Degrees of Freedom of Interference Channels with Hybrid Beamforming
Abstract
We study the sum degrees of freedom (DoF) of interference channels with hybrid beamforming in which each transmitter uses antennas and RF chains and each receiver uses antennas and RF chains, where and , , and hybrid beamforming composed of analog and digital precodings is employed at each node. For the twouser 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 beamforming 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 beamforming can increase the sum DoF of interference channel under certain conditions while it cannot improve the sum DoFs of pointtopoint channel, multiple access channel, and broadcast channel. The key insights on this gain is that hybrid beamforming enables users to manage interuser interference better, and thus each user can increase the dimension of interferencefree signal space for its own desired signals.
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 multipleinput multipleoutput (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 beamforming 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, millimeterwave (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 beamforming 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 beamforming, 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 beamforming structure, where analog beamforming 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 beamforming 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 beamforming structures, however, to the best of our knowledge, the degrees of freedom (DoF) gain from hybrid beamforming has not been analyzed before.
Ia Previous Works
The DoF, which is also known as a capacity prelog, gives the capacity approximation at high signal to noise ratio (SNR) regime. For example, for the pointtopoint (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., twouser 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 twouser interference channel, the sum DoF has been completely characterized, where zeroforcing 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.
IB Contributions
In this paper, our primary goal is to answer if hybrid beamforming 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 beamforming structure. In addition, we also derive a new upper bound on the sum DoF when hybrid beamforming is employed at each node. For the twouser 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 beamforming 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 beamforming cannot increase the sum DoF (see Section III). The key insight behind this gain is that hybrid beamforming enables users to manage interference better, and thus each user can increase the dimension of interferencefree signal space which can be used for its own desired signals.
IC 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 beamforming 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 beamforming in Section V. In Sections VI and VII, we provide the proofs of the main theorems. Finally, we conclude the paper in Section VIII.
ID 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 beamforming, 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.
Iia Channel Model
Similar to previous works [8, 10], in this paper we assume that transmitter utilizes transmit hybrid beamforming which consists of an analog precoder and an digital precoder as depicted in Fig. 1, where denotes the number of streams of user .^{1}^{1}1As compared to the hybrid beamforming 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 beamforming, the input signal of transmitter at time slot , , is assumed to be given by
where is the basebanddomain 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 RFdomain 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 beamforming 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 beamforming of transmitter and receive hybrid beamforming of receiver .
Finally, by applying the aforementioned hybrid beamforming 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 beamforming, i.e., , and , (1) becomes
(2)  
(3) 
IiB 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 sumcapacity prelog, provides the sum capacity approximation at high SNR as^{2}^{2}2In this paper, when we derive lower and upper bounds on the sum DoF, we restrict our attention on the cases in which the hybrid beamforming structure introduced in Section IIA is used.
Equivalently, the sum DoF can be defined as .
Iii Preliminary Discussion
To gain insights into the DoF gain from hybrid beamforming, 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 beamforming strategy (including digital precoder and analog precoder) for each channel is employed in a similar manner as in Section II.
Iiia PointtoPoint (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 beamforming, the DoF is given by .
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 RFdomain output vector at the receiver, is the channel matrix from the transmitter to the receiver, and are the RFdomain input vector and the basebanddomain 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 .
IiiB Multiple Access Channel (MAC) and Broadcast Channel (BC)
Now we consider the user MAC and BC with hybrid beamforming. 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 beamforming, the DoF is given by .
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 beamforming, the DoF is given by .
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 beamforming 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 interuser interferences, the sum DoF of an interference channel can be improved via hybrid beamforming for certain cases.
IiiC Interference Channel: Motivating Example
Now we provide a simple example where hybrid beamforming indeed improves the sum DoF. In the following example, we omit the time index for brevity.
Example 1
Consider the twouser 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 twouser full digital interference channel, which has the same number of RF chains as in the twouser 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 interferencefree 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 beamforming 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 beamforming cannot increase the sum DoF.
Example 2
Consider the twouser 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 twouser interference channel cannot be more than two. Note that the twouser 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 twouser 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 twouser 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.
Iva Twouser Case
For the twouser interference channel, we completely characterize the sum DoF as stated in the following theorem.
Theorem 1 (Twouser case)
For the twouser interference channel with hybrid beamforming, the sum DoF is given by
where and for .
See Section VI for the proof.
Remark 2
For the case where and , , the sum DoF becomes
which recovers the result for the twouser full digital interference channel in [15].
Remark 3
Note that when the condition
is satisfied, the sum DoF becomes
which is the sum DoF of the interferencefree 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 beamforming: 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 beamforming 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 interferencefree channel, when . The gain comes from the fact that hybrid beamforming 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].
IvB 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 beamforming, the following sum DoF is achievable:
where . For converse, the sum DoF is upper bounded by
See Section VII for the proof.
Remark 4
Similar to the twouser 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 nondecreasing 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 beamforming, we can get at most twofold DoF gain as compared to the full digital case in which the number of RF chains is the same as the hybrid beamforming case. {proof} Let and denote the sum DoFs with hybrid beamforming and full digital structures, respectively. For the twouser case, we have
In addition, for the general user case, we have
where . This completes the proof.
DoF gain due to hybrid beamforming: Consider the threeuser 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 beamforming, we can achieve a higher DoF and interestingly, it can reach up to the maximum DoF of six even when hybrid beamforming is applied at transmitters only. Furthermore, note that when achieving this DoF, interference alignment combined with hybrid beamforming is employed. From this point, we can see that hybrid beamforming can provide an improved capability not only nulling out interferences but also aligning interferences at RF domain.
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 beamforming case. From Fig. 4, we see that hybrid beamforming 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 beamforming 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 interferencefree 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 sumrate graphs, we plot the average sum rates as a function of .
Va Average Sum Rate for the Twouser 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 beamforming 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 interferencefree channel is four. When hybrid beamforming is employed, we can see by simulation that the sum DoF can be improved and even reach up to the interferencefree DoF, as shown in Theorem 1, and therefore the performance gap between hybrid beamforming and full digital cases dramatically increases as the SNR increases.
VB Average Sum Rate for the Threeuser Case
As in the previous subsection, the average sum rate is plotted as a function of in Fig. 6, where . When hybrid beamforming 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 alignment^{3}^{3}3Note 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 beamforming 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 interferencefree channel is six as shown in the figure. As in the twouser case, the sum DoF of the full digital scheme is only half of that of the interferencefree channel. When and , the hybrid beamforming 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
Via 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, beamforming vectors in can be decomposed into two parts:

denotes the th beamforming 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 beamforming vectors satisfying this condition is . Let denote the number of such vectors.

denotes the th beamforming 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 beamforming 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 FourierMotzkin elimination, we get the desired bound: