The considered scenario comprises 2 tiers, including G_{m} macrocells and G_{s} small cells with G=G_{m}+G_{s} cells. Each cell \(g\in \left \{1,2,\dots,G\right \}\) has an \(N_{BS_{g}}\)antenna base station (macrotier), or \(N_{AP_{g}}\)antenna access point (small tier), and user equipments with \(N_{UE_{g}}\) antennas. There is a single orthogonal frequencydivision multiplexing (OFDM) stream per user, and the intracell interference is assumed to be handled by means of resource block scheduling. Therefore, a single active user per cell is considered.
In such network, and assuming that a linear precoding/decoding technique is applied, the received signal at the input of receiver g is expressed by
$$\begin{array}{*{20}l} r_{g} &= u_{g}^{H}\mathbf{n}_{g} + i_{g}\mathbf{u}_{g}^{H}d_{gg}^{\alpha/2}\mathbf{H}_{gg}P_{BA_{g}}^{1/2}\mathbf{v}_{g}s_{g} + \left(1i_{g}\right)\mathbf{u}_{g}^{H}d_{gg}^{\alpha/2}\mathbf{H}_{gg}P_{UE_{g}}^{1/2}\mathbf{v}_{g}s_{g} \notag \\ &+\sum_{j=1,j\neq g}^{G}\left(i_{j}\mathbf{u}_{g}^{H}d_{gj}^{\alpha/2}\mathbf{H}_{gj}P_{BA_{j}}^{1/2}\mathbf{v}_{j}s_{j} + \left(1i_{j}\right)\mathbf{u}_{g}^{H}d_{gj}^{\alpha/2}\mathbf{H}_{gj}P_{UE_{j}}^{1/2}\mathbf{v}_{j}s_{j}\right) \quad\forall g \end{array} $$
(1)
where i_{g} is a Boolean variable indicating whether each cell g is in downlink (i_{g}=1) or in uplink (i_{g}=0), and s_{g} contains the information that transmitter g is sending to receiver g. H_{gj} is the MIMO channel from transmitter j to receiver g, and n_{g} is the noise at receiver g. Also, \(P_{BA_{g}}\) is the power level at the BS (AP) of macro (small) cell g, \(P_{UE_{g}}\) is the power level at the gth UE, and d_{gj} is the normalized distance from the jth transmitter to receiver g, whereas α denotes path loss exponent. Finally, v_{g} and u_{g} are the precoding and decoding vectors (i.e., beamformers) applied at transmitter and receiver g, respectively^{Footnote 1}.
As shown in Fig. 1, the scenario under evaluation contains G=3 cells with 1 macrocell and 2 small cells, where each cell includes a single active user. All the nodes in the network are equipped with 2 antennas, and as above mentioned, there is a single (OFDM) stream per cell. Therefore, according to the convention in [16], we are working with a set of multicarrier (2×2,1)^{3} interference channels (IC), each one associated to a different UL/DL combination. Consequently, for any given UL/DL setting, the signal model in (1) can be reformulated as
$$ \mathbf{r}_{g} = \mathbf{u}_{g}^{H}\left(d_{gg}^{\alpha/2}\mathbf{H}_{gg}P_{g}\mathbf{v}_{g}s_{g} + \sum_{j\neq g}^{G}d_{gj}^{\alpha/2}\mathbf{H}_{gj}P_{j}\mathbf{v}_{j}s_{j} + \mathbf{n}_{g}\right)\quad\forall g, $$
(2)
where \(P_{g} = i_{g}P_{BA_{g}} + \left (1i_{g}\right)P_{UE_{g}}\) is the power level at transmitter g∈{1,2,3}.
In order to characterize the impact of flexible duplexing on the network performance, we compare three different interference management strategies. Two of these techniques are based on multicell minimum mean square error (MMMSE) receivers and will be described in Section 2.1. Analogously, the IA scheme is introduced in Section 2.2.
Multicell MMSE
The MMMSE technique [17, 18], also called interference rejection combining (IRC) in the literature [19], is an extension of the wellknown MMSE technique that takes into account the knowledge of the channels from interferers to the intended receiver g, i.e. \(\{\mathbf {H}_{gj}\}_{j\neq g}^{G}\). From (2), the MMMSE filter at a given user g, \(\mathbf {u}_{g}^{\mathrm {MMMSE}}\), is calculated as
$$ \mathbf{u}_{g}^{\mathrm{MMMSE}} = \mathbf{v}_{g}^{H}\hat{\mathbf{H}}_{gg}^{H}\left(\hat{\mathbf{H}}_{gg}\mathbf{v}_{g} P_{g}d_{gg}^{\alpha}\mathbf{v}_{g}^{H}\hat{\mathbf{H}}_{gg}^{H}+\mathbf{R}_{g}+\mathbf{n}_{g}\right)^{1}\quad\forall g, $$
(3)
where, \(\mathbf {R}_{g} = \sum _{j\neq g}^{G}\hat {\mathbf {H}}_{gj}\mathbf {v}_{j} P_{j}d_{gj}^{\alpha }\mathbf {v}_{j}^{H}\hat {\mathbf {H}}_{gj}^{H}\) is the covariance matrix of the intercell interference, and \(\hat {\mathbf {H}}_{gj}\) is the estimated channel matrix between transmitter j and receiver g.
At the transmitter side, two different strategies are considered to calculate the beamforming vectors:

Complex random unit norm precoders, in such a way that a sufficient number of transmissions is statistically equivalent to an isotropic spatial distribution.

Dominant eigenmode transmission (DET), i.e., the precoder is the eigenvector associated to the maximum eigenvalue of the channel matrix between the transmitter and the intended receiver.
Spatial interference alignment
The main intuition behind the interference alignment technique is to confine interfering signals into a reduced dimensionality subspace at each receiver, allowing us to transmit the desired signals simultaneously over the remaining interferencefree subspace [20]. Although, multiple time and frequency dimensions can be exploited, we focus on the spatial dimension in our implementation.
Spatial IA builds on a set of precoders \(\{\mathbf {v}_{g}\}_{g=1}^{G}\) and decoders \(\{\mathbf {u}_{g}\}_{g=1}^{G}\), that must fulfill the following conditions for all transmitterreceiver pairs^{Footnote 2},
$$ \left\{\begin{array}{ll} \mathbf{u}_{g}^{H} \mathbf{H}_{gg} \mathbf{v}_{g} \neq 0 &\forall g \\ \mathbf{u}_{g}^{H}\mathbf{H}_{gj} \mathbf{v}_{j}=0,&\forall j\neq g. \end{array}\right. $$
(4)
Several IA algorithms are available in the existing literature for more general interference channels [21–23], as well as for more sophisticated topologies [24]. Nevertheless, for the (2×2,1)^{3} interference channel under evaluation, an analytical procedure to calculate the beamformers and filters satisfying (4) was proposed in [20]:

1.
The precoder for user 1, v_{1}, can be selected as any eigenvector of matrix
$$\mathbf{E} = \left(\mathbf{H}_{31}\right)^{1}\mathbf{H}_{32}\left(\mathbf{H}_{12}\right)^{1}\mathbf{H}_{13}\left(\mathbf{H}_{23}\right)^{1}\mathbf{H}_{21}. $$

2.
From v_{1}, precoders v_{2} and v_{3} can be calculated as
$$\mathbf{v}_{2} = \left(\mathbf{H}_{32}\right)^{1}\mathbf{H}_{31}\mathbf{v}_{1} $$
$$\mathbf{v}_{3} = \left(\mathbf{H}_{23}\right)^{1}\mathbf{H}_{21}\mathbf{v}_{1} $$

3.
The interference cancelation filters must be designed in such a way that the received signal is projected on the orthogonal subspace of the interference signal space. Equivalently, u_{1} is the eigenvector of [H_{12}v_{2} H_{13}v_{3}] associated to the zero eigenvalue, whereas u_{2} and u_{3} can be obtained analogously from [H_{21}v_{1} H_{23}v_{3}] and [H_{31}v_{1} H_{32}v_{2}].
As previously discussed in [25], several impairments can be found when IA is implemented in practice, limiting its performance when compared to the theoretical results:

While perfect channel knowledge is assumed in theory, channel estimation errors arise in practice. The impact of the resulting misalignment is taken into account for the theoretical studies in [9, 26], as well as in the experiments discussed in [27, 28].

Spatial collinearity between desired signal and interference subspaces could result in desired signal energy loss.

IA precoders and decoders are usually applied at symbol level in a persubcarrier fashion, i.e., after frame detection and time/frequency synchronization stages. However, in realworld systems, detection and synchronization are performed right after the RF demodulation and analogtodigital conversion, i.e., at sample level, thus they are affected by interference. The preFFT IA approach in [29, 30] overcomes this issue by operating at sample level, that is, in the time domain. Nevertheless, since the synchronization mismatches have been already studied in the aforementioned works, we rely on an external clock and oscillator for time and frequency synchronization (see Section 3), and apply IA precoders and decoders in the frequency domain.