Spatial degreesoffreedom in largearray fullduplex: the impact of backscattering
 Evan Everett^{1}Email author and
 Ashutosh Sabharwal^{1}
https://doi.org/10.1186/s1363801607813
© The Author(s) 2016
Received: 14 July 2016
Accepted: 26 November 2016
Published: 15 December 2016
Abstract
The key challenge for inband fullduplex wireless communication is managing selfinterference. Many designs have employed spatial isolation mechanisms, such as shielding or multiantenna beamforming, to isolate the selfinterference waveform from the receiver. Because such spatial isolation methods confine the transmit and receive signals to a subset of the available space, the full spatial resources of the channel may be underutilized, expending a cost that may nullify the net benefit of operating in fullduplex mode. In this paper, we leverage an antennatheorybased channel model to analyze the spatial degrees of freedom available to a fullduplex capable base station. We observe that whether or not spatial isolation outperforms timedivision (i.e., halfduplex) depends heavily on the geometric distribution of scatterers. Unless the angular spread of the objects that scatter to the intended users is overlapped by the spread of objects that backscatter to the base station, then spatial isolation outperforms time division, otherwise time division may be optimal.
1 Introduction
Currently deployed wireless communications equipment operates in halfduplex mode, meaning that transmission and reception are orthogonalized either in time (timedivisionduplex) or frequency (frequencydivisionduplex). Research in recent years [1–12] has investigated the possibility of wireless equipment operating in fullduplex mode, meaning that the transceiver will both transmit and receive at the same time and in the same spectrum. A potential benefit of fullduplex is illustrated in. User 1 wishes to transmit uplink data to a base station, and User 2 wishes to receive downlink data from the same base station. If the base station is halfduplex, then it must either service the users in orthogonal time slots or in orthogonal frequency bands. However, if the base station can operate in fullduplex mode, then it can enhance spectral efficiency by servicing both users simultaneously. The challenge to fullduplex communication, however, is that the base station transmitter generates highpowered selfinterference which potentially swamps its own receiver, precluding the detection of the uplink message.^{1}
For fullduplex to be feasible, the selfinterference must be suppressed. The two main approaches to selfinterference suppression are cancellation and spatial isolation, and we now define each. Selfinterference cancellation is any technique which exploits the foreknowledge of the transmit signal by subtracting an estimate of the selfinterference from the received signal. Cancellation can be applied at digital baseband, at analog baseband, at RF, or, as is most common, applied at a combination of these three domains [4–7, 11, 13, 14]. Spatial isolation is any technique to spatially orthogonalize the selfinterference and the signalofinterest. Some spatial isolation techniques studied in the literature are multiantenna beamforming [1, 15–19], directional antennas [20], shielding via absorptive materials [21], and crosspolarization of transmit and receive antennas [10, 21]. The key differentiator between cancellation and spatial isolation is that cancellation requires and exploits knowledge of the selfinterference, while spatial isolation does not. To our knowledge, all fullduplex designs to date have required both cancellation and spatial isolation in order for fullduplex to be feasible even at very short ranges (i.e., <10 m). For example, see designs such as [5, 6, 10, 11], each of which leverages cancellation techniques as well as at least one spatial isolation technique. Moreover, because cancellation performance is limited by transceiver impairments such as phase noise [22], spatial isolation often accounts for an outsized portion of the overall selfinterference suppression.
For example, in the fullduplex design of [21] which demonstrated fullduplex feasibility at WiFi ranges, of the 95 dB of selfinterference suppression achieved, 70 dB is due to spatial isolation, while only 25 dB is due to cancellation. Therefore, if fullduplex feasibility is to be extended from WiFitypical ranges to the ranges typical of femtocells or even larger cells, then excellent spatial isolation performance will be required, hence our focus is on spatial isolation in this paper.
In a previous work [21], we studied three passive techniques for spatial isolation: directional antennas, absorptive shielding, and crosspolarization, and measured their performance in a prototype base station both in an anechoic chamber that mimics free space, and in a reflective room. As expected, the techniques suppressed the selfinterference quite well (more than 70 dB) in an anechoic chamber, but scattering environments, the suppression was much less, (no more than 45 dB), due to the fact that passive techniques operate primarily on the direct path between the transmit and receive antennas, and do little to suppress paths that include an external backscatterer. The directpath limitation of passive spatial isolation mechanisms raises the question of whether or not spatial isolation can be useful in a backscattering environment. Another class of spatial isolation techniques called “active” or “channel aware” spatial isolation [23] can indeed suppress both direct and backscattered selfinterference. In particular, if multiple antennas are used and if the selfinterference channel response can be estimated, then the radiation pattern can be shaped adaptively to mitigate both directpath and backscattered selfinterference. However, this pattern shaping (i.e., beamforming) will consume spatial degreesoffreedom that could have otherwise been leveraged for spatial multiplexing. Thus, there is an important tradeoff between spatial selfinterference isolation and achievable degrees of freedom.
Question: Under what scattering conditions can spatial isolation be leveraged in fullduplex operation to provide a degreeoffreedom gain over halfduplex? More specifically, given a constraint on the size of the antenna arrays at the base station and at the user devices, and given a characterization of the spatial distribution of the scatterers in the environment, what is the uplink/downlink degreeoffreedom region when the only selfinterference mitigation strategy is spatial isolation?
 1.
When the base station arrays are larger than the corresponding user arrays, the base station has a larger signal space than is needed for spatial multiplexing and can leverage the extra signal dimensions to form beams that avoid selfinterference (i.e., selfinterference zeroforcing).
 2.
More interestingly, when the forward scattering intervals and the backscattering intervals are not completely overlapped, the base station can avoid selfinterference by signaling in the directions that scatter to the intended receiver, but do not backscatter to the basestation receiver. Moreover, the base station can also signal in directions that do cause selfinterference, but ensure that the generated selfinterference is incident on the basestation receiver only in directions in which uplink signal is not incident on the basestation receiver, i.e., signal such that the selfinterference and uplink signal are spatially orthogonal.
In [27], an experimental evaluation of a transmitbeamformingbased method for fullduplex operation called “SoftNull” is presented. Inspired by the achievability proof in Section 3.1, the SoftNull algorithm presented in [27] seeks to maximally suppress selfinterference for a given required number of downlinkdegreesoffreedom. This paper presents an information theoretic analysis of the performance limits of beamformingbased fullduplex systems, whereas [27] presents an experimental evaluation of a specific design. We would like to refer [27] to readers who may be interested in how the theoretical intuitions from this paper can guide the design and implementation of a beamformingbased fullduplex system.
Organization of the paper: Section 2 specifies the system model: we begin with an overview of the PBT model in Section 2.1 and then in Section 2.2 apply the model to the scenario of a fullduplex base station with uplink and downlink flows. Section 3 gives the main analysis of the paper, the derivation of the degreesoffreedom region. We start Section 3 by stating the theorem which characterizes the degreesoffreedom region and then give the achievability and converse arguments in Sections 3.1 and 3.2, respectively. In Section 4, we assess the impact of the degreesoffreedom result on the design and deployment of fullduplex base stations, and include an application example, that shows how the results of this paper are used to guide the design of a fullduplex base station in [27]. We give concluding remarks in Section 5.
2 System model
We now give a brief overview of the PBT channel model presented in [24]. We then extend the PBT model to the case of the threenode fullduplex topology of Fig. 1, and define the required mathematical formalism that will ease the degreesoffreedom analysis in the sequel.
2.1 Overview of the PBT model
2.2 Extension of PBT model to threenode fullduplex
Now we extend the PBT channel model in [24], which considers a pointtopoint topology, to the threenode fullduplex topology of Fig. 1. The antennatheorybased PBT channel model is built upon farfield assumptions, i.e., that the propagation path is much larger than a wavelength. We acknowledge that directpath selfinterference may not obey farfield behavior. However, the backscattered selfinterference, which will travel several wavelengths to reach an external scatterer and then return to the base station, is indeed a farfield signal. As discussed in the introduction, the intent of this paper is to understand the impact of backscattered selfinterference as a function of the size of antenna arrays and the geometric distribution of the scatterers. Since the backscattering is indeed a farfield phenomenon, the PBT model is a quite wellsuited model for our study.
As in [24], we consider continuous linear arrays of infinitely many infinitesimally small unipolarized antenna elements.^{2} Each of the two transmitters T _{ j }, j=1,2, is equipped with a linear array of length \(2{L}_{T_{j}}\), and each receiver, R _{ i }, i=1,2, is equipped with a linear array of length \(2{L}_{R_{i}}\). The lengths \({L}_{T_{j}}\) and \({L}_{R_{i}}\) are normalized by the wavelength of the carrier, and thus are unitless quantities. For each array, define a local coordinate system with origin at the midpoint of the array and zaxis aligned along the length of the array. Let \(\theta _{T_{j}} \in [0,\pi)\) denote the elevation angle relative to the T _{ j } array, and let \(\theta _{R_{i}}\) denote the elevation angle relative to the R _{ i } array. Denote the current distribution on the T _{ j } array as x _{ j }(p _{ j }), where \(p_{j}\in [{L}_{T_{j}},{L}_{T_{j}}]\phantom {\dot {i}\!}\) is the position along the lengths of the array, and \(\phantom {\dot {i}\!}x_{j}:[{L}_{T_{j}},{L}_{T_{j}}] \rightarrow \mathbb {C}\) gives the magnitude and phase of the current. The current distribution, x _{ j }(p _{ j }), is the transmit signal controlled by T _{ j }, which we constrain to be square integrable. Likewise, we denote the received current distribution on the R _{ i } array as \(\phantom {\dot {i}\!}y_{i}(q_{i}),\ q_{i}\in [{L}_{R_{i}},{L}_{R_{i}}]\).
where \(\boldsymbol {\hat {k}}\) is a unit vector that gives the direction of departure from the transmitter array, and \(\boldsymbol {\hat {\kappa }}\) is a unit vector that gives the direction of arrival to the receiver array. The transmit array response kernel, \(\phantom {\dot {i}\!}\boldsymbol {A}_{T_{j}}(\boldsymbol {\hat {k}}, {p})\), maps the current distribution along the T _{ j } array (a function of p) to the field pattern radiated from T _{ j } (a function of direction of departure, \(\boldsymbol {\hat {k}}\)). The scattering response kernel, \(H_{ij}(\boldsymbol {\hat {\kappa }}, \boldsymbol {\hat {k}})\), maps the fields radiated from T _{ j } in direction \(\boldsymbol {\hat {k}}\) to the fields incident on R _{ i } at direction \(\boldsymbol {\hat {\kappa }}\). The receive array response, \(\phantom {\dot {i}\!}{A}_{R_{i}}({q},\boldsymbol {\hat {\kappa }})\), maps the field pattern incident on R _{ i } (a function of direction of arrival, \(\boldsymbol {\hat {\kappa }}\)) to the current distribution excited on the R _{ i } array (a function of position q), which is the received signal.
2.3 Array responses
where \(\phantom {\dot {i}\!}\tau \equiv \cos \theta _{R_{i}} \in [1, 1]\) is the cosine of the elevation angle relative to the R _{ i } array. Note that the transmit and receive array response kernels are identical to the kernels of the Fourier transform and inverse Fourier transform, respectively, a relationship we will further explore in Section 2.5.
2.4 Scattering responses
The scattering response kernel, \(H_{ij}(\boldsymbol {\hat {\kappa }},\boldsymbol {\hat {k}})\), gives the amplitude and phase of the path departing from T _{ j } at direction \(\boldsymbol {\hat {k}}\) and arriving at R _{ i } at direction \(\boldsymbol {\hat {\kappa }}\). Since we are considering linear arrays which only resolve the cosine of the elevation angle, we can consider H _{ ij }(τ,t) which gives the superposition of the amplitude and phase of all paths emanating from T _{ j } with an elevation angle whose cosine is t and arriving at R _{ i } at an elevation angle whose cosine is τ.
As is done in [24], motivated by measurements showing that scattering paths are clustered with respect to the transmitter and receiver, we adopt a model that focuses on the boundary of the scattering clusters rather than the discrete paths themselves, as illustrated in Fig. 2.
Let \(\phantom {\dot {i}\!}\Theta _{T_{ij}}^{(k)}\) denote the angle subtended at transmitter T _{ j } by the k ^{th} cluster that scatters to R _{ i }, and let \(\Theta _{T_{ij}} = \bigcup _{k}\Theta _{T_{ij}}^{(k)}\) be the total transmit scattering interval from T _{ j } to R _{ i }. This scattering interval, \(\phantom {\dot {i}\!}\Theta _{T_{ij}}\), is the set of directions that when illuminated by T _{ j } scatters energy to R _{ i }. In Fig. 2, to avoid clutter, we illustrate the case in which \(\phantom {\dot {i}\!}\Theta _{T_{ij}}^{(k)}\) is a single contiguous angular interval, but in general, the interval will be noncontiguous and consist of several individual clusters. Similarly let \(\phantom {\dot {i}\!}\Theta _{R_{ij}}^{(k)}\) denote the corresponding angle subtended at R _{ i } by the k ^{th} cluster illuminated by T _{ j }, and let \(\phantom {\dot {i}\!}\Theta _{R_{ij}} = \bigcup _{k}\Theta _{R_{ij}}^{(k)}\) be set of directions from which energy can be incident on R _{ i } from T _{ j }.
Define the size of the transmit and receive scattering intervals as \( \Psi _{T_{ij}} = \int _{\Psi _{T_{ij}}} t\, dt\) and \( \Psi _{R_{ij}} = \int _{\Psi _{R_{ij}}} \tau \, d\tau. \)
 1)
H _{ ij }(τ,t)≠0 only if \((\tau,t) \in \Psi _{R_{ij}}\times \Psi _{T_{ij}}\).
 2)
\(\int H_{ij}(\tau,t)dt \neq 0\ \forall \ \tau \in \Psi _{R_{ij}}\).
 3)
\(\int H_{ij}(\tau,t)d\tau \neq 0\ \forall \ t \in \Psi _{T_{ij}}\).
 4)
The point spectrum of H _{ ij }(·,·), excluding 0, is infinite.
 5)
H _{ ij }(·,·) is Lebesgue measurable, that is \(\quad \int _{1}^{1} \int _{1}^{1} H_{ij}(\tau,t)^{2} \,d\tau \,dt < \infty.\)
The first condition means that the scattering response is zero unless the angle of arrival and angle of departure both lie within their respective scattering intervals. The second condition means that in any direction of departure, \(t \in \Psi _{T_{ij}}\), there exists at least one path from transmitter T _{ j } receiver R _{ i }. Similarly, the third condition implies that in any direction of arrival, \(\tau \in \Psi _{R_{ij}}\), there exists at least one path from T _{ j } to R _{ i }. The fourth condition means that there are many paths from the transmitter to the receiver within the scattering intervals, so that the number of propagation paths that can be resolved within the scattering intervals is limited by the length of the arrays and not by the number of paths. The final condition aids our analysis by ensuring the corresponding integral operator is compact, but is also a physically justified assumption since one could argue for the stricter assumption \(\int _{1}^{1} \int _{1}^{1} H_{ij}(\tau,t)^{2} \,d\tau \,dt \leq 1\), since no more energy can be scattered than is transmitted.
2.5 Hilbert space of wavevectors
The channel model of (7) and (8) is expressed in the array domain, that is the transmit and receive signals are expressed as the current distributions excited along the array. Just as one can simplify a signal processing problem by leveraging the Fourier integral to transform from the time domain to the frequency domain, we can leverage the transmit and receive array responses to transform the problem from the array domain to the wavevector domain. In other words, we can express the transmit and receive signals as field distributions over direction rather than current distributions over position along the array. In fact, for our case of the unipolarized linear array, the transmit and receive array responses are the Fourier and inverseFourier integral kernels, respectively.
where \(X_{j} \in \mathcal {T}_{j}\), for j=1,2 and \(Y_{i}, Z_{i} \in \mathcal {R}_{i}\) for i=1,2. The following lemma states key properties of the scattering operators in (12–13) that we will leverage in our analysis.
Lemma 1
 1.
The scattering operator, \(\mathsf {H}_{ij}: \mathcal {T}_{j}\rightarrow \mathcal {R}_{i}\), is a compact operator.
 2.
The dimension of the range of the scattering operator, dim R(H _{ ij })≡dim N(H _{ ij })^{⊥}, (i.e., the dimension of the space orthogonal to the operator’s nullspace) is given by \(\phantom {\dot {i}\!}\text {dim}\,R(\mathsf {H}_{ij}) = 2\,\text {min}\{{L}_{T_{j}} \Psi _{T_{ij}}, {L}_{R_{i}} \Psi _{R_{ij}} \}. \)
 3.
There exists a singular system \(\left \{\sigma _{ij}^{(k)}, U_{ij}^{(k)}, V_{ij}^{(k)}\right \}_{k=1}^{\infty }\) for scattering operator H _{ ij }, where the singular value \(\sigma _{ij}^{(k)}\) is nonzero if and only if \(k\leq 2\,\text {min}\{{L}_{T_{j}} \Psi _{T_{ij}}, {L}_{R_{i}} \Psi _{R_{ij}} \}\).
Proof
Property 1 holds because H _{ ij }(·,·), the kernel of integral operator H _{ ij }, is square integrable, and an integral operator with a square integrable kernel is compact (see Theorem 8.8 of [32]). Property 2 is just a restatement of the main result of [24]. Property 3 follows from the first two properties: The compactness of H _{ ij }, established in Property 1, implies the existence of a singular system, since there exists a singular system for any compact operator (see Section 16.1 of [32]). Property 2 implies that only the first \(2\,\text {min}\{{L}_{T_{j}} \Psi _{T_{ij}}, {L}_{R_{i}} \Psi _{R_{ij}} \}\) of the singular values will be nonzero, since the \(\left \{U_{ij}^{(k)}\right \}\) corresponding to nonzero singular values form a basis for R(H _{ ij }), which has dimension \(2\,\text {min}\{{L}_{T_{j}} \Psi _{T_{ij}}, {L}_{R_{i}} \Psi _{R_{ij}} \}\). See Lemma 5 in Appendix B for a description of the properties of singular systems for compact operators, or see Section 2.2 of [33] or Section 16.1 of [32] for a thorough treatment. □
3 Spatial degreesoffreedom analysis
We now give the main result of the paper: a characterization of the spatial degreesoffreedom region for the PBT channel model applied to a fullduplex base station with uplink and downlink flows.
Theorem 1
3.1 Achievability
 1.
First, the uplink transmitter, T _{1}, transmits the maximum number of data streams that the uplink channel will support, \(d_{1}^{\mathsf {max}} = 2\,\text {min} ({L}_{T_{1}} \Psi _{T_{11}}, {L}_{R_{1}} \Psi _{R_{11}})\) (illustrated by blue arrows in Fig. 5 b). The base station downlink transmitter, T _{2}, must then structure its transmit signal such that it does not interfere with the base station receiver’s reception of these \(d_{1}^{\mathsf {max}}\) data streams, as is described in the following steps.
 2.
Second, the base station transmitter, T _{2}, transmits as many data streams as can be supported in the interval \(\Psi _{T_{22}} \setminus \Psi _{T_{12}}\) (illustrated by red arrows in Fig. 5 b), which is the interval over which signal will couple to the downlink user R _{2}, but will not present any selfinterference to the base station’s own receiver R _{1}.
 3.
Third, the base station transmits as many data streams as possible in the interval \(\Psi _{T_{22}} \cap \Psi _{T_{12}}\) while ensuring that the selfinterference is only incident on the base station receiver, R _{1} over the interval \(\Psi _{R_{11}} \setminus \Psi _{R_{12}}\) (illustrated by green arrows in Fig. 5 b), which is the interval over which no uplink signal form T _{1} will be incident on receiver R _{1}. This step occupies a majority of the proof.
The final step in the achievability proof is to show that if the transmission strategies described in steps 1–3 are employed, that the receivers, R _{1} and R _{2}, can successfully recover the d _{1} and d _{2}dimensional data streams, respectively.
Full achievability proof: We establish achievability of \(\mathcal {D}_{\mathsf {FD}}\) by way of two lemmas. The first lemma shows the achievability of two specific spatial degreesoffreedom tuples, and the second shows that these tuples are indeed the corner points of \(\mathcal {D}_{\mathsf {FD}}\).
Lemma 2
where the terms \(\phantom {\dot {i}\!}d_{T_{2}}\), \(\phantom {\dot {i}\!}\delta _{T_{2}}d_{R_{1}}\), and \(\phantom {\dot {i}\!}\delta _{R_{1}}\) are given in (17–20) within the table at the bottom of the page.
Proof
Achievability of (d1′,d2′) in the \({L}_{T_{1}} \Psi _{T_{11}} < {L}_{R_{1}} \Psi _{R_{11}}\) case is analogous.
We now begin the steps to show achievability of (25) and (26). □
3.1.1 Defining key subspaces
We first define key subspaces of the transmit and receive wavevector spaces (\(\mathcal {T}_{1}\), \(\mathcal {T}_{2}\), \(\mathcal {R}_{1}\), and \(\mathcal {R}_{2}\)) that will be crucial in demonstrating achievability.
Definition 1
We say that Hilbert space \(\mathcal {A}\) is the orthogonal direct sum of Hilbert spaces \(\mathcal {B}\) and \(\mathcal {C}\) if any \(a\in \mathcal {A}\) can be decomposed as a=b+c, for some \(b\in \mathcal {B}\) and \(c \in \mathcal {C}\), where a and b are orthogonal. We use the notation \(\mathcal {A} = \mathcal {B} \oplus \mathcal {C}\) to denote that A is the orthogonal direct sum of \(\mathcal {B}\) and \(\mathcal {C}\).
thus any \(X_{2} \in \mathcal {T}_{2}\) can be written as \(\phantom {\dot {i}\!}X_{2} = X_{2_{\mathsf {Orth}}} + X_{2_{\mathsf {Int}}}\), for some \(\phantom {\dot {i}\!}X_{2_{\mathsf {Orth}}}\in \mathcal {T}_{22\setminus 12}\) and \(\phantom {\dot {i}\!}X_{2_{\mathsf {Int}}}\in \mathcal {T}_{12}\), such that \(\phantom {\dot {i}\!}X_{2_{\mathsf {Orth}}} \perp X_{2_{\mathsf {Int}}}.\) By the construction of \(\mathcal {T}_{22\setminus 12}\), \(\phantom {\dot {i}\!}\mathsf {H}_{12}X_{2_{\mathsf {Orth}}} = 0\), since \(H_{12}(\tau,t) = 0\forall \, t\notin \Psi _{T_{12}}\phantom {\dot {i}\!}\) and \(\phantom {\dot {i}\!}X_{2_{\mathsf {Orth}}}\in \mathcal {T}_{22\setminus 12}\) implies \(\phantom {\dot {i}\!}X_{2_{\mathsf {Orth}}}(t) = 0\ \forall \ t \in \Psi _{T_{12}}\). In other words, \(\phantom {\dot {i}\!}X_{2_{\mathsf {Orth}}}\in \mathcal {T}_{22\setminus 12}\) is zero everywhere the integral kernel H _{12}(τ,t) is nonzero. Thus, any transmitted field distribution that lies in the subspace \(\mathcal {T}_{22\setminus 12}\) will not present any interference to R _{2}.
Note that \(\mathcal {T}_{11}=\mathcal {T}_{1}\), since we have assumed \(\Psi _{T_{21}} = \emptyset \). Although \(\mathcal {T}_{11}\) is thus redundant, we define it for notational consistency.
Now that we have defined the relevant subspaces, we can show how these subspaces are leveraged in the transmission and reception scheme that achieves the spatial degreesoffreedom tuple (d1′,d2′).
3.1.2 Spatial processing at each transmitter/receiver
We now give the transmission schemes at each transmitter and the recovery schemes at each receiver.
Processing at uplink user transmitter, T _{ 1 } : Recall that \(d_{1}' = \text {dim} \mathcal {R}_{11}\) is the number of spatial degreesoffreedom we wish to achieve for Flow _{1}, the uplink flow. Let \(\left \{ \chi _{1}^{(k)} \right \}_{k=1}^{d_{1}'},\ \chi _{1}^{(i)} \in \mathbb {C},\) be the d1′ symbols that T _{1} wishes to transmit to R _{1}. We know from Lemma 1 that there exists a singular value expansion for H _{11}, so let \(\left \{\sigma _{11}^{(k)}, U_{11}^{(k)}, V_{11}^{(k)}\right \}_{k=1}^{\infty }\) be a singular system for the operator \(\mathsf {H}_{11}: \mathcal {T}_{1}\rightarrow \mathcal {R}_{1}\) (see Lemma 5 in Appendix B for the definition of a singular system).
Note that the functions \( \left \{ V_{11}^{(k)} \right \}_{k=1}^{\text {dim} \mathcal {T}_{1}}\) form an orthonormal basis for \(\mathcal {T}_{1}\), and since \(d_{1}' = \text {dim} \mathcal {R}_{11} \leq \text {dim} \mathcal {T}_{1}\), there are at least as many such basis functions as there are symbols to transmit.
Processing at the base station transmitter, T _{ 2 } :
Now since \(\phantom {\dot {i}\!}X_{2_{\mathsf {Int}}}\in \mathcal {T}_{12}\), \(\phantom {\dot {i}\!}\mathsf {H}_{12}X_{2_{\mathsf {Int}}}\) is nonzero in general, \(\phantom {\dot {i}\!}X_{2_{\mathsf {Int}}}\) will present interference to R _{1}. Therefore, we must construct \(\phantom {\dot {i}\!}X_{2_{\mathsf {Int}}}\) such that it communicates \(\phantom {\dot {i}\!}d_{2_{\mathsf {Int}}}'\) symbols to R _{2}, without impeding R _{1} from recovering the d1′ symbols transmitted from T _{1}. Thus, the construction of \(\phantom {\dot {i}\!}X_{2_{\mathsf {Int}}}\in \mathcal {T}_{12}\) will indeed depend on the structure of H _{12}.
Now that we have constructed X _{1}, the uplink wavevector signal transmitted on the the uplink user, and X _{2}, the wavevector signal transmitted on the dowlink by the base station, we show how the base station receiver, R _{1} and the downlink user R _{2} process their received signals to detect the original informationbearing symbols.
Processing at the base station receiver, R _{ 1 } : We need to show that R _{1} can obtain at least \(d_{1}' = \text {dim} \mathcal {R}_{11}\) independent linear combinations of the d1′ symbols transmitted from T _{1}, and that each of these linear combinations are corrupted only by noise, and not interference from T _{2}.
and thus obtains each of the \(d_{1}'=\text {dim} \mathcal {R}_{11}\) linear combinations of the intended symbols corrupted only by noise, as desired. Moreover, in this case the obtained linear combinations are already diagonalized, with the lth projection only containing a contribution from the lth desired symbol.
Thus, as desired, in all cases the base station receiver R _{1} is able to obtain d1′ interferencefree linear combinations of the d1′ symbols from the uplink user transmitter T _{1}. Now, we move to the processing at the downlink user receiver.
3.1.3 Reducing to parallel pointtopoint vector channels
respectively, where the linear combination coefficients, \(a_{1}^{(lm)}\) and \(a_{2}^{(lm)}\), are given in (77) and (81), respectively, and the additive noise on each of the recovered symbols, \(\zeta _{1}^{(l)}\) and \(\zeta _{2}^{(l)}\), are given in (78) and (82), respectively.
where χ _{1} and χ _{2} are the d1′×1 and d2′×1 vectors of input symbols for transmitters T _{1} and T _{2}, respectively, ζ _{1} and ζ _{2} are the d1′×1 and d2′×1 vectors of additive noise, respectively, and A _{1} and A _{2} are d1′×d1′ and d2′×d2′ square matrices whose elements are taken from \(a_{1}^{(lm)}\) and \(a_{2}^{(lm)}\), respectively. The matrices A _{1} and A _{2} will be full rank for all but a measurezero set of channel response kernels. Also, since each of the \(\zeta _{j}^{(l)}\)’s are linear combinations of Gaussian random variables, the the noise vectors, ζ _{1} and ζ _{2}, are Gaussian distributed. Therefore, the spatial processing has reduced the original channel to two parallel fullrank Gaussian vector channels: the first a d1′×d1′ channel and the second a d2′×d2′ channel, which are well known to have d1′ and d2′ degreesoffreedom, respectively [34]. Therefore, the spatial degreesoffreedom pair (d1′,d2′) is indeed achievable.
Lemma 3
Proof
Verifying that the corner points of inner and outer bounds coincide
Case  \(d_{2}' = \text {min}\left \{d_{2}^{\mathsf {max}}, d_{\mathsf {sum}}^{\mathsf {max}}  d_{1}^{\mathsf { max}}\right \}\) 

\({L}_{T_{1}}\Psi _{T_{11}} \geq {L}_{R_{1}}\Psi _{R_{11}},\)  
\(\text {min}\{d_{2}^{\mathsf {max}},\ 2{L}_{T_{2}}\Psi _{T_{22}}\cup \Psi _{T_{12}} \)  
\({\phantom {0000}}2{L}_{R_{1}}\Psi _{R_{11}}\cap \Psi _{R_{12}} \}\)  
\({L}_{T_{2}}\Psi _{T_{12}} \geq {L}_{R_{1}}\Psi _{R_{12}}\)  
\( {L}_{T_{1}}\Psi _{T_{11}} \geq {L}_{R_{1}}\Psi _{R_{11}},\)  
\(\text {min}\{d_{2}^{\mathsf {max}},\ 2{L}_{T_{2}}\Psi _{T_{22}}\setminus \Psi _{T_{12}}\)  
\(+ 2{L}_{R_{1}} \Psi _{R_{12}}\setminus \Psi _{R_{11}}\}\)  
\( {L}_{T_{2}}\Psi _{T_{12}} < {L}_{R_{1}}\Psi _{R_{12}}\)  
\( {L}_{T_{1}}\Psi _{T_{11}} < {L}_{R_{1}}\Psi _{R_{11}},\)  
\(\text {min}\{d_{2}^{\mathsf {max}},\ 2{L}_{T_{2}}\Psi _{T_{22}}\cup \Psi _{T_{12}} \)  
\(+ 2{L}_{R_{1}}\Psi _{R_{11}}\setminus \Psi _{R_{12}}  2{L}_{T_{1}}\Psi _{T_{11}} \}\)  
\( {L}_{T_{2}}\Psi _{T_{12}} \geq {L}_{R_{1}}\Psi _{R_{12}}\)  
\( {L}_{T_{1}}\Psi _{T_{11}} < {L}_{R_{1}}\Psi _{R_{11}},\)  
\(\text {min}\{d_{2}^{\mathsf {max}},\ 2{L}_{T_{2}}\Psi _{T_{22}}\setminus \Psi _{T_{12}}\)  
\(+ 2{L}_{R_{1}}\Psi _{R_{11}}\cup \Psi _{R_{12}}  2{L}_{T_{1}}\Psi _{T_{11}} \}\)  
\( {L}_{T_{2}}\Psi _{T_{12}} < {L}_{R_{1}}\Psi _{R_{12}}\) 
Lemmas 2 and 3 show that the corner points of \(\mathcal {D}_{\mathsf {FD}}\), (d1′,d2′) and (d1″,d2″) are achievable. And, thus, all other points within \(\mathcal {D}_{\mathsf {FD}}\) are achievable via time sharing between the schemes that achieve the corner points.
3.2 Converse
To establish the converse part of Theorem 1, we must show that the region \(\mathcal {D}_{\mathsf {FD}}\), which we have already shown is achievable, is also an outer bound on the degreesoffreedom, i.e., we want to show that if an arbitrary degreeoffreedom pair (d _{1},d _{2}) is achievable, then \((d_{1},d_{2}) \in \mathcal {D}_{\mathsf {FD}}\). It is easy to see that if (d _{1},d _{2}) is achievable, then the singeuser constraints on \(\mathcal {D}_{\mathsf {FD}}\), given in (14) and (15), must be satisfied as the degreesoffreedom for each flow cannot be more than the pointtopoint degreesoffreedom shown in [24]. Thus, the only step remaining in the converse is to establish an outer bound on the sum degreesoffreedom which coincides with \(d_{\mathsf {sum}}^{\mathsf {max}}\), the sumdegreesoffreedom constraint on the achievable region, \(\mathcal {D}_{\mathsf {FD}}\), given in (16).
Thus, to conclude the converse argument, we will now prove the following Genieaided outer bound on the sum degreesoffreedom which coincides with the sumdegreesoffreedom constraint on the achievable region.
Lemma 4

1) First, a genie expands the transmit scattering intervals \(\Psi _{T_{22}}\) and \(\Psi _{T_{12}}\) until the two intervals are fully overlapped, and likewise expands expands \(\Psi _{R_{11}}\) and \(\Psi _{R_{12}}\) until they are fully overlapped, as shown in Fig. 6. To ensure that the net manipulation of the genie can only enlarge \(\mathcal {D}_{\mathsf {FD}}\), the genie also increases the array lengths \({L}_{T_{2}}\) and \({L}_{R_{1}}\) sufficiently for any added interference due to the expansion of \(\Psi _{T_{12}}\) and \(\Psi _{R_{12}}\) to be compensated by the increased array lengths.

2) After the above genie manipulation is performed, the maximum of the T _{2} and R _{1} signaling dimensions are equal to \(d_{\mathsf {sum}}^{\mathsf {max}}\) in constraint (16), and since the scattering intervals are overlapped, the channel model becomes the Hilbert space equivalent of the wellstudied MIMO Zchannel [35, 36]. The Hilbert space analog to the bounding techniques employed in [35, 36] are then leveraged to conclude the converse proof.
Proof
We prove Lemma 4 by way of a Genie that aids the transmitters and receivers by enlarging the scattering intervals and lengthening the antenna arrays in a way that can only enlarge the degreesoffreedom region. Applying the pointtopoint bounds to the Genieaided system in a careful way then establishes the outer bound. Assume an arbitrary scheme achieves the degreesoffreedom pair (d _{1},d _{2}). Thus receivers R _{1} and R _{2} can decode their corresponding messages with probability of error approaching zero. We must show that the assumption of (d _{1},d _{2}) being achievable implies the constraint in Eq. (91).
We see in (99) that the Genie’s lengthening of T _{2}’s array to \(2{L'}_{T_{2}}\) increases the dimension of T _{2}’s transmit signal space by \(2{L}_{R_{1}} \Psi _{R_{11}}\setminus \Psi _{R_{12}}\), which is the worst case increase in the dimension of the subspace of R _{1}’s receive subspace vulnerable to interference from T _{2}. Therefore, T _{1} can leverage these extra \(2{L}_{R_{1}} \Psi _{R_{11}}\setminus \Psi _{R_{12}}\) dimensions to zero force to the subspace of R _{1}’s receive space that has become vulnerable to interference from T _{2} due to the expansion \(\Psi _{R_{12}}\) to \(\Psi '_{R_{1}}\). Thus, the net effect of the Genie’s expansion of T _{2}’s interference scattering interval, \(\Psi _{R_{12}}\), to \(\Psi '_{R_{1}}\) and lengthening of the T _{2} array to \(2{L'}_{T_{2}}\) can only enlarge the degreesoffreedom region.
then the converse is established. Because the Genieaided channel is now fully coupled, it is similar to the continuous Hilbert space analog of the fullrank discreteantennas MIMO Z interference channel. Thus, the remaining steps in the converse argument are inspired by the techniques used in [35–37] for outer bounding the degreesoffreedom of the MIMO interference channel.
Consider the case in which \(\text {dim} \,\mathcal {T}'_{2} \leq \text {dim}\,\mathcal {R}'_{1}\). Since our Genie has enforced \(\Psi '_{T_{22}} = \Psi '_{T_{12}}\) and we have assumed \(\text {dim}\, \mathcal {T}'_{2} \leq \text {dim}\,\mathcal {R}'_{1}\), receiver R _{1} has access to the entire signal space of T _{2}, i.e., T _{2} cannot zero force to R _{1}. Moreover, by our hypothesis that (d _{1},d _{2}) is achieved, R _{1} can decode the message from T _{1}, and can thus reconstruct and subtract the signal received from T _{1} from its received signal.
Since R _{1} has access to the entire signalspace of T _{2}, after removing the signal from T _{1} the only barrier to R _{1} also decoding the message from T _{2} is the receiver noise process. If it is not already the case, let a Genie lower the noise at receiver R _{1} until T _{2} has a better channel to R _{1} than R _{2} (this can only increase the capacity region since R _{1} could always locally generate and add noise to obtain the original channel statistics). By hypothesis, R _{2} can decode the message from T _{2}, and since T _{2} has a better channel to R _{1} than R _{2}, R _{1} can also decode the message from T _{1}.
thus showing that the sumdegreesoffreedom bound of Eq. (16) in Theorem 1 must hold for any achievable degreeoffreedom pair. □
Combining Lemma 4 with the trivial pointtopoint bounds establishes that the region \(\mathcal {D}_{\mathsf {FD}}\), given in Theorem 1, is an outer bound on any achievable degreesoffreedom pair, thus establishing the converse part of Theorem 1.
4 Impact on fullduplex design
We have characterized, \(\mathcal {D}_{\mathsf {FD}}\), the degreesoffreedom region achievable by a fullduplex basestation which uses spatial isolation to avoid selfinterference while transmitting the uplink signal while simultaneously receiving. Now, we wish to discuss how this result impacts the operation of fullduplex base stations. In particular, we aim to ascertain in what scenarios fullduplex with spatial isolation outperforms halfduplex, and are there scenarios in which fullduplex with spatial isolation achieves an ideal rectangular degreesoffreedom region (i.e., both the uplink flow and downlink flow achieving their respective pointtopoint degreesoffreedom).
where α∈[0,1] is the time sharing parameter. Obviously \(\mathcal {D}_{\mathsf {HD}}\subseteq \mathcal {D}_{\mathsf {FD}}\), but we are interested in contrasting the scenarios for which \(\mathcal {D}_{\mathsf {HD}}\subset \mathcal {D}_{\mathsf {FD}}\), and fullduplex spatial isolation strictly outperforms halfduplex time division, and the scenarios for which \(\mathcal {D}_{\mathsf {HD}}=\mathcal {D}_{\mathsf {FD}}\) and halfduplex can achieve the same performance as fullduplex. We will consider two particularly interesting cases: the fully spread environment, and the symmetric spread environment.
4.1 Overlapped scattering case
The following remark characterizes the scenarios for which fullduplex with spatial isolation beats halfduplex.
Remark
In the overlapped scattering case, \(\mathcal {D}_{\mathsf {HD}} \subset \mathcal {D}_{\mathsf {FD}}\) when 2L _{ BS }>2L _{ Usr }, else \(\mathcal {D}_{\mathsf {HD}} = \mathcal {D}_{\mathsf {FD}}\).
We see that fullduplex outperforms halfduplex only if the base station arrays are longer than the user arrays. This is because in the overlapped scattering case, the only way to spatially isolate the selfinterference is zero forcing, and zero forcing requires extra antenna resources at the base station. When 2L _{ BS }≤2L _{ Usr }, the base station has no extra antenna resources it can leverage for zero forcing, and thus, spatial isolation of the selfinference is no better than isolation via time division. However, when 2L _{ BS }>2L _{ Usr }, the base station transmitter can transmit (2L _{ BS }−2L _{ Usr })Ψ zeroforced streams on the downlink without impeding the reception of the the full 2L _{ Usr }Ψ streams on the uplink, enabling a sumdegreesoffreedom gain of (2L _{ BS }−2L _{ Usr })Ψ over halfduplex. Indeed when the base station arrays are at least twice as long as the user arrays, the degreesoffreedom region is rectangular, and both uplink and downlink achieve the ideal 2L _{ Usr }Ψ degreesoffreedom.
4.2 Symmetric spread
The previous overlapped scattering case is the worst case for full duplex operation. Let us now consider the more general case where the selfinterference backscattering and the signalofinterest forward scattering are not perfectly overlapped. This case illustrates the impact of the overlap of the scattering intervals on fullduplex performance. Once again, to reduce the number of variables, we will make following symmetry assumptions. Assume all the arrays in the network, the two arrays on the base station as well as the array on each of the user devices, are of the same length 2L, that is \(2{L}_{T_{1}} = 2{L}_{R_{1}} = 2{L}_{T_{2}} = 2{L}_{R_{2}} \equiv 2L.\) Also, assume that the size of the forward scattering intervals to/from the intended receiver/transmitter is the same for all arrays \(\Psi _{T_{11}} = \Psi _{R_{11}} = \Psi _{T_{22}} = \Psi _{R_{22}} \equiv \Psi _{\mathsf {Fwd}},\) and that the size of the backscattering interval is the same at the base station receiver as at the base station trasmitter \( \Psi _{T_{12}} = \Psi _{R_{12}} \equiv \Psi _{\mathsf {Back}}.\) Finally, assume the amount of overlap between the backscattering and the forward scattering is the same at the base station transmitter as at the base station receiver so that \(\Psi _{T_{22}}\cap \Psi _{T_{12}} = \Psi _{R_{11}}\cap \Psi _{R_{12}} \equiv \Psi _{\mathsf {Fwd}} \cap \Psi _{\mathsf {Back}} = \Psi _{\mathsf {Fwd}}  \Psi _{\mathsf {Fwd}} \setminus \Psi _{\mathsf {Back}}.\)
Remark
Comparing \(\mathcal {D}_{\mathsf {FD}}\) and \(\mathcal {D}_{\mathsf {HD}}\) above we see that in the case of symmetric scattering, \(\mathcal {D}_{\mathsf {HD}} = \mathcal {D}_{\mathsf {FD}}\) if and only if Ψ _{ Fwd }=Ψ _{ Back }, else \(\mathcal {D}_{\mathsf {HD}} \subset \mathcal {D}_{\mathsf {FD}}\) (we are neglecting the trivial case of L=0).
Thus, the fullduplex spatial isolation region is strictly larger than the halfduplex timedivision region unless the forward interval and the backscattering interval are perfectly overlapped. The intuition is that when Ψ _{ Fwd }=Ψ _{ Back } the scattering interval is shared resource, just as is time, thus trading spatial resources is equivalent to trading timeslots. However, if Ψ _{ Fwd }≠Ψ _{ Back }, there is a portion of space exclusive to each user which can be leveraged to improve upon time division. Moreover, inspection of \(\mathcal {D}_{\mathsf {FD}}\) above leads to the following remark.
Remark
Straightforward setalgebraic manipulation of condition (122) shows that it is equivalent to (121).
The intuition is that because Ψ _{ Back }∖Ψ _{ Fwd } are the set directions in which the base station couples to itself but not to the users, the corresponding 2LΨ _{ Back }∖Ψ _{ Fwd } dimensions are useless for spatial multiplexing, and therefore “free” for zero forcing the selfinterference, which has maximum dimension 2LΨ _{ Fwd }∩Ψ _{ Back }. Thus, when Ψ _{ Back }∖Ψ _{ Fwd }≥Ψ _{ Fwd }∩Ψ _{ Back }, we can zero force any selfinterference that is generated, without sacrificing any resource needed for spatial multiplexing to intended users.
The overall takeaway is that as the amount of backscattering increases, more degrees of freedom must be sacrificed to achieve sufficient selfinterference suppression, and less degrees of freedom are left for signaling to the desired users. Therefore, fullduplex operation, where selfinterference is suppressed by beamforming, is indeed feasible, when the antenna array at the base station is sufficiently large, and when the backscattering is sufficiently limited.
4.3 Simulation example
We now consider a simple simulation example that illustrates the results of Theorem 1. In particular, this example illustrates that as the angular spread of the backscattering increases, more transmit degrees of freedom must be sacrificed in order to sufficiently suppress selfinterference at the base station. Theorem 1 was derived under several theoretical assumptions which are relaxed in this simulation to show that the same trends still apply. The channel model of (2) focuses on backscattering only; in this simulation, we also consider the directpath selfinterference from the transmit array to the receive array. Moreover, Theorem 1 was derived using continuous linear arrays, but to make the simulation closer to practical implementations, we consider discrete arrays rather than continuous arrays.
where C _{12} is the continuous selfinterference channel response described in Eqs. (2)–(8). We generate channel realizations by drawing H(τ,t) from a twodimensional white gaussian process over \((\tau,t) \in \Psi _{R_{12}} \times \Psi _{T_{12}}\), and set H(τ,t)=0 for \((\tau,t) \notin \Psi _{R_{12}} \times \Psi _{T_{12}}\). As in the symmetricspread example, for convenience we let \(\Psi _{T_{12}} = \Psi _{R_{12}} = \Psi _{\mathsf {Back}}\). The total selfinterference channel is H _{ self }=C _{ direct }+α C _{ scat }., where α is a scalar chosen such that the backscattered selfinterference is 20 dB weaker (on average) than the directpath selfinterference. We assume the noise floor is 80 dB below the transmit signal power. We consider the case of no backscattering, as well as cases where the backscattering subtends angles of 15°, 45°, 90°, and the fully backscattered case where the backscattering subtends 180°.
We simulate a transmit beamforming scheme inspired by the degreesoffreedom achievability proof of section 3. Let d _{ T } denote the dimension of the base stations transmit signal (i.e., the number of data streams the base station wishes to transmit).^{4} In the achievability proof, the base station transmitter avoids selfinterference by projecting the d _{ T } transmit symbols onto the nullspace of the selfinterference channel. Here, we generalize this nullspaceprojection approach by having the base station transmitter project its d _{ T } transmit symbols onto the d _{ T } weakest singular vectors (i.e., the d _{ T } left singular vectors corresponding to the d _{ T } smallest singular values) of the selfinterference channel, H _{ self }. This beamforming approach, which we call “soft nulling” allows a flexible tradeoff between number of downlink dimensions, d _{ T }, and the amount of selfinterference generated: better selfinterference suppression can be achieved by sacrificing transmit dimension. This concept of soft nulling is explored in depth in [27].
5 Conclusions
Fullduplex operation presents an opportunity for base stations to as much as double their spectral efficiency by both transmitting downlink signal and receiving uplink signal at the same time in the same band. The challenge to fullduplex operation is highpowered selfinterference that is received both directly from the base station transmitter and backscattered from nearby objects. The receiver can be spatially isolated from the transmitter by leveraging multiantenna beamforming to avoid selfinterference, but such beamforming can also decrease the degreesoffreedom of the intended uplink and downlink channels. We have leveraged a spatial antennatheorybased channel model to analyze the spatial degreesoffreedom available to a fullduplex base station. The analysis has shown the fullduplex operation can indeed outperform halfduplex operation when either (1) the base station arrays are large enough for the base station to zeroforce the backscattered selfinterference or (2) the backscattering directions are not fully overlapped with the forward scattering directions, so that the base station can leverage the nonoverlapped intervals for interference free signaling to/from the intended users.
6 Endnotes
^{1} An additional challenge is the potential for the uplink user’s transmission to interfere with the downlink user’s reception, but in this paper we focus solely on the challenge of selfinterference.
^{2} We acknowledge that a continuous array which can support arbitrary current distributions may not be feasible to construct in practice due to the complications of feeding the array and achieving impedance match. However, as has been shown in the work of [24–26], a continuous array is nonetheless a very useful theoretical construct to develop performance bounds for any discrete antenna array subject to the same size constraint.
^{3} There is extensive ongoing research on scheduling algorithms to select uplink and downlink users such that the uplink user generates little interference to the downlink user [40–44] (and references within). Thus, we make the simplifying assumption that there is no channel from the uplink transmitter, T _{1}, to the downlink receiver, R _{2}. This assumption allows the analysis to focus on the challenge of backscattered selfinterference. An extension of this work, [45], which is outside the scope of this paper, focuses on the challenge of interuser interference in a fullduplex network, and provides analysis for the case where there is a nonzero channel from T _{1} to R _{2}.
^{4} We call d _{ T } the “dimension of the transmit signal” instead of “degrees of freedom”, because in this simulation, where SNR is finite, the term “degrees of freedom” is not correct by the rigorous definition used in the prior analysis.
7 Appendix A: Functional analysis definitions
Let \(\mathcal {X}\) be a Hilbert space, the orthogonal complement of \(\mathcal {S} \subseteq \mathcal {X}\), denoted \(\mathcal {S}^{\perp }\), is the subset \( \mathcal {S}^{\perp } \equiv \{x \in \mathcal {X}: \langle x,u\rangle =0\ \forall \ u\in \mathcal {S} \}.\) Let \(\mathcal {X}\) and \(\mathcal {Y}\) be vector spaces (e.g., Hilbert spaces) and let \(\mathsf {C}:\mathcal {X}\rightarrow \mathcal {Y}\) be a linear operator. Let \(\mathcal {S} \subseteq \mathcal {Y}\) be a subspace of \(\mathcal {Y}\). The nullspace of C, denoted N(C), is the subspace \(N(\mathsf {C}) \equiv \{x\in \mathcal {X}: \mathsf {C}x = 0 \}.\) The range of C, denoted R(C), is the subspace \(R(\mathsf {C}) \equiv \{\mathsf {C}x: x\in \mathcal {X}\}.\) The preimage of \(\mathcal {S}\) under C, \({\mathsf {C}}^{\leftarrow }(\mathcal {S})\), is the subspace (one can check that if \(\mathcal {S}\) is a subspace then \({\mathsf {C}}^{\leftarrow }(\mathcal {S})\) is a subspace also). \({\mathsf {C}}^{\leftarrow }(\mathcal {S}) \equiv \{x \in \mathcal {X}: \mathsf {C}x \in \mathcal {S} \}.\) The rank of C is the dimension of the range of C. A fundamental result in functional analysis is that the dimension of the range of C is also the dimension of the orthogonal complement of the nullspace of C (i.e. the coimage of C) so that we can write \( \mathop {\text {rank}} \mathsf {C} \equiv \text {dim} \,R(\mathsf {C}) = \text {dim}\, N(\mathsf {C})^{\perp }. \)
8 Appendix B: functional analysis lemmas
Lemma 5
Let \(\mathcal {X}\) and \(\mathcal {Y}\) be Hilbert spaces and let \(\mathsf {C}:\mathcal {X}\rightarrow \mathcal {Y}\) be a compact linear operator. There exists a singular system {σ _{ k },v _{ k },u _{ k }}, for C defined as follows. The set of functions {u _{ k }} form an orthonormal basis for \(\overline {R(\mathsf {C})}\), the closure of the range of C, and the set of functions {v _{ k }} form an orthonormal basis for N(C)^{⊥}, the coimage of C. The set of positive real numbers σ _{ k }, called the singular values of C, are the nonzero eigenvalues of (C ^{∗} C) arranged in decreasing order. The singular system diagonalizes C in the sense that for any (σ _{ k },v _{ k },u _{ k })∈{σ _{ k },v _{ k },u _{ k }}, C v _{ k }=σ _{ k } u _{ k }. Moreover, the operation of C on any \({x}\in \mathcal {X}\) can be expanded as \( \mathsf {C} {x} = \sum _{k} \sigma _{k} \langle {x}, {v}_{k} \rangle {u}_{k}, \) which is called the singular value expansion of C x. See Section 16.1 and 16.2 of [32] for a proof.
Lemma 6
Let \(\mathcal {X}\) and \(\mathcal {Y}\) be Hilbert spaces and let \(\mathsf {C}:\mathcal {X}\rightarrow \mathcal {Y}\) be a linear operator with closed range. There exists a unique linear operator C ^{+}, called the MoorePenrose pseudoinverse of C, with the following properties: (i) C ^{+} C x=x ∀x∈N(C)^{⊥} (ii) CC ^{+} y=y ∀y∈R(C) (iii) R(C ^{+})=N(C)^{⊥} (iv) N(C ^{+})=R(C)^{⊥}.
See Definition 2.2 and Proposition 2.3 of [33] for a proof.
Lemma 7
Proof
Note that \(\mathcal {B}\) is a subspace of \(\mathcal {X}\) since the intersection of any collection of subspaces is itself a subspace (see Thm. 1 on p. 3 of [46]). Every \(x\in {\mathsf {C}}^{\leftarrow }(\mathcal {S})\) can be expressed as x=w+u for some w∈N(C) and \(u \in \mathcal {B}\), and 〈w,u〉=0 for any w∈N(C) and \(u \in \mathcal {B}\). Thus, we can say that the preimage, \({\mathsf {C}}^{\leftarrow }(\mathcal {S})\), is the orthogonal direct sum of subspaces N(C) and \(\mathcal {B}\) ([32] Def. 4.26), a relationship we note we denote as \( {\mathsf {C}}^{\leftarrow }(\mathcal {S}) = N(\mathsf {C}) \oplus \mathcal {B}. \)
and since there are \(d_{\mathcal {B}}\) elements in \(\left \{\mathsf {C} e_{i}\right \}_{i=1+d_{N}}^{d_{N}+d_{\mathcal {B}}}\), it must be that \(d_{\mathcal {B}} \leq d_{R\cap \mathcal {S}}.\) Substituting the above inequality into Eq. (130) gives \(d_{P} \leq d_{N} + d_{R\cap \mathcal {S}}.\)
By property (iv) in Lemma 6, we have that C ^{+} s _{ i }∈N(C)^{⊥} for each \(\mathsf {C}^{+} s_{i}\in \left \{\mathsf {C}^{+} s_{i}\right \}_{i=1}^{d_{R\cap \mathcal {S}}}\). Since s _{ i }∈R(C), we have that C(C ^{+} s _{ i })=s _{ i } by property (ii) of the pseudoinverse, and since \(s_{i} \in \mathcal {S}\), we have that \(\mathsf {C} \mathsf {C}^{+} s_{i} = s_{i} \in \mathcal {S}\) for each \(\mathsf {C}^{+} s_{i}\in \left \{\mathsf {C}^{+} s_{i}\right \}_{i=1}^{d_{R\cap \mathcal {S}}}\). Thus, each element of \(\{\mathsf {C}^{+} s_{i}\}_{i=1}^{d_{R\cap \mathcal {S}}}\) is also in \({\mathsf {C}}^{\leftarrow }(\mathcal {S})\), the preimage of \(\mathcal {S}\) under C. Thus we have that each element of \( \left \{\mathsf {C}^{+} s_{i}\right \}_{i=1}^{d_{R\cap \mathcal {S}}}\) is in \(N(\mathsf {C})^{\perp } \cap {\mathsf {C}}^{\leftarrow }(\mathcal {S})\) which justifies the claim of Eq. (132). Now, Eq. (132) implies that \( d_{R\cap \mathcal {S}} \leq d_{\mathcal {B}} \). Substituting the above inequality into Eq. (130) gives \(d_{P} \geq d_{N} + d_{R\cap \mathcal {S}},\) concluding the proof. □
Corollary 1
Let \(\mathcal {X}\) and \(\mathcal {Y}\) be finitedimensional Hilbert spaces and let \(\mathsf {C}:\mathcal {X}\rightarrow \mathcal {Y}\) be a linear operator with closed range. Let \(\mathcal {S} \subseteq R(\mathsf {C})\subseteq \mathcal {Y}\) be a subspace of the range of C. Then, the dimension of the preimage of \(\mathcal {S}\) under C is \( \text {dim}\, {\mathsf {C}}^{\leftarrow }(\mathcal {S}) = \text {dim}\, N(\mathsf {C}) + \text {dim}\,(\mathcal {S}). \)
Proof
The proof follows trivially from Lemma 7 by noting that since \(\mathcal {S} \subseteq R(\mathsf {C})\), \(R(\mathsf {C})\cap \mathcal {S} = \mathcal {S}\), which we substitute into Eq. 125 to obtain the corollary. □
Declarations
Acknowledgements
This work was partially supported by National Science Foundation (NSF) Grants CNS 0923479, CNS 1012921, CNS 1161596 and NSF Graduate Research Fellowship 0940902.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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