A General Framework for MIMO Receivers with Low-Resolution Quantization

# A General Framework for MIMO Receivers with Low-Resolution Quantization

Stefano Rini1, Luca Barletta2, Yonina C. Eldar3, and Elza Erkip4
1 National Chiao Tung University, Hsinchu, Taiwan
2 Politecnico di Milano, Milano, Italy
3 Technion, Haifa, Israel
4 NYU Tandon School of Engineering, New York, USA
###### Abstract

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.

*[subfigure]position=bottom

## I 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 [1].

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 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.111 In the literature, the term “one-bit quantization” most often refers to sign quantization of the antenna outputs. Here, as in [2], we prefer the term “sign quantization” since we distinguish between sign and threshold quantization. The authors in [3] 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 [4], 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 [5]. The high signal-to-noise ratio (SNR) asymptotics for complex MIMO channels with sign quantization are studied are [6]. For the SISO channel with threshold quantization, [2] 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. II introduces the channel model. Sec. III reviews the results available for the case of sign quantization of the channel outputs. The main results are given in Sec. IV. Numerical evaluations are provided in Sec. V. Sec. VI concludes the paper.

### Notation

We adopt the standard notation for and . All logarithms are taken in base two. For the SISO model, we set w.l.o.g., for the MISO and SIMO models we denote the channel matrix as and respectively. For the MIMO case, the vector contains the eigenvalues of the matrix . The identity matrix of size is indicated as , the all-zero/all-one matrix of size as /. Finally, indicates the set of all permutation matrices.

## Ii Channel Model

### Problem Formulation

We consider a discrete-time real-valued MIMO channel with transmit antennas and receive antennas. At the channel use, the antenna output vector , is obtained from the channel input vector as

 Wn=HXn+Zn,n∈[1…N], (1)

where is a full rank matrix of size 222This condition guarantees the existence of a right pseudo-inverse for and holds with high probability in a richly scattering environment. and 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 . The channel input vector is subject to the average power constraint where indicates the 2-norm.

The antenna output vector is processed through sign quantizers, each receiving a linear combination of the antenna output vector plus a constant,333It 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.

 Yn=sign(VWn+t),n∈[1…N], (2)

where is the analog combining matrix of size ,  is a threshold vector of length and is the function producing the sign of each component of the vector as plus or minus one, so that . For a given choice of combining matrix and threshold vector , the capacity of the model in (2) is given by

 C(V,t)=maxPX(x), E[|X|22]≤PI(X;Y), (3)

where we have explicitly expressed the dependency of the capacity on the parameters .444The 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 (3) over , namely

 C(F)=max{V,t}∈FC(V,t). (4)

### Relevant Architectures

The formulation in (4) 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 . Note that -level multilevel quantization can be obtained by using sign quantizers and appropriate thresholds , resulting in 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 (4) is selected as

 Fa ={V=[INSQ,0NSQ×(Nr−NSQ)]Pπ, Pπ∈Pπ, t=0NSQ×1}, (5)

that is, each sign quantizer is connected to one of the channel outputs. Figure 0(a) represents this model for and .

(b) Single antenna selection and multilevel quantization: For this receiver architecture, the sign quantizers are used to construct an -level quantizer:

 Fb ={V=[1NSQ×1,0–NSQ×(Nr−1)]Pπ, Pπ∈Pπ, t∈RNSQ}, (6)

Figure 0(b) shows this model for and .

(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

 Fc ={V s.t. Vij∈{0,1}, Nr∑j=1Vij=1, t∈RNSQ}. (7)

This receiver architecture encompasses those in Fig. 0(a) and Fig. 0(b) as special cases. Figure 0(c) again shows this model for and .

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

## Iii 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. 0(a) for , 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.

###### Lemma III.1.

[4, Th. 2]: The capacity of the SISO channel with sign quantization of the antenna output with is

 CSISO=1−H2(Q(√P)). (8)

The capacity of the MISO channel with sign output quantization is obtained from the result in Lem. III.1 by transforming this model into a SISO channel through transmitter beamforming, thus yielding

 CMISO=1−H2(Q(|h|√P)). (9)

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

###### Lemma III.2.

[6, Prop. 1]. The capacity of the SIMO channel with sign quantization of the antenna output with at high SNR satisfies

 log(Nr)≤CSNR→∞SIMO,a≤log(Nr+1). (10)
###### Lemma III.3.

[7, Prop. 3]. The capacity of the MIMO channel with sign quantization and , and for which satisfies a general position condition (see [7, Def. 1]), is bounded at high SNR as

 12log(K(NSQ,Nt))≤CSNR→∞MIMO,a≤12log(K(NSQ,Nt)+1)

if , where

 K(NSQ,Nt)=2Nt−1∑k=0(2NSQ−1k). (11)

If , then .

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 [7, Sec. V.A].

## Iv Main Results

We begin by considering the capacity of the SISO channel for the receiver architectures in Sec. II. Capacity for the architecture (a) is provided in Lem. III.1 (necessarily ) while the architectures (b), (c) and (d) all correspond to the same model in which the channel output is quantized through an -level quantizer. The capacity for this latter model can be bounded to within a small additive gap as shown in the next proposition.

###### Proposition 1.

The capacity of the SISO channel with multi-level output quantization, , is upper-bounded as

 CSISO≤12log(min{P+1,(NSQ+1)2}), (12)

and capacity is to within bits-per-channel-use () from the upper bound in (12).

###### Proof:

The upper bound (12) 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 -PAM signal for

 M=min{⌊√P⌋,NSQ+1}, (13)

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 -PAM constellation points. The full proof is in App. A. ∎

For the SIMO and MIMO cases, given the generality of the formulation in (4), rather than attempting to find the exact capacity for each architecture in Sec. II, we instead focus on approximate characterization in the spirit of Prop. 1, 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. 1 to the SIMO and MIMO cases.555Note that the MISO case follows from the SISO case as in (9).

### Iv-1 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. 1.

###### Proposition 2.

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

 CSIMO,b≤12log(min{1+h2maxP,(NSQ+1)2}), (14)

where and the upper bound in (14) can be attained to within .

###### Proof:

The proof is provided in App. B

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.

###### Proposition 3.

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

 maxK12log(min{1+|h(K)|22P,(NSQK+1)2})−2 (15a) ≤CSIMO,c≤12log(1+|h|22P,(NSQ+1)2), (15b)

where is the vector of the largest channel gains.

###### Proof:

The upper bound is derived similarly to Prop. 1. 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 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 (15). The full proof is provided in App. C. ∎

###### Proposition 4.

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

 CSIMO,d≤12log(min{1+|h|22P,(NSQ+1)2), (16)

and the upper bound in (16) can be attained to within .

###### Proof:

With this architecture, the maximal ratio combining at the receiver results in the equivalent SISO channel with channel gain . The result in Prop. 1 can then be used to obtain the approximate capacity. ∎

The results in Prop. 2, Prop. 3 and Prop. 4 are related as follows. The results for the architecture (a) in Lem. III.2 and the architecture (b) in Prop. 2 show that the two architectures yield the same high-SNR behaviour when . When , 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. 3 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 (15a) dominates the channel performance.

###### Proposition 5.

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

 CSIMO,c≤12log(1+|h|22P), (17)

and the upper bound in (17) can be attained to within when and .

###### Proof:

Under these assumption, the minimum in (15a) is attained by setting , in which case the trivial outer bound of (15b) can be attained to within . ∎

### Iv-2 MIMO case

For the architecture (a), inner and outer bounds are derived in [7, Sec. V.A]; for the architecture (b), an upper bound is derived in the next proposition.

###### Proposition 6.

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

 CMIMO,b≤12log(min{1+|hTmax|22P,(NSQ+1)2}), (18)

where is the row of with the largest norm and the upper bound in (18) can be attained to within .

###### Proof:

The proof is provided in App. D. ∎

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 parallel channel with gains . Capacity is then obtained as

 max K∑i=112log(min{1+λ2iPi,(NSQ,i+1)2}), (19)

where the maximization is over , and . By relaxing the integer constraint on the parameters , we obtain to the outer bound

 C≤R⋆(λ,P,NSQ)= ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩∑min{Nr,Nt}i=112log(1+λiPi)if  ∑min{Nr,Nt}i=1(√1+λiPi−1)≤NSQKlog(NSQK+1)otherwise, (20)

where are chosen as and is the smallest value for which and . The approximate capacity for the architecture (d) is obtained by showing that a rate sufficiently close to (20) 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 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.

###### Proposition 7.

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

 CMIMO,d≤R⋆(λ,P,NSQ), (21)

and capacity is to within a gap of from the upper bound in (21) for and in (20).

###### Proof:

The proof is provided in App. E. ∎

The result in Prop. 7 shows that sign quantization is sufficient to attain the optimal performance in the high SNR regime since yields the largest rate in (20) when . 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.

## V Numerical Evaluations

In the following, we evaluate the results in Sec. IV by considering the expected value of capacity in (4) when the channel gains 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. 2 and with linear combining in Prop. 4. Figure 1(a) shows the upper bound expressions in (14) and (16) as a function of the number of receiver antennas and for a fixed transmit power and number of sign quantizers . For , the performance of the two architectures is the same as the SISO channel in Prop. 1, while, when increases, the performance approaches , 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 0(b), 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 1(b): in this figure, we plot the upper bound in Prop. 2 and Prop. 4 together with those in Prop. 5. From Figure 1(b) 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. 1(c): in this figure, we show the performance difference between the architectures (a) from [7, Sec. V.A]. 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 as grows while sign quantization converges to . It is interesting to observe that these two simple receiver architectures, together, are able to closely approach the performance in Prop. 7.

## Vi 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 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.

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## Appendix A Proof of Prop. 1

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

 CSISO≤12log(P+1), (22)

is obtained as the capacity of the channel without quantization constraints. The upper bound

 CSISO≤log(NSQ+1), (23)

is obtained as the capacity of the channel without additive noise. The intersection of the outer bounds in (22) and (23) yields the outer bound in (12). In the following we refer to this upper bound as the trivial upper bound for brevity.

Achievability: If , then capacity is provided by Lem. III.3 for any .

Let us first consider the case is which and : in this parameter regime it can be verified through numerical evaluations that the capacity expression in (8) is to within from the infinite quantization capacity in (22). This implies that the achievability proof in Lem. III.3 is sufficient to show the approximate capacity in this parameter regime.

For and , consider the achievable scheme in which the channel input is an equiprobable -PAM constellation while, at the receiver, the sign quantizers thresholds are chosen as the midpoints of the transmitted constellation points.

The parameter is chosen according to whether performance is limited by the transmit power or by the number of available quantizers. When , the number of available sign quantizers dominates the performance and is chosen as , which is the largest number of channel inputs that can be distinguished at the receiver. When , then the available transmit power dominates the performance and is chosen as .

Following these reasoning, we define

 M=min{NSQ+1,⌊√P⌋}≥3, (24)

and denote support of the input as

 X={x1,…,xM}, (25)

for are in increasing order. For is even, we choose as

 X =Δ⋅([−M/2+1,…,+M/2]−1/2), (26)

while, for odd, we let be equal to

 X=Δ[−M−12,…,M−12], (27)

for

 Δ =√12PM2−1. (28)

For in either (26) or (27), let the channel input be uniformly distributed on the set ; note that, by construction, the power constraint is attained with equality, i.e. .

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

 tm=12(xm+xm+1),m∈[1,…,M−1]. (29)

Note that, by definition, so that the constraint on the number of available sing quantizers is respected. In particular, for the case in which , 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 of the available quantizers.

For convenience of notation, we express in (2) through the random variable with support defined as

 P[ˆX=xm]=⎧⎪⎨⎪⎩P[W≤t1]m=1P[tm−1tM−1]m=M. (30)

The mapping in (30) is a one-to-one mapping since is of the form

 Yi=[−1…−1M−,+1,…+1M+]T, (31)

with and , so that the sign quantizer outputs have a one-to-one correspondence with possible values of .

With the definition in (30) and for the channel input uniformly distributed over the support in (26) and (27), we obtain the inner bound

 RIN =H(ˆX)−H(ˆX|X), (32)

where

 P[ˆX=ˆx|X=x] =P[|Z−(ˆx−x)|<Δ2] (33a) P[ˆX=ˆx] =1MM∑m=1P[|Z−(ˆx−x)|<Δ2], (33b) w

here and .

The entropy term in (32) is lower-bounded as

 H(ˆX)≥Mminˆx∈X−PˆX(ˆx)logPˆX(ˆx), (34)

and, given the symmetry in the input constellation, we have that the minimum is obtained at for even, and at for odd. Note moreover that, the minimum is at most : for , the function is a positive increasing in , so that a lower bound on produces a lower bound to the RHS of (34). For this reason, when is even, we lower bound as

 PˆX(+Δ/2) =1M⎛⎝(1−2Q(Δ/2))++M/2∑k=2(Q((k−2)Δ+Δ/2)−Q((k−1)Δ+Δ/2))+ ++M/2∑k=+1(Q((k−1)Δ+Δ/2)−Q(kΔ+Δ/2))⎞⎠ =1M(1−2Q(Δ/2)+(Q(Δ/2)−Q((M−1)Δ/2))+(Q(Δ/2)−Q((M+1)Δ/2))) =1M(1−Q((M−1)Δ/2)−Q((M+1)Δ/2)) ≥1M(1−2Q((M−1)Δ/2)). (35)

Similarly, for the case of odd, we have

 PˆX(0) =1M⎛⎝(1−2Q(Δ/2))+2+(M−1)/2∑k=1(Q((k−1)Δ+Δ/2)−Q(kΔ+Δ/2))⎞⎠ =1M(1−2Q(MΔ/2)). (36)

By plugging (35) and (36) in (34), depending on the value of , we obtain the bound

 minˆx∈XPˆX(ˆx)≥1M(1−2Q((M−1)Δ/2)). (37)

Let for convenience of notation and further bound (37) as

 H(ˆX) ≥−M1M(1−2˜Q)log(1M(1−2˜Q)) =logM−(1−2˜Q)log(1−2˜Q)−2˜Qlog(M) ≥logM−2˜Qlog(M) (38a) ≥logM−0.2, (38b) w

here (38a) follows from the fact that the function is positive defined while (38b) from the bound

 ˜Q =Q(12(M−1)√12PM2−1) (39a) =Q(√(M−1)2M2−1√3P) (39b) ≤Q(√3P), (39c) s

o that

 2˜Qlog(M)≤2Q(√3P)log(√P) ≤0.02, (40)

where (40) follows from the fact that is a decreasing function for .

Accordingly, we conclude that

 H(ˆX)≥logM−0.02. (41)

Next, we wish to upper bound the entropy term in (32). Note that, for each , , corresponds to the entropy of a Gaussian random variable with mean and unitary variance which is quantized with -level uniform quantization of step . From the “grouping rule for entropy” [8, Prob. 2.27] 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 the infinite quantization of a Gaussian variable with step ; more specifically, is defined as the random variable with support and for which is obtained as

 P[NΔ=0] =P[−Δ2≤X<+Δ2] (42a) P[NΔ=k] =P[(k−1)Δ+Δ2≤X

The entropy can be expressed as

 H(NΔ)=−(1−2Q(Δ/2))log(1−2Q(Δ/2)) (43a) ≤0.15−2∞∑k=0(Q(kΔ+Δ/2)−Q((k+1)Δ+Δ/2))log(Q(kΔ+Δ/2)−Q((k+1)Δ+Δ/2)). (43b) F

or in (28), we necessarily have , and thus

 Q(kΔ+Δ/2)−Q((k+1)Δ+Δ/2) (44a)

Using the bound in (44), together with the fact that is an increasing function of for , we have that an upper bound on the term results in an upper bound on the quantity in (43b).

Next, note that for , we have

 Q(kΔ+Δ/2)−Q((k+1)Δ+Δ/2)≤Q(kΔ)−Q(2kΔ)≤e−12−k2Δ2−e−2k2Δ2, (45)

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

 −2∞∑k=1(Q(kΔ+Δ/2)−Q((k+1)Δ+Δ/2)) ≤0.03+∫∞x=1(e−12−k2Δ2−e−2k2Δ2)dx≤0.25. (46)

Plugging the bound (46) in (43b) we obtain

 H(NΔ)≤0.15+0.25=0.4 (47)

Finally, combining (41) and (47)

 I(X;ˆX)≥log(M)−12, (48)

which is the desired result.

## Appendix B Proof of Prop. 2

When only one antenna can be selected, the result in Prop. 1 can be used to bound the capacity maximization in (4) to within from the trivial outer bound

 C(F)≤maxk12log(1+h2kP,(NSQ+1)2). (49)

The function on the RHS of (49) is increasing in when are ordered in increasing order, thus yielding the desired result.

## Appendix C Proof of Prop. 3

The outer bound in (15b) is the trivial outer bound as defined in App. A while the inner bound in (15a) is derived in the following. In the remainder of this appendix, the channel coefficients 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 are in descending order.

Achievability: If or , then

 12log(min{1+|h(K)|22P,(NSQ+1)2}) ≤12log(min{1+|h|22P,(NSQ+1)2})≤2 (50)

from which we conclude that (15a) is less than zero in this parameter subset. Since the rate zero is trivially achievable, the inequality in (50) proves that (15a) is achievable.

If and , the achievability of the bound in (15a) is shown by letting the channel input be the sum of an -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 (24), let us we define as

 M=⌊min{NSQK,|h(K)|2√P}−1⌋. (51)

For in (51), note that

 (???) =12log(min{1+|h(K)|22P,(NSQK+1)2}) (52) ≤log(M+2),

so that when , the expression in (15a) is less than zero which is trivially achievable.

For , let the channel input be obtained as

 X=S+U. (53)

where is an -PAM signal for in (51), with support as in (26) for even, or as (27) for odd but where is chosen as

 Δ=√12αPM2−1. (54)

The variable in (53) is quantization dither, that is and . Since , the power constraint is satisfied with equality by setting

 αP=P−Δ212, (55)

which yields

 Δ2=12PM2. (56)

At the receiver, the antennas with the best SNR are each quantized with an -level quantizer. More specifically, the antenna output, , is quantized with thresholds for chosen as

 t(k)0 =hi(x1−Δ2) (57a) t(k)m =hi2(xm+xm+1),m∈[1,…,M−1] (57b) t(k)M =hi(xM+Δ2). (57c)

Note that, although channel input has possible values but the receiver uses an -level quantizer to quantize each of the best antenna outputs: two additional quantization levels are used to detect whether the channel output is below or above (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 , so that the constraint on the total number of available sign quantizers is satisfied. As for the proof in App. A, it is possible that not all the sign quantizer are utilized in this achievable scheme.

Next, similarly to (30), we define for as

 P[ˆX(k)=xm]=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩P[Wk≤t(k)0]m=0P[t(k)m−1t(k)M]m=xM+1, (58)

where for is as in (25) while we additionally let