Degrees of Freedom of Interference Channels with Hybrid Beam-forming

Degrees of Freedom of Interference Channels with Hybrid Beam-forming

Sung Ho Chae,  and Cheol Jeong,  The authors are with the DMC R&D Center, Samsung Electronics, Suwon, Republic of Korea.
Abstract

We study the sum degrees of freedom (DoF) of interference channels with hybrid beam-forming in which each transmitter uses antennas and RF chains and each receiver uses antennas and RF chains, where and , , and hybrid beam-forming composed of analog and digital precodings is employed at each node. For the two-user case, we completely characterize the sum DoF for an arbitrary number of antennas and RF chains by developing an achievable scheme optimized for the hybrid beam-forming structure and deriving its matching upper bound. For a general -user case, we focus on a symmetric case where , , , and , , and obtain lower and upper bounds on the sum DoF, which are tight when is an integer. The results show that hybrid beam-forming can increase the sum DoF of interference channel under certain conditions while it cannot improve the sum DoFs of point-to-point channel, multiple access channel, and broadcast channel. The key insights on this gain is that hybrid beam-forming enables users to manage inter-user interference better, and thus each user can increase the dimension of interference-free signal space for its own desired signals.

Degrees of freedom, hybrid beam-forming, interference alignment, interference channel

I Introduction

Mobile data traffic has been growing dramatically as the number of mobile smart devices is increasing rapidly in recent years [1]. To accommodate tremendous demand on mobile data traffic, the cell capacity can be largely increased by deploying a very large number of antennas at base stations (BSs), often referred to as a massive multiple-input multiple-output (MIMO) system [2, 3]. The massive MIMO system, however, has hardware constrains that come from using a few hundred antennas. For a conventional antenna array structure, each antenna needs to have a dedicated RF chain. This naturally leads to an increment in the circuit size, power consumption, and device cost proportionally to the number of antennas, and hence it can be a serious problem in a practical point of view especially for massive MIMO systems. Therefore, to resolve this problem, a hybrid beam-forming structure with a lower number of RF chains than the number of antenna elements has been recently introduced as a practical solution [4, 5].

As an alternative approach to increase the cell capacity, millimeter-wave (mmWave) communications have attracted great attention recently [6]. The mmWave band from 30 to 300 GHz provides abundant contiguous frequency resources while frequency bands under 5 GHz used for legacy cellular communications are very crowded and fragmented. The main advantage in mmWave communications is that a very high data rate can be supported using a very large bandwidth at mmWave bands. However, one of major drawbacks is the high induced path loss due to the propagation loss and absorption loss at mmWave bands [7]. Fortunately, this high path loss can be effectively compensated by a high beam-forming gain obtained from a large number of antenna elements that can be packed into a small form factor due to the small wavelength in mmWave bands. To support a single stream only, the analog beam-forming, which is simply implemented by controlling attenuators and phase shifters of the antenna array to steer a directional beam, is enough to be considered. However, to transmit multiple streams, the hybrid beam-forming structure, where analog beam-forming is performed at RF domain and antenna arrays are connected to a relatively small number of digital paths, should be considered to get the multiplexing gain [8, 9].

As mentioned above, the hybrid beam-forming architecture can play a key role in the next generation communications (e.g., massive MIMO and/or mmWave communications) and hence has been widely studied recently [8, 9, 10, 11, 12]. In [8], precoders and combiners are designed using a sparse reconstruction approach. In [10], baseband and RF beams are designed for multiuser downlink spatial division multiple access (SDMA). In addition, a hybrid precoding algorithm based on a hierarchical codebook is proposed in [11]. Furthermore, a hybrid precoder is proposed for massive multiuser MIMO systems in [12]. While there are some works on hybrid beam-forming structures, however, to the best of our knowledge, the degrees of freedom (DoF) gain from hybrid beam-forming has not been analyzed before.

I-a Previous Works

The DoF, which is also known as a capacity pre-log, gives the capacity approximation at high signal to noise ratio (SNR) regime. For example, for the point-to-point (PTP) channel with transmit antennas and receive antennas, it is well known that the capacity increases with the growth rate at high SNR [13, 14]. Since exact capacity characterization is generally still unknown even for simple networks (e.g., two-user interference channel), instead of obtaining an exact capacity, approximate characterization by finding the optimal DoF has been studied in many networks recently [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31].

Specifically, for the two-user interference channel, the sum DoF has been completely characterized, where zero-forcing precoding has been shown to be enough to achieve the optimal DoF [15]. For a general -user interference channel, a novel interference management technique called interference alignment has been proposed in [16, 19], which achieves the optimal sum DoF of . Later this scheme has been extended to MIMO configurations both for rich scattering environment [23, 22] and poor scattering environment [29, 30, 31]. Furthermore, beyond the interference channels, the idea of interference alignment has been successfully adapted to various networks, e.g., see [17, 18, 19, 20, 21, 24, 25, 26, 27, 28] and references therein.

I-B Contributions

In this paper, our primary goal is to answer if hybrid beam-forming can increase the sum DoF of interference channels. To this end, motivated by the aforementioned previous works, we propose zero forcing and interference alignment schemes optimized for the hybrid beam-forming structure. In addition, we also derive a new upper bound on the sum DoF when hybrid beam-forming is employed at each node. For the two-user case, this upper bound coincides with the achievable sum DoF of the proposed scheme, thereby completely characterizing the sum DoF. For a general -user case, our proposed scheme can achieve the upper bound when is an integer, where and denote the number of antennas at each transmitter and receiver, respectively. As a consequence of the result, we show that hybrid beam-forming can indeed improve the sum DoF of the -user interference channel under certain conditions. This is in contrast to the PTP channels, multiple access channel (MAC), and broadcast channel (BC) cases in which hybrid beam-forming cannot increase the sum DoF (see Section III). The key insight behind this gain is that hybrid beam-forming enables users to manage interference better, and thus each user can increase the dimension of interference-free signal space which can be used for its own desired signals.

I-C Organization

The rest of this paper is organized as follows. In Section II, we describe the system model and sum DoF metric considered in this paper. In Section III, we give an intuition as to how hybrid beam-forming can increase the sum DoF through motivating examples. In Section IV, we present and discuss about the main results of this paper. In addition, we provide numerical results which show the performance improvement from hybrid beam-forming in Section V. In Sections VI and VII, we provide the proofs of the main theorems. Finally, we conclude the paper in Section VIII.

I-D Notations

Throughout the paper, we will use , , and to denote a matrix, a vector, and a scalar, respectively. For a rational number , the notation denotes the integer part of . For matrix , let , , and denote the transpose, the complex conjugate transpose, and the norm of , respectively. In addition, let and denote the determinant and the rank of , respectively. The notations and denote the identity matrix and zero matrix, respectively. We write if .

Ii System Model

Consider a -user interference channel with hybrid beam-forming, as shown in Fig. 1. Transmitter wishes to communicate with receiver only, while causing interference to all the other receivers. In addition, transmitter uses antennas and RF chains and receiver uses antennas and RF chains, where and , . Specially, when and , , we call the corresponding channel as a full digital channel.

Fig. 1: The -user interference channel with hybrid beam-forming

Ii-a Channel Model

Similar to previous works [8, 10], in this paper we assume that transmitter utilizes transmit hybrid beam-forming which consists of an analog precoder and an digital precoder as depicted in Fig. 1, where denotes the number of streams of user .111As compared to the hybrid beam-forming structure introduced in [8, 10], in this paper, coefficients in can have different norms by relaxing the constraint that all entries are of equal norm. In practical point of view, this is feasible since we can implement by using both attenuators and analog phase shifters rather than using analog phase shifters only. In addition, based on this hybrid beam-forming, the input signal of transmitter at time slot , , is assumed to be given by

where is the baseband-domain input vector and

is the symbol vector of transmitter . Here, denotes the th symbol of user at time slot . Then the input and output relationship at RF domain is given by

where is the channel matrix from transmitter to receiver , is the RF-domain received signal vector at receiver , and is the Gaussian noise vector at receiver whose entries are drawn from . We assume that all channel coefficients are independent and identically distributed (i.i.d) from a continuous distribution and known to all nodes.

After receiving , receiver applies receive hybrid beam-forming which consists of an analog precoder and a digital precoder as shown in Fig. 1. Specifically, by applying the analog precoder to the received signal at RF domain, we can obtain the input and output relationship at baseband domain as

where , , and . If we further apply the digital precoder to the received signal at baseband domain, we finally get

where , , and . Note that is the effective channel matrix which can be obtained after applying transmit hybrid beam-forming of transmitter and receive hybrid beam-forming of receiver .

Finally, by applying the aforementioned hybrid beam-forming strategy and assuming Gaussian signaling , the following average sum rate is achievable for a given transmit power  [32]:

(1)

where . Specifically, when all the interferences are eliminated via hybrid beam-forming, i.e., , and , (1) becomes

(2)
(3)

Ii-B Encoding, Decoding, and Sum DoF

There are independent messages . For each transmitter , a message is mapped to an length codeword . To send the message , at time , transmitter sends . Here, we assume that each transmitter should satisfy the average power constraint , i.e., for . Then receiver decodes its desired message , based on its received signal.

A rate tuple is said to be achievable for the channel if there exists a sequence of codes such that the average probability of decoding error tends to zero as the code length goes to infinity. The capacity region of this channel is the closure of the set of achievable rate tuples . The sum DoF , which is also known as a sum-capacity pre-log, provides the sum capacity approximation at high SNR as222In this paper, when we derive lower and upper bounds on the sum DoF, we restrict our attention on the cases in which the hybrid beam-forming structure introduced in Section II-A is used.

Equivalently, the sum DoF can be defined as .

Iii Preliminary Discussion

To gain insights into the DoF gain from hybrid beam-forming, we begin with examining PTP channel, MAC, and BC cases. Note that the PTP channel, the -user MAC, and the -user BC can be obtained from the -user interference channel by allowing full cooperation among all the transmitters and among all the receivers, full cooperation among all the receivers only, and full cooperation among all the transmitters only, respectively. Here, we assume that hybrid beam-forming strategy (including digital precoder and analog precoder) for each channel is employed in a similar manner as in Section II.

Iii-a Point-to-Point (PTP) Channel

Consider the PTP channel in which the transmitter uses RF chains and antennas and the receiver uses RF chains and antennas. The DoF of this channel is stated in the following lemma.

Lemma 1

For the PTP channel with hybrid beam-forming, the DoF is given by .

{proof}

We first provide a converse proof. Following a similar way described in Section II, we can write the input and output relationship of the PTP channel at time slot as

where is the RF-domain output vector at the receiver, is the channel matrix from the transmitter to the receiver, and are the RF-domain input vector and the baseband-domain input vector at the transmitter, respectively, is the analog precoder of the transmitter, and is the Gaussian noise vector.

Now focus on the input and output relationship at baseband domain. By applying receive analog precoding at the receiver, we can get

where is the analog precoder of the receiver, , and . Since , we see that .

For achievability, we only use transmit antennas out of antennas of the transmitter and receive antennas out of antennas of the receiver to equivalently create a conventional full digital PTP channel with transmit antennas and receive antennas. Therefore, is achievable [13, 14], which completes the proof.

It is well known that the DoF of the full digital PTP channel with transmit antennas and receive antennas is equal to  [13, 14]. Therefore, from the result of Lemma 1, we see that adding more antennas only cannot increase the DoF of a PTP channel without increasing the number of RF chains, regardless of the values of and .

Iii-B Multiple Access Channel (MAC) and Broadcast Channel (BC)

Now we consider the -user MAC and BC with hybrid beam-forming. For the MAC case, each transmitter uses RF chains and antennas and the receiver uses RF chains and antennas. For the BC case, the transmitter uses RF chains and antennas and each receiver uses RF chains and antennas. The DoFs of these channels are stated in the following lemmas.

Lemma 2

For the -user multiple access channel (MAC) with hybrid beam-forming, the DoF is given by .

{proof}

For a converse proof, we allow full cooperation among all the transmitters to form PTP channel. Then, from the result of Lemma 1, the sum DoF of this network is equal to . Since allowing cooperation does not reduce the capacity region [17], this is an upper bound of the original network, and thus

For achievability, we use only antennas out of antennas of transmitter , , and antennas out of antennas of the receiver to form a conventional full digital MAC in a similar manner as in Lemma 1. Then, is achievable [15], which completes the proof.

Lemma 3

For the -user broadcast channel (BC) with hybrid beam-forming, the DoF is given by .

{proof}

We can easily prove Lemma 3 by following similar proof steps in Lemma 2 except the fact that we now allow full cooperation among all the receivers instead of transmitters for a converse proof. For brevity, we omit the rest of the proof steps.

From the results of Lemmas 2 and 3, adding more antennas only without more RF chains cannot increase the sum DoFs of MAC and BC, as in the PTP case. Therefore, we can see that when full cooperation is already allowed at either transmitter side or receiver side of the -user interference channel, hybrid beam-forming cannot further improve the DoF. However, as we will show in the following example, for the case in which full cooperation is not allowed so that there exist inter-user interferences, the sum DoF of an interference channel can be improved via hybrid beam-forming for certain cases.

Iii-C Interference Channel: Motivating Example

Now we provide a simple example where hybrid beam-forming indeed improves the sum DoF. In the following example, we omit the time index for brevity.

Example 1

Consider the two-user interference channel where and , . We first set the analog precoder to satisfy for and . Since is the matrix and all channel coefficients are generic, we can easily find that satisfies these conditions. In addition, for the digital precoder of transmitter , we set , . Then, the received signal at each receiver is given by

where is the transmitted symbol vector of user and . Since , we can achieve for each user, thus achieving . Note that for the two-user full digital interference channel, which has the same number of RF chains as in the two-user interference channel, only the sum DoF of two can be achieved. This shows that for some cases, the sum DoF of an interference channel can actually be increased by adding more antennas only without increasing the number of RF chains.

Remark 1

As shown in Example 1, by using more antennas, we can have a better ability to null out interferences from/to other users at RF domain. This enables users to secure more interference-free dimensions, and as a result, a higher sum DoF is achievable without any additional RF chains for some cases. However, despite this improved capability dealing with interferences, hybrid beam-forming does not always increase the DoF of an interference channel. For instance, as will be demonstrated in the next example, if all the interferences can be eliminated without the need to add more antennas, hybrid beam-forming cannot increase the sum DoF.

Example 2

Consider the two-user interference channel where , , , and , . By allowing full cooperation among transmitters and among receivers, we can get the PTP channel. Since the DoF of this channel is given by two from Lemma 1 and allowing full cooperation does not reduce the capacity region, the sum DoF of the two-user interference channel cannot be more than two. Note that the two-user full digital interference channel can also achieve the sum DoF of two [15]. Therefore, unlike in Example 1, adding antennas only cannot increase the sum DoF in this case. In fact, in this case, to achieve a higher DoF, we need to use more RF chains as well as more antennas. For example, if we use additional one RF chain and two RF chains at each transmitter and receiver, respectively, the channel becomes the two-user full digital interference channel, and we can now achieve the improved DoF of 4.

Iv Main Results

In this section, we state and discuss about the main results of this paper. For the two-user case, the sum DoF is completely characterized for any antenna configuration. When , we focus on a symmetric case where , , , and , , and derive lower and upper bounds on the sum DoF. It is shown that two bounds are matched under a certain condition.

Iv-a Two-user Case

For the two-user interference channel, we completely characterize the sum DoF as stated in the following theorem.

Theorem 1 (Two-user case)

For the two-user interference channel with hybrid beam-forming, the sum DoF is given by

where and for .

{proof}

See Section VI for the proof.

Remark 2

For the case where and , , the sum DoF becomes

which recovers the result for the two-user full digital interference channel in [15].

Fig. 2: Sum DoFs of the two-user case with respect to when and .
Remark 3

Note that when the condition

is satisfied, the sum DoF becomes

which is the sum DoF of the interference-free channel. Therefore, we can see that by adding enough number of antennas at each node, all the users can utilize their full DoFs as if there is no interference.

DoF gain due to hybrid beam-forming: Consider a symmetric case where and . We plot the sum DoF as a function of with fixed in Fig. 2. For comparison, we also plot the sum DoF of the full digital case where the number of RF chains is the same as the hybrid beam-forming case. As can be seen in Fig. 2, although we add antennas only, we can achieve a higher DoF and it reaches up to the maximum value of , the sum DoF of the interference-free channel, when . The gain comes from the fact that hybrid beam-forming can null out more interferences without increasing the number of RF chains, as well as enhancing the capacity of PTP channel as reported in [8, 10].

Iv-B -user Case

When , we focus on a symmetric case where , , , and , . Under this configuration, we obtain lower and upper bounds on the sum DoF as stated in the following theorem, which are tight when is an integer.

Theorem 2 (-user case)

For the symmetric -user interference channel with hybrid beam-forming, the following sum DoF is achievable:

where . For converse, the sum DoF is upper bounded by

{proof}

See Section VII for the proof.

Remark 4

Similar to the two-user case explained in Remark 2, for the case where and , Theorem 2 recovers the result for the -user full digital interference channel in [23].

Remark 5

It is easy to see that is a non-decreasing function of and . Intuitively, this is clear since having more antennas does not reduce the capacity region. Moreover, when each user can achieve the maximum DoF of as if there is no interference.

Corollary 1

By employing hybrid beam-forming, we can get at most two-fold DoF gain as compared to the full digital case in which the number of RF chains is the same as the hybrid beam-forming case. {proof} Let and denote the sum DoFs with hybrid beam-forming and full digital structures, respectively. For the two-user case, we have

In addition, for the general -user case, we have

where . This completes the proof.

Fig. 3: Sum DoFs of the three-user cases with respect to when .

DoF gain due to hybrid beam-forming: Consider the three-user case where . First, we set and plot the sum DoF as a function of with fixed and in Fig. 3. In addition, we consider another scenario in which additional antennas are employed only at transmitters, i.e., , and again plot the sum DoF as a function of . As can be seen in the figure, by using hybrid beam-forming, we can achieve a higher DoF and interestingly, it can reach up to the maximum DoF of six even when hybrid beam-forming is applied at transmitters only. Furthermore, note that when achieving this DoF, interference alignment combined with hybrid beam-forming is employed. From this point, we can see that hybrid beam-forming can provide an improved capability not only nulling out interferences but also aligning interferences at RF domain.

Fig. 4: Sum DoFs with respect to when .

Now, we examine a tendency of the sum DoF with respect to with the fixed number of antennas and RF chains at each node. Specifically, we set and plot the sum DoFs when and in Fig. 4. For comparison, we also plot the sum DoF of the full digital case where the number of RF chains is the same as the hybrid beam-forming case. From Fig. 4, we see that hybrid beam-forming can improve the sum DoF for all values of , and moreover, the slope also increases as the number of additional antennas at each receiver increases.

V Numerical Simulation

In this section, we numerically evaluate the average sum rate performance of the proposed hybrid beam-forming schemes for and cases to show that the sum DoFs stated in Theorems 1 and 2 are indeed achievable. For comparison, the sum DoFs of the full digital and the interference-free cases are also plotted. Here, we assume Rayleigh fading environment where each channel coefficient is drawn i.i.d from . In addition, we assume that all the noise power is normalized to unity and thus . Furthermore, to clearly capture the sum DoFs from the sum-rate graphs, we plot the average sum rates as a function of .

Fig. 5: Average sum rates of the two-user case when and .

V-a Average Sum Rate for the Two-user Case

In Fig. 5, the average sum rates are plotted as a function of , where and . Note that the sum DoFs can be observed from the slopes in the figure. We can see that the sum DoFs obtained by the simulation are well matched with the sum DoFs stated in Lemma 1 and Theorem 1. Here, when the simulation is performed, the number of streams of hybrid beam-forming for each user is set by and for and for by following Theorem 1.

As shown in the figure, the full digital scheme can only achieve the sum DoF of two, while the sum DoF of the interference-free channel is four. When hybrid beam-forming is employed, we can see by simulation that the sum DoF can be improved and even reach up to the interference-free DoF, as shown in Theorem 1, and therefore the performance gap between hybrid beam-forming and full digital cases dramatically increases as the SNR increases.

Fig. 6: Average sum rates of the three-user case when .

V-B Average Sum Rate for the Three-user Case

As in the previous subsection, the average sum rate is plotted as a function of in Fig. 6, where . When hybrid beam-forming is used, we consider the two different scenarios in which additional antennas are employed only at transmitters, i.e., for and 6, and additional antennas are employed both at transmitters and receivers, i.e., for and 6. Here, we adopt the distributed interference alignment333Note that the achievable scheme proposed in Theorem 2 requires an arbitrary large number of symbol extension. Therefore, in this subsection, instead of adopting the achievable scheme in Theorem 2 directly, we employ the DIA algorithm to numerically show that the sum DoF stated in Theorem 2 is indeed feasible. Here, Theorem 2 provides theoretical guidance when selecting a suitable number of streams for each user. (DIA) algorithm proposed in [33] for numerical simulation and the number of streams of hybrid beam-forming used for the simulation is given by Theorem 2. The slopes in the figure show that the sum DoFs stated in Theorem 2 is indeed achievable.

The full digital scheme can only achieve the sum DoF of three, while the sum DoF of the interference-free channel is six as shown in the figure. As in the two-user case, the sum DoF of the full digital scheme is only half of that of the interference-free channel. When and , the hybrid beam-forming can achieve the maximum sum DoF of six as if there is no interference between users. Interestingly, for the case in which additional antennas are employed only at transmitters , the sum DoF can also be increased as compared to the full digital case, and the performance gain over the full digital case increases as the number of additional antennas increases.

Vi Proof of Theorem 1

Vi-a Achievability

In our achievable scheme, we will use only transmit RF chains out of RF chains of transmitter and receive RF chains out of RF chains of receiver , for all . Hence, from now on, we can equivalently consider the interference channel instead of the original channel, the interference channel. In addition, since our achievable scheme operates in a single time slot, we omit the time index for brevity.

We design the input signal of transmitter as

where is the transmit analog precoder, is the transmit digital precoder, and is the vector of transmitted Gaussian symbols of user . To be specific, beam-forming vectors in can be decomposed into two parts:

  • denotes the th beam-forming vector in such that and , where . Note that since the size of is given by and channel matrices are drawn i.i.d from a continuous distribution, the maximum number of linearly independent beam-forming vectors satisfying this condition is . Let denote the number of such vectors.

  • denotes the th beam-forming vector in whose coefficients are randomly generated from a continuous distribution and , where has a finite value. Hence, and for with probability one. Let denote the number of such vectors. In addition, we further restrict and to satisfy .

In summary, we choose , , , , , and to satisfy the following conditions.

(4)
(5)
(6)
(7)
(8)
(9)

Then the received signal at receiver at RF domain is given by

(10)

where (10) is due to the properties of and .

Now we explain the beam-forming matrix at receiver . Denote as the receive analog precoder and as the receive digital precoder. We set such that and . Since we have

we can find satisfying these conditions. Therefore, after applying receive analog precoding, we obtain

Recall that . Now, we set and as the left and right singular matrices of the matrix , respectively. Then we get parallel AWGN channels for user after applying the receive digital precoding as follows:

where is the diagonal matrix with the singular values of on the diagonal and . Therefore, we can see that each user achieves DoF via the proposed scheme, and thus the achievable total DoF is given by .

Finally, by evaluating the conditions (4)–(9) using the Fourier-Motzkin elimination, we get the desired bound: