A General Framework for Low-Resolution Receivers for MIMO Channels
Stefano Rini Luca Barletta Yonina C. Eldar Elza Erkip


Simplified proposal for realizing multiqubit tunable phase gate in circuit QED
    Wen-An Li[E-mail: liwenan@126.com] and Yuan Chen
    December 30, 2023
==============================================================================


The capacity of a discrete-time multi-input multi-output (MIMO) Gaussian channel with output quantization is investigated for different
 receiver architectures.

A general formulation of this problem is proposed in which the antenna outputs are processed by analog combiners while
sign quantizers are used for analog-to-digital conversion.




To exemplify this approach, four analog receiver architectures of varying generality and complexity are considered:
(a) multiple antenna selection and sign quantization of the antenna outputs,
(b) single antenna selection and multilevel quantization,
(c) multiple antenna selection and multilevel quantization, and
(d) linear combining of the antenna outputs and multilevel quantization.

Achievable rates are studied as a function of the number of
available sign quantizers and compared among different architectures.

In particular, it is shown that architecture (a) is sufficient to attain the optimal high signal-to-noise ratio performance for  a MIMO receiver in which the number of antennas is larger than the number of sign quantizers.

Numerical evaluations of the average performance are presented for the case in which the channel gains
are i.i.d. Gaussian.












§ INTRODUCTION


Low-resolution quantization is an important technology for massive MIMO and millimeter-wave communication systems as it allows the transceivers to operate at low power levels
<cit.>.




Although the performance of MIMO receivers with large antenna arrays and low-resolution quantizers has been investigated in the literature
under different assumptions on the hardware limitations and antenna architectures,



a complete fundamental information theoretic understanding

is currently not available.


In this paper, we propose a unified framework to analyze and compare low-resolution receiver architectures.



More specifically, we assume that the receiver is comprised of  N_SQ sign quantizers that process  antenna outputs.

Each sign quantizer is connected to the antenna outputs via an analog combining circuit with limited processing capabilities.

Through this general formulation, we study
the effects of limited processing and low-resolution quantization
on the capacity of MIMO channels.








Op-amp voltage comparators are employed
in nearly all analog-to-digital converters to obtain multilevel
quantization. Given the receiver's ability to partially reconfigure its circuitry depending on the channel realization, it is of interest to determine which
configuration of the comparators  yields the
largest capacity.



  §.§.§ Literature Review



Quantization in MIMO systems is a well-investigated topic in the literature:
for the sake of brevity we focus here on the results
regarding sign quantization.[
In the literature, the term “one-bit quantization” most often refers to sign quantization of the antenna outputs.

Here, as in <cit.>,  we prefer the term “sign quantization”  since we distinguish between sign and threshold quantization.]

The authors in <cit.> are perhaps the first to point out that the capacity loss in MIMO channels due to coarse quantization is surprisingly small, although this observation is supported mostly through numerical evaluations.






In <cit.>, the authors derive fundamental properties of the capacity-achieving
distribution for a single-input single-output (SISO) channel with output quantization.

A lower bound on the capacity of sign-quantized MIMO channels with Gaussian inputs
based on the Bussgang decomposition is derived in <cit.>.

The high signal-to-noise ratio (SNR) asymptotics for complex MIMO channels with sign quantization are studied are <cit.>.

For the SISO channel with  threshold quantization, <cit.> shows that, in the limit of vanishing SNR,
asymmetric quantizers outperform symmetric ones.























  §.§.§ Contributions


We focus, in the following, on four analog receiver architectures with different levels of complexity: (a) multiple antenna selection and sign quantization, (b) single antenna selection and multilevel quantization, (c) multiple antenna selection and multilevel quantization, and (d) linear combining and multilevel quantization.

The architecture (c) is more general than both (a) and (b), and (d) is the most general one.

We study the case of a SIMO channel and a MIMO channel and provide capacity bounds of each architecture as a function of the


number of sign quantizers.



For the SIMO channel, our results suggest conditions under which the capacity of the architecture with multiple antenna selection and multilevel quantization
closely approaches that of the architecture with linear combining and multilevel quantization.

For the MIMO channel with linear combining and multilevel quantization, we derive an approximatively optimal usage of the sign quantizers
as a variation of the classic water-filling power allocation scheme.


This solution shows that, if the number of antennas at the receiver is larger than the number of sign quantizers,
sign quantization is sufficient to attain the optimal performance in the high SNR regime.




Numerical evaluations are provided for the case in which the channel gains are i.i.d. Gaussian distributed.





  §.§.§ Paper Organization




Sec. <ref> introduces the channel model.
Sec. <ref> reviews the results available for the case of sign quantization of the channel outputs.



The main results  are given in Sec. <ref>.

Numerical evaluations are provided in Sec. <ref>.
Sec. <ref> concludes the paper.




  §.§.§ Notation

We adopt the standard notation for H_2(x)=-xlog x-(1-x)log(1-x) and Q(x)=1/√(2 π)∫_x^+∞exp(-u^2/2) u.

All logarithms are taken in base two.
For the SISO model, we set  =1 w.l.o.g.,  for the MISO and SIMO models we denote the channel matrix
as  and ^T respectively.


For the MIMO case, the vector =[_1 …_min{,}] contains the eigenvalues of the matrix ^T.

The  identity matrix of size n× n  is indicated as _n, the  all-zero/all-one matrix of size n × m
as _n × m/_n × m.

Finally,  _π indicates the set of all permutation matrices.






§ CHANNEL MODEL





  §.§.§ Problem Formulation



We consider a discrete-time real-valued MIMO channel with  transmit antennas and  receive antennas.

At the n^ th channel use, the antenna output vector _n=[W_1,n… W_,n]^T,
is obtained from the channel input vector _n=[X_1,n… X_,n]^T as


_n=_n+_n,   n∈[1 …N],


where  is a full rank matrix of size ×

[This condition guarantees the existence of a right pseudo-inverse for  and holds with high probability in a richly scattering environment.]

 and _n is an -vector of i.i.d. additive Gaussian noise samples with zero mean and unitary variance.
 
 The channel matrix  is assumed to be known at both transmitter and receiver and to be fixed throughout the transmission block-length N.
 



The channel input vector is subject to the average power constraint ∑_n=1^N [ |_n|_2^2]≤  N P where |_n|_2 indicates the 2-norm.


The antenna output vector is processed through N_SQ sign quantizers, each receiving a linear combination of the antenna output vector plus a constant,[It must be noted that generating a precise voltage reference is another major hurdle in analog-to-digital conversion. Although possible in our framework, in the following we do not consider such limitation.] i.e.

_n = (_n+),   n∈[1 …N],


where  is the analog combining matrix of size N_SQ×,  is a threshold vector of length N_SQ and
 () is the function producing the sign of each component of the vector  as plus or minus one, so that _n ∈{-1,+1}^N_SQ.

For a given choice of combining matrix  and threshold vector , the capacity of the model in (<ref>) is given by

(,)=max_P_(),  [ ||_2^2]≤P I(;),


where we have explicitly expressed the dependency of the capacity on the parameters {,}.[The capacity (,)
is also a function of the channel matrix , although not explicitly indicated.
 
 ]

The analog processing capabilities at the receiver are modeled as a set of feasible values of {,}, denoted as  .


Our goal is to maximize the capacity expression in (<ref>) over , namely


()=max_{,} ∈ (,).





  §.§.§ Relevant Architectures




The formulation in (<ref>) attempts to capture the tension  between
the quantization of few antennas with high precision versus the quantization of many antennas with low precision.

This is accomplished by treating the sign quantizers as a resource to be allocated optimally among a set of possible configurations F.

Note that   multilevel quantization  can be obtained by using M-1 sign quantizers and appropriate thresholds , resulting in log(M) information bits.



It follows that sign quantization produces the most information bits per sign quantizer and increasing the
number of quantization levels increases the information bits only logarithmically.


To exemplify the insights provided by our approach,


we study four analog receiver architectures:




(a) Multiple antenna selection and sign quantization:

Here 
  in (<ref>) is selected as

_a    ==_N_SQ,_N_SQ ×(-N_SQ) P_π,  P_π ∈_π, 

            =_N_SQ×1 ,

 

that is, each sign quantizer is connected to one of the channel outputs.

Figure <ref> represents this model for =4 and N_SQ=3.





(b) Single antenna selection and multilevel quantization:




For this receiver architecture, the sign quantizers are used to construct an (N_SQ+1)-level quantizer:

_b    =
= _N_SQ  ×1 , _ N_SQ ×(-1) P_π,  P_π ∈_π, 

              ∈^N_SQ ,
 


Figure <ref> shows this model for =4 and N_SQ=3.



(c) Multiple antenna selection and multilevel quantization:

Here, each sign quantizer can select an antenna output and a voltage offset before performing quantization.


This is obtained by choosing

_c     =
V_ij ∈{0,1},  _j=1^ V_ij=1, 
    
    ∈^N_SQ .
 



This receiver architecture encompasses those in Fig. <ref> and Fig. <ref> as special
cases.
Figure <ref> again shows this model for =4 and N_SQ=3.

(d) Linear combining and multilevel quantization: Corresponds to the set of all possible choices of  and .




§ SIGN QUANTIZATION



The effect of quantization on the capacity of the MIMO channel  has been investigated thoroughly in the literature.

For conciseness, we review only the results on sign quantization of the channel outputs, corresponding to the architecture in Fig. <ref> for N_SQ=,  which will be relevant in the remainder of the paper.





The capacity of SISO channel with sign quantization of the outputs is attained by antipodal signaling.


<cit.>:

The capacity of the SISO channel with sign quantization of the antenna output with N_SQ= is

_SISO=1-H_2 Q √(P) .




The capacity of the MISO channel with sign output quantization is obtained from the result in Lem. <ref>
by transforming this model into a SISO channel through transmitter beamforming, thus yielding

_MISO=1-H_2 Q ||√(P) .



For the SIMO and MIMO channel, capacity with sign quantization is known in the high-SNR regime.

<cit.>.

The capacity of the  SIMO  channel with sign quantization of the antenna output   with N_SQ= at high SNR satisfies

 log() ≤_SIMO,a^SNR ∞ ≤log(+1).



<cit.>.



The capacity of the  MIMO  channel with   sign quantization and N_SQ=, and
for which  satisfies a general position condition  (see <cit.>), is bounded at high SNR as

12 log(K(N_SQ,) ) ≤_MIMO,a^SNR ∞ ≤12log(K(N_SQ,)+1)

if <N_SQ, where

K(N_SQ,) = ∑_k=0^2 -1 2N_SQ-1k.

If ≥ N_SQ, then _ MIMO,a^ SNR ∞   = N_SQ.





At finite SNR, upper and lower bounds on the capacity of the MIMO channel with sign quantization are known but are not tight in general <cit.>.






















§ MAIN RESULTS



We begin by considering the capacity of the SISO channel for the receiver architectures in Sec. <ref>.

Capacity for the architecture (a) is provided in Lem. <ref> (necessarily N_SQ=1) while the architectures  (b), (c) and (d) all correspond to the same model in which the channel output is quantized
through an (N_SQ+1)-level quantizer.

The capacity for this latter model can be bounded to within a small additive gap as shown in the next proposition.





The capacity of the SISO channel with multi-level output quantization, N_SQ>1,

is upper-bounded as



_SISO ≤12 logminP+1,(N_SQ+1)^2 ,


and capacity is to within  1 bits-per-channel-use () from the upper bound in (<ref>).




The upper bound (<ref>) is the  minimum between the capacity of the model without quantization constraints
and the capacity of the channel without additive noise.



For the achievability proof, the input is chosen as an equiprobable M-PAM signal for


M= min⌊√(P) ⌋, N_SQ +1 ,



in which the distance between the constellation points is

such that the power constraint is met with equality.

At the receiver, the quantization thresholds are selected as the midpoints of the M-PAM constellation points.

The full proof is in  App. <ref>.


For the SIMO and MIMO cases, given the generality of the formulation in (<ref>), rather than attempting to find the exact capacity C(F) for each architecture in Sec. <ref>,





we instead focus on approximate characterization  in the spirit of Prop. <ref>,

that is:
(i) the upper bound  is obtained  as the minimum among   two simple upper bounds 

and (ii) the achievability proof relies on a transmission scheme whose performance can be easily compared to the upper bound
to show a small gap  between the two bounds.



This approach provides an approximate characterization of capacity which is useful in comparing the performance of different architectures.

In the following, we extend the result in Prop. <ref> to the SIMO and MIMO cases.[Note that the MISO case follows from the SISO case as in (<ref>).]






  §.§.§ SIMO case


The capacity for the architecture (a) is obtained by selecting the antenna with the largest gain; for the architecture (b) the capacity

is a rather straight-forward extension of the result in
Prop. <ref>.


The capacity of the SIMO channel with single antenna selection and multilevel quantization is upper-bounded as

_SIMO, b ≤12 logmin1+h_max^2 P, (N_SQ+1)^2 ,


where h_max=max_i h_i and the upper bound in (<ref>) can be attained to within 1/2.



The proof is provided in App. <ref>


For the architecture (c), sampling  more antennas allows the
receiver to collect more information on the input but  reduces
the  number of samples that can be acquired from each antenna.



The capacity of the SIMO channel with multiple antenna selection and multilevel quantization
for


P>log(N_SQ)>2 and h_i^2 >1  is bounded as






    max_K 12 logmin1+ |^(K)|_2^2  P,  N_SQ K +1 ^2 -2
 

         ≤_SIMO, c  ≤12 log1+||_2^2P, (N_SQ+1)^2 ,


where ^(K)  is the vector of the K largest channel gains.






The upper bound is derived similarly to Prop. <ref>.

The achievable rate with finite uniform output quantization is  related to the achievable rate with infinite uniform output
quantization by bounding the largest difference between these two quantities under the conditions P>log(N_SQ) and   .


In the model with infinite output quantization, a dither can be used to make the quantization noise
independent of the channel input and of the additive noise,





so that the worst additive noise lemma

may then be used to  lower bound
the attainable rate as in (<ref>).

The full proof is provided in App. <ref>.





The capacity of the SIMO channel with linear combining and multilevel quantization is upper-bounded as

_SIMO, d ≤12 logmin1+||_2^2P, (N_SQ+1)^2 ,


and the upper bound in (<ref>) can be attained to within 1/2.



With this architecture, the maximal ratio combining at the receiver results in the equivalent SISO channel with channel gain ||_2.
The result in Prop. <ref> can then be used to obtain the approximate capacity.







The results in Prop. <ref>, Prop. <ref> and Prop. <ref> are related as follows.

The results for the architecture (a) in Lem. <ref> and the architecture (b) in Prop. <ref>
show that the two architectures yield the same high-SNR behaviour when ≥ N_SQ.

When < N_SQ, though, the architecture in (b) can attain higher performance at high SNR.

The architectures (c) and (d) differ as follows: in the former, the estimate of the transmitted message is implicitly obtained by combining the quantized information while, in the latter, combining occurs before quantization.

From Prop. <ref> we gather the conditions under which  combining after quantization roughly attains the same performance
as combining before quantization: this occurs when the number of quantizers is sufficiently large so that the first term in the minimum in (<ref>) dominates the channel performance.


The capacity of the SIMO channel with multiple antenna selection and multilevel quantization
is upper-bounded as

_SIMO, c ≤12 log1+||_2^2P ,


and the upper bound in (<ref>) can be attained to within 1
when N_SQ> √(||_2^2 P+1) and  h_i^2 >1.




Under these assumption, the minimum in (<ref>)  is attained by setting K=N_r, in which case the trivial outer bound of (<ref>) can be attained to within 2.








  §.§.§ MIMO case




For the architecture (a),  inner and outer bounds are derived in <cit.>; for the architecture (b), an upper bound is derived in the next proposition.


The capacity of the MIMO channel with single antenna selection and multilevel quantization is upper-bounded as
 
_MIMO,b≤12 logmin1+|_max^T|_2^2P,(N_SQ+1)^2 ,

where _max^T is the row of  with the largest norm and the upper bound in (<ref>)
can be attained to within 2.



The proof is provided in App. <ref>.











For the architecture (d), the approximate capacity can be obtained as a variation of the classic water-filling solution.

By decomposing the channel matrix through singular value decomposition, the channel can be transformed in  K=min{N_t,N_r} parallel channel with gains
 {_i}.





Capacity is then obtained as

max ∑_i=1^K  12 logmin1+_i^2 P_i, ( N_SQ,i+1)^2,


where the maximization is over P_i ∈^+,  ∑_i P_i = P, N_SQ,i∈,  ∑_i N_SQ,i =N_SQ and K ∈ [0,min{N_t,N_r}].

By relaxing the integer constraint on the parameters N_SQ,i, we obtain to the outer bound

    ≤R^⋆(,P,N_SQ)= 

    
∑_i=1^min{,}12 log(1+_i P_i)  

      if    ∑_i=1^min{,} √(1+_i P_i)-1 ≤N_SQ 

 K  logN_SQ K  +1 

        otherwise,



where P_i are chosen as P_i=(μ-_i^-2)^+  and μ is the smallest value for which  ∑_i P_i=P  and
K=∑_i 1_{P_i>0}.

The approximate capacity for the architecture (d) is obtained by showing that a rate sufficiently close to (<ref>) is achievable.




The capacity approaching transmission strategy is interpreted as follows: the classic water-filling
solution is approximatively optimal as long as each channel output can be quantized using  N_SQ,i≈√(1+_i P_i)-1 quantizers. 

If this condition is not satisfied, then the optimal solution is to uniformly assign the quantizers to all the active antennas. This leads to the next proposition.




The capacity of a MIMO channel with linear combining and multilevel quantization is upper-bounded as

    _MIMO,d ≤R^⋆(,P,N_SQ),




and capacity  is to within a gap of  3/2 K from the upper bound in (<ref>) for R^⋆(,P,N_SQ) and K in (<ref>).
 
 
The proof is provided in App. <ref>.



The result in Prop. <ref> shows that sign quantization is sufficient to attain the optimal performance in the high SNR regime since K=N_SQ yields the largest rate in (<ref>) when P ∞.

This follows from the fact that sign quantization, among all possible architectures, yields the largest number of information bits.

The optimality of this solution arises from the fact that the number of sign quantizer is  a fixed resource that limits, at the receiver side, the largest attainable rate.








§ NUMERICAL EVALUATIONS



In the following, we evaluate the results in Sec. <ref> by considering the expected value of capacity () in
(<ref>) when the channel gains H_ij are drawn from a Gaussian distribution with mean zero and variance one.





We begin by numerically evaluating the performance for the SIMO channel with single antenna and multilevel quantization
selection in Prop. <ref> and with linear combining in Prop. <ref>.

Figure <ref>  shows the upper bound expressions in (<ref>) and (<ref>) as a function of the
number of receiver antennas  and for a fixed transmit
power P and number of sign quantizers N_SQ.

For =1, the performance of the two architectures is the same as the SISO channel in Prop. <ref>, while, when
  increases, the performance approaches log(N_SQ+1), albeit at a slower rate for the single antenna selection case.

As the power increases, the transition between these two regimes requires fewer  antennas.
Consequently, the performance loss of the receiver
architecture in Figure <ref>, in comparison with linear combining receiver, decreases as the transmit power grows large.


The performance of multiple antenna selection for the SIMO case is shown in Figure <ref>: in this figure, we plot
the upper bound in Prop. <ref> and Prop. <ref> together with those in Prop. <ref>.

From Figure <ref> we observe how increasing the number of antennas that are selected impacts the achievable rate, reducing the gap from the performance of the architecture with linear combining and multilevel quantization.


The performance for the MIMO case is presented in Fig. <ref>: in this figure, we show the performance difference
between the architectures (a) from <cit.>.



Single antenna selection with multilevel quantization performs well when the number of receive antennas is small but its performance is surpassed by multi-antenna selection and sign quantization
as the number of receiver antennas grows.

This follows from the fact that the attainable rate with single antenna selection converges to log(N_SQ+1) as  grows while
sign quantization converges to N_SQ.




It is interesting to observe that these two simple receiver architectures, together, are able to closely approach the performance in Prop. <ref>.




§ CONCLUSION



A general approach to model receiver architectures for MIMO channels with low-resolution output quantization has been proposed.

In our formulation, the antenna outputs undergo analog processing before being quantized using N_SQ sign quantizers.

Analog  processing is embedded in the channel model description while the channel output corresponds to the output of the sign quantizers.

Through this formulation, it is then possible to optimize the capacity expression over the set of feasible analog processing operations while keeping the number of sign quantizers fixed.








IEEEtran






§ PROOF OF PROP. <REF>




∙ Converse:

The capacity of the SISO channel with multilevel quantization is necessarily dominated by the capacity of the AWGN channel without quantization constraints and by the capacity of the channel with channel with output quantization but no additive noise.


The upper bound

_SISO ≤12 logP+1,


is obtained as the capacity of the channel without quantization constraints.

The upper bound

_SISO ≤logN_SQ+1 ,


is obtained as the capacity of the channel without additive noise.

The intersection of the outer bounds in (<ref>) and (<ref>) yields the outer bound in (<ref>).

In the following we refer to this upper bound as the trivial upper bound for brevity.

∙ Achievability:
If N_SQ=1, then capacity is provided by Lem. <ref> for any  P>0.


Let us first consider the case is which  P≤ 6 and N_SQ>1: in this parameter regime

it can be verified through numerical evaluations that the capacity expression in (<ref>) is to within 1/2 from the infinite quantization capacity in (<ref>).

This implies that the achievability proof in Lem. <ref> is sufficient to show the approximate capacity in this parameter regime.




For P>6 and N_SQ≥  2, consider the achievable scheme in which the channel input is an equiprobable M-PAM constellation while, at the receiver,
the M-1 sign quantizers thresholds are chosen as the midpoints of the transmitted constellation points.




The parameter M is chosen according to whether performance is limited by the transmit power or by the number of available quantizers.

When 1/2 log(P+1) ≥log(N_SQ+1), the number of available sign quantizers dominates the performance and M is chosen as N_SQ+1, which is the largest number
of channel inputs that can be distinguished at the receiver.

When log(N_SQ+1) > 1/2 log(P+1), then the available transmit power dominates the performance and M is chosen as ⌊√(P)⌋.




Following these reasoning, we define

M=min{N_SQ+1,⌊√(P) ⌋} ≥3,


and denote support of the input X as

={x_1,…,x_M},


for {x_m}_1^M are in increasing order.




For M is even, we choose  as

    =·-M/2+1, …,+M/2  -1/2,



while, for M odd, we let  be equal to

= -M-1 2, …, M-1 2 ,


for

    =√(12 PM^2-1).


For  in either (<ref>) or (<ref>), let the  channel input be uniformly distributed on the set ; note that, by construction, the power constraint is attained with equality, i.e. [X^2]= P.


At the receiver, the channel output is quantized using M-1 sign quantizers, each with threshold t_k obtained as

t_m= 12 x_m + x_m+1 ,   m ∈[1,…,M-1].


Note that, by definition, M-1≤ N_SQ so that the constraint on the number of available sing quantizers is respected.

In particular, for the case in which N_SQ+1>√(P), we have that not all the sign quantizers are employed at the receiver.

In this scenario a better performance can be attained by employing all the available quantizer: for simplicity in the analysis, we only consider the sub-optimal strategy which employs M-1 of the  N_SQ available quantizers.






For convenience of notation,


we express   in (<ref>) through the random variable  with support  defined as





[=x_m]=
[W ≤t_1]        m=1  

[t_m-1<W ≤t_m]     m∈[2,…, M-1] 

[W > t_M-1]        m=M.




The mapping in (<ref>) is a one-to-one mapping since _i is of the form

_i=[-1 …-1_M^-, +1, …+1_M^+]^T,


with M^-,M^+≥ 0 and  M^-+M^+=M,  so that the M-1 sign quantizer outputs have a one-to-one correspondence with M possible values of .






With the definition in (<ref>) and for the channel input uniformly distributed over the support in (<ref>) and (<ref>),
we obtain the inner bound

R^IN
    = H()-H(|X),


where

[=|X=x]     = |Z-(-x)|<  2

[=]    =1 M∑_m=1^M Z-(-x)< 2,

where Z ∼(0,1) and x,∈.


The entropy term H() in (<ref>) is lower-bounded as

H() ≥M min_∈   -P_()  logP_(),


and, given the symmetry in the input constellation,

we have that the minimum P_() is obtained at =±/2 for M even,  and at =0 for  M odd.

Note moreover that, the minimum P_() is at most 1/M≤ 1/3 < e^-1: for x<e^-1, the function -x log(x) is a positive increasing in x, so that a lower bound
on P_() produces a lower bound to the RHS of (<ref>).











For this reason, when M is even,  we lower bound P_(+/2)=P_(-/2) as

P_(+/2)

     = 1 M 1-2 Q(/2)+ ∑_k=2^+M/2 Q((k-2)+/2)-Q((k-1)+/2) +  

        +∑_k=+1^+M/2 Q((k-1) +/2)-Q( k +/2) 

    = 1 M 1-2 Q(/2)+ Q(/2)-Q((M-1)/2)+Q(/2)-Q((M+1) /2)

    =1 M  1-Q((M-1)/2)-Q((M+1) /2) 

    ≥1 M  1-2Q((M-1) /2) .




Similarly, for the case of M odd, we have

P_(0)

     = 1 M 1-2 Q(/2)+ 2 ∑_k=1^+(M-1)/2 Q((k-1)+/2)-Q(k+/2) 

    =  1 M  1-2Q( M /2)  .


By plugging (<ref>) and (<ref>) in (<ref>), depending on the value of M, we obtain the bound

min_∈ P_() ≥1 M  1-2Q((M-1) /2) .



Let =Q((M-1) /2) for convenience of notation and further bound (<ref>) as


H()     ≥- M 1 M   1-2log1 M  1-2



    = logM  - (1-2)log(1-2 )-2 log(M) 


    ≥logM  -2 log(M)  


    ≥logM -0.2 ,

where (<ref>) follows from the fact that the function -log(1-2 )-2 log   (1-2 ) is positive defined while
(<ref>) from the bound

     = Q 1 2 (M-1) √( 12 P M^2-1) 

     = Q  √((M-1)^2M^2-1) √(3 P) 

 
     ≤Q(√(3 P)),
 
 

so that

2log(M)  ≤2 Q(√(3 P)) log(√(P))     ≤0.02,


where (<ref>) follows from the fact that Q(√(3 P)) log(√(P)) is a decreasing function for P>6.


Accordingly, we conclude that

H() ≥logM -0.02.































































Next, we wish to upper bound the entropy term H(|X) in (<ref>).

Note that, for each X=x_m,  H(|X=x_m),  corresponds to  the entropy of a Gaussian random variable with mean x_m and unitary variance which is quantized with M-level uniform quantization of step .


From the “grouping rule for entropy” <cit.>

we have that the value of this entropy  is smaller than the entropy of a Gaussian variable
 with infinite uniform quantization of step .


Let us denote as N^ the infinite quantization of a Gaussian variable with step ; more specifically,
 N^ is defined as the random variable with support  and for which [N^=z],  z ∈ is obtained as




[N^=0]      = - 2 ≤X <  +2  

[N^=k]    =(k-1) +2 ≤X <  k+ 2  ,   k ∈∖{0}.


The entropy H(N^) can be expressed as

   H(N^) = -1-2 Q(/2) log1-2 Q(/2) 

    ≤0.15 - 2 ∑_k=0^∞ Q(k +/2 )-Q((k+1)+ /2) logQ(k +/2 )-Q((k+1)+ /2) .


For  Δ in (<ref>), we necessarily have >2√(3), and thus

    Q(k +/2 )-Q((k+1)+ /2)  

   <  Q(/2)- Q(3/2) 

    < Q(/2) < e^-1.
 
Using the bound in (<ref>), together with the fact that -xlog (x) is an increasing function of x for x≤ e^-1,
we have that an upper bound on the term
Q(k +/2 )-Q((k+1)+ /2) results in an upper bound on the quantity in (<ref>).


Next, note that for k>1, we have

Q(k +/2 )-Q((k+1)+ /2)
≤Q(k )-Q(2 k ) ≤e^-1 2 -k^2 ^2-e^ -2 k^2 ^2,

so that, by numerical integration methods, we obtain the bound

    - 2 ∑_k=1^∞ Q(k +/2 )-Q((k+1)+ /2) 

    ≤0.03+  ∫_x=1^∞ e^-1 2 -k^2 ^2-e^ -2 k^2 ^2 x ≤0.25.


Plugging the bound (<ref>) in (<ref>)
we obtain

H(N^) ≤0.15+0.25=0.4


Finally, combining (<ref>) and (<ref>)

I(X;)≥log(M)-12,

which is the desired result.






§ PROOF OF PROP. <REF>



When only one antenna can be selected, the result in Prop. <ref> can be used to bound the capacity maximization in (<ref>) to within 1/2 from the trivial outer bound

() ≤max_k 12 log1+h_k^2 P, (N_SQ+1)^2.




The function on the RHS of (<ref>) is increasing in k when h_k are ordered in increasing order, thus yielding the desired result.





§ PROOF OF PROP. <REF>



The outer bound in (<ref>) is the trivial outer bound as defined in App. <ref> while the inner bound in (<ref>) is derived in the following.

In the remainder of this appendix, the channel coefficients h_i are taken positive: this assumption is without loss of optimality as the noise distribution is symmetric.

Also, in the following, we assume without loss of generality that the terms h_k are in descending order.

Achievability:

If ||_2^2 P ≤ 15 or  N_SQ≤ 3, then

	    12 logmin1+ |^(K)|_2^2  P, N_SQ  +1 ^2 

	    ≤  12 logmin1+ ||_2^2  P, N_SQ  +1 ^2 ≤2
	


from which we conclude that (<ref>) is less than zero in this parameter subset.

Since the rate zero is trivially achievable, the inequality in (<ref>) proves that (<ref>) is achievable.




If ||_2^2 P > 15 and  N_SQ> 3,  the achievability of the bound in (<ref>) is shown by letting the channel input
 be the sum of an M-PAM signal plus a dither.

For this receiver architecture dithered quantization is necessary to evaluate the performance of the combining of the sampled channel outputs.


Similarly to (<ref>), let us we define M as

M= ⌊minN_SQ K,  |^(K)|_2 √(P)  -1 ⌋.




	
For M in (<ref>), note that

(<ref>)     = 12 logmin1+ |^(K)|_2^2  P,  N_SQ K +1 ^2 

    ≤log(M+2), 
so that when M ≤ 2, the expression in  (<ref>) is less than zero which is trivially achievable.



For M ≥ 3,  let the channel input be obtained as



X=S+U.


where S is an M-PAM signal for M in (<ref>),  with support as in (<ref>)  for M even, or as (<ref>)
for M odd but where   is chosen as

=√( 12 P M^2-1 ).

The variable U in (<ref>) is quantization dither, that is U ∼([-/2,+/2]) and U ⊥ S.



Since [U^2] = ^2 /12,  the power constraint is satisfied with equality by setting

P = P-^2 12,

which yields

^2=12 PM^2.








At the receiver, the K antennas with the best SNR are each quantized with an (M+1)-level quantizer.

More specifically, the k^ th antenna output, k ∈ [1 … K], is quantized with thresholds  t_i^(k) for i∈ [0 … M]  chosen as

t_0^(k)     = h_i x_1- 2  

t_m^(k)    =  h_i 2 x_m + x_m+1 ,   m ∈[1,…,M-1] 

t_M^(k)     = h_i x_M+ 2  .

Note that, although channel input has M possible values but the receiver uses an (M+1)-level quantizer to quantize each of the K best antenna outputs:

 two additional quantization levels are used to detect  whether the channel output is below h_i(x_1-/2)  or  above h_i(x_M+/2)
  (as specified at the beginning of the appendix,  the channel coefficients are assumed to be positive and with decreasing magnitude without loss of generality).





Note that  the total number of quantizers employed at the receiver is K (M+1) ≤   N_SQ, so that the constraint on the total number of available sign quantizers
is satisfied.

As for the proof in App. <ref>, it is possible that not all the sign quantizer are utilized in this achievable scheme.


Next, similarly to (<ref>), we define ^(k) for k∈[1,…,K] as

[^(k)=x_m]=
[W_k ≤t_0^(k)]        m=0 

[t_m-1^(k)<W_k ≤t_k^(k)]     m ∈[1,…, M] 

[W_k > t_M^(k)]        m=x_M+1,



where x_m for m∈ [1,M] is as in (<ref>) while we additionally let x_0=x_1- and x_M+1=x_M+.

As for the mapping in (<ref>), the mapping in (<ref>) is a one-to-one  correspondence between W_k and ^(k).

Finally, let ^(k)=^(k)-U and =[^(k), …, ^(K)].





























We next lower bound the achievable rate as follows: first (i) we show that the capacity of the channel with finite quantization levels is to within a
constant gap from the channel with infinite quantization levels, successively (ii) we lower bound  that the capacity of the model with infinite quantization levels.

This lower bound minus with the gap between the capacity of the model with finite and infinite quantization corresponds to the achievable rate in (<ref>).

Define ^(k) as the quantization of  W_k for k ∈ [1, … ,K] with infinite quantization levels and with step  as in (<ref>).

Similarly, let ^(k)=^(k)-U and =[^(1)…^(K)].

The rate 	achievable with the transmission strategy described above is lower bounded as



R^IN

    ≥I(; X) 


    =  H(,)-H(|)-H(|X)


    = I(; X)-H(|) 

    ≥I(; X)-∑_k=1^K H(^(k)|^(k)).



The expression in (<ref>) is interpreted as follows: I(; X) is the attainable rate for the model with infinite output quantization while
∑_k=1^K H(^(k)|^(k)) is an upper bound to the performance gap between the attainable rate with infinite and finite quantization.





Let us first bound the performance gap between the channel with finite and infinite output quantization:

for each term H(^(k)|^(k)), we observe that, if W_k/h_k ∈ [x_1-/2,x_M+/2], then  ^(k)=^(k):
using this observation and given the symmetry of the input and noise distributions, we write

-H(^(k)|^(k))
    = -H(^(k)|^(k)=x_M+1)  [^(k)_i=x_M+1] -H(^(k)|^(k)=x_0)  [^(k)=x_0] 

    = -2 H(^(k)|^(k)=x_M+1)  [^(k)_i=x_M+1].




If i ∈ [0,M+1], then

[^(k)=x_i|^(k)=x_M+1]=0,

on the other hand, for i >M+1, we have

[^(k)=x_i|^(k)=x_M+1]

    =∑_m=1^M [^(k)=x_i|^(k)=x_M+1, X=x_m] [X=x_m] 

     ≤1 M ∑_m=1^M [Xt^(k)=x_i| ^(k)=x_M+1, X=x_M] 

    ≤[^(k)=x_i|^(k)=x_M+1,X=x_M] 

    ≤Q h_k((i-1)+/2)-Q h_k(i+/2)) Q( h_k M +/2)  




    ≤Q h_k((i-1)+/2Q( h_k M +/2),
 

    ≤1 -1 h_k (M +/2) ^-1 e^- h_k^2(i^2-M^2)^2  


    ≤1 -1 h_k M  ^-1 e^- h_k^2(i-M)^2^2
	

The case for i<0, can be bounded in a symmetric matter to yield

[^(k)=x_i|^(k)=x_M+1] ≤Q h_k((i-1)+/2Q( h_k M +/2) ≤1 -1 h_k M  ^-1 e^- h_k^2(i-M)^2^2.


Since M^2 ^2=12 P and P h_k>1 by assumption, we have

h_k M ≥2 √(3),   k∈[1, …, K],

which implies Q((i h_k )) ≤ e^-1 for all k.



Since -x log x is positive increasing function in x for x ∈ [0,1/e], we can write

    H(^(k)=x_m|^(k)=x_M+1) 

    =  ∑_i=M^∞[^(k)=x_i|^(k)=x_M+1]  log[^(k)=x_i|^(k)=x_M+1] 

        + ∑_i=0^-∞  [^(k)=x_i|^(k)=x_M+1] [^(k)=x_i|^(k)=x_M+1] log

    ≤2 ∑_i=M^∞Q(k h_i )  logQ(k  h_i )   



    ≤∑_k=M^∞∑_i=M^∞[^(k)=x_i|^(k)=x_M+1]  log[^(k)=x_i|^(k)=x_M+1] 






    ≤∑_j=0^∞6 5 h_k^2 j^2^2 e^- h_k^2 j^2^2


≤0.15,



where we have used the fact that

	√(15 P) ≤M ≤√(18 P),

and, similarly, √(12)≤≤√(15) for P>6.

Plugging the bound in (<ref>)  in (<ref>) yields

∑_k H(^(k)|^(k))  ≤0.3,

which shows that the capacity of the channel with infinite quantization is at most 0.3 from the capacity of the channel with finite output quantization.







Having bounded the performance gap between finite infinite quantization, we next lower bound the rate attainable in the model with infinite quantization of the K antenna outputs with the highest SNR.






















































































For this model, the attainable rate can be lower bounded using the results that Gaussian distributed noise is the worst additive noise under a covariance
constraint in <cit.>.

More specifically, let us define






    =1 |^(K)|_2^2 ∑h_k^2 ^k-U.

Note that, from properties of dithered quantization <cit.>, we have

^(k)    =S + Z_k h_k+ N_k,

where N_k∼( /2, /2) and independent from S and Z_k.

Using this observation, we have

I(;X)     = I(;X|U) 

     ≥I(;S) 

    ≥I(S+Z^;S),

where Z^∼(0,) for

=1 |^(K)|_2^4  ∑_k=1^K h_k  Z_k + ∑_k=1^K h_k^2  N_k 

Note that, from the achievability proof in Prop. in <ref>, we  have

 I(S+Z^;S) ≥log(M)-0.6-log().
 

A bound on  in (<ref>) is obtained as follows:

∑_k=1^K h_k Z_k  =|^(K)|_2^2


and

∑_k h_k^2 N_k  ≤1 12 |^(K)|_4 + 2 12 ∏_i>j h_i^2 h_j^2 ≤|^(K)|_4^2 12




so that,  h_i>1, as by assumption

    ≤|^(K)|_2^2+|^(K)|_4^2 |^(K)|_2^4  

    ≤1+|^(K)|_4 |^(K)|_2^2^2 ≤2.


Substituting M in (<ref>) and bounding  as in (<ref>) in
(<ref>) finally yields (<ref>).




§ PROOF OF PROP. <REF>


With single antenna selection, the capacity maximization in (<ref>) can be rewritten as

() ≤max_k 12 log1+|_k|_2^2 P, (N_SQ+1)^2,


where _k is the k^ th row of .

In other words, the capacity is the maximum among the capacity of the MISO channels between the transmitter and each of the antennas at the receiver.

For each MISO channel, the capacity can be attained using the result in Prop. <ref> since transmitter pre-coding can be used to turn the MISO channel into a SISO channel.













§ PROOF OF PROP. <REF>




Through the classic VBLAST architecture, the channel can be equivalently written as a set of parallel channels

_i=_i _i+_i,   i=1,…, min{,},


where [_1, …, _min{,}] are the eigenvalues of  and _i ∼(0,1).


Since the capacity of the parallel of  channels is obtained as the sum of the capacity of each channel, we have that an upper bound to capacity is

R^OUT=max∑_i=1^min{,} 12 logmin{ _i^2 P_i+1,(N_SQ,i+1)^2} ,


where the maximization is over ∑ P_i=P and ∑ N_SQ,i=N_SQ as P_i is the input power  and N_SQ,i the number of sign quantizers allocated to the i^ th equivalent channel.

Additionally, the upper bound in (<ref>) can be attained to within min{,} following the result in Prop. <ref>.





We next wish to determine an approximate expression for the solution of the optimization in  (<ref>) as a function of the available power and number of sign quantizers.

To simplify this analysis, we relax this optimization problem and let N_SQ take values in ^+.

Under this relaxation of the optimization problem in (<ref>), we have that the term  min{_i^2 P_i+1,(N_SQ,i+1)^2}
must be attained by either the power or the sign quantizer allocation on all channels simultaneously.

This can be shown by contradiction: assume that there exist two subchannels j and k such that

min{ _j^2 P_j^*+1,(N_SQ,j^*+1)^2}     = _j^2 P_j^*+1 

min{ _k^2 P_k^*+1,(N_SQ,k^*+1)^2}     = (N_SQ,k^*+1)^2,

in the optimal solution, then there must exist _1,_2>0 such that

min{ _j^2 (P_j^*+_2)+1,(N_SQ,j^*-_1+1)^2
}     = _j^2 (P_j^*+_2)+1>  _j^2 P_j^*+1

min{ _k^2 (P_k^*-_2)+1,(N_SQ,k^*+_1+1)^2}     = (N_SQ,k^*+_1+1)^2> (N_SQ,k^*+1)^2,

which contradicts the claim of optimality.
For the case in which the power constraint is active, the optimal solution corresponds to the classical waterfilling solution
in the channel with infinite quantization levels.

For the case in which the constraint on the quantization is active, then maximization problem becomes

max_∑N_SQ,i=N_SQ  ∑_i=1^min{,}logN_SQ,i+1 .


The optimization problem in (<ref>) is equivalent to the waterfilling problem with equal channel gains and thus
 the uniform allocation of quantizers across all sub-channels is optimal.





1+_i^2 P_i=N_SQK+1 ^2


where K is the number of active channels.





Since N_SQ+1≤ 2 N_SQ, the assignment ⌊ N_SQ⌋ provides a loss of at most 1 per each channel, so that the overall gap between inner and upper bound is 2 min{,}.




































































































































































































































