12848178
Optimality of Simple Layered Superposition Coding in the 3 User MISO BC With Finite Precision CSIT
We study the <inline-formula> <tex-math notation="LaTeX">$K=3$ </tex-math></inline-formula> user multiple input single output (MISO) broadcast channel (BC) with <inline-formula> <tex-math notation="LaTeX">$M=3$ </tex-math></inline-formula> antennas at the transmitter and 1 antenna at each receiver, from the generalized degrees of freedom (GDoF) perspective, under the assumption that the channel state information at the transmitter (CSIT) is limited to the finite precision. In particular, our goal is to identify a parameter regime where a simple layered superposition (SLS) coding scheme achieves the entire GDoF region. With <inline-formula> <tex-math notation="LaTeX">$\alpha _{ij}$ </tex-math></inline-formula> representing the channel strength parameter for the link from the <inline-formula> <tex-math notation="LaTeX">$j^{th}$ </tex-math></inline-formula> antenna of the transmitter to the <inline-formula> <tex-math notation="LaTeX">$i^{th}$ </tex-math></inline-formula> receiver, we prove that the SLS is GDoF optimal without the need for time-sharing if <inline-formula> <tex-math notation="LaTeX">$\max (\alpha _{ki},\alpha _{im})\leq \alpha _{ii}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\alpha _{ki}+\alpha _{im}\le \alpha _{ii}+\alpha _{km}$ </tex-math></inline-formula> for all <inline-formula> <tex-math notation="LaTeX">$i,k\in [{3}],m\in [M]$ </tex-math></inline-formula>. The GDoF region under this condition is a convex polyhedron. The result generalizes to arbitrary <inline-formula> <tex-math notation="LaTeX">$M\geq 3$ </tex-math></inline-formula>.
1558248
On the Capacity of Fading MIMO Broadcast Channels with Imperfect Transmitter Side-Information
A fading broadcast channel is considered where the transmitter employs two antennas and each of the two receivers employs a single receive antenna. It is demonstrated that even if the realization of the fading is precisely known to the receivers, the high signal-to-noise (SNR) throughput is greatly reduced if, rather than knowing the fading realization \emph{precisely}, the trasmitter only knows the fading realization \emph{approximately}. The results are general and are not limited to memoryless Gaussian fading.
6520431
DoF analysis of the K-user MISO broadcast channel with hybrid CSIT
We consider a K-user multiple-input single-output (MISO) broadcast channel (BC) where the channel state information (CSI) of user i(i = 1, 2, ..., K) may be either instantaneously perfect (P), delayed (D) or not known (N) at the transmitter with probabilities λ<sub>P</sub><sup>i</sup>, λ<sub>D</sub><sup>i</sup> and λ<sub>N</sub><sup>i</sup>, respectively. In this setting, according to the three possible CSIT for each user, knowledge of the joint CSIT of the K users could have at most 3<sup>K</sup> states. Although the results by Tandon et al. show that for the symmetric two user MISO BC (i.e., λ<sub>Q</sub><sup>i</sup> = λ<sub>Q</sub>, ∀<sub>i</sub> ∈ {1, 2}, Q ∈ {P, D, N}), the Degrees of Freedom (DoF) region depends only on the marginal probabilities, we show that this interesting result does not hold in general when K ≥ 3. In other words, the DoF region is a function of all the joint probabilities. In this paper, given the marginal probabilities of CSIT, we derive an outer bound for the DoF region of the K-user MISO BC. Subsequently, we investigate the achievability of the outer bound in some scenarios. Finally, we show the dependence of the DoF region on the joint probabilities.
6148489
On the Degrees of Freedom of Finite State Compound Wireless Networks
We explore the degrees of freedom (DoF) of three classes of finite state compound wireless networks in this paper. First, we study the multiple-input single-output (MISO) finite state compound broadcast channel (BC) with arbitrary number of users and antennas at the transmitter. In prior work, Weingarten have found inner and outer bounds on the DoF with 2 users. The bounds have a different character. While the inner bound collapses to unity as the number of states increases, the outer bound does not diminish with the increasing number of states beyond a threshold value. It has been conjectured that the outer bound is loose and the inner bound represents the actual DoF. In the complex setting (all signals, noise, and channel coefficients are complex variables), we solve a few cases to find that the outer bound - and not the inner bound - of Weingarten is tight. For the real setting (all signals, noise, and channel coefficients are real variables), we completely characterize the DoF, once again proving that the outer bound of Weingarten is tight. We also extend the results to arbitrary number of users. Second, we characterize the DoF of finite state scalar (single antenna nodes) compound X networks with arbitrary number of users in the real setting. Third, we characterize the DoF of finite state scalar compound interference networks with arbitrary number of users in both the real and complex setting. The key finding is that scalar interference networks and (real) X networks do not lose any DoF due to channel uncertainty at the transmitter in the finite state compound setting. The finite state compound MISO BC does lose DoF relative to the perfect CSIT scenario. However, what is lost is only the DoF benefit of joint processing at transmit antennas, without which the MISO BC reduces to an X network.
2136825
On the degrees of freedom of the compound MISO broadcast channels with finite states
We consider compound multiple-antenna broadcast channels with M transmit antennas and K single-antenna receivers, where the channel of receiver r takes one of Jr complex vectors. We show that for any finite Jr, the degrees of freedom (DoF) of MK over M+K−1 is achievable. It is in contrary to the commonly believed conjecture that the DoF collapses to one for large Jr's. We also establish the optimality of this result, where Jr ≥ M, r = 1, …, K. The achievable scheme relies on using a number theoretic approach of interference alignment, and ignoring the possibility of cooperation among the transmit antennas. Moreover, we show that if Jr's for some of the users are relatively small, then both transmit cooperation and interference alignment are needed. Unlike the conventional schemes, here the transmit precoders are not full rank.
11059919
On the Synergistic Benefits of Alternating CSIT for the MISO BC
The degrees of freedom (DoF) of the two-user multiple-input single-output (MISO) broadcast channel (BC) are studied under the assumption that the form, Ii;i = 1; 2; of the channel state information at the transmitter (CSIT) for each user’s channel can be either perfect (P ), delayed (D) or not available (N), i.e., I1;I2 2 fP;N;Dg, and therefore the overall CSIT can alternate between the 9 resulting states I1I2. The fraction of time associated with CSIT state I1I2 is denoted by the parameter I1I2 and it is assumed throughout that I1I2 = I2I1, i.e., PN = NP; PD = DP; DN = ND. Under this assumption of symmetry, the main contribution of this paper is a complete characterization of the DoF region of the two user MISO BC with alternating CSIT. Surprisingly, the DoF region is found to depend only on the marginal probabilities ( P; D; N ) = P I2 PI2; P I2 DI2; P I2 NI2 , I22 fP;D;Ng, which represent
16316364
Blind Interference Alignment
The main contribution of this paper is the insight that the transmitters' knowledge of channel coherence intervals alone (without any knowledge of the values of channel coefficients) can be surprisingly useful in a multiuser setting, illustrated by the idea of blind interference alignment that is introduced in this work. Specifically, we explore five network communication problems where the possibility of interference alignment, and consequently the total number of degrees of freedom (DoF) with channel uncertainty at the transmitters, are unknown. These problems share the common property that in each case the best known outer bounds are essentially robust to channel uncertainty and represent the outcome with interference alignment, but the best inner bounds-in some cases conjectured to be optimal-predict a total collapse of DoF, thus indicating the infeasibility of interference alignment under channel uncertainty at transmitters. For each of these settings we show that even with no knowledge of channel coefficient values at the transmitters, under certain heterogeneous block fading models, i.e., when certain users experience smaller coherence time/bandwidth than others, blind interference alignment can be achieved. In each case we also establish the DoF optimality of the blind interference alignment scheme.
52830856
Aligned Image Sets Under Channel Uncertainty: Settling Conjectures on the Collapse of Degrees of Freedom Under Finite Precision CSIT
A conjecture made by Lapidoth et al. at Allerton 2005 (also an open problem presented at ITA 2006) states that the degrees of freedom (DoF) of a two user broadcast channel, where the transmitter is equipped with two antennas and each user is equipped with one antenna, must collapse under finite precision channel state information at the transmitter (CSIT). That this conjecture, which predates interference alignment, has remained unresolved, is emblematic of a pervasive lack of understanding of the DoF of wireless networks—including interference and $X$ networks—under channel uncertainty at the transmitter(s). In this paper, we prove that the conjecture is true in all non-degenerate settings (e.g., where the probability density function of unknown channel coefficients exists and is bounded). The DoF collapse even when perfect channel knowledge for one user is available to the transmitter. This also settles a related recent conjecture by Tandon et al. The key to our proof is a bound on the number of codewords that can cast the same image (within noise distortion) at the undesired receiver whose channel is subject to finite precision CSIT, while remaining resolvable at the desired receiver whose channel is precisely known by the transmitter. We are also able to generalize the result along two directions. First, if the peak of the probability density function is allowed to scale as $O((\sqrt {P})^\alpha )$ , representing the concentration of probability density (improving CSIT) due to, e.g., quantized feedback at rate $({\alpha }/{2})\log (P)$ , then the DoF is bounded above by $1+\alpha $ , which is also achievable under quantized feedback. Second, we generalize the result to arbitrary number of antennas at the transmitter, arbitrary number of single-antenna users, and complex channels. The generalization directly implies a collapse of DoF to unity under non-degenerate channel uncertainty for the general $K$ -user interference and $M\times N$ user $X$ networks as well.
53041837
GDoF Region of the MISO BC: Bridging the Gap Between Finite Precision and Perfect CSIT
For the <inline-formula> <tex-math notation="LaTeX">$K=2$ </tex-math></inline-formula> user MISO BC, i.e., the wireless broadcast channel where a transmitter equipped with <inline-formula> <tex-math notation="LaTeX">$K=2$ </tex-math></inline-formula> antennas sends independent messages to <inline-formula> <tex-math notation="LaTeX">$K=2$ </tex-math></inline-formula> receivers each of which is equipped with a single antenna, the generalized degrees of freedom (GDoFs) region are characterized for arbitrary channel strength and channel uncertainty levels for each of the channel coefficients. The result is extended to K>2 users under additional restrictions which include the assumption of symmetry.
632321
K-User Symmetric M×N MIMO Interference Channel under Finite Precision CSIT: A GDoF perspective
Generalized Degrees of Freedom (GDoF) are characterized for the symmetric K-user Multiple Input Multiple Output (MIMO) Interference Channel (IC) under the assumption that the channel state information at the transmitters (CSIT) is limited to finite precision. In this symmetric setting, each transmitter is equipped with M antennas, each receiver is equipped with N antennas, each desired channel (i.e., a channel between a transmit antenna and a receive antenna belonging to the same user) has strength ∼ P , while each undesired channel has strength ∼ P, where P is a nominal SNR parameter. The result generalizes a previous GDoF characterization for the SISO setting (M = N = 1) and is enabled by a significant extension of the Aligned Image Sets bound that is broadly useful. GDoF per user take the form of a W -curve with respect to α for fixed values of M and N . Under finite precision CSIT, in spite of the presence of multiple antennas, all the benefits of interference alignment are lost.
35248009
Aligned Image Sets and the Generalized Degrees of Freedom of Symmetric MIMO Interference Channel With Partial CSIT
The generalized degrees of freedom of the two-user symmetric multiple input multiple output interference channel are characterized as a function of the channel strength levels and the level of channel state information at the transmitters. In this symmetric setting, each transmitter is equipped with <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> antennas, each receiver is equipped with <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> antennas, and both cross links have the same strength parameter <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> and the same channel uncertainty parameter <inline-formula> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula>. The main challenge resides in the proof of the outer bound which is accomplished by a generalization of the aligned image sets approach.
7467283
Topological Interference Management Through Index Coding
This paper studies linear interference networks, both wired and wireless, with no channel state information at the transmitters except a coarse knowledge of the end-to-end one-hop topology of the network that only allows a distinction between weak (zero) and significant (nonzero) channels and no further knowledge of the channel coefficients' realizations. The network capacity (wired) and degrees of freedom (DoF) (wireless) are found to be bounded above by the capacity of an index coding problem for which the antidote graph is the complement of the given interference graph. The problems are shown to be equivalent under linear solutions. An interference alignment perspective is then used to translate the existing index coding solutions into the wired network capacity and wireless network DoF solutions, as well as to find new and unified solutions to different classes of all three problems.
15545336
Interference Networks With no CSIT: Impact of Topology
We consider partially connected K -user interference networks, where the transmitters have no knowledge about the channel gain values, but they are aware of network topology. We introduce several linear algebraic and graph theoretic concepts to derive new topology-based outer bounds and inner bounds on the symmetric degrees-of-freedom of these networks. We evaluate our bounds for two classes of networks to demonstrate their tightness for most networks in these classes, quantify the gain of our inner bounds over benchmark interference management strategies, and illustrate the effect of network topology on these gains.
9199308
Partially Connected Interference Networks with No CSIT: Symmetric Degrees of Freedom and Multicast Across Alignment Blocks
We explore the symmetric degrees of freedom (DoF) of partially connected interference networks for the case in which transmitters have no knowledge of their channel states (channel coefficients and coherence times) but are aware of the network topology. In particular, we consider interference networks which have no internal conflicts in their alignment graph. This class of networks can be used to model various interference scenarios in modern wireless systems. A sufficient condition is established for which half DoF can be achieved per message, which coincide with the best known outer bound, thereby establishing the optimal symmetric DoF. For networks in this class that do not satisfy the sufficient condition, a systematic achievable scheme, termed multicast across alignment blocks, is derived and an analytical expression is obtained for the achieved symmetric DoF. The new scheme is based on our insight that an interference alignment-based scheme may not utilize all available signalling dimensions, and the unused signalling dimensions can be further exploited through multicast transmission.
19027946
Sum-set inequalities from aligned image sets: Instruments for robust GDoF bounds
We present sum-set inequalities specialized to the generalized degrees of freedom (GDoF) framework. These are information theoretic lower bounds on the entropy of bounded density linear combinations of discrete, power-limited dependent random variables in terms of the joint entropies of arbitrary linear combinations of new random variables that are obtained by power level partitioning of the original random variables. The bounds are useful instruments to obtain GDoF characterizations for wireless interference networks, especially with multiple antenna nodes, subject to arbitrary channel strength and channel uncertainty levels.
6942867
Optimality of Orthogonal Access for One-Dimensional Convex Cellular Networks
It is shown that a greedy orthogonal access scheme achieves the sum degrees of freedom (DoF) of all one-dimensional (all nodes placed along a straight line) convex cellular networks (where cells are convex regions) when no channel knowledge is available at the transmitters except the knowledge of the network topology. In general, optimality of orthogonal access holds neither for two-dimensional convex cellular networks nor for one-dimensional non-convex cellular networks, thus revealing a fundamental limitation that arises when both one-dimensional and convex properties are simultaneously enforced. The result also establishes the sum capacity of the corresponding class of index coding problems.
3429571
TDMA is Optimal for All-Unicast DoF Region of TIM if and only if Topology is Chordal Bipartite
The main result of this paper is that an orthogonal access scheme, such as time division multiple access achieves the all-unicast degrees of freedom (DoF) region of the topological interference management problem if and only if the network topology graph is chordal bipartite, i.e., every cycle that can contain a chord, does contain a chord. The all-unicast DoF region includes the DoF region for any arbitrary choice of a unicast message set, so e.g., the results of Maleki and Jafar on the optimality of orthogonal access for the sum-DoF of one-dimensional convex networks are recovered as a special case. The result is also established for the corresponding topological representation of the index coding problem.
47283085
On the optimality of treating interference as noise
It is shown that in the K-user interference channel, if for each user the desired signal strength is no less than the sum of the strengths of the strongest interference from this user and the strongest interference to this user (all values in dB scale), then the simple scheme of using point to point Gaussian codebooks with appropriate power levels at each transmitter and treating interference as noise at every receiver (in short, TIN scheme) achieves all points in the capacity region to within a constant gap. The generalized degrees of freedom (GDoF) region under this condition is a polyhedron, which is shown to be fully achieved by the same scheme, without the need for time-sharing. The results are proved by first deriving a polyhedral relaxation of the GDoF region achieved by TIN, then providing a dual characterization of this polyhedral region via the use of potential functions, and finally proving the optimality of this region in the desired regime.
2837877
ITLinQ: A New Approach for Spectrum Sharing in Device-to-Device Communication Systems
We consider the problem of spectrum sharing in device-to-device communication systems. Inspired by the recent optimality condition for treating interference as noise, we define a new concept of information-theoretic independent sets (ITISs), which indicates the sets of links for which simultaneous communication and treating the interference from each other as noise is information-theoretically optimal (to within a constant gap). Based on this concept, we develop a new spectrum sharing mechanism, called information-theoretic link scheduling (ITLinQ), which at each time schedules those links that form an ITIS. We first provide a performance guarantee for ITLinQ by characterizing the fraction of the capacity region that it can achieve in a network with sources and destinations randomly located within a fixed area. Furthermore, we demonstrate how ITLinQ can be implemented in a distributed manner, using an initial two-phase signaling mechanism that provides the required channel state information at all the links. Through numerical analysis, we show that distributed ITLinQ can outperform similar state-of-the-art spectrum sharing mechanisms, such as FlashLinQ, by more than 100% of sum-rate gain, while keeping the complexity at the same level. Finally, we discuss a variation of the distributed ITLinQ scheme, which can also guarantee fairness among the links in the network and numerically evaluate its performance.
9478531
Optimality of Treating Interference as Noise: A Combinatorial Perspective
For single-antenna Gaussian interference channels, we reformulate the problem of determining the generalized degrees of freedom (GDoF) region achievable by treating interference as Gaussian noise (TIN) derived by Geng et al. from a combinatorial optimization perspective. We show that the TIN power control problem can be cast into an assignment problem, such that the globally optimal power allocation variables can be obtained by well-known polynomial time algorithms (e.g., centralized Hungarian method or distributed Auction algorithm). Furthermore, the expression of the TIN-achievable GDoF region (TINA region) can be substantially simplified with the aid of maximum weighted matchings. We also provide conditions under which the TINA region is a convex polytope that relax those by Geng et al. For these new conditions, together with a channel connectivity (i.e., interference topology) condition, we show TIN optimality for a new class of interference networks that is not included, nor includes, the class found by Geng et al. Building on the above insights, we consider the problem of joint link scheduling and power control in wireless networks, which has been widely studied as a basic physical layer mechanism for device-to-device communications. Inspired by the relaxed TIN channel strength condition as well as the assignment-based power allocation, we propose a low-complexity GDoF-based distributed link scheduling and power control mechanism (ITLinQ+) that improves upon the ITLinQ scheme proposed by Naderializadeh and Avestimehr and further improves over the heuristic approach known as FlashLinQ. It is demonstrated by simulation that ITLinQ+ without power control provides significant average network throughput gains over both ITLinQ and FlashLinQ, and yet still maintains the same level of implementation complexity. Furthermore, when ITLinQ+ is augmented by power control, it provides an energy efficiency substantially larger than that of ITLinQ and FlashLinQ, at the cost of additional complexity and some signaling overhead.
11467126
A new achievable rate region for the interference channel
A new achievable rate region for the general interference channel which extends previous results is presented and evaluated. The technique used is a generalization of superposition coding to the multivariable case. A detailed computation for the Gaussian channel case clarifies to what extent the new region improves previous ones. The capacity of a class of Gaussian interference channels is also established.
2800189
Wireless Network Information Flow: A Deterministic Approach
In a wireless network with a single source and a single destination and an arbitrary number of relay nodes, what is the maximum rate of information flow achievable? We make progress on this long standing problem through a two-step approach. First, we propose a deterministic channel model which captures the key wireless properties of signal strength, broadcast and superposition. We obtain an exact characterization of the capacity of a network with nodes connected by such deterministic channels. This result is a natural generalization of the celebrated max-flow min-cut theorem for wired networks. Second, we use the insights obtained from the deterministic analysis to design a new quantize-map-and-forward scheme for Gaussian networks. In this scheme, each relay quantizes the received signal at the noise level and maps it to a random Gaussian codeword for forwarding, and the final destination decodes the source's message based on the received signal. We show that, in contrast to existing schemes, this scheme can achieve the cut-set upper bound to within a gap which is independent of the channel parameters. In the case of the relay channel with a single relay as well as the two-relay Gaussian diamond network, the gap is 1 bit/s/Hz. Moreover, the scheme is universal in the sense that the relays need no knowledge of the values of the channel parameters to (approximately) achieve the rate supportable by the network. We also present extensions of the results to multicast networks, half-duplex networks, and ergodic networks.
61751279
Transmitter Cooperation under Finite Precision CSIT: A GDoF Perspective
The benefits of partial and full transmitter cooperation are evaluated for a two user interference channel under finite precision channel state information at the transmitters (CSIT), using the generalized degrees of freedom (GDoF) metric. Under finite precision CSIT, the benefits of interference alignment are completely lost, so that the X channel obtained by partial transmitter cooperation does no better than the underlying interference channels. Full transmitter cooperation produces a vector broadcast channel (BC) which has a strict GDoF advantage over partial cooperation (X channel) and whose GDoF are fully achieved by interference enhancement.
27708452
Network Coherence Time Matters—Aligned Image Sets and the Degrees of Freedom of Interference Networks With Finite Precision CSIT and Perfect CSIR
This paper obtains the first degrees of freedom (DoFs) bound that is provably sensitive to <italic>network</italic> coherence time, i.e., coherence time in an interference network, where all channels experience the same coherence patterns. This is accomplished by a novel adaptation of the aligned image sets bound and settles various open problems noted previously by Naderi and Avestimehr and by Gou <italic>et al.</italic> For example, a necessary and sufficient condition is obtained for the optimality of 1/2 DoF per user in a partially connected interference network, where the channel state information at the receivers (CSIRs) is perfect, the channel state information at the transmitters (CSITs) is instantaneous but limited to finite precision, and the network coherence time is <inline-formula> <tex-math notation="LaTeX">$T_{c}=1$ </tex-math></inline-formula>. The surprising insight that emerges is that even with perfect CSIR and instantaneous finite precision CSIT, the network coherence time matters, i.e., it has a DoF impact.
213220633
Toward an Extremal Network Theory—Robust GDoF Gain of Transmitter Cooperation Over TIN
Significant progress has been made recently in Generalized Degrees of Freedom (GDoF) characterizations of wireless interference channels (IC) and broadcast channels (BC) under the assumption of finite precision channel state information at the transmitters (CSIT), especially for smaller or highly symmetric network settings. A critical barrier in extending these results to larger and asymmetric networks is the inherent combinatorial complexity of such networks. Motivated by other fields such as extremal combinatorics and extremal graph theory, we explore the possibility of an extremal network theory, i.e., a study of extremal networks within particular regimes of interest. As our test application, we study the GDoF benefits of transmitter cooperation in a <italic>K</italic> user IC over the simple scheme of power control and treating interference as Gaussian noise (TIN) for three regimes of interest – a TIN regime identified by Geng et al. where TIN was shown to be GDoF optimal for the <italic>K</italic> user interference channel, a CTIN regime identified by Yi and Caire where the GDoF region achievable by TIN is <underline>c</underline>onvex without time-sharing, and an SLS regime identified by Davoodi and Jafar where a simple layered superposition (SLS) scheme is shown to be optimal in the <italic>K</italic> user MISO BC, albeit only for <inline-formula> <tex-math notation="LaTeX">$ {K}\leq 3$ </tex-math></inline-formula>. The SLS regime includes the CTIN regime, and the CTIN regime includes the TIN regime. As our first result, we show that under finite precision CSIT, TIN is GDoF optimal for the <italic>K</italic> user IC throughout the CTIN regime. Furthermore, under finite precision CSIT, appealing to extremal network theory we obtain the following results. In the TIN regime as well as the CTIN regime, we show that the extremal GDoF gain from transmitter cooperation over TIN is bounded regardless of the number of users. In fact, the gain is exactly a factor of 3/2 in the TIN regime, and <inline-formula> <tex-math notation="LaTeX">$2-1/{K}$ </tex-math></inline-formula> in the CTIN regime, for arbitrary number of users <inline-formula> <tex-math notation="LaTeX">${K}>1$ </tex-math></inline-formula>. However, in the SLS regime, the gain is <inline-formula> <tex-math notation="LaTeX">$\Theta (\log _{2}({K}))$ </tex-math></inline-formula>, i.e., it scales logarithmically with the number of users.
6319349
Generalized Degrees of Freedom of the Symmetric $K$ User Interference Channel Under Finite Precision CSIT
The generalized degrees of freedom (GDoF) characterization of the symmetric <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> user interference channel is obtained under finite precision channel state information at the transmitters (CSIT). The symmetric setting is where each cross channel is capable of carrying <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> DoF, while each direct channel is capable of carrying 1 DoF. Remarkably, under finite precision CSIT the symmetric <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> user interference channel loses all the GDoF benefits of interference alignment. The GDoF per user diminish with the number of users everywhere except in the very strong (optimal for every receiver to decode all messages) and very weak (optimal to treat all interference as noise) interference regimes. The result stands in sharp contrast to prior work on the symmetric setting under perfect CSIT, where the GDoF per user remain undiminished due to interference alignment. The result also stands in contrast to prior work on a subclass of asymmetric settings under finite precision CSIT, i.e., the topological interference management problem, where interference alignment plays a crucial role and provides substantial GDoF benefits.