# Capacity Analysis of One-Bit Quantized MIMO Systems with Transmitter Channel State Information

## Abstract

With bandwidths on the order of a gigahertz in emerging wireless systems, high-resolution analog-to-digital convertors (ADCs) become a power consumption bottleneck. One solution is to employ low resolution one-bit ADCs. In this paper, we analyze the flat fading multiple-input multiple-output (MIMO) channel with one-bit ADCs. Channel state information is assumed to be known at both the transmitter and receiver. For the multiple-input single-output channel, we derive the exact channel capacity. For the single-input multiple-output and MIMO channel, the capacity at infinite signal-to-noise ratio (SNR) is found. We also derive upper bound at finite SNR, which is tight when the channel has full row rank. In addition, we propose an efficient method to design the input symbols to approach the capacity achieving solution. We incorporate millimeter wave channel characteristics and find the bounds on the infinite SNR capacity. The results show how the number of paths and number of receive antennas impact the capacity.

## 1Introduction

Bandwidth and antennas are growing in next generation wireless systems. A main reason is due to the use of millimeter wave (mmWave) carrier frequencies in personal area networks [2], local area networks [3], and likely even cellular networks [4]. The channel bandwidths in mmWave systems are larger than those used in lower frequency UHF (ultra high frequency) systems. For example, in IEEE 802.11ad the bandwidth is GHz while in potential mmWave cellular applications, bandwidths of MHz or more are being considered [5]. Antenna arrays are important for mmWave systems. They provide array gain that helps mmWave systems achieve a favorable link margin. Due to the small carrier wavelength, a large number of co-located antennas can be deployed within a fixed antenna area in mmWave systems. For example, some IEEE 802.11ad chipsets use antennas [6], while mmWave celular applications envision hundreds of antennas at the base station and perhaps a dozen on the handset [5]. This motivates the study of MIMO systems with large numbers of antennas and higher channel bandwidths.

High bandwidth channels introduce new challenges in system design compared with design at lower frequencies [7]. One major issue is the power consumption associated with the analog-to-digital conversion. In conventional receiver designs, the are expected to have high resolution (e.g., more than bits) and act as transparent waveform preservers. In channels with larger bandwidths, the corresponding sampling rate of the scales up. Unfortunately, high-speed (e.g., more than GSample/s), high-resolution ADCs are costly and power-hungry for portable devices[8]. For example, in an ideal -bit ADC with flash architecture, there are comparators and therefore the power consumption grows exponentially with the resolution [8]. At present, commercially available with high-speed and high-resolution consume power on the order of several Watts[11]. Furthermore, because most communication systems exploit some form of operation, multiple ADCs will be needed to quantize the received signals from multiple antennas separately if conventional digital baseband processing of all antenna outputs is assumed. Therefore, the total power assumption can be excessive, especially at the mobile station.

The most direct solutions to the power assumption bottleneck are to reduce the sampling rate and/or the quantization resolution of ADCs. The sampling rate can be reduced by employing a special ADC structure called the time-interleaved ADC (TI-ADC) where a number of low-speed, high-resolution ADCs operate in parallel. The main drawback of the TI-ADC is the mismatch among the sub-ADCs in gain, timing and voltage offset which can cause error floors in receiver performance (though it is possible to compensate the mismatch at the price of higher complexity of the receiver [12]). An alternative to high resolution ADCs is to live with ultra low resolution ADCs (1-3 bits), which reduces power consumption and cost.

The use of low resolution and especially one-bit ADCs radically changes both the theory and practice of communication. For example, the capacity maximizing transmit signals are discrete [14]. This is in contrast with unquantized MIMO systems where the optimal input distribution is continuous. Although it is finite-dimensional, finding the optimal discrete input distribution is a challenging problem that depends on the and . In [15], the capacity of the real-valued channel with and was considered. For one-bit quantization, binary antipodal signaling was found to be optimal. It was also shown that at low SNR, the use of low-resolution ADCs incurs a surprisingly small loss in spectral efficiency compared to unquantized observations. In the block fading SISO channel without and , it was proven in [16] that the capacity is achieved by on-off QPSK signaling where the on-off probability depends on the coherence time. The results in [15] and [16], however, do not readily extend to the MIMO channel.

The most related work to our contribution is [17] where the one-bit quantized channel with only was considered. It was found that under the constraint that each antenna transmits signals independently and with equal power, independent QPSK signaling across different antennas is optimal in the low SNR regime. In addition, there is a reduction of low SNR channel capacity by a factor ( dB) due to one bit quantization ^{1}

In [19], the capacity of the quantized MIMO channel was derived under different assumptions. The input symbols, though, were only BPSK or QAM symbols since they assumed there is no at the transmitter with which to optimize the constellation. Therefore, what is called *capacity* in [19] is actually *the mutual information* or *an achievable rate* since the input distributions are not optimized. Though reasonable for practical implementation, without a carefully designed input distribution, the mutual information of quantized MIMO channel may achieve its maximum at a finite SNR (this phenomenon is called *stochastic resonance*). With CSIT, the transmitter can implement beamforming to obtain array gain and can also design the transmitted constellation to achieve higher achievable rates. In this paper, we consider the quantized channel with both perfect and . Algorithms for CSI estimation in the one-bit framework have been considered in related work [22] and our work [30]. Incorporating estimation error and studying CSI feedback are left to future work.

For low-resolution ADCs with more than one-bit resolution, i.e., 2-3 bits, besides the difficulty of finding the optimal input distribution, another major challenge is selecting the appropriate quantizer thresholds. Although the well-known uniform quantizer and nonuniform Lloyd-Max quantizer are commonly used for receiver design and rate analysis in related work (e.g., [19]), they are not necessarily optimal for maximizing the channel capacity. For example, assuming Lloyd-Max quantizer is used and the quantization error is treated as Gaussian noise, a lower bound of the channel capacity is derived in [35]. The lower bound is tight only at low SNR. In addition, the two challenges are coupled, which makes this problem even difficult. Namely, the optimal input distribution depends on the thresholds of ADCs; the thresholds of ADCs will also affect the choice of input distribution. Furthermore, the input distribution and the thresholds may change greatly with SNR (see Fig. 5 in [15] as an example). In the narrowband channel, a numerical approach is to iteratively optimize the input distribution and the ADC thresholds [15]. For the SISO frequency-selective channel, vector quantization may be adopted to better exploit the correlation in the received sequence [36]. Related work [37] on the SISO frequency-selective channel also optimized the thresholds numerically, but based on a bit-error-rate criterion.

In this paper, we focus on the MIMO channel with one-bit quantization. The main advantage of this architecture is that the one-bit ADC is just a simple comparator and can be implemented with very low power consumption [39]. The architecture also simplifies the overall complexity of the circuit, for example automatic gain control (AGC) may not be required [40]. Compared to our initial work [1], a tighter bound on the MIMO channel capacity at infinite SNR was provided in this paper. In addition, new bounds on capacity at finite SNR are included in this paper.

The main contributions of this paper are summarized as follows.

We find the capacity of the channel with one-bit quantization in the whole SNR regime.

We derive the infinite SNR capacity of the channel with one-bit quantization. We also find a closed-form expression when the receiver has a large number of antennas.

We provide bounds on the infinite SNR capacity of the MIMO channel with one-bit quantization. The decoding process at the receiver is similar to finding the transmitted symbols satisfying a series of linear inequalities. Based on this observation, we develop accurate bounds on the infinite SNR capacity by relating it to a problem in classical combinatorial geometry. A computationally efficient method based on convex optimization is proposed to design the input alphabet such that the infinite SNR capacity is approached.

We provide a new upper bound for the channel with one-bit quantization at finite SNR. The bound is tight when the channel is row full rank. A simple lower bound by using channel inversion strategy is derived. We also prove that the lower bound obtained by treating the quantization noise as Gaussian noise is loose at high SNR.

We find the infinite SNR capacity of a sparse mmWave channel. We show that the capacity is mainly limited by the number of paths in the mmWave propagation environment. In a special case when there is only one single path, we propose a capacity-achieving transmission strategy.

The paper is organized as follows. In Section II, we describe a MIMO system with one-bit quantization. In Section III, we present the SISO and MISO capacities of the one-bit quantized MIMO channel. In Section IV and V, we analyze the capacity of MIMO channel at infinite and finite SNR, respectively. Numerical methods to optimize the distribution of input symbols are shown in Section VI. We then consider the quantized mmWave MIMO channel in Section VII. Simulation results are shown in Section VIII, followed by the conclusions in Section IX.

Notation

: is a scalar, is a vector and is a matrix. represents the phase of a complex number . is the vector consisting of {, }. , and represent the trace, transpose and conjugate transpose of a matrix , respectively. stands for the Hadamard product of and . represents a square diagonal matrix with the elements of vector on the diagonal. and stand for the real and imaginary part of , respectively. denotes the probability.

## 2System Model

Consider a MIMO system with one-bit quantization, as shown in Fig. ?. There are antennas at the transmitter and antennas at the receiver. Assuming perfect synchronization and a narrowband channel, the baseband received signal in this MIMO system is,

where is the channel matrix, is the signal sent by the transmitter, is the received signal before quantization, and is the circularly symmetric complex Gaussian noise.

In our system, there are a total of one-bit resolution quantizers that separately quantize the real and imaginary part of each received signal. The output after the one-bit quantization is

where is the signum function applied componentwise and separately to the real and imaginary parts. Therefore, the quantization output at the th antenna for .

In this paper, we assume that there is both and . Consequently, the channel capacity with one-bit quantization is

where

and is the average power constraint at the transmitter.

## 3SISO and MISO Channel Capacities with One-bit Quantization

The mutual information in is in the form of multiple integrals and sums. Obtaining a closed form expression for the optimization problem is a challenge. For the simple and cases where there is only one receive antenna, we find the capacity and the capacity-achieving strategy. We start with the case then move on to the case.

### 3.1SISO Channel with One-Bit Quantization

First, we deal with the very special case when . The channel coefficient now is a scalar denoted by .

Note that the capacity of the one-bit quantized SISO channel is similar to that of the binary symmetric channel with crossover probability , which is [42].

### 3.2MISO Channel with One-Bit Quantization

In this subsection, we consider the MISO channel with one bit quantization. The received signal is

where is the channel vector. If ADCs with infinite resolution are used at the receiver (i.e., there is no quantization noise), the capacity-achieving transmission strategy is to use beamforming and Gaussian signaling. In a system with one-bit ADCs, Gaussian signaling, it turns out, is not optimal.

Similar to the SISO case, converges to the upper bound 2 bps/Hz in the high SNR regime. In the low SNR regime,

where (a) and (b) follow from and , respectively. Note that in the low SNR regime, the capacity of MISO channel without quantization is . Therefore, in the MISO channel with CSIT, the one-bit quantization results in dB power loss. A similar result was reported in [17] but under the assumption of only CSIR.

When there is only , as shown in [17], the achievable rate with independent QPSK signaling on each transmitter antenna is

Comparing the first terms in and , we can see that independent QPSK signaling results in power loss compared to the optimal strategy. The reason is that the optimal strategy has the array gain from beamforming.

## 4SIMO and MIMO Channel Capacities at Infinite SNR with One-Bit Quantization

For SIMO and MIMO channel with one-bit quantization, the exact capacity is unknown. In this section, we consider the special case when the SNR is infinite, i.e., there is no additive white Gaussian noise. In the next section, we will provide bounds on the capacity at finite SNR.

### 4.1SIMO Channel with One-Bit Quantization

First we consider the SIMO channel. With antennas at the receiver, there are at most possible quantization outputs. Therefore, bps/Hz is a simple upper bound on the channel capacity. This upper bound, unfortunately, cannot be approached when is larger than one. We provide a precise characterization in the following proposition.

In Fig. ?, we plot the infinite SNR capacity obtained by numerically maximizing the mutual information function . The lower bound and upper bound are also plotted. It is shown that the high SNR capacity converges to when .

In communication systems, the zero symbol is often not included as part of the constellation due to the peak-to-average power ratio (PAPR) issue. Therefore, in the high SNR regime, only distinguishable symbols are employed as the channel inputs and there will be different quantization outputs corresponding to each input symbol. The resulting achievable rate will converge to as transmission power increases.

### 4.2MIMO Channel Capacity with One-Bit Quantization

In the infinite SNR regime, the decoding process at the receiver is as follows:

where is the set containing all the input symbols and the equation in is applied componentwise and separately to the real and imaginary parts. Therefore, the decoding process is similar to finding the input symbols satisfying a system of linear inequalities,

where the Hadamard product and inequality are applied componentwise and separately to the real and imaginary parts. If there is only one input symbol satisfying the system of linear inequalities, it can be correctly decoded at infinite SNR. It turns out that the infinite SNR capacity of the quantized MIMO channel is closely related to a problem in classic combinatorial geometry. We first give a related lemma and its dual statement; the proof is available in several references, e.g., [43].

We first give two examples of the lemma. First, a line passing through the origin divides any dimensional space into regions. Second, distinct lines passing through the origin divide the 2-D plane into regions.

If we put the vectors, denoted as , into a matrix . These vectors are *in general position* in -dimensional space if and only if every submatrix has a nonzero determinant [45]. Note that general position is a strengthened rank condition^{2}

Based on Lemma ?, we provide our results on the MIMO channel capacity with one-bit quantization.

If the coefficients of are independently generated from a continuous distribution, for example, , the condition of *general position* is satisfied with probability one.

In practice, the condition of general position may not be satisfied. Next, we relate the infinite SNR capacity to the rank of the channel .

The function has the following properties:

;

.

Property 1 implies that the high SNR capacities of SISO and MISO channel are bps/Hz. Combining Property 2 and Proposition ?, we obtain that the infinite SNR capacity of SIMO channel is between and , which is in Proposition ?.

In Fig. ? and Fig. ?, we plot , which is very close to the high SNR capacity as shown in . In Fig. ?, is a strictly increasing function of . In Fig. ?, we see that increases fast with and quickly becomes saturated when . When , . Moreover, we see that in general. This means that the capacity of a quantized channel is different from that of a quantized channel. This is in striking contrast with unquantized MIMO systems with .

## 5Bounds of MIMO Channel Capacity with One-Bit Quantization at Finite SNR

In this section, we turn to the channel capacity at finite SNR. We will propose a new upper bound and discuss two lower bounds.

### 5.1Upper Bound of MIMO Channel Capacity at finite SNR

The upper bound is achieved when has same singular values (or equivalently, ). The transmission strategy is the simple channel inversion which will be discussed later.

This bound is loose when has less than nonzero singular values (or equivalently, ). For example, by Proposition ?, the SIMO channel capacity at infinite SNR is around . But the upper bound in approaches at high SNR, which is larger than when .

At low SNR, we obtain

The unquantized MIMO channel capacity with CSIT is at low SNR. Therefore, the one-bit quantization results in *at least* dB power loss.

### 5.2Lower Bounds on MIMO Channel Capacity at Finite SNR

#### Channel Inversion

When is invertible, a simple transmission strategy is to use channel inversion (CI) precoding and QPSK signaling [35]. The transmitted symbol is

where is a vector with independent QPSK entries that satisfies . The expected transmission power is mounted as . The output of the quantizer with this choice of precoder is

The channel decomposes into parallel one-bit quantized SISO channels with same channel gain. According to Lemma ?, the achievable rate is

If , we find that is equal to . This implies that the channel inversion transmission strategy is *capacity-achieving* when the channel has identical singular values.

Denote the eigenvalues of as . We have

Note that , we conclude that the power loss of the channel inversion strategy compared to the optimum is at most dB. Note that is the condition number of the matrix . Therefore, the channel inversion strategy is nearly optimum when the channel matrix has a small condition number.

At low SNR, we have

In [17], the achievable rate of MIMO channel with independent QPSK signaling and one-bit ADCs at low SNR is shown to be

When , because of Jensen’s inequality . This means that with relative small transmitter antenna array, the channel inversion method does not provide gain over the simple QPSK signaling. However, if , then . The reason is that the channel inversion strategy has array gain which increases with the number of transmitter antennas.

#### Additive Quantization Noise Model (AQNM)

It is known that Gaussian distributed noise minimises the mutual information [42]. A lower bound of the capacity can be derived by assuming the quantization error as Gaussian distributed noise. Such a lower bound was given in [31] as

where is the -th row of . As pointed out in [35], this lower bound is quite tight at low SNR when the additive white Gaussian noise is dominating. As shown below, however, we find that this lower bound is loose at high SNR when the quantization noise is dominating.

where is the distortion factor (see Table 1 in [31] for the value of ), is obtained by normalizing each row of , and is the eigenvalue of the matrix .

Since each row of has unit norm, then

Therefore,

where follows from that is concave in . For one-bit quantization, the distortion factor is . Therefore, we have,

If is invertible, we know that the achievable rate of channel inversion strategy in approaches at high SNR. Therefore, we conclude that this AQNM lower bound is loose at high SNR.

## 6Numerical Input Optimization Methods for the MIMO Channel

The channel inversion strategy only applies when is invertible (or when has IID Gaussian entries). In this section, we propose a heuristic method that achieves the high SNR capacity without requiring being invertible. The basic idea is: for each possible quantization output , find the input signal such that .

The input signals are obtained by solving the following optimization problem,

where the inequalities in and are applied componentwise. The objective is to maximize the minimum distance between the the noiseless received signal and the threshold of the one-bit ADCs, which is zero. If , then and imply .

Using the notation given in , we rewrite Problem **P1** in a compact form as,

For fixed , the inequality constraint is linear and thus convex. Therefore, the problem is convex and can be solved by software solver such as CVX [47].

There are a total of possible values of and thus different optimization problems. Denote the optimal value of Problem **P2** as . Note that if , the value of the objective function in Problem **P2** is zero regardless of . Therefore, a lower bound of is zero. When , the corresponding is put in the transmitter constellation. There may be many optimization problems with , for example in the SIMO channel, out of the problems has objective value . When is large, it is inefficient to solve each of the convex problem.

We now provide a method to reduce the complexity. We can see that if and only if there exists satisfying

Therefore, it is efficient to check the feasibility of before solving the optimization problem. Actually, is a system of linear inequalities and can be solved by a methd called ‘Fourier-Dines-Motzkin Elimination’ [48] (see [48] for more details of the method).

The transmitter constellation is composed of the nonzero solutions of the convex optimization problems and the zero symbol. To reduce the PAPR, the zero symbol is often not included in the constellation. Therefore, in our simulations, the transmitter constellation only contains the nonzero solutions of the convex optimization problems. Instead of transmitting the symbols with equal probability, we can also optimize the probabilities of each symbol using the well-known Blahut-Arimoto algorithm [50]. The performance improvement is shown in our simulation results. We find that optimization of the transmission probability of the symbols only provides small gain over a uniform distribution in the low and medium SNR regime.

In the appendix, we prove that a lower bound of the achievable rate of this heuristic method is

where is a constant depending on the channel and

where is number of the convex optimization with nonzero objective function value and .

At high SNR, this lower bound converges to . Note that when the condition of general position is satisfied, . This implies that at high SNR, the rate of this convex optimization based method approaches the infinite SNR capacity.

## 7MmWave Channel with One-Bit Quantization

In mmWave communications, the channel matrix is usually assumed to be low rank due to sparse scattering in the channel [51] and therefore does not satisfy the strict condition of *general position* in Proposition ?. As a result, we cannot directly obtain the infinite SNR capacity of the mmWave channel.

In this section, the mmWave MIMO channel is modeled using a ray-based model with paths. We also assume that uniform planar arrays (UPA) in the -plane are deployed at the transmitter and receiver. Denote , , as the strengths, the azimuth (elevation) angles of arrival and the angle of departure of the th path, respectively. The array response vectors at the transmitter or receiver is given by [52]

where and are the and indices of an antenna element respectively. Herein, where is the wavelength and is the inter-element spacing. Hence, the channel matrix is,

where , and .

The number of multipaths tends to be lower in the mmWave band compared with lower frequencies. Meanwhile, large antenna arrays are usually deployed to obtain array gain for combatting the path loss. Hence, we suppose that in mmWave MIMO channels.

Corollary ? shows that multipath is helpful in improving the infinite SNR capacity in mmWave MIMO systems.

As is not invertible in mmWave systems, the channel inversion approach cannot be used to design the transmitted symbols. Instead, the convex optimization approach proposed in Section 6 has to be used.

Next, to provide intuition, we consider a mmWave MIMO channel with only one path and propose a capacity-achieving transmission strategy. If , the channel in degenerates to

Following the same logic in the MISO setting, matched filter beamforming is used at the transmitter to obtain the array gain. The resulting channel is equivalent to a SIMO channel with channel coefficients as . Then the cutting plane method (details of the method are presented in the next section) can be used to design the transmitted symbols. Therefore, the transmitted symbols will be

where is the symbol obtained by the cutting plane method in the equivalent SIMO channel.

## 8Simulation Results

In this section, we first present simulation results of the capacities or achievable rates of the SIMO, MISO and the MIMO channel at high and low SNR. The performance of the proposed convex optimization method will also be shown. Last we consider the mmWave channel model and evaluate the achievable rates.

### 8.1MISO Channel with One-Bit Quantization

In Fig. ?, we plot the MISO channel capacity with and without one-bit quantization for different values of . First, we see that the channel capacity with one-bit quantization approaches bps/Hz in the high SNR regime. Second, the transmitter antenna array provides power gain as shown in the figure. There is about dB power loss in the low and medium SNR regimes, which verifies our analysis in . Third, when , the capacity is close to the upper bound when SNR is larger than dB. We see that the “high SNR” in our analysis can be very low in the practice thanks to the array gain provided by the multiple antennas.

### 8.2SIMO Channel with One-Bit Quantization

In the SIMO channel, we can obtain the capacity-achieving input distribution using the cutting plane method proposed in [53] and used in [54]. For this method, we take a fine quantized discrete grid on the region, for example, , as the possible inputs and optimize their probabilities. The algorithm is an iterative algorithm which converges quite fast in several tens of iterations. Each iteration is a convex problem which can be solved efficiently.

In Fig. ?, we show a simple case when . It is interesting to find that the optimal input constellation contains the rotated 8-PSK symbols and the symbol zero. For other general channels, the optimal constellation may not be regular. For example, in Fig. ?, we show the optimal input distribution when .

### 8.3MIMO Channel with One-Bit Quantization

In this part, we illustrate the average achievable rates of MIMO system with one-bit quantization. The channel coefficients are generated from distribution independently. The input alphabet, which contains input symbols, are constructed by two methods, i.e., channel inversion method and convex optimization method. For the convex optimization method, these symbols are transmitted with equal probabilities or with probabilities optimized by the Blahut-Arimoto algorithm (denoted as “BA” in the figures).

In Figs. ?, ? and ?, we plot the achievable rates in the medium and high SNR regimes when is , and , respectively. Note that the channel inversion method can be used in the first two cases but not in the third case.

First, the achievable rates of convex optimization method, denoted as ‘Convex Opt.’, converge to the upper bound bps/Hz in Fig. ? and Fig. ? and bps/Hz in Fig. ?. This verifies that the convex optimization method can approach the infinite SNR capacity in Proposition ?. Its lower bound given in denoted as ‘Convex Opt., Lower Bound’ is also plotted in these figures. We can see that the bound is tight at high SNR.

Second, let us examine the bounds at finite SNR. The theoretical upper bound given in is quite tight in Fig. ? and Fig. ? but loose in Fig. ?. The reason is that the channel in Fig. ? does not has full row rank. Namely, the rank of the channel is 2, while there are receiver antennas. The channel inversion method works well at high SNR but has worse performance in the medium and low SNR regimes compared to the convex optimization method. But when , the gap between the performances of these two methods is negligible. As expected, the AQNM lower bound given in is loose at high SNR in all three figures.

Third, the rates of the channel without quantization are computed using the usual SVD precoding and waterfilling approach. We can see that the power loss of the quantized systems compared to the unquantized systems is less than dB at medium SNR.

Last, let us evaluate the performance of independent QPSK signaling across the transmitter antennas. For the case of independent QPSK signaling and one-bit quantization, we find that although its rate is larger than the AQNM lower bound, its performance is worse than the two proposed methods at high SNR. We also simulate the case of QPSK signaling without quantization where the input is discrete and the output is continuous (see for example [55] for the computation of the rate). In the medium SNR, we see that the rate is close to the cases with one-bit quantization. In the high SNR regime, however, the rate will converge to bps/Hz, instead of bps/Hz in the cases of one-bit quantization.

In Figs. ?, ? and ?, the achievable rates in the low SNR regime (below dB) are plotted versus the SNR in linear scale. The slopes of these curves verify our analysis. For independent QPSK signaling, we plot the low SNR capacity approximation given by . When , we find that the channel inversion method is worse than the QPSK signaling. However, the channel inversion method is better when . This verifies our analysis in Section ?.

As shown in Figs. ?- ?, optimizing the probabilities by the Blahut-Arimoto algorithm does not improve the achievable rates at the high SNR but provides gain at low and medium SNR. In addition, the convex optimization method with probabilities optimized by the Blahut-Arimoto algorithm achieves the largest rate in the cases with one-bit quantization.

### 8.4MmWave Channel with One-Bit Quantization

We show the achievable rates in a channel with varying number of paths in Fig. ?. The azimuth angles and elevation angles are uniformly distributed over and , respectively. The complex path gains are complex Gaussian variables. The inter-element spacing of the receiver antenna array is set to one half of the wavelength. It is shown that as increases from to , the achievable rate at high SNR increases. The rates converge to bps/Hz, which is , and bps/Hz when , , , respectively. In addition, the rates only converge at very high SNR which is larger than 60 dB in the figure. The rates of the mmWave channel without quantization are also shown. We see that the loss incurred by the use of one-bit ADCs increases with the number of paths .

## 9Conclusion

In this paper, we analyzed the capacity of point-to-point MIMO channel with one-bit quantization at the receiver. CSI was assumed to be known at both the transmitter and receiver. We obtained the MISO channel capacity in closed-form. For the SIMO and MIMO channel, we derived bounds on the capacities at infinite and finite SNR. A convex optimization based method was also proposed to design the transmitter constellation. Last, we considered the mmWave MIMO channel with limited path and showed the capacity is limited by the number of paths.

From the main results present in this paper, we draw several conclusions. First, when there is single antenna at the receiver, the capacity is achieved by precoding and QPSK signaling. Second, the MIMO channel capacity at infinite SNR is related to a classic combinatorial geometric problem. Our proposed input design method can approach the infinite SNR capacity. Third, when the channel matrix has full row rank and a small condition number, channel inversion precoding is close to the optimum. Last, treating the quantization error as Gaussian noise is unsuitable at high SNR.

There are many potential directions of future work. Perhaps the most critical assumption is that the transmitter and receiver have complete and perfect CSI. Channel estimation with one-bit quantization has been considered in [22] at lower frequencies. Due to the sparse structure in mmWave channels, it is of interest to develop efficient compressed channel estimation techniques at the receiver and also methods for feeding back CSI to the transmitter. Initial work on channel estimation using the one-bit compressive sensing framework has been reported in [56] and our work in [30]. Another possible direction is to consider the effects of imperfect CSI. With imperfect CSI, the design of the transmitter constellation should be robust to the CSI error term. Studying the effect of imperfect CSI is left to future work.

## 10Bounds of the Achievable Rate of Convex Optimization Method

We provide a lower bound for the achievable rate of the convex optimization strategy. Assume that there are convex optimization problems having nonzero objective function values. Denote the quantization vectors, optimal solutions and optimal objective function values of these problems as , and , , respectively. Denote the minimum as . Assume that these symbols are transmitted with equal probability . The conditional entropy of is,

where follows from that the noises across different antennas are independent, follows from that the in-phase and quadrature parts of the noises are independent and Gaussian distributed with variance , follows from that .

For , we have

We denote . We now consider the following problem,

Since the entropy function is concave [42], the minimum is achieved at the extreme points. There are two kinds of extreme points.

The first kind of extreme points:

The second kind of extreme points:

The corresponding entropies of are,

respectively.

Therefore, the mutual information is

In , we can see that where is a constant depending on the channel .

Last, an upper bound of can be derived as follows,

Hence,

### Footnotes

- This type of quantization loss is also well known to the error control coding community. At low SNR, there exists roughly a 2-dB difference between the performance of hard and soft decision decoding. See, for example, [18] for details.
- For example, the matrix has full column rank but does not satisfy the condition of general position.

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