Analog Spatial Cancellation for Tackling the NearFar Problem in Wirelessly Powered Communications (Extended Version)
Abstract
The implementation of wireless power transfer in wireless communication systems opens up a new research area, known as wirelessly powered communications (WPC). In nextgeneration heterogeneous networks where ultradense smallcell base stations are deployed, simultaneouswirelessinformationandpowertransfer (SWIPT) is feasible over short ranges. One challenge for designing a WPC system is the severe nearfar problem where a user attempts to decode an informationtransfer (IT) signal in the presence of extremely strong SWIPT signals. Jointly quantizing the mixed signals causes the IT signal to be completely corrupted by quantization noise and thus the SWIPT signals have to be suppressed in the analog domain. This motivates the design of a framework in this paper for analog spatial cancellation in a multiantenna WPC system. In the framework, an analog circuit consisting of simple phase shifters and adders, is adapted to cancel the SWIPT signals by multiplying it with a cancellation matrix having unitmodulus elements and full rank, where the full rank retains the spatialmultiplexing gain of the IT channel. The unitmodulus constraints render the conventional zeroforcing method unsuitable. Therefore, the paper presents a novel systematic approach for constructing cancellation matrices. For the singleSWIPTinterferer case, the matrices are obtained as truncated Fourier/Hadamard matrices after compensating for propagation phase shifts over the SWIPT channel. For the more challenging multipleSWIPTinterferer case, it is proposed that each row of the cancellation matrix is constructed as a Kroneckerproduct of component vectors, with each component vectors designed to null the signal from a corresponding SWIPT interferer similarly as in the preceding case.
Wireless power transfer, wirelessly powered communications, PTIT nearfar problem, analog spatial cancellation.
1 Introduction
Recent years have seen a series of breakthroughs in wireless communication technologies, such as millimeter wave communications, massive multipleinputandmultipleoutput (MIMO), and smallcell networks, which jointly provide a solution for coping with exponential growth of mobile data traffic. In contrast, wireless power transfer (WPT) using microwaves remains a relatively stagnant field and the current low transfer efficiencies due to severe propagation loss prevents its extensive commercialization. This key challenge, however, may be tackled by implementing WPT using nextgeneration wireless networks as a platform where the increasing network densification reduces the transmission distances to merely tens of meters [1] and the deployment of largescale arrays enable sharp beamforming to suppress dispersion of radiated energy [2]. This vision has motivated active research on seamless integration between WPT and wireless communications, opening a new area called wirelessly powered communications (WPC). In WPC networks, the ranges of power transfer (PT) (e.g., tens of meters) and information transfer (IT) (up to several kilometers) can be drastically different, leading to a severe nearfar problem [3]. This results in the coexisting of PT and IT signals with the power difference of many orders of magnitude. Note that a practical issue incurred by the nearfar problem is that the joint quantization of the strong and weak signals at the latter to be corrupted by quantization noise. This issue is particularly severe in the context of WPC system compared with that in the conventional uplink multiuser access scenario, since the power difference between the PT and IT signals could be many orders of higher than that between the IT signals from different users [3]. The solution for this issue, to the best of the authors’ knowledge, has not been reported in the existing WPC literature. This paper addresses this issue by presenting a framework for spatial cancellation of PT signals using an analog circuit prior to quantization, referred to as analog spatial cancellation.
1.1 Prior Work
For shortrange transmission, the same carrier can be used for both PT an IT, which is commonly known as simultaneous wireless information and power transfer (SWIPT). The idea was first explored in [4, 5] from the information theoretic perspective and the fundamental tradeoff between harvested energy and information capacity in a SWIPT system is quantified. The results are based on an ideal assumption that the receiver is able to harvest energy and decode information from the same received signal. The difficulty of realizing this assumption in practice motivated the design of powersplitting SWITP receiver in [6] where the received signal is split for separate energy harvesting and information decoding and the rateenergy tradeoff of a multiuser MIMO SWIPT system was characterized based on this receiver architecture. The idea of SWIPT opens up a rich set of interesting research opportunities having a similar theme of revamping communication theory and techniques to incorporate the feature of WPT. Various types of SWIPT systems have been proposed and studied recently including broadband SWIPT [7, 8], relayassisted SWIPT [9, 10, 11], cognitive SWIPT networks [12, 13], and interference channels with SWIPT [14, 15]. In future dense heterogeneous networks, SWIPT links will coexist with much weaker IT links, creating the mentioned nearfar problem [3]. Surveys of latest advancements in this active area can be found in [16, 17, 18, 19].
Besides SWIPT, other configurations of WPC systems are also developed in the literature. A WPC network was proposed in [20] where base stations power uplink mobiles by downlink PT. The throughput maximization problem was formulated and solved in this paper. The work has been extended to multiantenna system with energy beamforming [21], fullduplex systems [22], massive MIMO systems [23] and largescale communication networks [24]. However, it is impractical to rely on only existing base stations for achieving network coverage of PT as current intercell distances are much longer than PT ranges. One practical solution as proposed in [25] is to densely deploy power stations dedicated for PT, called power beacons. power beacons have low complexity and require no backhaul, allowing dense deployment to increase PT coverage. Moreover, those power beacons with Internet access can double as ultradense smallcell BSs.
In addition to the aforementioned design of WPC systems and techniques, recent information theoretic research building on the initial work in [4, 5] has been reported in [26, 27, 28]. In [26], it was shown that by exploiting the channel state information available at the wireless charger, the capacity of the WPC system can be significantly improved by performing transmitpower adaption, and the fundamental capacity limit under different levels of side information sharing was characterized from the information theoretic perspective. On the other hand, coding schemes have been designed in [27, 28] to optimize the tradeoffs between the IT rate and the PT efficiency. Despite this theoretical research, transforming WPC from theory to practice still faces many unsolved practical issues, and the said nearfar problem remains one of the key challenges for designing WPC systems.
As mentioned, given the scale of their power difference, jointly quantizing the received PT and IT signals renders the latter completely corrupted by quantization noise and it is impractical to solve this problem by increasing the ADC resolution. The conventional approaches to avoid this problem is to perform PT and IT in separate frequency subchannels (see e.g., [7]) or by time sharing [29] but they are not without drawbacks. Time shared PTandIT reduces their efficiency/rate and furthermore requires strict synchronization between users. For frequency division PTandIT, the suppression of ultrastrong PT signal at an information decoder requires a sharp analog bandpass filter plus sufficiently large frequency separation between PT and IT signals. SWIPT using the same spectrum does not have the drawbacks mentioned above but requires analog spatial cancellation of the IT signal prior to quantizing the PT signal, which is a largely uncharted area and the theme of the paper.
1.2 Contributions and Organization
We consider WPC system where a macrocell BS, called a IT BS, performing IT to a single user who also receives strong intended/unintended SWIPT signals transmitted by ultradense smallcell BSs (or power beacons with Internet access), called SWIPT BSs. All nodes are equipped with multiantennas. On one hand, the shortrange SWIPT with sharp beamforming over sparse scattering is modeled as a freespace channel. The SWIPT BSs are assumed to have different anglesofarrival at the user. It is worth pointing out freespace channels are essential for efficient WPT and thus widely assumed in the literature of WPT (see e.g., [30, 31, 3]). On the other hand, the longrange IT channel with rich scattering is modeled as an independent and identically distributed (i.i.d.) Rayleigh fading channel. This enables spatial multiplexing over the MIMO IT channel. Our work focuses on the user’s decoding of the data streams in the IT signal. In addition, besides harvesting energy from the SWIPT signal, the user can also decode information in the signal if it is intended for the user. The nearfar problem is irrelevant for the processing of the SWIPT signal that is thus neglected in our work.
In this paper, we identify the said nearfar problem and present a novel framework of analog spatial cancellation implemented at the user for suppressing the strong SWIPT signal prior to quantizing the IT signal. The operation is implemented using an analog circuit comprising simple RF components including phase shifters and adders, which is attached to the receive antenna array. The circuit implements multiplication of the observation vector of receive antennas by a matrix with unitmodulus elements, which is called a cancellation matrix and whose rows cancellation vectors. The design problem is formulated as the optimization of maximizing the row rank of the cancellation matrix under zeroforcing (ZF) constraints for nulling the SWIPT signals and unitmodulus constraints for individual elements, where rank maximization maximizes the spatial multiplexing gain for IT. The unitmodulus constraints render the conventional ZF cancellation technique based on linear algebra inapplicable. The focus of the work is to develop a systematic approach for solving the design problem. Essentially, the approach finds a set of independent cancellation vectors, each of which is orthogonal to and thus cancel the set of SWIPT channel vectors. The independence between the vectors ensures that the cancellation matrix has full row rank. The contributions are summarized as follows.

Consider the simple case of single SWIPT BS. The proposed design of the analog cancellation circuit, which solves the design problem, comprises two sequential parts. The first part performs phase derotation to compensate for propagation phase shifts of observations from different receive antennas, reducing the SWIPT channel to be equivalent to an allone vector. The second part transforms the mixed input signal by a truncated Fourier/Hadamard matrix with the allone row eliminated. Leveraging the row orthogonality of the matrix, the signal from the effective SWIPT channel (an allone vector) is nulled. The unitmodulus properties of the Fourier/Hadamard matrix elements facilitate analog implementation of the transform using phase shifters. Furthermore, the full row rank of the transform matrix ensures maximum spatial multiplexing gain of the IT signal under the SWIPT interference cancellation constraints.

Consider the general and more complex case of multiple SWIPT BSs. The preceding design cannot be easily extended to cancel multiple SWIPT signals arriving from different angles. A more sophisticated design is proposed for the current case. Specifically, by exploiting its Vandermonde structure, each freespace SWIPT channel can be decomposed into the Kronecker product of component phaseshift vectors. This motivates the proposed construction of each individual cancellation vector as a Kronecker product of component phaseshift vectors. The orthogonality between an arbitrary pair of cancellation and SWIPT channel vectors can be achieved by the orthogonality between any pair of their components vectors. According to this property, a single cancellation vector can be constructed, using a Fourierbased construction method, to null all multiple SWIPT interference channels based on a mapping between the components of the former to targeted SWIPT channels. Repeating the construction for all possible mappings generates a mother set of cancellation vectors. Then a subset of independent vector can be selected from the mother set by a greedy/random search, yielding the desired cancellation matrix solving the design problem for the current case.
The remainder of the paper is organized as follows: Section II introduces the system model. Section III formulates the design problem for implementing the analog spatial cancellation. Two simple but optimal solutions based on Fourier or Hadamard matrices targeting for the single SWIPT BS case are proposed in Section IV. Then, a novel systematic construction framework tackling the more challenging multiple SWIPT BSs case is developed in Section V and VI for two different subcases. Numerical results and discussions are presented in Section VII, followed by conclusion in Section VIII.
2 System Model
As shown in Fig. 1, we consider a WPC system where a user attempts to retrieve spatial multiplexed data streams transmitted by an IT BS (marco BS) over a MIMO channel, in the presence of strong SWIPT signals beamed by SWIPT BSs ( in Fig. 1). The SWIPT signals can be intended or unintended to the considered user. It is assumed that the IT BS is equipped with antennas and the user is equipped with a linear array of antennas, while all the SWIPT BSs are provisioned with arrays for freespace beamforming.
Assumption 1.
The shortrange SWIPTs are over narrowband freespace channels
Based on this assumption, a SWIPT channel can support only a single data stream, while up to streams can be spatially multiplexed in the IT channel. Let vector represents the effective SIMO channel (after transmit beamforming) between the th SWIPT BS and the user. The signal received at the user can be represented by a vector given as
(1) 
where the matrix represents the IT channel, denotes the IT signal transmitted by the IT BS, is the signal transmitted by the th SWIPT BS, and represents the additive white Gaussian noise. Assuming that the incident SWIPT signal from the th SWIPT BS can be approximated as a plane wave with the angleofarrival , the effective channel vector can be written as where scalar captures the path loss as well as the beamforming gain of , and vector represents the phase response for the linear receive array. Specifically,
(2) 
where denotes the constant phase difference between the signal observed by two adjacent receive antennas with being the antenna separation distance and representing the carrier wavelength. The received signal is split for energy harvesting (EH), SWIPT information decoding (ID) and IT ID as illustrated in Fig. 1.
3 Problem Formulation
The mentioned severe nearfar problem leads to a extremely low SQNR when quantizing the mixed SWIPTIT signal, making it difficult if not impossible to decode the weak IT signal. Specifically, in the quantization process illustrated in Fig. 2, the SWIPT signal is scaled to span the full dynamic range of the ADC. This reduces the peak magnitude of IT signal to be smaller than the quantization step size. The corresponding SQNR for IT signal can be calculated as follows [32],
(3) 
where represents the given ADC resolution in bit, denotes the power ratio between the received SWIPT and IT signals, and the constant depends on the distribution of the input signal. For example, given and bits, the SQNR can be computed to be approximately which makes it impossible to recover the data in the IT signal. Therefore, to tackle the nearfar problem, it is essential to decouple the IT and SWIPT signals in the analog domain and then quantize them separately.
To reliably decode the IT signal, we design an analog circuit that implements the multiplication the received signal with the cancellation matrix denoted as to null the strong SWIPT signal. Under the ZF constraints, the row rank of is maximized such that maximum multiplex gain can be achieved for the effective MIMO fading channel, defined as . The proposed simple design comprises an array of interconnected adjustable phase shifters. For the conventional digital spatial cancellation using a DSP processor, both the magnitudes and phases of signals can be varied. In contrast, for the proposed design, we can only adjust the phases of signals via phase shifters, which introduces the unitmodulus constraints to the elements of the ZF combining matrix . Based on the above discussion, the design problem is formulated as follows:
(4)  
s.t.  
where is the full row rank cancellation matrix needs to be designed, represents the row rank of , is the element of . To ensure the existence of the cancellation matrix, it is assumed that .
The main challenge for solving the above optimization problem is to satisfy the unitmodulus constraints, i.e., . In particular, the unitmodulus constraints make the feasible set, denoted as , for problem P1 no longer a traditional Euclidean vector/space. The reason is that the vectors in do not satisfy the closure properties for addition and scalar multiplication, i.e., , and if . As a result, the conventional approach of computing ZF vectors as those lying in the null space of SWIPT channel vectors, e.g., singular value decomposition (SVD), do not solve problem P1 since the elements of such vectors have different norms. Therefore, a new systematic solution approach based on new mathematical tools is developed in the sequel.
4 Analog Spatial Cancellation with a Single SWIPT BS
In this section, we consider the simple case where only a single SWIPT BS is employed to wirelessly power the user. Then, problem P1 in (5) reduces to,
(6)  
s.t.  
Note that, without the unitmodulus constraints, the maximum rank of the desired matrix should be equal to the dimension of the null space of , i.e., , since one degreeoffreedom (DoF) is used to suppress the SWIPT signal. Therefore, given the constraints, it is interesting to investigate whether a rank() solution can still be obtained. To this end, two simple but optimal analog spatial cancellation schemes are proposed in the following subsections.
4.1 The Fourier Based Scheme
Design
As illustrated in Fig. 3, for the processing of the SWIPT signal, if it is intended for the user, it can be easily decoded by using a simple coherent combiner
(7) 
where .
Next, the truncated Fourier transform multiplies the input with a truncated Fourier matrix with the first row removed:
where . Since the rows of the Fourier matrix are orthogonal, the multiplication suppresses the strong SWIPT signal and the IT signal will be extracted from the mixed signal for further decoding.
Mathematically, the transformed signals, denoted as , is
(8) 
Note that the allone column vector is exactly the transpose of the first row in the Fourier matrix, yielding in (4.1.1) due to the orthogonality between rows of a Fourier matrix. This suppresses the SWIPT signal at the output of the truncated Fourier transform.
After phase compensation and truncated Fourier transform, the equivalent MIMO fading channel for the IT signal is given by , which is a rank matrix. It means that, with one DoF used to suppress the SWIPT signal, the remaining DoF can still be exploited in the effective MIMO fading channel for spatial multiplexing. Note that SWIPT channel can provide one DoF for data transmission, which interestingly implies that maximum multiplexing gain of can be achieved in the considered system with analog spatial cancellation.
The above discussion leads to the following main result.
Proposition 1.
The solution of the optimization problem in (6) can be obtained as follows:
(9) 
where .
Proof: According to the preceding analysis presented above, the optimal analog cancellation matrix can be obtained as a truncated Fourier matrix times a phase compensation matrix as shown in (1). The optimality of (1) can be proven as follows. Firstly, note that the phase compensation matrix is an unitary matrix and the truncated Fourier matrix has a rank of . Thus, it is easy to verified that which achieves the desired maximum rank. Next, as shown in (4.1.1), we have which enforces the zero forcing constraint in (6). Finally, it can be observed from (1) that each element in involves only phase shift and the unitmodulus constraints are also satisfied. This completes the proof.
Remark 1.
A close observation reveals that the proposed is a scaled paraunitary matrix, i.e., . In other words, the rows in are mutually orthogonal and have the same norm. Such a property is quite important in the perspective of system stability, which can be indicated by the condition number. Generally, the smaller the condition number is, the more stable the system can be. Condition number of can be calculated as , where and are maximal and minimal singular values of matrix . It is easy to show that the minimal condition number of is one, given the is a scaled paraunitary matrix.
Implementation and complexity
Since a Fourier matrix for an arbitrary exists, the design in Fig. 3 is valid for any . The implementation requires phase shifters for the phase compensation array and shifters for the truncated Fourier transform. As a result, the total number of the required phase shifters is , which may lead to a high implementation cost if is large. This motivates an alternative lowcost scheme proposed in the following subsection.
4.2 The Hadamard Based Scheme
Design
The architecture of the Hadamard based scheme is similar to that of the Fourier based one shown in Fig. 3. The only difference is that the matrix is replaced by a truncated Hadamard transform , i.e., a Hadamard matrix with the first row removed. A Hadamard matrix is a square matrix whose entries are either or and whose rows are mutually orthogonal. For example, a Hadamard matrix is
(10) 
A necessary condition for a Hadamard matrix to exist is that is equal to or a positive multiple of , and the corresponding constructing methods can be found in [33].
Extending Proposition 1 gives the following result.
Proposition 2.
An alternative solution of the problem in (6) is given by
(11) 
Remark 1 also applies here with replaced with .
Implementation and complexity
Compared with the Fourier based scheme, the drawback of the Hadamard based scheme is that must be or a multiple of . However, the latter incurs a much lower implementation cost. In particular, since the entries of a Hadamard matrix are either or , the implementation of the truncated Hadamard transform requires no phase shifter but just adders. Fig. 4 gives an example for the practical circuit of truncated Hadamard transform with . As a result, the total number of required phase shifters reduces from for the Fourier based design to , corresponding to nearly times cost reduction. A comparison including implementation requirement and complexity for different proposed schemes are given in Table 1.
Fourier based scheme  Hadamard based scheme  

Number of required receive antennas  or  
Number of required phase shifters  
Number of required adders  
Rank of or  
Condition number of or 
5 Analog Spatial Cancellation with the Maximum Number of SWIPT BSs
In this section, we consider the case of multiple SWIPT BSs. For ease of exposition, let where denotes the maximum number of SWIPT BSs such that their signals can be cancelled by the user in the analog domain and is derived in the sequel. The design of analog spatial cancellation presented in the preceding section for the case of single SWIPT BS cannot be directly applied to the current case since the multiple SWIPT signals with different anglesofarrival cannot be simply aligned and cancelled using the scheme of phase compensation plus truncated Fourier/Hadamard transform. To address this issue, a more sophisticated systematic approach, called Kronecker based construction, for computing the phases of the phase shifters in the analog cancellation circuit (or equivalently the cancellation matrix) is proposed in the current section.
It can be observed from Problem P1 that the maximum row rank of the cancellation matrix is . Under the fullrank constraint, can be written as , where are linearly independent cancellation vectors. Then problem P1 can be rewritten to explicitly reflect the design goal of constructing linearly independent cancellation vectors as follows:
(12) 
It is important to note that unlike traditional interference cancellation, the condition does not guarantee the existence of a solution for problem P2. In other words, given an arbitrary setup, although we have more receive antennas than interferers, there still may not be an analog cancellation vector that can cancel all the interferences from different SWIPT BSs due to the unitmodulus constraints. Responding to this, a sufficient condition on the maximum number of SWIPT BSs guaranteeing the feasibility of the formulated problem is analyzed in the sequel.
5.1 Summary of Kronecker Based Construction
Basically, the proposed Kronecker based construction is motivated by a key observation that each of the SWIPT channel vectors possesses a special Vandermonde structure due to the freespace propagation environment, which enables it to be decomposed into a series of subvector components connected by Kronecker product (this decomposition is referred to as Kronecker decomposition). Then, by exploiting the mixedproduct property of the Kronecker product, the multipleSWIPT suppression constraints targeting at simultaneously nulling multiple SWIPT channels can be translated to several individual singleSWIPT suppression subconstraints targeting at only one specific SWIPT channel, which can be easily handled by utilizing the Fourier based construction method proposed in the case of single SWIPT BS. In particular, the said Kronecker based construction approach has the procedure as summarized below and is elaborated in the sequel.

Offline Construction

Step [Generation of a single cancellation vector]: A cancellation vector is designed to be a Kronecker product of a set of component phaseshift vectors, each component of which targets for suppressing only one specific SWIPT signal according to a predefined order, e.g., , called a SWIPTcancellation order. See details in subsection B.1).

Step [Generation of the cancellation vector mother set]: A mother set of cancellation vectors, denoted as , is generated using the same construction framework in Step with two specific techniques, called Fourier based cancellation and cancellation order rearranging. The constructed mother set consists of a series of orthonormal subsets where the cancellation vectors are mutually orthogonal and have the same norm. See details in subsection B.2).

Step [Selection of linearly independent cancellation vectors]: linearly independent vectors are selected from the mother set, , obtained in Step using a simple greedy/random search algorithm, giving the desired solution for the optimization problem in (4). See details in subsection B.3).


Adaptive Analog Spatial Cancellation
The preceding procedure of offline construction constructs a set of linear functions mapping the parameters to the phase shift elements of . Interestingly, the linear independence of the selected cancellation vectors in the step 3 of the above offline construction is observed from simulation to be almost invariant to the changes of the parameters .
^{3} Consequently, in the presence of mobility (time varying ), adaptive analog spatial cancellation is simple and involves computing the cancellation matrix using and adjusting the phase shifts in the analog circuit accordingly. The offline construction advantage of the proposed Kronecker based construction framework makes the analog spatial cancellation technique suitable for implementation at lowcomplexity mobile devices.
An example illustrating the above procedure of Kronecker based construction is provided in Subsection C. Before we proceed to the discussion of the detailed procedure design, an unified flow chart showing the whole picture including both the case in this section and the case in the next section is provided in Fig. 5 to facilitate the reader to gain a better overview of the proposed Kronecker based construction.
5.2 Detailed Design for Kronecker Based Construction
Generation of a single cancellation vector
As mentioned earlier, the Vandermonde structure of the SWIPT channel vectors makes it possible to be decomposed into a Kronecker product of component phaseshift vectors. Moreover, it can be shown that the Kronecker decomposition of is highly related to the factorization of its length . To be specific, we have the following key result showing the relationship between the factorization of and the Kronecker decomposition of .
Proposition 3.
given , with being positive integers, the Kronecker decomposition of can be given by
(13) 
where denotes the left Kronecker product operation.
Proof: It can be easily verified according to the definition of left Kronecker product [34].
Proposition 3 provides a general solution for performing Kronecker decomposition to a SWIPT channel vector given an arbitrary length and its factorization. The resultant Kronecker decomposition of the SWIPT channel vector is the most important step of the whole Kronecker based construction framework which motivates the construction of the corresponding cancellation vectors as specified in the sequel.
Motivated by the Kronecker structure of the SWIPT channel vectors in Proposition 3, the desired cancellation vectors are designed to have the same structure as follows.
For ease of notation, write the Kronecker decomposition of as according to (13), where represents the th component phaseshift vector of . Then, a typical cancellation vector, denoted by , can be constructed using the same Kronecker structure of as shown in Fig. 6. Specifically, consists of component phaseshift vectors connected by Kronecker product, i.e., , where denotes the th component phaseshift vector of . For each component, the length of is set to be the same as that of , denoted by , satisfying . Thus, the multipleSWIPT suppression constraints can be rewritten as follows.
(14) 
Note that each one of the constraints in (14) related to a specific can be enforced if one of , holds. In other words, each component phaseshift vector can be designed individually for tackling only one of the SWIPT channels among all total ones, then Kronecker combining all the component phaseshift vector leads to a desired cancellation vector being orthogonal to all . Based on this key observation, the multipleSWIPT suppression constraints can then be translated to singleSWIPT subconstraints given a predefined SWIPTcancellation order which indicates the mapping between the phaseshiftvector components and the targeted SWIPT channels. For example, given that the SWIPTcancellation order is (as depicted in Fig. 6), the th component phaseshift vector of should be designed targeting at the th SWIPT channel vector, then (14) can be translated to
(15) 
For each singleSWIPT suppression subconstraint in (15), it can be easily enforced by following the Fourier based construction method proposed in Section 4. Specifically, a single component phaseshift vector can be constructed as a product of a Fourier row vector and a phase compensation matrix as shown in the following.
(16) 
where represents the th row vector of a truncated Fourier matrix, denoted by , with being an arbitrary value among ; and is the phase compensation matrix, which is a diagonal matrix with the diagonal elements being exactly the elements of . To this end, by connecting all phaseshiftvector components via Kronecker product, the desired cancellation vector satisfying the multipleSWIPT suppression constraints in (14) can be obtained.
The above discussion also sheds light on the maximum number of cancellable SWIPT BSs that can be achieved by the proposed construction. To be specific, it can be noted that the number of cancellable SWIPT BSs is determined by the number of component phaseshift vectors that can be decomposed to. In other words, is achieved when the number of component phaseshift vectors in the Kronecker decomposition of is maximized or, equivalently, when the number of factors in the facterization of is maximized. Note that prime decomposition gives the maximum number of factors as pointed out by the following well known result from number theory.
Lemma 1 (Fundamental Theorem of Arithmetic).
Every positive integer can be represented in exactly one way as a product of prime powers
(17) 
where are primes and are positive integers, represents the maximum number of prime factors of integer , and this representation is called the canonical representation.
Accordingly, is obtained in the following proposition.
Proposition 4 (Maximum number of SWIPT BSs).
The maximum number of SWIPT BSs that can be supported by analog spatial cancellation with receive antennas using the Kronecker based construction approach is given by
(18) 
where denotes the th power in the canonical representation of which is given as follows based on Lemma 1
(19) 
Proposition 4 indicates the maximum number of cancellable SWIPT BSs, which is also served as a sufficient condition on the number of SWIPT BSs ensuring the feasibility of problem P2 as mentioned eariler. A close observation on this key result reveals that, from the perspective of achievable number of SWIPT BSs that can be supported, it is more “efficient” to set as an th power of two, i.e., , . This is because that it requires the fewest number of receive antennas to perform analog spatial cancellation for a given compared with other values of comprising other prime factors larger than two. To this end, for the special case of , we have the following corollary.
Corollary 1.
The maximum number of SWIPT BSs that can be supported by analog spatial cancellation with receive antennas is given as .
Generation of the cancellation vector mother set
Having obtained a single cancellation vector according to Step 1, next, Step 2 of the systematic procedure generates the mother set of cancellation vectors based on the construction framework in Step 1 following the proposed Fourier based cancellation (FBC) and cancellation order rearranging (COR) approaches as follows.
Specifically, given a certain cancellation order , FBC generates a set of cancellation vectors by selecting different Fourier row vectors to enforce the singleSWIPT suppression subconstraints in (15). Note that there are candidate Fourier row vectors that can be selected to achieve the construction of each component phaseshift vector as indicated in (16), where denotes the length of vector as shown in Fig. 6. Repeating Step 1 for all possible combinations yields cancellation vectors, where represents the number of the cancellation vectors provided by the FBC approach. Specifically, we have
(20) 
where are the factors of satisfying . Moreover, we have the following key property for the FBC construction approach.
Proposition 5.
All those cancellation vectors generated by the FBC construction approach following the same cancellation order are mutually orthogonal.
Proof.
See Appendix 9.
Note that the above construction involves only a single SWIPTcancellation order. Actually, for each SWIPTcancellation order, the same FBC approach can be repeated so that more cancellation vectors can be generated, which is the key idea of the COR approach. In particular, by repeating the construction for all SWIPTcancellation orders, the COR approach expands the vector set generated by the FBC approach to a times larger mother set consisting of total candidate cancellation vectors. To this end, based on Proposition 5, we have the following key property for the proposed Kronecker based construction framework.
Proposition 6.
Given , the mother set generated by the proposed Kronecker based construction procedure is composed of subsets with each containing mutually orthogonal vectors.
Proposition 6 points out the partial orthogonal property of the constructed mother set, which can be further exploited to develop a more efficient greedy search algorithm in the sequel for finding the linearly independent cancellation vectors. Furthermore, with the help of the Lemma 2 shown in the following, it can be proven that . This reveals that the proposed systematic procedure can always generate a sufficiently large mother set with the number of cancellation vectors being no less than the required ones.
Selection of linearly independent cancellation vectors
Last, a set of linearly independent vectors can be easily chosen from using one of the wellknown search methods such as exhaust search, greedy search and random search. Specifically, exhaust search exhausts a complete search of all possible vectors combinations from the generated mother set to guarantee a global optimal solution in terms of condition number, which also leads to the highest implementation complexity. Greedy search does not pursue a global optimal solution, instead, it looks for a local optimal solution in each iteration (in our case, it is based on a criterion that the incremental matrix formed by the target vector and all those selected vectors should have the minimum condition number). Simulation results show that it can achieve almost the same performance of the exhaust search method with moderate complexity. Last, random search adopts the simplest search strategy which randomly picks cancellation vectors from the mother set in each iteration and stops when the selected vectors are linearly independent. From simulations, it is observed that random search can always achieve a much lower complexity than the other two methods at the expense of moderate performance loss (in terms of condition number). Since mobile users are usually resourceconstrained, implementation complexity should be given the highest priority when selecting a search algorithm. In this regard, random search method may be more preferred in mobiles from the practical perspective. Furthermore, it is worth pointing out that a subset of linearly independent vectors can always be found out by any one of the three mentioned search methods, which also indicates that there always exists a desired linearly independent subset of the mother set.
5.3 An Example
To demonstrate the proposed systematic construction procedure, an example is provided below.
Example 1.
Let’s consider the simple case with and . According to Proposition 4, it is noted that and the corresponding prime factorization of is , which gives the Kronecker decomposition of and based on Proposition 3 as follows.
(22) 
where .
Thus, the first cancellation vector can be constructed as a product of two component phaseshift vectors, denoted by and respectively. Without loss of generality, assuming the SWIPTcancellation order is , and using (16), and can be given by (23) (24) where is constructed using the first row of a truncated Fourier matrix. Then, it follows that
(25) 
Alternatively, can be constructed using the second row of a truncated Fourier matrix, which gives (26) It follows that can be constructed as
(27) 
It is easy to verify that these two cancellation vectors are orthogonal, i.e., , which agrees with the result in Proposition 5.
Next, by following another SWIPTcancellation order , and repeating the above construction procedure, we can obtain the other two cancellation vectors as follows,
(28) 
(29) 
To this end, combining all the generated cancellation vectors yields the desired cancellation matrix as shown in (30).
(30) 
It can be easily verified that is the desired solution of the problem in (5) for the case of , since , given , and the unitmodulus constraints are also perfectly satisfied.
Remark 2.
In this simple example, the size of mother set is equal to that of the final cancellation vector set, i.e., and the linear independency of the generated vectors can be easily verified. For the more complex case where , it can be shown that the required linear independent vectors can still be selected from by using the said greedy or random search algorithm (see the example in Fig. 7).
Last, adapting analog spatial cancellation to timevarying involves recomputing using (30) upon changes on the parameters.
6 Analog Spatial Cancellation for The Case of
We now look into the case that . The main difference between this case and the prior case is that the less number of SWIPT signals that needs to be analog decoupled leads to an nonunique Kronecker decomposition strategy of or, equivalently, an nonunique factorization strategy of . For example, given and , we have two factorization strategies for tackling this case, i.e., and , which respectively correspond to two Kronecker decomposition strategies of as follows,
(31)  
(32) 
Note that (31) and (32) correspond to different cancellation vector construction processes as indicated in Section 5, which results in different mother set with different size and . This raises additional design problem for selecting the proper factorization strategy in this case. To be specific, (31) leads to a mother set containing cancellation vectors with , while (32) results in another mother set comprising vectors with . Apparently, the linearly independent subset of cancellation vectors selected form the latter would achieve a lower condition number than that of those selected from the former due to the fact that the latter is larger mother set with more mutually orthogonal cancellation vectors. In this regard, it is more desired to choose a factorization strategy leading to the largest . Mathematically, assuming the factorization of can be expressed as , the optimal factorization strategy selection problem, given a certain and , can be formulated as follows.
(33)  
s.t.  
Note that it is a challenging integer programming problem with polynomial objective function and constraint. To tackle this problem, we have the following Lemma.
Lemma 3.
For any , , , , the following inequality holds,
(34) 
where the equality holds if and only if .
Proof: See Appendix 11.
Lemma 3 reveals that the possible maximum value of the objective function in (33) is
(35) 
and it is achieved only when is evenly factorized, i.e., . However, this solution may not lie in the feasible set of the problem in (33) due to the integer factors constraints. Alternatively, the optimal solution for the problem in (33) requires all factors of to be as even as possible. To this end, the following Algorithm 1 is proposed to attain the optimal factorization solution for the problem (33).
Having obtained the optimal factorization solution for problem in (33), we can design the optimal Kronecker decomposition strategy according to Proposition 3, and then following the Kronecker based construction framework proposed in Section 5, a corresponding mother set can be generated. Finally, by employing one of the search algorithms mentioned in Section 5.2.3, we can select the desired linearly independent cancellation vectors from the mother set to form a full row rank phase shift matrix as a solution for the problem P2.
Furthermore, the above discussion also implies an interesting tradeoff between and as expressed in the following proposition.
Proposition 7.
Given a certain , a smaller leads to a larger .
Proof: Without loss of generality, assume that the optimal factorization strategy of for a given is , where . Then, using (20), the corresponding can be computed as follows
(36) 
Next, consider the case that the number of SWIPT BS is . According to Algorithm 1, the optimal factorization strategy can be expressed as , where and , . Then we have
(37) 
Note that , since . To this end, the desired result can be proven by induction.
Remark 3.
Proposition 7 characterizes the tradeoff between and , which also implicitly reflects the relationship between and the achievable condition number of the constructed cancellation matrix . In general, given a certain , a larger number of SWIPT BSs leads to a smaller , which in turn results in a degradation of the condition number performance. Simulation result will be provided in Fig. 14 to verify this analytical result.
7 Simulation Results
In this section, the effectiveness of the proposed Kronecker based construction framework for finding the desired full row rank analog spatial cancellation matrix is first examined. Then, simulation results for further evaluating the performance of analog spatial cancellation are presented. The simulation settings for Fig.8–13 are summarized as follows. We set , , the average received SNR for the IT signal as . Moreover, the number of data streams for IT is fixed at for Fig. 8–10 and Fig. 13, and fixed at for Fig. 11 and 12, and that for SWIPT is per SWIPT BS, all modulated using QAM. It is also assumed that all the SWIPT signals are intended for the considered user, and the receiver at the user decouples the received mixed signals using the proposed analog spatial cancellation technique and decodes the SWIPT and IT signals separately. Perfect analog spatial cancellation is assumed for the results in Fig. 9–12, and the impact of imperfect cancellation is investigated in Fig. 13. Finally, the impact of the number of SWIPT BSs on the condition number of the constructed cancellation matrix is illustrated in Fig. 14.
7.1 Condition Number of Cancellation Matrix
Considering the case of and with and uniformly distributed within , the empirical probability distributions of the condition number of the constructed cancellation matrix obtained by the three mentioned search algorithms are plotted in Fig. 7 respectively based on realizations. It is observed that the selected by the exhaust search and greedy search can achieve nearly minimum condition number (e.g., close to ) for most realizations, and the probability decays sharply as the condition number increases, leaving only negligible portion of the realizations resulting in a condition number larger than a moderate value (e.g., ). It is also noted that, all realizations lead to finite condition numbers which verifies the claim that there always exists a set of () linearly independent vectors in the generated mother set as mentioned in Section 5.2.3. However, the iterations required by the exhaust search and greedy search in the considered case are and respectively, which may be too computationconsuming to be implemented in some resourcelimited mobiles. Alternatively, random search algorithm provides us with a low complexity solution requiring only about iterations in average, which implies that in most cases, the initial random selection is enough for finding a linearly independent subset solution and no further iteration is required. However, the low complexity is achieved at the expense of relatively bad condition numbers distribution as shown in Fig. c, where the distribution looks quite even with a relatively smooth decay rate. Nevertheless, it is still an effective solution since the linear independence of the selected vectors can be guaranteed for all realizations of and .
7.2 The PTIT NearFar Problem in Digital Spatial Cancellation
Consider the use of the traditional digital spatial cancellation of IT and SWIPT signals in the presence of a single SWIPT BS. Fig. 8 shows the symbol error rate (SER) for IT transmission versus the power ratio between the received SWIPT and IT signals, denoted as . The SER is observed to depend on both the ADC resolution and . Specifically, given a required SER, increasing (corresponding to a more severe nearfar problem) requires higher ADC resolution so as to regulate the quantization noise in the weak IT signal. For high (e.g., or higher), even a bit ADC is insufficient for achieving a low SER (e.g., 0.01). The results show that digital spatial cancellation is incapable of coping with the nearfar problem, for which has to rely on using an ADC with an impractically high resolution.
7.3 SER Comparison between Analog and Digital Spatial Cancellation
Fig. 9 compares the (IT transmission) SER performance of the proposed analog spatial cancellation and the traditional digital spatial cancellation in the case of single SWIPT BS. The channel noise variance and the average received SNR for the IT signal are assumed fixed and the ADC resolution is bits. It is observed that, for the digital cancellation, as the received SNR for the SWIPT signal increases, the SER for the IT signal grows and saturates at the worst point, i.e., , which agrees with the SQNR analysis in (3) where the quantization noise for the IT signal is shown to be proportional to . In contrast, when the proposed analog spatial cancellation is used, the (IT) SER performance is observed to be independent of the received SWIPT signal and attain a constant low SER throughout the whole SWIPT SNR range with only a bit ADC. This demonstrates the robustness of analog spatial cancellation against the nearfar problem. In addition, it is also noted that the SER curve for the SWIPT signal has a floor due to quantization noise independent of the received SNR and determined only by the ADC resolution.
SER comparison in the case of multiple SWIPT BS is illustrated in Fig. 10, where the system setup with two SWIPT BSs transmitting two independent SWIPT data streams is assumed, and the ADC resolution is also bits. As expected, similar trends can be observed in this case, which verifies the effectiveness of the proposed systematic solution for analog cancellation matrix in the case of multiple SWIPT BSs. Also, it is observed that, in terms of SER performance, singleSWIPTBS case can achieve a better IT performance than the multipleSWIPTBS case given the number of supported IT data streams is the same. It is intuitive since additional DoF will be used to suppress the SWIPT signals for IT signals decoding in the case of multiple SWIPT BSs.
7.4 Throughput Comparison between Analog and Digital Spatial Cancellation
The throughput comparison between the proposed analog spatial cancellation and the traditional digital spatial cancellation for the case of single SWIPT BS is illustrated in Fig. 11. We consider the effective throughput in this comparison, which is defined by , where