Capacity of Multiple One-Bit Transceivers in a Rayleigh Environment

04/21/2018 ∙ by Kang Gao, et al. ∙ University of Notre Dame 0

We analyze the channel capacity of a system with a large number of one-bit transceivers in a classical Rayleigh environment with perfect channel information at the receiver. With M transmitters and N=α M receivers, we derive an expression of the capacity per transmitter C, where C≤(1,α), as a function of α and signal-to-noise ratio (SNR) ρ, when M→∞. We show that our expression is a good approximation for small M, and provide simple approximations of C for various ranges of α and SNR ρ. We conclude that at high SNR, C reaches its upper limit of one only if the ratio α>1.24. Expressions for determining when C "saturates" as a function of α and ρ are given.

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I Introduction

In an effort to save power and cost in wideband wireless transceiver systems, low-resolution (especially one-bit) analog-to-digital converters (ADCs) [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] and digital-to-analog converters (DACs) [14, 15, 16] are being considered in transmitter and receiver chains, especially in systems involving many such chains. The nonlinearity introduced by coarse quantization becomes a limiting factor in the achievable throughput of such a wireless system. Channel capacity is one measure of this throughput.

There is a rich literature on the subject of capacity with coarse quantization. The capacity of a system with one-bit ADCs at the receiver is analyzed in [1, 2, 3, 4, 6, 7, 5, 8, 9]

with various assumptions about the channel, the channel information, and communication schemes. Communication techniques including channel estimation and signal detection for a multiple-input multiple-output (MIMO) system with one-bit ADCs at the receiver are studied in

[10, 11, 12, 13]. A communication system with one-bit DACs at the transmitter is studied in [14, 15, 16].

While many of the efforts consider low-resolution quantization effects at the transmitter or receiver, a few consider low-resolution quantization on both, including [17] and [18]. A linear minimum-mean-squared-error precoder design is proposed for a downlink massive MIMO scenario to mitigate the quantization distortion in [17] and the performance analysis of a system with a small number of one-bit transceivers is studied in [18]. We focus on a model where one-bit quantization is considered at both the transmitter and receiver:

(1)

where and are the number of transmitters and receivers, and are the transmitted and received signals, is the channel matrix known to the receiver, is the additive Gaussian noise with and is independent of and , is the expected received SNR at each receive antenna. The function provides the sign of the input as its output. The channel is modeled as real-valued since only the in-phase (I) information is used and the quadrature (Q) phase is ignored at the receiver. A Rayleigh channel is assumed, with each element to be independent Gaussian . This assumption appears to hold for non-line of sight (NLOS) channels in many frequency bands [19], and also appears in the analysis in [6, 7, 8, 9, 10, 11, 14].

Our contribution is a large and analysis, where the ratio is constant, of the capacity for the model (1). Analytical expressions are derived that can be used to explain the behavior of the system in various limiting regimes of operation in and .

Ii Capacity for a large number of transmitters and receivers

The capacity of the channel in (1) as a function of , , and is

(2)

where we have normalized by , and where is the input distribution independent of , and is the mutual information notation. When with a ratio , the capacity is defined as

(3)

We can readily see that because each transmitter can transmit at most one bit of information, and each receiver can decode at most one bit of information. Therefore, for all and .

Limiting capacities such as (3) are difficult to compute in closed form, but the “replica method” [20] can be brought to bear on the problem. Some details of how to apply the method are presented in Section IV. For now, we present the result:

(4)

where is the capacity of a single transceiver with SNR , which is defined as

(5)

where is the binary entropy function, and are the solutions of

(6)
(7)
(8)

Equation (4) gives the capacity for any SNR and , and some limiting situations are readily analyzed, including: (i) high SNR, ; (ii) low SNR, ; (iii) many more receivers than transmitters, ; (iv) many more transmitters than receivers, . These are now presented, with only limited proofs.

Ii-a High SNR ()

When , the system is effectively becoming “noise-free”, and we might expect , but as we show this does not happen for all . For , , and (4) becomes

(9)

where is defined in (5), (7) can be simplified as

(10)

and are the solution of (6) and (10).

The expression (9) is not especially intuitive, but it is not difficult to solve. We show some numerical examples in Section III. It turns out that in this case, solving (9

) is essentially equivalent to solving for the “quenched entropy” for Gibbs learning of the Ising perceptron (Section 7.2 in

[21]).

Ii-B Low SNR ()

When , (6), (7), and (8) become

(11)

For small , we use a Taylor series expansion to obtain

Then

(12)

to second order in . It turns out that this result matches the expression in [6], with a difference in factor of that comes from the fact that [6] considers two bits per transmission, and ‘nats’ instead of ‘bits’.

Ii-C ()

When , (6)-(8) becomes

(13)

and

(14)

More will be said about this in Section III.

Ii-D ()

When , the first order approximations of and based on (6)-(8) become

(15)

Therefore,

(16)

where is the capacity of a single transceiver, defined in (5).

Iii numerical evaluation of capacity

We first compare in (2) with in (4) for and , to show how the large- and limit (4) can be used to approximate the exact capacity. Figure 1 displays that the approximation is accurate for small for a wide range of SNR (from 0 dB to 30 dB) when . When is larger than 0.7, is not big enough and a larger is required to get a valid approximation. We can see that can saturate at 1 when with SNR smaller than 30 dB, and an SNR higher than 30 dB is required to achieve the maximum for , but cannot achieve the maximum when . We will show later that is required to achieve the maximum.

Fig. 1: Comparison between (2) and (4) with for with SNR varying from 0 dB to 30 dB. The accuracy of the approximation in a wide range of SNR shows that is big enough to get a valid approximation with .

Figure 2 displays in (4) for and varying from 0.1 to 10 with step 0.1. We can see that increases linearly with and when and are small, and the rate of increase slows down dramatically as and grow and nears saturation at . When is small but is large, saturates at (its upper bound). When is small but is large, we can see that increases with and reaches its maximum value 1. We show that for any , when .

Contours of constant for and are shown in Figure 3. We can observe that there is generally a sharp tradeoff between and , and that operating near the knee in the curve is generally desirable for a given capacity since both and are small.

Furthermore, the contours are dense when and start becoming sparse when , thus showing that has started to “saturate” at 0.8 and improves only slowly with further increases in either or .

The contours allow us to explore optimal operating points. For example, given a cost function where for some constant , we find an approximately optimal point to achieve is and . Attempts to make smaller will require significant increase in , and attempts to make smaller will require significant increase in .

Fig. 2: (4) versus and with and . increases linearly with and when and are small but “saturates” quickly.
Fig. 3: Contour plot of (4) with and . The gap between the adjacent contours becomes large as increases, which shows the increasing demands on and to increase .

Figure 4 shows the accuracy of the approximations of at high and low SNR. Plotted are examples when SNR is large (10 dB to 30 dB) and SNR is low (-10 dB, ) of the actual capacity (4) and the corresponding approximations (9) and (12). Of particular interest is the approximately linear growth in (9) with until it reaches the saturation point when . This hard limit value of 1.24 receive antennas for every transmit antenna is perhaps surprising.

The curves for low SNR show that at is close to the low SNR approximation in (12) with a simple second order expression when . In general, when , we need for an accurate low SNR approximation according to (11).

Fig. 4: Comparison between (4) at high SNR (10 dB to 30 dB), low SNR (-10 dB, ), and their corresponding approximations in (9) and (12). The curve at (10 dB) is already close to the noise-free case, which increases nearly linearly with before saturation at . The low SNR approximation (12) is similarly accurate when for . In general, we need for an accurate low SNR approximation when .

Figure 5 presents a comparison of (4) with the large , small approximations in (14) and (16). We obtain excellent agreement for even the modest values and over a wide range of SNR. Moreover, according to (13), when , we have for any , and thus . This differs from the high SNR case, where when even as .

Iii-a Tradeoff between and for fixed

We are interested in characterizing the contours in Figure 3 analytically, and we use the large approximation for in (14). Since in (14) is just a function of , to achieve some target capacity , we solve for numerically, and denote the result as . With , (13) then provides the relationship between and . To simplify the relationship, (13) can be further approximated as

(17)

with good accuracy when . The relationship between and is then

(18)
Fig. 5: The comparison between the capacity (4) with large (), small () and their corresponding approximations in (14) and (16). The good approximations over a wide range of SNR shows that is big enough and is small enough to use (14) and (16) for accurate approximations.

To verify the approximation in (18), we compare the actual SNR with the approximated (18) in Figure 6. Shown are contours for and . The solid lines are the contour plot of (4), and the dashed lines are (18). We see good agreement over a wide range of .

Fig. 6: Comparison between the actual SNR and the approximated SNR (18) for the contours . The solid lines are contour plots of (4), and the dashed lines present the approximation of (18) as a function of .

Clearly, there are many other comparisons and tradeoffs we can analyze using (4) and its approximations. We briefly describe how the replica method was applied to obtain (4).

Iv Replica Analysis

The replica method, a tool used in statistical mechanics and applied to the theory of spin glass [20], has been applied in many communication system contexts [22, 23, 24, 25, 26]

, neural networks

[27, 21], error-correcting codes [28], and image restoration [29]. A mathematically rigorous justification of the replica method is elusive, but the success of the method maintains its popularity. We apply the method to solve for a closed-form answer to (3). We omit many details, and present only the primary steps.

Because the channel is unknown to the transmitter, according to [18], the optimal input distribution is , and then (2) becomes

where is the standard definition of entropy.

Since the elements in are i.i.d. , and , we have , and

(19)

where is defined in (5).

Then, (3) becomes

(20)

The replica method is used to compute the limit, and the processes are similar to that used in [22, 23, 24, 25, 26].

We start with the identity

which holds for . The idea of the replica method is to compute as a function of by treating as a positive integer, and then assume the result to be valid for .

We assume the limit of and can commute, then

(21)

where .

Now, we regard as a positive integer and we have

where is the th replica of (), and

are i.i.d. uniform distributed in

.

Based on (1), we further have

(22)

where is the th element of , is the well-known Q-function, and is the th row of ,

(23)

with .

Let and . Then , where is the covariance matrix with elements for and . Then only depends on :

(24)

and (22) becomes

where

(25)

We can consider as the distribution of a random symmetric matrix , and we have

Similarly to [24], we apply Varadhan’s theorem and Gartner-Ellis theorem[30] and obtain

(26)

where is an matrix with as its elements, and is defined as

(27)

Based on the distribution of in (25), we further have

(28)

where are independent uniform distributed in .

and that achieve the optimal value described in (26) are called the saddle point[21]. The saddle point either stays on the boundary ( or ) or satisfies , i.e.

(29)

with and expressed in (24) and (28).

Here, we further assume that permutations among the replicas with index will not change the saddle point. This assumption is called the “replica symmetry” (RS) assumption. At the saddle point, we let

(30)

which are called RS saddle points.

Based on (20), (21), (26), and (30), we can get (4), where is the solution when the saddle point is on the boundary (). The remaining expressions in (4) are obtained when the saddle point satisfies (29), and the corresponding RS saddle point is the solution of (6)-(8).

Iv-a Extension to complex signals

The real-valued model (1) is now extended to both I and Q phase at the transmitter and receiver, and hence

(31)

where are defined as

where are the real and imaginary parts of the channel, and are the real and imaginary parts of the transmitted and received signal, and and are additive noise. The elements in , , , and are independent Gaussian , and is the expected received SNR at each receive antenna.

Since the channel is known only to the receiver, the uniform input is optimal and the channel capacity is

When with a ratio , the capacity is defined as

(32)

Similarly to the analysis for real signal, we have

(33)
(34)

Using the replica method with the RS assumption, we obtain

(35)

and therefore

(36)

Consequently, the I-Q model capacity is twice the I-only capacity.

V Conclusion

We have presented the capacity per transmitter in the limit where the number of single-bit transmitters and receivers is large, and where was fixed. A flat Rayleigh fading channel was considered, and we assumed the channel was only known by the receiver. We were able to derive a variety of approximations using the analytical results, and showed that the large-system formulas are useful even for a small numbers of transmitters and receivers. We examined how saturated with either large or , and gave formulas for exploring the contours of fixed as a function of and . Further work in expanding these results to different channel models would be of great interest.

Acknowledgment

The authors are grateful for the support of the National Science Foundation, grants ECCS-1731056, ECCS-1509188, and CCF-1403458.

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