Downlink precoding has been regarded as a key technology in multi-user multiple-input and multiple-output (MIMO) communications. With the channel state information (CSI) available at the base station, the multi-user interference can be calculated prior to transmission. In this way, the interference mitigation (IM)-based precoder techniques have been extensively investigated to strictly suppress the interference. The dirty-paper coding (DPC) scheme was proposed in Costa (1983) by pre-subtracting the interference prior to transmission for achieving capacity, which however assumes infinite alphabet input and incurs high computational cost. Although the Tomlinson-Harashima precoding (THP) Sun and Lei (2013) and vector perturbation (VP) Hochwald et al. (2005) precoders aim to reduce the computational complexity over the DPC approach, they still need a sophisticated sphere-search algorithm for algorithm implementation. Hence, low-complexity linear precoders, such as zero-forcing (ZF) and minimum mean squared error (MMSE) Tse and Viswanath (2005), Peel et al. (2005), have attracted much attention in practices due to their low-complexity. On the other hand, optimization-based precoding has been a popular research topic. For example, signal-to-interference-plus-noise ratio (SINR) balancing aims to maximize the minimum SINR subject to a total power constraint (Wiesel et al., 2006); transmission power minimization problem aims to reduce the transmission power at the base station, subject to user’s minimum SINR requirements (Schubert and Boche, 2004).

The above designs treat the input as infinite Gaussian signal. Hence, they only exploit the channel correlation for the precoding design. In practice, modulation size is finite, and the input is not Gaussian signal. In this case, there is scope to jointly exploit the correlation among the channels and transmitted data, so that the multi-user interference is possible to make constructive at each receiver, termed as interference utilization (IE) precoding (Masouros and Alsusa, 2007). The concept of constructive interference (CI) has been applied for anonymous communications (Wei et al., 2021a), cognitive radio (Law and Masouros, 2018), large-scale MIMO (Amadori and Masouros, 2016, 2017b), constant envelope (Amadori and Masouros, 2017a; Liu et al., 2017), hybrid beamforming (Hegde et al., 2019), multi-cell coordination (Wei et al., 2020b), rate-splitting (Salem et al., 2019), physical layer security (Khandaker et al., 2019; Wei et al., 2020a; Wei and Masouros, 2020), directional modulation (Wei et al., 2021b), and integrated sensing and communication systems (Liu et al., 2018). In the following section, we briefly discuss the IE-based precoder design.

For comparison, let us first consider a classic power minimization problem subject to per-device’s signal-to-interference-plus-noise ratio (SINR) requirement. Assume that the transmitter is equipped with N antennas for serving K devices (N ≥ K). Define w i ∈ C ( N × 1 ) as the precoder vector for the i-th device’s intended signal s _i . Write the transmitted symbol vector s = [ s 1 , … , s K ] T ∈ C ( K × 1 ) , the signal received by the i-th user can be written as y i = h i [ w 1 , … , w K ] s + n i (1) where h i ∈ C ( 1 × N ) is the multiple-input and single-output (MISO) channel spanning from the transmitter to the i-th device, while n _i denotes the receiver’s noise, following a Gaussian distribution C N ( 0 , σ 2 ) . A generic power minimization problem can be formulated as P 1 : min w 1 , … , w K ∑ i = 1 K w i 2 ,  s.t.  ( C 1 ) : h i w i 2 ∑ j ≠ i h i w j 2 + σ 2 ≥ Γ i , ∀ i ∈ K , (2) where Γ _i is the i-th device’s SINR requirement. The problem P1 represents a non-convex second-order cone programming (SOCP) exercise. By defining W i = w i w i H ∈ C N × N , P1 can be equivalently transformed into P 2 : min w 1 , … , w K ∑ i = 1 K tr W i ,  s.t.  ( C 1 ) : h i W i h i H ≥ Γ i ∑ j ≠ i h i W j h i H + σ 2 , ∀ i ∈ K , ( C 2 ) : W i ⪰ 0 , ∀ i ∈ K , ( C 3 ) : rank W i = 1 , ∀ i ∈ K , (3) which can be readily solved as a standard convex semi-definite programming (SDP) problem after dropping constraint (C3).

Different from IM-based precoding that needs to strictly suppress interference, the IE-based precoder is able to exploit the multi-device interference as a constructive element. Multi-device interference can be achieved by exploiting geometrical interpretation shown in Figure 1. Explicitly, we first rotate the signal y _i by the angle of ∠s _i , and then the rotated signal can be mapped onto real axis and imaginary axis respectively. As can be seen, the received signal falls into a constructive region (in Figure 1B) if and only if the trigonometry below is ensured Re y i s i ∗ − σ Γ ⋅ tan π M ≥ Im y i s i * , ∀ k ∈ K , (4) where M represents constellation size. s i * denotes the conjugate of s _i , where s _i is the intended symbol for the i-th user. In particular, Γ physically represents the Euclidean distance in the signal constellation between the constructive region and the decision thresholds, which also directly relates to SINR performance of the received signal. The above discussion can be extended into any order M-PSK and multi-level modulations Kabir et al. (2018). Now, we are able to give the interference utilization-based power minimization precoder such as P 1 : min w 1 , … , w K ∑ i = 1 K w i 2 ,  s.t.  ( C l ) : Re h i w 1 , … , w K s s i * − σ Γ ⋅ tan π M ≥ Im h i w 1 , … , w K s s i * , ∀ k ∈ K . (5) Evidently, the precoder optimization is convex in nature, which can be solved directly. Then, we further discuss the IE-based precoder for SINR balancing optimization. When formulating SINR balancing for IE precoder, its problem formulation can be written as P 2 : min w 1 , … , w K ∑ i = 1 K w i 2 ,  s.t.  ( C 1 ) : Re h i w 1 , … , w K s s i * − σ Γ ⋅ tan π M ≥ Im h i w 1 , … , w K s s i ∗ , ∀ k ∈ K , ( C 2 ) : w 1 , … , w K s 2 ≤ P max , (6) where P _max denotes the power budget. It has been proved in Li and Masouros (2018) that, the closed-form of such an IE-based precoder is given as W = 1 K H H H H H − 1 diag ( Λ ) s s ̂ , (7) where s ̂ is given as s ̂ = 1 S 1 , 1 S 2 , … , 1 S k . Λ is an auxiliary matrix, whose value can be calculated by a low-complexity iterative algorithm in Li and Masouros (2018). It can be seen that regardless of power minimization or SINR balancing IE-based precoders, they always have linear structure and can be solved directly, without the need of calling SDP optimization.

Here, we illustrate BER performance of the IE-based precoder, compared against the ZF and MMSE designs as shown in Figure 2. It is observed that as the SNR increases, the BER performance of the IE-based precoder shows rapid improvement. Furthermore, the performance of the IE-based precoder is always superior to the conventional ZF, and outperforms the MMSE at moderate/high SNR regions, which is in line with the analysis of this section.

In multi-cluster IoT systems shown in Figure 3, the APs are connected with high-speed optical fiber for joint signal processing. Generally, there are two different coordination mechanisms, i.e., partially-coordinated IE and fully-coordinated IE -based precoder designs. By the former design, the APs only share CSI with others for inter-cluster interference suppression. Since transmission data is not shared among the APs, each AP only serves its associated users, and at the same time suppresses inter-cluster interference. Assume there are M APs for corporation. Define y _im and n _im as the received signal and noise at the i-th device belonging to the m-th cluster. h i m ∈ C ( 1 × N ) is the MISO channel spanning from the m-th AP to the i-th device. W _m and s _m denote the precoder matrix and transmitted symbol vector at the m-th AP, respectively. The received signal can be calculated as y i m = h i m W m s m + ∑ m ′ ≠ m h i m , W m ′ s m ′ + n i m . (8) When formulating the optimization for the partially-coordinated IE precoder, the CI constraint is rewritten as Re y i m s im ∗ − σ 2 + Δ i m Γ ⋅ tan π M ≥ Im y i m s im ∗ , ∀ k ∈ K . (9) In particular, the term Δ _im represents the inter-cluster interference at the i-th user, which needs to be carefully suppressed. The IE-based power minimization problem is re-formulated as P 3 : min W 1 , … , W M ∑ m = 1 M W m 2 ,  s.t.  ( C 1 ) : Re y i m s im ∗ − σ 2 + Δ i m Γ ⋅ tan π M ≥ Im y i m s im ∗ s i ∗ , ∀ k ∈ K , ( C 2 ) : Δ i m ≥ ∑ m ′ ≠ m h i m , W m s m 2 . (10) In a similar vein, the IE-based SINR balancing precoder in multi-cluster scenario can be formulated as P 4 : max w 1 , … , w K Γ ,  s.t.  ( C 1 ) : Re y i m s im ∗ − σ 2 + Δ i m Γ ⋅ tan π M ≥ Im y i m s i ∗ , ∀ k ∈ K , ( C 2 ) : Δ i m ≥ ∑ m ′ ≠ m h i m ′ , W m ′ s m ′ 2 , ∀ i a n d m , ( C 3 ) : W m s m 2 ≤ P max , ∀ m . (11) By contrast, the fully-coordinated IE design shares both the CSI and the data to be transmitted among the APs, where the APs jointly serve the downlink users in a similar vein of distributed antenna systems. In fact, the fully-coordinated IE makes no much difference compared to the classic IE-based precoder, as the distributed APs can be seen as a virtual multiple transmission antennas.

The topic of the IE-based precoder is still broadly open for research and could be extended in many interesting directions:

In this review paper, we have briefly introduced the concept of IE-based precoding, two generic optimizations, i.e., power minimization and SINR balancing optimizations, are formulated. Then, we have examined the IE-based precoder design in multi-cluster IoT scenario. Furthermore, open challenges related to emerging applications are present, where the gap between theory and implementations should be bridged. In a nutshell, there are still essential works for the research of the IE-based precoder, which holds the promise of exciting research in the years to come.