{\mathrm{r}}}=4$, both with optimized (solid lines) and
with uniform (dashed lines) quartet probabilities. Inset: spectral efficiency
as a function of $\eta$ for ${\mathsf{SNR}}=5$ dB. The channel is LOS and the
arrays are broadside with $d_{\rm t}=d_{\rm r}$.

###  VII-C IID Rayleigh Fading

For ${\bm{H}}$ having IID complex Gaussian entries, we resort to the ergodic
interpretation. Shown in Fig. 8 is the evolution of $\frac{E_{\rm
b}}{N_{0}}_{\rm min}$ with the number of antennas for the optimum strategy
(beamforming on every channel realization) as well as for equiprobable
signaling. The bounds in (39) provide an effective characterization of the
optimum $\frac{E_{\rm b}}{N_{0}}_{\rm\min}$. Moreover, $\lambda_{0}$ and
${\bm{v}}_{0}$ are independent [80, lemma 5] and, although
$\mathbb{E}[\lambda_{0}]$ does not lend itself to a general characterization,
for growing ${N_{\mathrm{t}}}$ and ${N_{\mathrm{r}}}$ it approaches
$(\sqrt{{N_{\mathrm{t}}}}+\sqrt{{N_{\mathrm{r}}}})^{2}$. Thus, beamforming
achieves

$\displaystyle\frac{\pi}{2\,\big{(}\sqrt{{N_{\mathrm{t}}}}+\sqrt{{N_{\mathrm{r}}}}\big{)}^{2}\log_{2}e}$
$\displaystyle\lesssim\frac{E_{\rm b}}{N_{0}}_{\rm\min}$ (48)
$\displaystyle\lesssim\frac{\pi^{3}{N_{\mathrm{t}}}}{16\,\big{(}\sqrt{{N_{\mathrm{t}}}}+\sqrt{{N_{\mathrm{r}}}}\big{)}^{2}\,\mathbb{E}\big{[}\|{\bm{v}}_{0}\|^{2}_{1}\big{]}}$

which sharpens with ${N_{\mathrm{t}}}$ and ${N_{\mathrm{r}}}$. For
${N_{\mathrm{t}}}={N_{\mathrm{r}}}=64$, for instance, (48) gives $\frac{E_{\rm
b}}{N_{0}}_{\rm\min}\in[-21.77,-23,71]$ dB, correctly placing the actual value
of $-22.21$ dB. The term $\mathbb{E}\big{[}\|{\bm{v}}_{0}\|^{2}_{1}\big{]}$ is
readily computable for given values of ${N_{\mathrm{t}}}$ and we note, as a
possible path to taming it anaytically, that ${\bm{v}}_{0}$ is a column of a
standard unitary matrix, uniformly distributed over an
${N_{\mathrm{t}}}$-dimensional sphere.

Figure 8: Minimum $\frac{E_{\rm b}}{N_{0}}$ vs
${N_{\mathrm{t}}}={N_{\mathrm{r}}}$ for an IID Rayleigh-faded channel: 1-bit
beamforming (exact values in solid, interval spanned by the bounds in shaded)
vs equiprobable signaling. Also shown is the performance with full resolution.

With equiprobable signaling, $\frac{E_{\rm b}}{N_{0}}_{\rm\min}$ is given by
(40) and we can further characterize $S_{0}$. Starting from (29) and using

$\displaystyle\big{(}\text{nondiag}({\bm{H}}{\bm{H}}^{*})\big{)}^{2}$
$\displaystyle=({\bm{H}}{\bm{H}}^{*})^{2}-{\bm{H}}{\bm{H}}^{*}\,\text{diag}({\bm{H}}{\bm{H}}^{*})$
$\displaystyle\quad-\text{diag}({\bm{H}}{\bm{H}}^{*})\,{\bm{H}}{\bm{H}}^{*}$
$\displaystyle\quad+\big{(}\text{diag}({\bm{H}}{\bm{H}}^{*})\big{)}^{2}$ (49)

in conjuntion with [80, lemma 4]

$\displaystyle\mathbb{E}\big{[}{\rm
tr}\big{(}({\bm{H}}{\bm{H}}^{*})^{2}\big{)}\big{]}$
$\displaystyle={N_{\mathrm{t}}}{N_{\mathrm{r}}}\,({N_{\mathrm{t}}}+{N_{\mathrm{r}}})$
(50) $\displaystyle\mathbb{E}\big{[}{\rm
tr}\big{(}{\bm{H}}{\bm{H}}^{*}\,\text{diag}({\bm{H}}{\bm{H}}^{*})\big{)}\big{]}$
$\displaystyle={N_{\mathrm{t}}}{N_{\mathrm{t}}}({N_{\mathrm{t}}}+1)$ (51)
$\displaystyle\mathbb{E}\big{[}{\rm
tr}\big{(}\text{diag}({\bm{H}}{\bm{H}}^{*})\,{\bm{H}}{\bm{H}}^{*}\big{)}\big{]}$
$\displaystyle={N_{\mathrm{t}}}{N_{\mathrm{t}}}({N_{\mathrm{t}}}+1)$ (52)
$\displaystyle\mathbb{E}\big{[}{\rm
tr}\big{(}(\text{diag}({\bm{H}}{\bm{H}}^{*}))^{2}\big{)}\big{]}$
$\displaystyle={N_{\mathrm{t}}}{N_{\mathrm{t}}}({N_{\mathrm{t}}}+1)$ (53)

we have that

$\mathbb{E}\Big{[}{\rm
tr}\big{(}(\text{nondiag}({\bm{H}}{\bm{H}}^{*}))^{2}\big{)}\Big{]}={N_{\mathrm{t}}}{N_{\mathrm{t}}}\,({N_{\mathrm{r}}}-1).$
(54)

In turn,

$\mathbb{E}\big{[}\|{\bm{H}}{\bm{x}}\|^{4}_{4}\big{]}=6N^{2}_{\rm
t}{N_{\mathrm{r}}}$ (55)

and, altogether,

$S_{0}=\frac{2{N_{\mathrm{t}}}{N_{\mathrm{r}}}}{(\pi-1){N_{\mathrm{t}}}+{N_{\mathrm{r}}}-1},$
(56)

which is an increasing function of both ${N_{\mathrm{t}}}$ and
${N_{\mathrm{r}}}$.

Figure 9: Ergodic spectral efficiency vs ${\mathsf{SNR}}$ for
${N_{\mathrm{t}}}={N_{\mathrm{r}}}=2$ and $4$ both with optimized (solid
lines) and with equiprobable (dashed lines) signaling. The channel is IID
Rayleigh-faded.

At intermediate SNRs, equiprobable signaling is remarkably effective (see Fig.
9). At the same time, the complexity of computing the mutual information—for
equiprobable signaling, let alone with optimized quartet probabilities—is
compounded by the need to expect it over the distribution of ${\bm{H}}$, to
the point of becoming unwieldy even for very small antenna counts. Analytical
characterizations are thus utterly necessary, and it is shown in Appendix B
that

$\displaystyle\mathbb{E}_{\bm{H}}\big{[}\mathcal{I}({\mathsf{SNR}},{\bm{H}})\big{]}$
$\displaystyle=\mathbb{E}_{\bm{H}}\big{[}\mathcal{H}({\bm{y}})\big{]}$ (57)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-\frac{2{N_{\mathrm{r}}}}{\sqrt{2\pi}}\\!\int_{-\infty}^{\infty}\\!\mathcal{H}_{\rm
b}\\!\left(Q\\!\left(\\!-\sqrt{{\mathsf{SNR}}}\,\xi\\!\right)\\!\right)e^{-\xi^{2}/2}\,\mathrm{d}\xi$

with

$\displaystyle
2{N_{\mathrm{r}}}\geq\mathbb{E}_{\bm{H}}\big{[}\mathcal{H}({\bm{y}})\big{]}\geq
2{N_{\mathrm{t}}}-2{N_{\mathrm{r}}}$
$\displaystyle\qquad\;\;\,-\log_{2}\\!\Bigg{[}\sum_{i=0}^{N_{\mathrm{t}}}\left(\\!\\!\begin{array}[]{c}{N_{\mathrm{t}}}\\\
i\end{array}\\!\\!\right)^{\\!\\!2}\left(\frac{1}{4\pi^{2}}\arccos^{2}\\!\left(\frac{2i}{{N_{\mathrm{t}}}}-1\right)\\!\right)^{\\!{N_{\mathrm{r}}}}$
(60)
$\displaystyle\qquad\;\;\,+2\sum_{i=0}^{{N_{\mathrm{t}}}}\sum_{j=i+1}^{{N_{\mathrm{t}}}}\left(\\!\\!\begin{array}[]{c}{N_{\mathrm{t}}}\\\
i\end{array}\\!\\!\right)\\!\left(\\!\\!\begin{array}[]{c}{N_{\mathrm{t}}}\\\
j\end{array}\\!\\!\right)P^{N_{\mathrm{r}}}_{\cap}(i,j)\Bigg{]}$ (65)

where $P_{\cap}(i,j)$ is given by (66).

The bounds specified by (57)–(66) are readily computable even for very large
numbers of antennas. For ${N_{\mathrm{t}}}={N_{\mathrm{r}}}=64$, for instance,
a direct evaluation of
$\mathbb{E}_{\bm{H}}\big{[}\mathcal{I}({\mathsf{SNR}},{\bm{H}})\big{]}$ would
require the $\mathcal{I}({\mathsf{SNR}},{\bm{H}})$ for many realizations of
${\bm{H}}$, with each such mutual information calculation involving over
$10^{75}$ terms. In contrast, the bounds entail the single SNR-dependent
integral in (57) along with (65), which does not depend on the SNR and can be
precomputed; Table I provides such precomputation for a range of antenna
counts. Also of interest is that the upper bound becomes exact for
${\mathsf{SNR}}\to 0$.

$\displaystyle\\!\\!P_{\cap}(i,j)$ $\displaystyle=\frac{1}{2\pi}\int_{0}^{\infty}\\!\\!\\!\\!\int_{0}^{\infty}\text{erfc}\\!\left(-\frac{\gamma({N_{\mathrm{t}}}-i-j)+\xi(i-j)}{\sqrt{i({N_{\mathrm{t}}}-i)+j({N_{\mathrm{t}}}-j)}}\right)\text{erfc}\\!\left(\frac{\gamma(i-j)-\xi({N_{\mathrm{t}}}-i-j)}{\sqrt{i({N_{\mathrm{t}}}-i)+j({N_{\mathrm{t}}}-j)}}\right)e^{-2\left(\gamma^{2}+\xi^{2}\right)}\,\mathrm{d}\gamma\,\mathrm{d}\xi$ (66) TABLE I: Lower bound on $\mathbb{E}_{\bm{H}}\\!\big{[}\mathcal{H}({\bm{y}})\big{]}$ as a function of ${N_{\mathrm{t}}}$ and ${N_{\mathrm{r}}}$. ${N_{\mathrm{r}}}\downarrow$ ${N_{\mathrm{t}}}\to$ | 1 | 2 | 4 | 8 | 16 | 32
---|---|---|---|---|---|---
1 | $2$ | $2$ | $2$ | $2$ | $2$ | 2
2 | $2$ | $3.02$ | $3.65$ | $3.85$ | $3.92$ | $3.96$
4 | $2$ | $3.81$ | $6.07$ | $7.15$ | $7.57$ | $7.78$
8 | $2$ | $4$ | $7.79$ | $12.35$ | $14.14$ | $15.03$
16 | $2$ | $4$ | $8$ | $15.87$ | $24.81$ | $28.14$
32 | $2$ | $4$ | $8$ | $16$ | $31.96$ | $49.6$
Figure 10: Ergodic spectral efficiency as a function of ${\mathsf{SNR}}$ with
equiprobable signaling: the shaded areas are the bounding regions, the solid
lines are the actual values for ${N_{\mathrm{t}}}={N_{\mathrm{r}}}=2$ and
${N_{\mathrm{t}}}={N_{\mathrm{r}}}=4$. The channel is IID Rayleigh-faded.
Figure 11: Ergodic spectral efficiency as a function of ${\mathsf{SNR}}$ with
equiprobable signaling: the shaded areas are the bounding regions, the solid
lines are the actual values for ${N_{\mathrm{t}}}=4,{N_{\mathrm{r}}}=1$ and
${N_{\mathrm{t}}}=8,{N_{\mathrm{r}}}=2$. The channel is IID Rayleigh-faded.

The range specified by the bounds is illustrated in Fig. 10 for various values
of ${N_{\mathrm{t}}}={N_{\mathrm{r}}}$, alongside the actual spectral
efficiencies (obtained via Monte-Carlo) for
${N_{\mathrm{t}}}={N_{\mathrm{r}}}=2$ and
${N_{\mathrm{t}}}={N_{\mathrm{r}}}=4$.

For ${N_{\mathrm{t}}}>{N_{\mathrm{r}}}$, the lower bound approaches its upper
counterpart and, for ${N_{\mathrm{t}}}\gg{N_{\mathrm{r}}}$,

$\displaystyle\mathbb{E}_{\bm{H}}\big{[}\mathcal{I}({\mathsf{SNR}},{\bm{H}})\big{]}$
(67) $\displaystyle\;\approx
2{N_{\mathrm{r}}}\left(1-\frac{1}{\sqrt{2\pi}}\\!\int_{-\infty}^{\infty}\\!\mathcal{H}_{\rm
b}\\!\left(Q\\!\left(\\!-\sqrt{{\mathsf{SNR}}}\,\xi\\!\right)\\!\right)e^{-\xi^{2}/2}\,\mathrm{d}\xi\right)\\!.$

Indeed, as detailed in Appendix B, this approximation becomes an exact result
for ${N_{\mathrm{r}}}=1$ or for ${N_{\mathrm{t}}}\to\infty$ with
${N_{\mathrm{r}}}$ arbitrary.

Some examples for ${N_{\mathrm{t}}}=4{N_{\mathrm{r}}}$, presented in Fig. 11,
confirm how precisely the ergodic spectral efficiency is determined when the
antenna counts are somewhat skewed.

##  VIII Conclusion

A host of issues that are thoroughly understood for full-resolution settings
must be tackled anew for 1-bit MIMO communication. In particular, the
computation of the capacity becomes unwieldy for even very modest
dimensionalities and the derivation of general precoding solutions becomes a
formidable task, itself power-consuming. Fortunately, in the single-user case
such general precoding can be circumvented via a judicious switching between
beamforming and equiprobable signaling, with the added benefits that these
transmissions strategies are much more amenable to analytical
characterizations and that their requirements in terms of channel-state
information at the transmitter are minimal:
$\log_{2}4^{{N_{\mathrm{t}}}-1}=4\,({N_{\mathrm{t}}}-1)$ bits for beamforming,
none for equiprobable signaling.

The transition from beamforming to equiprobable signaling could be finessed by
progressively activating quartets as the SNR grows, but the results in this
paper suggest that there is a small margin of improvement: a direct switching
at some appropriate point suffices to operate within a few dB of capacity at
both low and intermediate SNRs. It would be of interest to gauge this
shortfall for more intricate channel models such as those in [81, 82, 83].

Channel estimation at the receiver is an important aspect, with the need for
procedures that avoid having to painstakingly gauge all the transition
probabilities between ${\bm{x}}$ and ${\bm{y}}$ to deduce ${\bm{H}}$. Of much
interest would be to extend existing results for channel estimation with full-
resolution DACs and 1-bit ADCs [84, 24, 25, 85, 86] to the complete 1-bit
realm. Equally pertinent would be to establish the bandwidths over which a
frequency-flat representation suffices for each channel model, and to extend
the respective analyses to account for intersymbol interference. This is
acutely important given the impossibility of implementing OFDM with 1-bit
converters.

In those multiuser settings where orthogonal (time/frequency) multiple access
is effective, switching between beamforming and equiprobable signaling is also
enticing. In other cases, chiefly if the antenna numbers are highly
asymmetric, orthogonal multiple access is decidedly suboptimum, and there is
room for more general schemes. We hope that the results in this paper can
serve as a stepping stone to such schemes.

## Appendix A

Let $\sigma_{0},\ldots,\sigma_{{N_{\mathrm{t}}}-1}$ be the singular values of
${\bm{H}}$, ordered from largest to smallest, while ${\bm{u}}_{m}$ and
${\bm{v}}_{m}$ are the left and right singular vectors corresponding to
$\sigma_{m}$. From the singular value decomposition

${\bm{H}}=\\!\\!\\!\sum_{m=0}^{{N_{\mathsf{min}}}-1}\\!\\!\\!\sigma_{m}{\bm{u}}_{m}{\bm{v}}^{*}_{m}$
(68)

we have that

$\|{\bm{H}}{\bm{x}}\|^{2}=\\!\\!\sum_{m=1}^{{N_{\mathsf{min}}}-1}\\!\\!\sigma^{2}_{m}\left|{\bm{v}}^{*}_{m}{\bm{x}}\right|^{2},$
(69)

which is the quantity to maximize. Under full-resolution transmission, (69) is
maximized by ${\bm{x}}\propto{\bm{v}}_{0}$: complete projection on the
dimension exhibiting the largest gain and zero projection elsewhere [72, sec.
5.3]. With 1-bit transmission, perfect alignment with ${\bm{v}}_{0}$ is
generally not possible, and the goal becomes to determine which ${\bm{x}}$
best aligns. If ${\bm{H}}$ is rank-1, then such ${\bm{x}}$ is sure to maximize
(69). If the rank is plural, however, optimality cannot be guaranteed from
best alignment with ${\bm{v}}_{0}$ because some other ${\bm{x}}$ leaning
further away could have a more favorable projection across the rest of
dimensions. Suppose, for instance, that the rank is $3$; if the ${\bm{x}}$
best aligned with ${\bm{v}}_{0}$ does not further project on ${\bm{v}}_{1}$,
but only on ${\bm{v}}_{2}$, there could be another ${\bm{x}}$ aligning
slightly less with ${\bm{v}}_{0}$ but projecting also on ${\bm{v}}_{1}$ in a
way that yields a higher metric in (69). This possibility may arise when the
largest singular value is not very dominant. Even then, though, the ${\bm{x}}$
that projects maximally on ${\bm{v}}_{0}$ is bound to perform well.

Values of ${\bm{x}}$ that align well with ${\bm{v}}_{0}$ can be obtained as
${\bm{x}}=\text{sgn}\big{(}e^{\mathrm{j}\varphi}{\bm{v}}_{0}\big{)}$ where
$\varphi$ allows setting the absolute phase arbitrarily before quantization.
Letting $\varphi$ run from $0$ to $2\pi$, every entry of the quantized
${\bm{x}}$ changes four times and a subset of $4{N_{\mathrm{t}}}$ vectors
${\bm{x}}$ is obtained. These $4{N_{\mathrm{t}}}$ vectors actually belong to
${N_{\mathrm{t}}}$ quartets because, if ${\bm{\mathsf{x}}}_{k}$ is in the
$k$th subset, $\mathrm{j}{\bm{\mathsf{x}}}_{k}$ is sure to be there too. Since
identifying one representative per quartet suffices for our purposes,
attention can be restricted to those values of $\varphi$ that trigger a change
in $\text{sgn}\big{(}e^{\mathrm{j}\varphi}{\bm{v}}_{0}\big{)}$, i.e.,
$\varphi=\angle(v_{0,m})$ for $m=0,\ldots,{N_{\mathrm{t}}}-1$. Letting
$\varphi_{m}=\angle(v_{0,m})+\epsilon$ with $\epsilon$ a small quantity, we
obtain the subset of ${N_{\mathrm{t}}}$ quartet representatives as

${\bm{\mathsf{x}}}_{k}=\text{sgn}\big{(}e^{\mathrm{j}\varphi_{k-1}}{\bm{v}}_{0}\big{)}\qquad\quad
k=1,\ldots,{N_{\mathrm{t}}}.$ (70)

The sign of $\epsilon$ is irrelevant, it merely changes which representative
is selected for each quartet. Confirming the intuition that the
${N_{\mathrm{t}}}$ quartets in (70) are good choices, it is proved in [47]
that the quartet that best aligns with ${\bm{v}}_{0}$ is sure to be in this
subset. Thus, searching a subset of ${N_{\mathrm{t}}}$ candidates suffices to
beamform optimally in rank-1 channels, and quasi-optimally in higher-rank
channels, without having to search the entire field of
$4^{{N_{\mathrm{t}}}-1}$ possibilities.

Let us now turn to the performance. An upper bound on
$\|{\bm{H}}{\bm{\mathsf{x}}}_{k^{\star}}\|^{2}$ can be obtained by assuming
that, on every channel realization, there is a value of ${\bm{x}}$ that aligns
perfectly with ${\bm{v}}_{0}$. From (69), this gives

$\|{\bm{H}}{\bm{x}}\|^{2}\leq 2{N_{\mathrm{t}}}\sigma^{2}_{0},$ (71)

which, along with (36), yields the lower bound in (39).

In turn, a lower bound on $\|{\bm{H}}{\bm{\mathsf{x}}}_{k^{\star}}\|^{2}$ is
obtained for any choice of ${\bm{x}}$, and in particular for
${\bm{x}}=\text{sgn}\big{(}e^{\mathrm{j}\varphi}{\bm{v}}_{0}\big{)}$ with
$\varphi\in[-\pi/4,\pi/4]$, such that [47]

$\displaystyle\left|{\bm{v}}^{*}_{0}{\bm{\mathsf{x}}}_{k^{\star}}\right|$
$\displaystyle\geq\max_{\varphi\in[-\pi/4,\pi/4]}\left|{\bm{v}}^{*}_{0}\,\text{sgn}\big{(}e^{\mathrm{j}\varphi}{\bm{v}}_{0}\big{)}\right|$
(72)
$\displaystyle\geq\mathbb{E}_{\varphi}\\!\\!\left[\left|{\bm{v}}^{*}_{0}\,\text{sgn}\big{(}e^{\mathrm{j}\varphi}{\bm{v}}_{0}\big{)}\right|\right]$
(73)
$\displaystyle=\mathbb{E}_{\varphi}\\!\\!\left[\left|\sum_{m=0}^{{N_{\mathrm{t}}}-1}v^{*}_{0,m}\,\text{sgn}\big{(}e^{\mathrm{j}\varphi}v_{0,m}\big{)}\right|\right]$
(74)
$\displaystyle=\mathbb{E}_{\varphi}\\!\\!\left[\left|\sum_{m=0}^{{N_{\mathrm{t}}}-1}\\!\left|v_{0,m}\right|e^{-\mathrm{j}\phi_{m}}\,\text{sgn}\big{(}e^{\mathrm{j}(\varphi+\phi_{m})}\big{)}\right|\right]$
$\displaystyle=\mathbb{E}_{\varphi}\\!\\!\left[\left|\sum_{m=0}^{{N_{\mathrm{t}}}-1}\\!\left|v_{0,m}\right|e^{\mathrm{j}\varphi}e^{-\mathrm{j}(\varphi+\phi_{m})}\,\text{sgn}\big{(}e^{\mathrm{j}(\varphi+\phi_{m})}\big{)}\right|\right]$
$\displaystyle=\mathbb{E}_{\varphi}\\!\\!\left[\left|\sum_{m=0}^{{N_{\mathrm{t}}}-1}\\!\left|v_{0,m}\right|e^{-\mathrm{j}(\varphi+\phi_{m})}\,\text{sgn}\big{(}e^{\mathrm{j}(\varphi+\phi_{m})}\big{)}\right|\right].$

For any $\theta\in[0,2\pi]$, the phase of
$e^{-\mathrm{j}\theta}\text{sgn}\big{(}e^{\mathrm{j}\theta}\big{)}$ is within
$[-\pi/4,\pi/4]$ while
$\left|e^{-\mathrm{j}\theta}\text{sgn}\big{(}e^{\mathrm{j}\theta}\big{)}\right|=\sqrt{2}$.
Hence, letting $\theta_{m}=\varphi+\phi_{m}$,

$\displaystyle\left|{\bm{v}}^{*}_{0}{\bm{\mathsf{x}}}_{k^{\star}}\right|$
$\displaystyle\geq\sqrt{2}\,\mathbb{E}_{\theta_{0},\ldots,\theta_{{N_{\mathrm{t}}}-1}}\\!\\!\left[\left|\sum_{m=0}^{{N_{\mathrm{t}}}-1}\\!\left|v_{0,m}\right|e^{-\mathrm{j}\theta_{m}}\right|\right]$
(75)
$\displaystyle\geq\sqrt{2}\,\,\mathbb{E}_{\theta_{0},\ldots,\theta_{{N_{\mathrm{t}}}-1}}\\!\\!\left[\left|\sum_{m=0}^{{N_{\mathrm{t}}}-1}\\!\left|v_{0,m}\right|\cos(\theta_{m})\right|\right]$
(76)
$\displaystyle=\sqrt{2}\,\sum_{m=0}^{{N_{\mathrm{t}}}-1}\\!\left|v_{0,m}\right|\mathbb{E}\big{[}\\!\cos(\theta_{m})\big{]}$
(77)
$\displaystyle=\sqrt{2}\,\sum_{m=0}^{{N_{\mathrm{t}}}-1}\\!\left|v_{0,m}\right|\frac{2}{\pi}\int_{-\pi/4}^{\pi/4}\\!\\!\cos(\xi)\,\mathrm{d}\xi$
(78)
$\displaystyle=\frac{4}{\pi}\sum_{m=0}^{{N_{\mathrm{t}}}-1}\\!\left|v_{0,m}\right|$
(79) $\displaystyle=\frac{4}{\pi}\,\|{\bm{v}}_{0}\|_{1},$ (80)

where (77) holds because $\cos(\theta_{m})>0$ for
$\theta_{m}\in[-\pi/4,\pi/4]$. Disregarding
$\left|{\bm{v}}^{*}_{m}{\bm{\mathsf{x}}}_{k^{\star}}\right|$ for $m>0$ in
(69),

$\|{\bm{H}}{\bm{\mathsf{x}}}_{k^{\star}}\|^{2}\geq\frac{16}{\pi^{2}}\,\sigma^{2}_{0}\,\|{\bm{v}}_{0}\|^{2}_{1},$
(81)

which, along with (36), yields the upper bound in (39).

## Appendix B

In (23), for every $n$ and $k$,
$\Re\\{{\bm{h}}_{n}{\bm{\mathsf{x}}}_{k}\\}\sim\mathcal{N}(0,{N_{\mathrm{t}}})$
and
$\Im\\{{\bm{h}}_{n}{\bm{\mathsf{x}}}_{k}\\}\sim\mathcal{N}(0,{N_{\mathrm{t}}})$.
Thus, letting $r\sim\mathcal{N}(0,1)$,

$\displaystyle\mathbb{E}_{\bm{H}}\big{[}\mathcal{H}({\bm{y}}|{\bm{x}})\big{]}$
$\displaystyle=2{N_{\mathrm{r}}}\,\mathbb{E}_{r}\\!\left[\mathcal{H}_{\rm
b}\\!\left(Q\\!\left(\\!-\sqrt{{\mathsf{SNR}}}\,r\\!\right)\\!\right)\\!\right]$
(82)
$\displaystyle=\frac{2{N_{\mathrm{r}}}}{\sqrt{2\pi}}\\!\int_{-\infty}^{\infty}\\!\mathcal{H}_{\rm
b}\\!\left(Q\\!\left(\\!-\sqrt{{\mathsf{SNR}}}\,\xi\\!\right)\\!\right)e^{-\xi^{2}/2}\,\mathrm{d}\xi.$

In turn,

$\mathbb{E}_{\bm{H}}\big{[}\mathcal{H}({\bm{y}})\big{]}\leq 2{N_{\mathrm{r}}}$
(83)

with equality for ${\mathsf{SNR}}\to 0$, when the receiver observes only noise
and ${\bm{y}}$ is equiprobably binary on $2{N_{\mathrm{r}}}$ real dimensions.
As the removal of noise can only decrease it, $\mathcal{H}({\bm{y}})$
diminishes as the SNR grows, being lower-bounded by its value for
${\mathsf{SNR}}\to\infty$. The expectation of such noiseless lower bound over
${\bm{H}}$ can be elaborated by generalizing to our complex setting a clever
derivation in [46], starting from

$\displaystyle\mathbb{E}_{\bm{H}}\big{[}\mathcal{H}({\bm{y}})\big{]}=\mathbb{E}_{\bm{H}}\\!\\!\left[\sum_{\ell=1}^{4^{{N_{\mathrm{r}}}}}p_{\bm{y}}({\bm{\mathsf{y}}}_{\ell})\log_{2}\frac{1}{p_{\bm{y}}({\bm{\mathsf{y}}}_{\ell})}\right]$
(84)
$\displaystyle\qquad=\mathbb{E}_{\bm{H}}\\!\\!\left[\sum_{\ell=1}^{4^{{N_{\mathrm{r}}}}}\mathbb{E}_{\bm{x}}\big{[}p_{{\bm{y}}|{\bm{x}}}({\bm{\mathsf{y}}}_{\ell}|{\bm{x}})\big{]}\log_{2}\frac{1}{\mathbb{E}_{\bm{x}}\big{[}p_{{\bm{y}}|{\bm{x}}}({\bm{\mathsf{y}}}_{\ell}|{\bm{x}})\big{]}}\right]$
$\displaystyle\qquad=\mathbb{E}_{\bm{H}}\\!\\!\left[\sum_{\ell=1}^{4^{{N_{\mathrm{r}}}}}\mathbb{E}_{\bm{x}}\big{[}1\\{\text{sgn}({\bm{H}}{\bm{x}})={\bm{y}}_{\ell}\\}\big{]}\right.$
$\displaystyle\qquad\qquad\quad\cdot\log_{2}\frac{1}{\mathbb{E}_{\bm{x}}\big{[}1\\{\text{sgn}({\bm{H}}{\bm{x}})={\bm{y}}_{\ell}\\}\big{]}}\Bigg{]}$
(85)

where $1\\{\cdot\\}$ is the indicator function. Since ${\bm{H}}$ is isotropic
and ${\bm{x}}$ is equiprobable, no ${\bm{\mathsf{y}}}_{\ell}$ is favored over
the rest in terms of the probability of $\text{sgn}({\bm{H}}{\bm{x}})$
equalling such ${\bm{\mathsf{y}}}_{\ell}$. Hence, (85) can be evaluated for
any specific ${\bm{\mathsf{y}}}_{\ell}$, say ${\bm{\mathsf{y}}}_{1}$ whose
entries all equal $1+\mathrm{j}$. This gives

$\displaystyle\mathbb{E}_{\bm{H}}\big{[}\mathcal{H}({\bm{y}})\big{]}$
$\displaystyle=4^{N_{\mathrm{r}}}\,\mathbb{E}_{\bm{H}}\\!\Bigg{[}\mathbb{E}_{\bm{x}}\big{[}1\\{\text{sgn}({\bm{H}}{\bm{x}})={\bm{\mathsf{y}}}_{1}\\}\big{]}$
$\displaystyle\quad\cdot\log_{2}\frac{1}{\mathbb{E}_{\bm{x}}\big{[}1\\{\text{sgn}({\bm{H}}{\bm{x}})={\bm{\mathsf{y}}}_{1}\\}\big{]}}\Bigg{]}.$
(86)

Likewise, the probability that
$\text{sgn}({\bm{H}}{\bm{x}})={\bm{\mathsf{y}}}_{1}$ is common to every
realization of ${\bm{x}}$ and thus

$\displaystyle\mathbb{E}_{\bm{H}}\big{[}\mathcal{H}({\bm{y}})\big{]}$
$\displaystyle=-4^{N_{\mathrm{r}}}\,\mathbb{E}_{\bm{H}}\\!\Big{[}1\\{\text{sgn}({\bm{H}}{\bm{\mathsf{x}}}_{1})={\bm{\mathsf{y}}}_{1}\\}$
$\displaystyle\quad\cdot\log_{2}\mathbb{E}_{\bm{x}}\big{[}1\\{\text{sgn}({\bm{H}}{\bm{x}})={\bm{\mathsf{y}}}_{1}\\}\big{]}\Big{]}$
(87)

where all entries of ${\bm{\mathsf{x}}}_{1}$ equal $1+\mathrm{j}$ and where it
is convenient to retain the second expectation over ${\bm{x}}$ in order to
later solve its counterpart over ${\bm{H}}$. Then,

$\displaystyle\mathbb{E}_{\bm{H}}\big{[}\mathcal{H}({\bm{y}})\big{]}$
$\displaystyle=-4^{N_{\mathrm{r}}}\,\mathbb{E}_{{\bm{H}}|\text{sgn}({\bm{H}}{\bm{\mathsf{x}}}_{1})={\bm{\mathsf{y}}}_{1}}\\!\Big{[}1\\{\text{sgn}({\bm{H}}{\bm{\mathsf{x}}}_{1})={\bm{\mathsf{y}}}_{1}\\}$
$\displaystyle\quad\cdot\log_{2}\mathbb{E}_{\bm{x}}\big{[}1\\{\text{sgn}({\bm{H}}{\bm{x}})={\bm{\mathsf{y}}}_{1}\\}\big{]}\Big{]}$
$\displaystyle\quad\cdot\mathbb{P}[\text{sgn}({\bm{H}}{\bm{\mathsf{x}}}_{1})={\bm{\mathsf{y}}}_{1}]$
(88)

and, since
$\mathbb{P}[\text{sgn}({\bm{H}}{\bm{\mathsf{x}}}_{1})={\bm{\mathsf{y}}}_{1}]=\frac{1}{4^{N_{\mathrm{r}}}}$
and the factor
$1\\{\text{sgn}({\bm{H}}{\bm{\mathsf{x}}}_{1})={\bm{\mathsf{y}}}_{1}\\}$
becomes immaterial once the expectation over ${\bm{H}}$ has been conditioned
on $\text{sgn}({\bm{H}}{\bm{\mathsf{x}}}_{1})$,

$\displaystyle\mathbb{E}_{\bm{H}}\big{[}\mathcal{H}({\bm{y}})\big{]}$ (89)
$\displaystyle\;=-\mathbb{E}_{{\bm{H}}|\text{sgn}({\bm{H}}{\bm{\mathsf{x}}}_{1})={\bm{\mathsf{y}}}_{1}}\\!\Big{[}\log_{2}\mathbb{E}_{\bm{x}}\big{[}1\\{\text{sgn}({\bm{H}}{\bm{x}})={\bm{\mathsf{y}}}_{1}\\}\big{]}\Big{]}$
$\displaystyle\;=-\mathbb{E}_{{\bm{H}}|\text{sgn}({\bm{H}}{\bm{\mathsf{x}}}_{1})={\bm{\mathsf{y}}}_{1}}\\!\\!\left[\log_{2}\frac{1}{4^{N_{\mathrm{t}}}}\sum_{k=1}^{4^{N_{\mathrm{t}}}}1\\{\text{sgn}({\bm{H}}{\bm{\mathsf{x}}}_{k})={\bm{\mathsf{y}}}_{1}\\}\\!\right]$
$\displaystyle\;=2{N_{\mathrm{t}}}-\mathbb{E}_{{\bm{H}}|\text{sgn}({\bm{H}}{\bm{\mathsf{x}}}_{1})={\bm{\mathsf{y}}}_{1}}\\!\\!\left[\log_{2}\sum_{k=1}^{4^{N_{\mathrm{t}}}}1\\{\text{sgn}({\bm{H}}{\bm{\mathsf{x}}}_{k})={\bm{\mathsf{y}}}_{1}\\}\\!\right]$
$\displaystyle\;\geq
2{N_{\mathrm{t}}}-\log_{2}\sum_{k=1}^{4^{N_{\mathrm{t}}}}\mathbb{E}_{{\bm{H}}|\text{sgn}({\bm{H}}{\bm{\mathsf{x}}}_{1})={\bm{\mathsf{y}}}_{1}}\big{[}1\\{\text{sgn}({\bm{H}}{\bm{\mathsf{x}}}_{k})={\bm{\mathsf{y}}}_{1}\\}\big{]}$

where the last step follows from Jensen’s inequality. Since the expectation of
an indicator function yields the probability of the underlying event,

$\displaystyle\mathbb{E}_{\bm{H}}\big{[}\mathcal{H}({\bm{y}})\big{]}\geq
2{N_{\mathrm{t}}}$ (90)
$\displaystyle\qquad\quad-\log_{2}\sum_{k=1}^{4^{N_{\mathrm{t}}}}{\mathbb{P}}\big{[}\text{sgn}({\bm{H}}{\bm{\mathsf{x}}}_{k})={\bm{\mathsf{y}}}_{1}|\text{sgn}({\bm{H}}{\bm{\mathsf{x}}}_{1})={\bm{\mathsf{y}}}_{1}\big{]}$

with

$\displaystyle{\mathbb{P}}\big{[}\text{sgn}({\bm{H}}{\bm{\mathsf{x}}}_{k})={\bm{\mathsf{y}}}_{1}|\text{sgn}({\bm{H}}{\bm{\mathsf{x}}}_{1})={\bm{\mathsf{y}}}_{1}\big{]}$
(91)
$\displaystyle\qquad\quad=\frac{{\mathbb{P}}\big{[}\text{sgn}({\bm{H}}{\bm{\mathsf{x}}}_{k})={\bm{\mathsf{y}}}_{1}\,\cap\,\text{sgn}({\bm{H}}{\bm{\mathsf{x}}}_{1})={\bm{\mathsf{y}}}_{1}\big{]}}{{\mathbb{P}}\big{[}\text{sgn}({\bm{H}}{\bm{\mathsf{x}}}_{1})={\bm{\mathsf{y}}}_{1}\big{]}}$
$\displaystyle\qquad\quad=4^{N_{\mathrm{r}}}\,{\mathbb{P}}\big{[}\text{sgn}({\bm{H}}{\bm{\mathsf{x}}}_{k})={\bm{\mathsf{y}}}_{1}\,\cap\,\text{sgn}({\bm{H}}{\bm{\mathsf{x}}}_{1})={\bm{\mathsf{y}}}_{1}\big{]}.$

As the channel has IID entries, letting ${\bm{h}}$ be an arbitrary row of
${\bm{H}}$,

${\mathbb{P}}\big{[}\text{sgn}({\bm{H}}{\bm{\mathsf{x}}}_{k})={\bm{\mathsf{y}}}_{1}\,\cap\,\text{sgn}({\bm{H}}{\bm{\mathsf{x}}}_{1})={\bm{\mathsf{y}}}_{1}\big{]}=(P_{\cap})^{N_{\mathrm{r}}}$
(92)

with

$\displaystyle P_{\cap}$
$\displaystyle={\mathbb{P}}\big{[}\text{sgn}({\bm{h}}{\bm{\mathsf{x}}}_{k})=(1+\mathrm{j})\,\cap\,\text{sgn}({\bm{h}}{\bm{\mathsf{x}}}_{1})=(1+\mathrm{j})\big{]}$
$\displaystyle={\mathbb{P}}\big{[}\text{sgn}(\Re\\{{\bm{h}}{\bm{\mathsf{x}}}_{k}\\})=1\,\cap\,\text{sgn}(\Im\\{{\bm{h}}{\bm{\mathsf{x}}}_{k}\\})=1$
$\displaystyle\quad\,\cap\,\text{sgn}(\Re\\{{\bm{h}}{\bm{\mathsf{x}}}_{1}\\})=1\,\cap\,\text{sgn}(\Im\\{{\bm{h}}{\bm{\mathsf{x}}}_{1}\\})=1\big{]}$
(93)
$\displaystyle={\mathbb{P}}\big{[}\Re\\{{\bm{h}}{\bm{\mathsf{x}}}_{k}\\}>0\,\cap\,\Im\\{{\bm{h}}{\bm{\mathsf{x}}}_{k}\\}>0$
$\displaystyle\quad\,\cap\,\Re\\{{\bm{h}}{\bm{\mathsf{x}}}_{1}\\}>0\,\cap\,\Im\\{{\bm{h}}{\bm{\mathsf{x}}}_{1}\\}>0\big{]}.$
(94)

Defining $a_{k}={\bm{h}}{\bm{\mathsf{x}}}_{k}$ and
$a_{1}={\bm{h}}{\bm{\mathsf{x}}}_{1}$,

$\displaystyle P_{\cap}$
$\displaystyle=\int_{0}^{\infty}\\!\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\\!\int_{0}^{\infty}f_{\Re\\{a_{k}\\}\Im\\{a_{k}\\}\Re\\{a_{1}\\}\Im\\{a_{1}\\}}(\alpha,\beta,\gamma,\xi)$
$\displaystyle\quad\;\cdot\mathrm{d}\alpha\,\mathrm{d}\beta\,\mathrm{d}\gamma\,\mathrm{d}\xi$
(95)

with $\Re\\{a_{k}\\}$, $\Im\\{a_{k}\\}$, $\Re\\{a_{1}\\}$, and
$\Im\\{a_{1}\\}$ jointly Gaussian with mean zero and covariance

$\bm{\Sigma}_{k}=\left[\begin{array}[]{cc}{N_{\mathrm{t}}}{\bm{I}}&{\bm{R}}_{k}\\\
{\bm{R}}_{k}&{N_{\mathrm{t}}}{\bm{I}}\end{array}\right]$ (96)

where ${\bm{I}}$ is the $2\times 2$ identity matrix while

${\bm{R}}_{k}=\left[\begin{array}[]{cc}{N_{\mathrm{t}}}-i-j&j-i\\\
i-j&{N_{\mathrm{t}}}-i-j\end{array}\right].$ (97)

with $i$ and $j$ the respective number of entries of $\Re\\{a_{k}\\}$ and
$\Im\\{a_{k}\\}$ that are $-1$, the rest of their entries (along with all the
entries of $\Re\\{a_{1}\\}$ and $\Im\\{a_{1}\\}$) being $+1$. Most
importantly, because the entries of ${\bm{h}}$ are IID, the position of those
$-1$ values is immaterial and only their totals $i$ and $j$ matter.
Altogether, the joint distribution of $\Re\\{a_{k}\\}$, $\Im\\{a_{k}\\}$,
$\Re\\{a_{1}\\}$ and $\Im\\{a_{1}\\}$ is as in (105) and, plugging it in (95)
and tediously integrating over two of the dimensions, what emerges is (66)
with the dependence on $i$ and $j$ made explicit and with $\text{erfc}(\cdot)$
the complementary error function.

Returning to (90), and accounting for the number of indices $k$ that map to
each $i$ and $j$,

$\displaystyle\mathbb{E}_{\bm{H}}\big{[}\mathcal{H}({\bm{y}})\big{]}$
$\displaystyle\geq 2{N_{\mathrm{t}}}-2{N_{\mathrm{r}}}$ (98)
$\displaystyle\quad-\log_{2}\sum_{i=0}^{{N_{\mathrm{t}}}}\sum_{j=0}^{{N_{\mathrm{t}}}}\left(\\!\\!\begin{array}[]{c}{N_{\mathrm{t}}}\\\
i\end{array}\\!\\!\right)\\!\left(\\!\\!\begin{array}[]{c}{N_{\mathrm{t}}}\\\
j\end{array}\\!\\!\right)P^{N_{\mathrm{r}}}_{\cap}(i,j).$ (103)

For $i=j$, (66) can be integrated into

$P_{\cap}(i,i)=\frac{1}{4\pi^{2}}\arccos\\!\left(\frac{2i}{{N_{\mathrm{t}}}}-1\right)$
(104)

with $P_{\cap}(0,0)=1/4$ and $P_{\cap}({N_{\mathrm{t}}},{N_{\mathrm{t}}})=0$.
Furthermore, $P_{\cap}(i,j)=P_{\cap}(j,i)$ with
$P_{\cap}(0,{N_{\mathrm{t}}})=P_{\cap}({N_{\mathrm{t}}},0)=0$. With these
relationships accounted for, (98) yields (65).

$\displaystyle
f_{\Re\\{a_{k}\\}\Im\\{a_{k}\\}\Re\\{a_{1}\\}\Im\\{a_{1}\\}}(\alpha,\beta,\gamma,\xi)=\frac{1}{8\pi^{2}}\frac{1}{i({N_{\mathrm{t}}}-i)+j({N_{\mathrm{t}}}-j)}\exp\\!\left(-\frac{1}{4}\frac{[\alpha\;\beta\;\gamma\;\xi]\,\bm{\Sigma}^{-1}_{k}\,[\alpha\;\beta\;\gamma\;\xi]^{\rm
T}}{i({N_{\mathrm{t}}}-i)+j({N_{\mathrm{t}}}-j)}\right)$ (105)

Let us now consider some special cases of interest. For ${N_{\mathrm{t}}}=1$,
(65) reduces to

$\mathbb{E}_{\bm{H}}\big{[}\mathcal{H}({\bm{y}})\big{]}\geq
2-2{N_{\mathrm{r}}}-\log_{2}\\!\left(4^{-{N_{\mathrm{r}}}}+4P_{\cap}(0,1)\right)$
(106)

and, since $P_{\cap}(0,1)=0$, further to
$\mathbb{E}_{\bm{H}}\big{[}\mathcal{H}({\bm{y}})\big{]}\geq 2$. In fact, in
this case $\mathcal{H}({\bm{y}})=2$ for every channel realization and thus
$\mathbb{E}_{\bm{H}}\big{[}\mathcal{H}({\bm{y}})\big{]}=2$.

For ${N_{\mathrm{r}}}=1$, the scalar quantized signal $y$ takes four
equiprobable values—again, not only on average, but for every channel
realization—and thus $\mathbb{E}_{\bm{H}}\big{[}\mathcal{H}(y)\big{]}=2$.

For fixed ${N_{\mathrm{t}}}$ and ${N_{\mathrm{r}}}\to\infty$, the key is the
observation that $P_{\cap}(i,j)$ achieves its largest value for $i=j=0$,
namely $P_{\cap}(0,0)=1/4$. For $i>0$ and/or $j>0$, $P_{\cap}(i,j)<1/4$
because any negative sign in either the real and imaginary parts of
${\bm{\mathsf{x}}}_{k}$ reduces the probability in (94). The largest term in
the summations within the logarithm in (65) equals $4^{-{N_{\mathrm{r}}}}$
and, as ${N_{\mathrm{r}}}\to\infty$, every other term vanishes faster and the
lower bound on $\mathbb{E}_{\bm{H}}\big{[}\mathcal{H}({\bm{y}})\big{]}$
converges towards $2{N_{\mathrm{t}}}$.

Finally, for fixed ${N_{\mathrm{r}}}$ and ${N_{\mathrm{t}}}\to\infty$, the
rows of ${\bm{H}}$ become asymptotically orthogonal [72, sec. 5.4.2] and
hence, for every realization of ${\bm{H}}$, ${\bm{y}}$ consists of IID complex
components. Again,
$\mathbb{E}_{\bm{H}}\big{[}\mathcal{H}({\bm{y}})\big{]}=2{N_{\mathrm{r}}}$.

For ${N_{\mathrm{r}}}=1$ and for ${N_{\mathrm{t}}}\to\infty$ with fixed
${N_{\mathrm{r}}}$, the above observations reveal that the lower and upper
bounds coincide, fully determining, as per (67), the ergodic spectral
efficiency with equiprobable signaling and IID Rayleigh fading.

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