Achievable Rate Region of Quantized Broadcast and MAC Channels

Achievable Rate Region of Quantized Broadcast and MAC Channels

Suresh Chandrasekaran, Saif K. Mohammed, and A. Chockalingam
Department of ECE, Indian Institute of Science, Bangalore 560012, India
Communication Systems Division, Department of Electrical Engg., Linköping University, Sweden
Abstract

In this paper, we study the achievable rate region of Gaussian multiuser channels with the messages transmitted being from finite input alphabets and the outputs being quantized at the receiver. In particular, we focus on the achievable rate region of Gaussian broadcast channel (GBC) and Gaussian multiple access channel (GMAC). First, we study the achievable rate region of two-user GBC when the messages to be transmitted to both the users take values from finite signal sets and the received signal is quantized at both the users. We refer to this channel as quantized broadcast channel (QBC). We observe that the capacity region defined for a GBC does not carry over as such to QBC. We show that the optimal decoding scheme for GBC (i.e., high SNR user doing successive decoding and low SNR user decoding its message alone) is not optimal for QBC. We then propose an achievable rate region for QBC based on two different schemes. We present achievable rate region results for the case of uniform quantization at the receivers. Next, we investigate the achievable rate region of two-user GMAC with finite input alphabet and quantized receiver output. We refer to this channel as quantized multiple access channel (QMAC). We derive expressions for the achievable rate region of a two-user QMAC. We show that, with finite input alphabet, the achievable rate region with the commonly used uniform receiver quantizer has a significant loss compared to the achievable rate region without receiver quantization. We propose a non-uniform quantizer which has a significantly larger rate region compared to what is achieved with a uniform quantizer in QMAC.

KeywordsGaussian broadcast channel, Gaussian multiple access channel, finite input alphabet, quantized receiver, achievable rate region, successive decoding, discrete memoryless channel.

I Introduction

Communication receivers are often based on digital signal processing, where the analog received signal is quantized into finite number of bits using analog-to-digital converters (ADC) whose outputs are then processed in digital domain. These ADCs are expected to operate at high speeds in order to meet the increasing throughput and bandwidth requirements. However, at high conversion speeds, the precision of ADCs is typically low which results in loss of system performance [1]. For example, low-precision receiver quantization can cause floors in the bit error performance [2],[3]. Also, it has been shown that in a single-input single-output (SISO) point-to-point single user system with additive white Gaussian noise (AWGN), low-precision receiver quantization results in significant loss of capacity when compared to an unquantized receiver [4]. Motivated by the increasing need to investigate the effect of receiver quantization in high-throughput communication, we, in this paper, address the issue of characterizing the achievable rate region of two different Gaussian multiuser channels, namely,

  1. Gaussian broadcast channel (GBC) with finite input alphabet and quantized receiver output; we refer to this channel as the Quantized broadcast channel (QBC),

  2. Gaussian multiple access channel (GMAC) with finite input alphabet and quantized receiver output; we refer to this channel as Quantized multiple access channel (QMAC),

and report some interesting results.

GBC comes under the class of degraded broadcast channels, for which capacity is known. For a two-user GBC, it is known that the capacity is achieved when superposition coding is done at the transmitter assuming that the users’ messages are from Gaussian distribution, and, at the receiver, the high SNR user does successive decoding and the low SNR user decodes its message alone considering the other user’s message as noise [5]. However, the capacity region of two-user QBC is not known. Recently, achievable rate region for two-user GBC when the input messages are from finite signal sets and the received signals are unquantized has been studied in [6], and it is referred to as the constellation constrained (CC) capacity of GBC [7].

Our present contribution first gives achievable rate region for two-user QBC in Section II. The main results on QBC are summarized as follows.

  • The capacity region defined for a GBC does not carry over as such to QBC.

  • Once quantization is done at the receiver in a GBC, the channel is no more degraded. Therefore, the optimal decoding scheme for GBC (i.e., high SNR user alone doing successive decoding) does not necessarily result in achievable rate pairs for QBC.

  • We then propose achievable rate region for QBC based on two different schemes (scheme 1 and scheme 2). In scheme 1, user 1 will do successive decoding and user 2 will not, whereas, in scheme 2, user 2 will do successive decoding and user 1 will not. In addition to this, in both the schemes, the message for the user which does not do successive decoding is coded at such a rate that the message of that user can be decoded error free at both the receivers.

  • Rotation of one of the user’s input alphabet with respect to the other user’s alphabet marginally enlarges the achievable rate region of QBC when almost equal powers are allotted to both the users.

Next, in Section III, we address the achievable rate region of two-user QMAC. With finite input alphabets and an unquantized receiver, the two-user GMAC rate region has been studied in [8]. In [8], in terms of the achievable rate region, it was shown that, compared to having both the users transmit using the same finite signal set, it is better to have the second user transmit using a rotated version of the first user’s signal set. We refer to the two-user GMAC system model in [8] (with finite input alphabet and no output quantization) as constellation constrained MAC (CCMAC).

In this paper, instead of assuming an unquantized receiver as was done in [8], we consider quantized receiver. Since uniform quantizers are commonly used in communication receivers, we first consider uniform quantization at the receiver, and show that with uniform quantization, there is a significant reduction in the achievable rate region compared to the CCMAC rate region. This is due to the fact that the received analog signal is densely distributed around the origin, and is therefore not efficiently quantized with a uniform quantizer. This then motivates us to propose a non-uniform quantizer with finely spaced quantization intervals near the origin. We show that the proposed non-uniform quantizer results in enlargement of the achievable rate region of two-user QMAC compared to that achieved with a uniform quantizer. It is further observed that, with increasing number of users, the probability distribution of the received analog signal is more and more dense around the origin. Hence, it is expected that with increasing number of users, larger enlargement in rate region of QMAC may be achieved with non-uniform quantization compared to uniform quantization.

The rest of this paper is organized as follows. Achievable rate region of two-user QBC is studied in Section II. Achievable rate region of two-user QMAC is presented in Section III. Conclusions are given in Section IV.

Ii Quantized Broadcast Channel

In this section, we propose achievable rate region for two-user QBC. We show achievable rate region results when the users employ uniform receiver quantization.

Ii-a System Model

We consider a two-user GBC as shown in Fig. 1. Let and denote the messages to be transmitted to the users 1 and 2, respectively. Let and take values from finite signal sets and , respectively. The sets and contain and equi-probable complex entries, respectively. Let the sum signal set of and be defined as

(1)

Let and be defined as

(2)

where and represent the real and imaginary components of , respectively.

Fig. 1: (a) Two-user Gaussian broadcast channel with receiver quantization. (b) Equivalent discrete memoryless channel.

Let be the message sent by the transmitter to the users 1 and 2 with an average power constraint . We further assume that the average power constraint on is and the average power constraint on is , where . Let and denote the additive white Gaussian noise at receivers 1 and 2, respectively. The signal-to-noise ratio (SNR) at user 1 (SNR1) is and the SNR at user 2 (SNR2) is . The received signal at user 1 is then given by

(3)

Similarly, the received signal at user 2 is given by

(4)

The received analog signals, at user 1 and at user 2, are quantized independently, resulting in outputs at user 1 and at user 2. The complex quantizer at each user is composed of two similar quantizers acting independently on the real and imaginary components of the received analog signal. The real and imaginary components of the quantized output for the users 1 and 2 are then given by

(5)
(6)

where the functions and model the quantizers having a resolution of and bits, respectively. The function defines a mapping from the set of real numbers to a finite alphabet set of cardinality , i.e.,

(7)

Similarly,

(8)

Thus, the quantized received signals at user 1 and at user 2 take values from the sets and , respectively, where

(9)
(10)

Henceforth, we refer to the above system model as quantized broadcast channel (QBC).

Ii-B Achievable Rate Region of QBC

In this subsection, we derive analytical expressions for the achievable rate region of two-user QBC.

The capacity region of a two-user GBC is known [9],[10], and is given by the set of all rate pairs satisfying

(11)
(12)

assuming , where and represent the rates achieved by user 1 and user 2, respectively. The optimal input distribution that attains the capacity is known to be Gaussian. The optimal decoding scheme is that, user 1 does successive decoding (i.e., user 1 first decodes user 2’s message assuming its own message as noise and subtracts the decoded user 2’s message from its received signal , and then decodes its own message from the subtracted signal ), and user 2 decodes its message alone by considering user 1’s message as noise. This GBC belongs to a class of broadcast channel, degraded broadcast channel, which satisfies the condition

(13)

i.e., (Markov). However, observe that, in QBC,

(14)

i.e., is not true. Hence, the effective channel is no more degraded. Thus, the capacity region expressions given for GBC can not be carried over to QBC.

Through simulations, we observed that in QBC, even in presence of a Gaussian noise with , is not always greater than . Table I shows a listing of the mutual information for a two-user QBC when both the users use a 1-bit uniform quantizer and the input messages for both the users are from 4-QAM input alphabet at SNR1 = 10 dB and SNR2 = 7 dB. Observe that at and , . Hence, user 1 can not decode user 2’s message when and the rate of user 2’s message is , which, in turn, implies that user 1 can not do successive decoding. However, if we set the rate of user 2 to , then it is guaranteed that both user 1 and user 2 can decode user 2’s message and user 1 can do successive decoding.

Mutual Information
0.08083 0.37272 0.93188 1.59350
0.00893 0.15668 0.71584 1.52160
0.03572 0.20718 0.60551 1.19670
1.52160 0.71584 0.15668 0.00893
1.19670 0.60551 0.20718 0.03572
1.31920 0.82872 0.43039 0.15825
TABLE I: Mutual information for a two-user QBC when both the users use a 1-bit uniform quantizer and the input messages for both the users are from a 4-QAM alphabet at SNR1= 10 dB and SNR2 = 7 dB.

Based on the above observation, we now propose an achievable rate region for two-user QBC. We consider two schemes characterizing two different coding/decoding procedures to arrive at the proposed achievable rate region of QBC.

Scheme 1: User 1 does successive decoding and user 2 decodes its message alone.

User 1 can achieve a rate of by successive decoding (i.e., user 1 will cancel the interference due to user 2’s message and then it will decode its own message) only when it can decode user 2’s message error free. From the observations made in Table I, we know that is not always greater than and hence, for user 1 to decode user 2’s message error free, user 2’s information must be restricted to a rate of . Thus, the set of achievable rate pairs when user 1 does successive decoding and user 2 decodes its message alone, is given by

(15)
(16)

Scheme 2: User 2 does successive decoding and user 1 decodes its message alone.

Similarly, user 2 can achieve a rate of by successive decoding only when the information to user 1 is restricted to a rate of . Thus, the set of achievable rate pairs , when user 2 does successive decoding and user 1 decodes his message alone, is given by

(17)
(18)

Since any line joining a pair of achievable rate pairs in the above two schemes is also achievable by time sharing, we propose the achievable rate region of QBC, , as the set of all rate pairs which are in the convex hull [11] of the union of the achievable rate pairs of the above two schemes. The proposed achievable rate region, , is then given by

(19)

where denotes convex hull, and satisfies (15),(16) and satisfies (17),(18).

(20)
(21)
(22)
(23)

The mutual information in the expressions (15), (16), (17), (18) are calculated using the probability distribution

(24)

where , and , and refer to the th element of sets , and , respectively. The region is defined as

(25)

and . From (24), the marginal probability distributions , and are calculated as

(26)
(27)
(28)

Similarly, the probability distributions , , and can be calculated. Using the above probability distributions, the final expressions of (15)-(18) are given by Eqns. (20)-(23), which are listed above.

In the illustration of numerical results, we plot the boundary of the achievable rate region of two-user QBC by varying the proportion of power () allocated to each user from to and finding the achievable rate pairs using (19). When both and take values from the same signal set, we consider rotation of the second user’s signal set by an angle with respect to the first user’s signal set for further enlargement of the achievable rate region, i.e.,

(29)

where is the rotation angle. We observe that, the rate expressions now become a function of , and hence they are explicitly denoted as , , and . The achievable rate region of QBC with rotation, , is then given by

(30)

Ii-C QBC with Uniform Quantizer

In this subsection, we study the achievable rate region of two-user QBC with uniform receiver quantization.

Ii-C1 Uniform Quantizer

A uniform -bit quantizer, acting on the real component of the analog received signal is given by

(31)

where and is defined in (2). Similarly,

(32)

where and is defined in (2).

We assume that the user 1 uses a -bit uniform quantizer and user 2 uses a -bit uniform quantizer. Applying the above uniform quantizer to the analog received signal at the users 1 and 2, their quantized outputs on the real and imaginary components are given by

(33)
(34)

With the uniform quantizer defined in (33) and (34), we numerically evaluate the proposed achievable rate region of two-user QBC using (30) or (19), the results of which are discussed in the following subsection.

(a) SNR1 = 13 dB, SNR2 = 15 dB
(b) SNR1 = 15 dB, SNR2 = 13 dB
(c) SNR1 = 15 dB, SNR2 = 15 dB
Fig. 2: Plots of the boundary of the proposed achievable rate region of two-user QBC when the input alphabet for user 1 is 16-QAM and the input alphabet for user 2 is a rotated 16-QAM with different SNR combinations at the two users. The users use -bit uniform receiver quantizer.

Ii-C2 Results and Discussion

In Fig. 2, we first illustrate the significance of using the two schemes instead of assuming that the user with high SNR alone does successive decoding. Figures 2(a), 2(b) and 2(c) show the proposed achievable rate region of two-user QBC when the input alphabet for user 1 is 16-QAM and the input alphabet for user 2 is a rotated 16-QAM, and both the users use 4-bit uniform receiver quantization. In Fig. 2(a), SNR1 = 13 dB and SNR2 = 15 dB. In Fig. 2(b), SNR1 = 15 dB, SNR2 = 13 dB, and in Fig. 2(c), SNR1 = SNR2 = 15 dB. We observe that most of the contribution to the proposed achievable rate region of QBC is due to the scheme of the user with high SNR doing successive decoding and the user with low SNR decoding his message alone. For example, observe the performance of scheme 2 in Fig. 2(a) and scheme 1 in Fig. 2(b). However, there is an appreciable contribution to the proposed achievable rate region of QBC when the user with low SNR performs successive decoding and the user with high SNR decodes his message alone, especially when the proportion of the total transmit power allotted to that user (the one with low SNR) is more than that of the other. For instance, observe the performance in the circled regions of scheme 1 in Fig. 2(a) and scheme 2 in Fig. 2(b). When the SNR of both the users are same, equal contribution is made by the two schemes to the proposed achievable rate region of QBC, which is illustrated in Fig. 2(c).

Figure 3 shows the significance of rotation on the proposed achievable rate region of QBC when both the users use uniform quantizer of same resolution, i.e., at SNR1 = 10 dB and SNR2 = 7 dB. The input alphabet for user 1 is 4-QAM and the input alphabet for user 2 is a rotated 4-QAM. We observe that there is no increase in the achievable rate region for a 1-bit uniform quantizer due to rotation compared to that of the achievable rate region without rotation. For or bit uniform quantizers, there is a small increase in the achievable rate region due to rotation compared to the achievable rate region without rotation only when is around 0.5. The reason could be that rotation gives significant enlargement in the achievable rate region only when the sum signal set is not uniquely decodable, i.e., when there is no one-to-one correspondence between the elements in the set to the elements in the set . This happens more only when is around 0.5. For instance, when , and thus the set is not uniquely decodable. Hence, when , rotation by even a small angle makes the set to be uniquely decodable resulting in an increase in the achievable rate region of QBC. Finally, we have computed the proposed achievable rate region for QBC with asymmetric quantizers also, i.e., with .

Fig. 3: Comparison of the proposed achievable rate region of two-user QBC when both the users use uniform quantizer of same resolution i.e., at SNR1 = 10 dB and SNR2 = 7 dB. The input alphabet for user 1 is 4-QAM and the input alphabet for user 2 is a rotated 4-QAM.

Iii Quantized Multiple Access Channel

In this section, we study the achievable rate region of two-user QMAC [12].

Iii-a System Model

Consider a two-user Gaussian MAC channel. Let and be the symbols transmitted by the first and second user, respectively. Let and , where and are finite signal sets with and equi-probable complex entries, respectively. Let be the additive white Gaussian noise at the receiver. The analog received signal is then given by

(35)

The analog received signal, , is quantized by a complex quantizer , resulting in the output , as shown in Fig. 4.

Fig. 4: (a) Two-user Gaussian MAC model with quantized output. (b) Equivalent DMC.

The quantizer is composed of two similar quantizers acting independently on the real and imaginary components of the received analog signal, . The real and imaginary components of the quantized output are then given by

(36)

where the function models a receiver quantizer having a resolution of bits. is a mapping from the set of real numbers to a finite alphabet set of cardinality , i.e.,

(37)

Let be defined as

(38)

Thus the quantized output, , takes values from the set . Henceforth, we refer to the above system model as quantized MAC (QMAC).

Iii-B Achievable Rate Region of QMAC

In this subsection, we derive analytical expressions for the rate region of a two-user QMAC. From the Fig. 4, we observe that the effective multiple-access channel after receiver quantization, , is a discrete memoryless channel (DMC) with the transition probabilities derived and given in (44). Let and represent the rates achieved by user 1 and user 2, respectively. Since QMAC is a discrete memoryless multiple-access channel, the achievable rate region [9] is the set of all rate pairs satisfying

(39)
(40)
(41)

The mutual information , are given by

(42)
(43)

where the entropies in (42) and (43) are calculated using the probability distribution function

(44)

where , and , and refer to the th element of sets , and , respectively. , and , are the real and imaginary parts of and , respectively.

The region is defined as

(45)

and . From (44), the probability distributions and are calculated as

(46)
(47)

On substituting (44), (46), (47) into (42) and (43), and can be computed. By symmetry, and can be computed in a similar manner. The final expressions for the achievable rate pairs are then given by (48), (49), and (50), presented on the top of the next page.

(48)
(49)