Cooperative Radar and Communications Signaling:The Estimation and Information Theory Odd Couple

Daniel W. BlissSchool of Electrical, Computer and Energy Engineering

Arizona State UniversityTempe, Arizona USA

Abstract—We investigate cooperative radar and communica-tions signaling. While each system typically considers the othersystem a source of interference, by considering the radar andcommunications operations to be a single joint system, the perfor-mance of both systems can, under certain conditions, be improvedby the existence of the other. As an initial demonstration, we focuson the radar as relay scenario and present an approach denotedmultiuser detection radar (MUDR). A novel joint estimationand information theoretic bound formulation is constructed fora receiver that observes communications and radar return inthe same frequency allocation. The joint performance bound ispresented in terms of the communication rate and the estimationrate of the system.

I. INTRODUCTION

Given the reality of the ever increasing strain on limitedspectral resources, radar and communications systems are insome cases being forced into an uneasy coexistence. Thetypical assumption is that the existence of one type of system(either a radar or a communications system) will degrade theperformance of the other system. Consequently, the systemsare usually isolated temporally, spectrally, or spatially in mostoperations.

A. Background

During the last decade, cognitive radio technologies [1], [2]have been considered that implement opportunistic spectrumsharing as they are able to sense under-utilized spectrum andadaptively allocate it to other users [3]. A similar coexistenceproblem is currently faced by radars as their performance de-teriorates due to coexisting wireless communications systems.Cognitive radars indicate initial attempts to adapt intelligentlyto complicated environments [4].

Current research on the spectral coexistence of radar andcommunications systems has mainly involved concepts similarto cooperative sensing [5]–[9]. Other methodologies that havebeen applied to the radar-communications coexistence probleminclude signal sharing [10], [11] and waveform shaping [12]–[15]. In other research and applied systems radars based oncommunication system waveforms have been considered. Asan example, operating the radar passively or parasitically byusing a broadcast communication system has been investigated

This work was sponsored in part by DARPA under the SSPARC program.The views expressed are those of the author and do not reflect the officialpolicy or position of the Department of Defense or the U.S. Government.

(for example in [16] and references therein). Also, radiosthat communicate with radar systems by modulating the radarwaveform have been considered [17].

B. Contributions

The principal contribution of the paper is that we develop anovel performance bound formulation to provide insight intothe limits of coexisting radar and communications systems.For joint decoding and radar channel estimation (which wedenote multiuser detection radar: MUDR), we allow the radarto demodulate and decode the communications signal jointlywith estimating its radar channel. Rather than have radar andcommunications system performance degraded, the perfor-mance of both systems is potentially enhanced by the systems’interactions. In its most general form, the coexisting radarand communications system becomes a large heterogenousmultistatic radars or statistical multiple-input multiple-output(MIMO) radar [18]–[20] and simultaneously a heterogenouscommunication network [2]. These jointly cooperative systemsare only possible under certain theoretical constraints that webegin to explore in this paper.

In this paper, as a preliminary exploration, we consider thelimited scenario of a joint radar and communications relay.In this case, the node traditionally denoted “radar” is also acommunications relay that jointly estimates the radar returnand receives a communication signal. The radar waveform isthen assumed to be a communications waveform. Becauseof the advantages of the radar power, the performance ofthe communications between two or more nodes is typicallyimproved by using the radar as a relay compared to directground-to-ground communications. The principal constraint inperformance of this system is in simultaneous reception of theradar return and communications signal, and is therefore themain thrust of this work.

II. JOINT ESTIMATION/COMMUNICATIONS BOUNDS

In general, much like network communications [2], exactbounds are challenging. However, in certain cases, such asthe multiuser base station, bounds are tenable. We developa generalization of the multiple-access receiver discussionfor the joint radar channel estimation and communicationsreception.

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A. Multiple-Access Communications Analogy

For reference, we review the multiple-access communica-tions system performance bound [2], [21]. In the multiple-access channel that we discuss here, we assume that twoindependent transmitters are communicating with a singlereceiver. The channel-attenuation-power product for the twotransmitters are given by a21P1 and a22P2, respectively. Theircorresponding rates are denoted R1 and R2. Assuming thatpower is normalized so that the noise variance is unity, thefundamental limits on rate are given by1

R1 ≤ log2(1 + a21P1)

R2 ≤ log2(1 + a22P2)

R1 +R2 ≤ log2(1 + a21P1 + a22P2) (1)

Vertices are found by jointly solving two bounds,

R2 = log2(1 + a22P2)

R1 +R2 −R2 = log2(1 + a21P1 + a22P2)− log2(1 + a22P2)

R1 = log2

(1 + a21P1 + a22P2

1 + a22P2

)

{R1, R2} ={log2

(1 +

a21P1

1 + a22P2

), log2(1 + a22P2)

},

(2)

and

{R1, R2} ={log2(1 + a21P1), log2

(1 +

a22P2

1 + a21P1

)}.

(3)

The region that satisfies these theoretical bounds is depictedin Figure 1.

Achievable Rate Region2 1

3

4

a) Parasitic Communications:ulation of the radar waveform, the radar can broadcast a

communications and the radar return, we can replace the rateof the second user by the information rate associated withthe time-varying parameters of a radar return. The solutionin the large communications signal power, low target returnpower, portion of the information bound region is to employsuccessive interference cancellation [2], [21]. If the communi-cations signal operates at a decodable rate and SINR observedby the radar receiver (where the interference here is the radarreturn), then the radar can decode the communications signal,remodulate the signal, and subtract it from the raw dataobserved by the radar. The resulting data stream is e�ectivelyfree of the communications signal contamination and thesystem can then estimate the parameters of the radar return.In addition, the channel estimated during the decoding of thecommunications signal may contain bi-static radar informationabout the radar target. This information can be combined withthe results of the monostatic radar processing. The radar can

a) Parasitic Communications: By employing mild mod-ulation of the radar waveform, the radar can broadcast acommunication signal parasitically with the radar waveform.Because the radar knows the transmitted signal, it can employthis reference in estimating the radar channel without lossin radar performance. In using more aggressive communica-tion/radar waveforms, an instantaneously wideband communi-cation waveform can be employed with essentially no loss inradar estimation and detection performance.

b) Parasitic Radar: Another simple example of a sym-biotic communication and radar systems is the passive radar.For example, a bistatic receiver employs a broadcast com-munications signal as an illuminating source. In this case,the communications system operates normally. The bistaticradar receiver observes the communication signal directly, andconstructs a reference of the transmission. It uses this referenceto search for targets. The benefits for the radar system aresignificant both in terms of spectral resource allocation (forwhich none was required), and in terms of hardware. No highpower transmitter was required; however many of the signalparameters may be suboptimal.

III. MUDR

A. Heterogeneous Radar/Communications Model

As a specific example, consider a joint air surveillance radarand mobile ad hoc network (MANet) system [2]. We assumea heterogeneous network of communication/radar nodes, suchthat all nodes can potentially decode signals. While all nodesmay not perform all functions, we assume in general thatthey can. We assume relatively capable radar/communicationnodes, and assume that they have a common control channel.We assume all radar/communications signals subtend the fullavailable bandwidth. Specifically, we assume an operatingscenario in which some small set of the nodes have high-power amplifies and high-gain antenna arrays. These nodescorrespond to those that would traditionally have been identi-fied as radars. While other nodes are relatively disadvantagedgeographically, in transmit power, and in antenna gain. Thesenodes correspond to those that would traditionally have beenidentified as MANet radios.

B. MUDR: Radar as a Relay

As a basic starting point in the analysis, we begin with athree node communications network with simultaneous radarfunctionality, as discussed in Section III-B. This is an ex-tension to the concepts discussed in Section II. The radarwill transmit its radar pulse that also contains an encodedcommunications signal to a destination radio. The radar willreceive the source radio signal and the radar returns forchannel estimation. Because the radar is typically sited in anadvantaged location, and because it has access to relativelyhigh transmit power and high gain compared to a typicalcommunications transmitter, it is well suited to act as acommunications relay. As an example, two nodes that simplycannot communicate at any reasonable rate because of theirgeometry, may be able to communicate via the radar relay.

Thus, the communication rate improvement could potentiallybe orders of magnitude.

To study the performance bounds associated with the radaras a relay concept, an interesting analogy to the informationtheoretic capacity of the multiuser receiver can be employed.In particular, in considering the two transmitter, single receivermultiuser problem, the capacity region is known [2], [21]. Fora given average transmit power, with axes defined by the datarate of User 1, and the data rate of User 2, the capacity regionis given by an irregular pentagon.

In our problem, for the simultaneous reception of the sourcecommunications and the radar return, we can replace the rateof the second user by the information rate associated withthe time-varying parameters of a radar return. The solutionin the large communications signal power, low target returnpower, portion of the information bound region is to employsuccessive interference cancellation [2], [21]. If the communi-cations signal operates at a decodable rate and SINR observedby the radar receiver (where the interference here is the radarreturn), then the radar can decode the communications signal,remodulate the signal, and subtract it from the raw dataobserved by the radar. The resulting data stream is effectivelyfree of the communications signal contamination and thesystem can then estimate the parameters of the radar return.In addition, the channel estimated during the decoding of thecommunications signal may contain bi-static radar informationabout the radar target. This information can be combined withthe results of the monostatic radar processing. The radar canthen encode the signal for the next radar pulse and, by usingthe radar waveform as a communications waveform, relayingthe data to the destination node.

IV. JOINT MULTIUSER ESTIMATION/COMMUNICATIONBOUNDS

In general, much like network communication [2], exactbounds are challenging. However, in certain cases, such asthe multiuser base station, bounds are tenable. We develop ageneralization of the relay discussion. Furthermore, we sketcha more general set of scaling bounds.

The fundamental limits on rate are given by1

R1 ≤ log2(1 + a21P1)

R2 ≤ log2(1 + a22P2)

R1 +R2 ≤ log2(1 + a21P1 + a22P2) (1)

Vertices are found by

R2 = log2(1 + a22P2)

R1 +R2 −R2 = log2(1 + a21P1 + a22P2)− log2(1 + a22P2)

R1 = log2

(1 + a21P1 + a22P2

1 + a22P2

)

{R1, R2} =

{log2

(1 +

a21P1

1 + a22P2

), log2(1 + a22P2)

}

(2)

1Note: We assume complex signals, so there are two degrees; thus, thereis no “1/2” before the log term

a) Parasitic Communications: By employing mild mod-ulation of the radar waveform, the radar can broadcast acommunication signal parasitically with the radar waveform.Because the radar knows the transmitted signal, it can employthis reference in estimating the radar channel without lossin radar performance. In using more aggressive communica-tion/radar waveforms, an instantaneously wideband communi-cation waveform can be employed with essentially no loss inradar estimation and detection performance.

b) Parasitic Radar: Another simple example of a sym-biotic communication and radar systems is the passive radar.For example, a bistatic receiver employs a broadcast com-munications signal as an illuminating source. In this case,the communications system operates normally. The bistaticradar receiver observes the communication signal directly, andconstructs a reference of the transmission. It uses this referenceto search for targets. The benefits for the radar system aresignificant both in terms of spectral resource allocation (forwhich none was required), and in terms of hardware. No highpower transmitter was required; however many of the signalparameters may be suboptimal.

III. MUDR

A. Heterogeneous Radar/Communications Model

As a specific example, consider a joint air surveillance radarand mobile ad hoc network (MANet) system [2]. We assumea heterogeneous network of communication/radar nodes, suchthat all nodes can potentially decode signals. While all nodesmay not perform all functions, we assume in general thatthey can. We assume relatively capable radar/communicationnodes, and assume that they have a common control channel.We assume all radar/communications signals subtend the fullavailable bandwidth. Specifically, we assume an operatingscenario in which some small set of the nodes have high-power amplifies and high-gain antenna arrays. These nodescorrespond to those that would traditionally have been identi-fied as radars. While other nodes are relatively disadvantagedgeographically, in transmit power, and in antenna gain. Thesenodes correspond to those that would traditionally have beenidentified as MANet radios.

B. MUDR: Radar as a Relay

As a basic starting point in the analysis, we begin with athree node communications network with simultaneous radarfunctionality, as discussed in Section III-B. This is an ex-tension to the concepts discussed in Section II. The radarwill transmit its radar pulse that also contains an encodedcommunications signal to a destination radio. The radar willreceive the source radio signal and the radar returns forchannel estimation. Because the radar is typically sited in anadvantaged location, and because it has access to relativelyhigh transmit power and high gain compared to a typicalcommunications transmitter, it is well suited to act as acommunications relay. As an example, two nodes that simplycannot communicate at any reasonable rate because of theirgeometry, may be able to communicate via the radar relay.

Thus, the communication rate improvement could potentiallybe orders of magnitude.

To study the performance bounds associated with the radaras a relay concept, an interesting analogy to the informationtheoretic capacity of the multiuser receiver can be employed.In particular, in considering the two transmitter, single receivermultiuser problem, the capacity region is known [2], [21]. Fora given average transmit power, with axes defined by the datarate of User 1, and the data rate of User 2, the capacity regionis given by an irregular pentagon.

In our problem, for the simultaneous reception of the sourcecommunications and the radar return, we can replace the rateof the second user by the information rate associated withthe time-varying parameters of a radar return. The solutionin the large communications signal power, low target returnpower, portion of the information bound region is to employsuccessive interference cancellation [2], [21]. If the communi-cations signal operates at a decodable rate and SINR observedby the radar receiver (where the interference here is the radarreturn), then the radar can decode the communications signal,remodulate the signal, and subtract it from the raw dataobserved by the radar. The resulting data stream is effectivelyfree of the communications signal contamination and thesystem can then estimate the parameters of the radar return.In addition, the channel estimated during the decoding of thecommunications signal may contain bi-static radar informationabout the radar target. This information can be combined withthe results of the monostatic radar processing. The radar canthen encode the signal for the next radar pulse and, by usingthe radar waveform as a communications waveform, relayingthe data to the destination node.

IV. JOINT MULTIUSER ESTIMATION/COMMUNICATIONBOUNDS

In general, much like network communication [2], exactbounds are challenging. However, in certain cases, such asthe multiuser base station, bounds are tenable. We develop ageneralization of the relay discussion. Furthermore, we sketcha more general set of scaling bounds.

The fundamental limits on rate are given by1

R1 ≤ log2(1 + a21P1)

R2 ≤ log2(1 + a22P2)

R1 +R2 ≤ log2(1 + a21P1 + a22P2) (1)

Vertices are found by

R2 = log2(1 + a22P2)

R1 +R2 −R2 = log2(1 + a21P1 + a22P2)− log2(1 + a22P2)

R1 = log2

(1 + a21P1 + a22P2

1 + a22P2

)

{R1, R2} =

{log2

(1 +

a21P1

1 + a22P2

), log2(1 + a22P2)

}

(2)

1Note: We assume complex signals, so there are two degrees; thus, thereis no “1/2” before the log term

a) Parasitic Communications: By employing mild mod-ulation of the radar waveform, the radar can broadcast acommunication signal parasitically with the radar waveform.Because the radar knows the transmitted signal, it can employthis reference in estimating the radar channel without lossin radar performance. In using more aggressive communica-tion/radar waveforms, an instantaneously wideband communi-cation waveform can be employed with essentially no loss inradar estimation and detection performance.

b) Parasitic Radar: Another simple example of a sym-biotic communication and radar systems is the passive radar.For example, a bistatic receiver employs a broadcast com-munications signal as an illuminating source. In this case,the communications system operates normally. The bistaticradar receiver observes the communication signal directly, andconstructs a reference of the transmission. It uses this referenceto search for targets. The benefits for the radar system aresignificant both in terms of spectral resource allocation (forwhich none was required), and in terms of hardware. No highpower transmitter was required; however many of the signalparameters may be suboptimal.

III. MUDR

A. Heterogeneous Radar/Communications Model

As a specific example, consider a joint air surveillance radarand mobile ad hoc network (MANet) system [2]. We assumea heterogeneous network of communication/radar nodes, suchthat all nodes can potentially decode signals. While all nodesmay not perform all functions, we assume in general thatthey can. We assume relatively capable radar/communicationnodes, and assume that they have a common control channel.We assume all radar/communications signals subtend the fullavailable bandwidth. Specifically, we assume an operatingscenario in which some small set of the nodes have high-power amplifies and high-gain antenna arrays. These nodescorrespond to those that would traditionally have been identi-fied as radars. While other nodes are relatively disadvantagedgeographically, in transmit power, and in antenna gain. Thesenodes correspond to those that would traditionally have beenidentified as MANet radios.

B. MUDR: Radar as a Relay

As a basic starting point in the analysis, we begin with athree node communications network with simultaneous radarfunctionality, as discussed in Section III-B. This is an ex-tension to the concepts discussed in Section II. The radarwill transmit its radar pulse that also contains an encodedcommunications signal to a destination radio. The radar willreceive the source radio signal and the radar returns forchannel estimation. Because the radar is typically sited in anadvantaged location, and because it has access to relativelyhigh transmit power and high gain compared to a typicalcommunications transmitter, it is well suited to act as acommunications relay. As an example, two nodes that simplycannot communicate at any reasonable rate because of theirgeometry, may be able to communicate via the radar relay.

Thus, the communication rate improvement could potentiallybe orders of magnitude.

To study the performance bounds associated with the radaras a relay concept, an interesting analogy to the informationtheoretic capacity of the multiuser receiver can be employed.In particular, in considering the two transmitter, single receivermultiuser problem, the capacity region is known [2], [21]. Fora given average transmit power, with axes defined by the datarate of User 1, and the data rate of User 2, the capacity regionis given by an irregular pentagon.

In our problem, for the simultaneous reception of the sourcecommunications and the radar return, we can replace the rateof the second user by the information rate associated withthe time-varying parameters of a radar return. The solutionin the large communications signal power, low target returnpower, portion of the information bound region is to employsuccessive interference cancellation [2], [21]. If the communi-cations signal operates at a decodable rate and SINR observedby the radar receiver (where the interference here is the radarreturn), then the radar can decode the communications signal,remodulate the signal, and subtract it from the raw dataobserved by the radar. The resulting data stream is effectivelyfree of the communications signal contamination and thesystem can then estimate the parameters of the radar return.In addition, the channel estimated during the decoding of thecommunications signal may contain bi-static radar informationabout the radar target. This information can be combined withthe results of the monostatic radar processing. The radar canthen encode the signal for the next radar pulse and, by usingthe radar waveform as a communications waveform, relayingthe data to the destination node.

IV. JOINT MULTIUSER ESTIMATION/COMMUNICATIONBOUNDS

In general, much like network communication [2], exactbounds are challenging. However, in certain cases, such asthe multiuser base station, bounds are tenable. We develop ageneralization of the relay discussion. Furthermore, we sketcha more general set of scaling bounds.

The fundamental limits on rate are given by1

R1 ≤ log2(1 + a21P1)

R2 ≤ log2(1 + a22P2)

R1 +R2 ≤ log2(1 + a21P1 + a22P2) (1)

Vertices are found by

R2 = log2(1 + a22P2)

R1 +R2 −R2 = log2(1 + a21P1 + a22P2)− log2(1 + a22P2)

R1 = log2

(1 + a21P1 + a22P2

1 + a22P2

)

{R1, R2} =

{log2

(1 +

a21P1

1 + a22P2

), log2(1 + a22P2)

}

(2)

1Note: We assume complex signals, so there are two degrees; thus, thereis no “1/2” before the log term

a) Parasitic Communications: By employing mild mod-ulation of the radar waveform, the radar can broadcast acommunication signal parasitically with the radar waveform.Because the radar knows the transmitted signal, it can employthis reference in estimating the radar channel without lossin radar performance. In using more aggressive communica-tion/radar waveforms, an instantaneously wideband communi-cation waveform can be employed with essentially no loss inradar estimation and detection performance.

b) Parasitic Radar: Another simple example of a sym-biotic communication and radar systems is the passive radar.For example, a bistatic receiver employs a broadcast com-munications signal as an illuminating source. In this case,the communications system operates normally. The bistaticradar receiver observes the communication signal directly, andconstructs a reference of the transmission. It uses this referenceto search for targets. The benefits for the radar system aresignificant both in terms of spectral resource allocation (forwhich none was required), and in terms of hardware. No highpower transmitter was required; however many of the signalparameters may be suboptimal.

III. MUDR

A. Heterogeneous Radar/Communications Model

As a specific example, consider a joint air surveillance radarand mobile ad hoc network (MANet) system [2]. We assumea heterogeneous network of communication/radar nodes, suchthat all nodes can potentially decode signals. While all nodesmay not perform all functions, we assume in general thatthey can. We assume relatively capable radar/communicationnodes, and assume that they have a common control channel.We assume all radar/communications signals subtend the fullavailable bandwidth. Specifically, we assume an operatingscenario in which some small set of the nodes have high-power amplifies and high-gain antenna arrays. These nodescorrespond to those that would traditionally have been identi-fied as radars. While other nodes are relatively disadvantagedgeographically, in transmit power, and in antenna gain. Thesenodes correspond to those that would traditionally have beenidentified as MANet radios.

B. MUDR: Radar as a Relay

As a basic starting point in the analysis, we begin with athree node communications network with simultaneous radarfunctionality, as discussed in Section III-B. This is an ex-tension to the concepts discussed in Section II. The radarwill transmit its radar pulse that also contains an encodedcommunications signal to a destination radio. The radar willreceive the source radio signal and the radar returns forchannel estimation. Because the radar is typically sited in anadvantaged location, and because it has access to relativelyhigh transmit power and high gain compared to a typicalcommunications transmitter, it is well suited to act as acommunications relay. As an example, two nodes that simplycannot communicate at any reasonable rate because of theirgeometry, may be able to communicate via the radar relay.

Thus, the communication rate improvement could potentiallybe orders of magnitude.

To study the performance bounds associated with the radaras a relay concept, an interesting analogy to the informationtheoretic capacity of the multiuser receiver can be employed.In particular, in considering the two transmitter, single receivermultiuser problem, the capacity region is known [2], [21]. Fora given average transmit power, with axes defined by the datarate of User 1, and the data rate of User 2, the capacity regionis given by an irregular pentagon.

In our problem, for the simultaneous reception of the sourcecommunications and the radar return, we can replace the rateof the second user by the information rate associated withthe time-varying parameters of a radar return. The solutionin the large communications signal power, low target returnpower, portion of the information bound region is to employsuccessive interference cancellation [2], [21]. If the communi-cations signal operates at a decodable rate and SINR observedby the radar receiver (where the interference here is the radarreturn), then the radar can decode the communications signal,remodulate the signal, and subtract it from the raw dataobserved by the radar. The resulting data stream is effectivelyfree of the communications signal contamination and thesystem can then estimate the parameters of the radar return.In addition, the channel estimated during the decoding of thecommunications signal may contain bi-static radar informationabout the radar target. This information can be combined withthe results of the monostatic radar processing. The radar canthen encode the signal for the next radar pulse and, by usingthe radar waveform as a communications waveform, relayingthe data to the destination node.

IV. JOINT MULTIUSER ESTIMATION/COMMUNICATIONBOUNDS

In general, much like network communication [2], exactbounds are challenging. However, in certain cases, such asthe multiuser base station, bounds are tenable. We develop ageneralization of the relay discussion. Furthermore, we sketcha more general set of scaling bounds.

The fundamental limits on rate are given by1

R1 ≤ log2(1 + a21P1)

R2 ≤ log2(1 + a22P2)

R1 +R2 ≤ log2(1 + a21P1 + a22P2) (1)

Vertices are found by

R2 = log2(1 + a22P2)

R1 +R2 −R2 = log2(1 + a21P1 + a22P2)− log2(1 + a22P2)

R1 = log2

(1 + a21P1 + a22P2

1 + a22P2

)

{R1, R2} =

{log2

(1 +

a21P1

1 + a22P2

), log2(1 + a22P2)

}

(2)

1Note: We assume complex signals, so there are two degrees; thus, thereis no “1/2” before the log term

a) Parasitic Communications: By employing mild mod-ulation of the radar waveform, the radar can broadcast acommunication signal parasitically with the radar waveform.Because the radar knows the transmitted signal, it can employthis reference in estimating the radar channel without lossin radar performance. In using more aggressive communica-tion/radar waveforms, an instantaneously wideband communi-cation waveform can be employed with essentially no loss inradar estimation and detection performance.

b) Parasitic Radar: Another simple example of a sym-biotic communication and radar systems is the passive radar.For example, a bistatic receiver employs a broadcast com-munications signal as an illuminating source. In this case,the communications system operates normally. The bistaticradar receiver observes the communication signal directly, andconstructs a reference of the transmission. It uses this referenceto search for targets. The benefits for the radar system aresignificant both in terms of spectral resource allocation (forwhich none was required), and in terms of hardware. No highpower transmitter was required; however many of the signalparameters may be suboptimal.

III. MUDR

A. Heterogeneous Radar/Communications Model

As a specific example, consider a joint air surveillance radarand mobile ad hoc network (MANet) system [2]. We assumea heterogeneous network of communication/radar nodes, suchthat all nodes can potentially decode signals. While all nodesmay not perform all functions, we assume in general thatthey can. We assume relatively capable radar/communicationnodes, and assume that they have a common control channel.We assume all radar/communications signals subtend the fullavailable bandwidth. Specifically, we assume an operatingscenario in which some small set of the nodes have high-power amplifies and high-gain antenna arrays. These nodescorrespond to those that would traditionally have been identi-fied as radars. While other nodes are relatively disadvantagedgeographically, in transmit power, and in antenna gain. Thesenodes correspond to those that would traditionally have beenidentified as MANet radios.

B. MUDR: Radar as a Relay

As a basic starting point in the analysis, we begin with athree node communications network with simultaneous radarfunctionality, as discussed in Section III-B. This is an ex-tension to the concepts discussed in Section II. The radarwill transmit its radar pulse that also contains an encodedcommunications signal to a destination radio. The radar willreceive the source radio signal and the radar returns forchannel estimation. Because the radar is typically sited in anadvantaged location, and because it has access to relativelyhigh transmit power and high gain compared to a typicalcommunications transmitter, it is well suited to act as acommunications relay. As an example, two nodes that simplycannot communicate at any reasonable rate because of theirgeometry, may be able to communicate via the radar relay.

Thus, the communication rate improvement could potentiallybe orders of magnitude.

To study the performance bounds associated with the radaras a relay concept, an interesting analogy to the informationtheoretic capacity of the multiuser receiver can be employed.In particular, in considering the two transmitter, single receivermultiuser problem, the capacity region is known [2], [21]. Fora given average transmit power, with axes defined by the datarate of User 1, and the data rate of User 2, the capacity regionis given by an irregular pentagon.

In our problem, for the simultaneous reception of the sourcecommunications and the radar return, we can replace the rateof the second user by the information rate associated withthe time-varying parameters of a radar return. The solutionin the large communications signal power, low target returnpower, portion of the information bound region is to employsuccessive interference cancellation [2], [21]. If the communi-cations signal operates at a decodable rate and SINR observedby the radar receiver (where the interference here is the radarreturn), then the radar can decode the communications signal,remodulate the signal, and subtract it from the raw dataobserved by the radar. The resulting data stream is effectivelyfree of the communications signal contamination and thesystem can then estimate the parameters of the radar return.In addition, the channel estimated during the decoding of thecommunications signal may contain bi-static radar informationabout the radar target. This information can be combined withthe results of the monostatic radar processing. The radar canthen encode the signal for the next radar pulse and, by usingthe radar waveform as a communications waveform, relayingthe data to the destination node.

IV. JOINT MULTIUSER ESTIMATION/COMMUNICATIONBOUNDS

In general, much like network communication [2], exactbounds are challenging. However, in certain cases, such asthe multiuser base station, bounds are tenable. We develop ageneralization of the relay discussion. Furthermore, we sketcha more general set of scaling bounds.

The fundamental limits on rate are given by1

R1 ≤ log2(1 + a21P1)

R2 ≤ log2(1 + a22P2)

R1 +R2 ≤ log2(1 + a21P1 + a22P2) (1)

Vertices are found by

R2 = log2(1 + a22P2)

R1 +R2 −R2 = log2(1 + a21P1 + a22P2)− log2(1 + a22P2)

R1 = log2

(1 + a21P1 + a22P2

1 + a22P2

)

{R1, R2} =

{log2

(1 +

a21P1

1 + a22P2

), log2(1 + a22P2)

}

(2)

1Note: We assume complex signals, so there are two degrees; thus, thereis no “1/2” before the log termFig. 1. Pentagon that contains communications multiple-access achievable

rate region.

Unfortunately, this discussion serves only as a motivationbecause radar returns do not satisfy the fundamental commu-nications assumption that they are drawn from a countabledictionary. Consequently, we do not expect that this form isdirectly applicable. However, by using a formalism similarto the communications multiple-access bound, we can gain

1Note: We assume complex baseband signals, so there are two degrees offreedom; thus, there is no “1/2” before the log term

insight into the simultaneous channel use by communicationsand radar.

B. Joint Radar-Communications Notation

Because there is a significant quantity of notation in dis-cussing this topic, in Table I we present an overview of theimportant notation employed.

TABLE ISURVEY OF NOTATION.

Variable Description〈·〉 Expectation‖ · ‖ L2-norm or absolute valueB Total system bandwidthz(t) Observed signal including radar and communicationsz̃(t) Observed signal with predicted radar return removed

zradar(t) Observed radar returnsradar(t) Unit-variance transmitted radar signalPradar Radar powerτm Time delay to mth targetτ(k)m kth observation of delay for mth target

τm,pre Predicted time delay to mth targetam Combined antenna gain, cross-section, and propagationN Number of targetsT Radar pulse durationN Number of targetsTpri Pulse repetition intervalδ Radar duty factor

scom(t) Unit-variance transmitted communication signalPcom Total communications powerb Communications propagation loss

n(t) Receiver thermal noiseσ2noise Thermal noise powerkB Boltzmann constantTtemp Temperaturenint+n Interference plus noise for communications receiver

θ Set of nonspecific system and target parametersBrms Root-mean-squared radar bandwidthγ Radar spectral shape parameter

Bcom Communications-only subbandBmix Mixed radar and communications subbandα Fraction of bandwidth for communications onlyβ Power fraction used by communications-only subband

µcom Channel of communications-only subbandµmix Channel of mixed use subband

C. Joint Radar-Communications Channel Model

In this section, we consider bounds for the multiple-accesscommunications and radar return channel. We employ a num-ber of simplifying assumptions for the sake of exposition;however, generalizations are possible. As an example, weestimate the range, but assume that the target cross-section isknown. We assume that the targets are well separated and thethat return is modeled well by a Gaussian distribution beforepulse compression. We assume that the range of any giventarget is predictable up to some Gaussian random processvariation (not be confused with estimation error). We consideronly the portion of time during which the radar return overlapswith the communications signal. We assume that temporaluncertainty of the random target process is within one overthe bandwidth.

For N targets, the observed radar return zradar(t) as afunction of t is given by

zradar(t) =

N∑

m=1

am sradar(t− τm)√Pradar + n(t) . (4)

The zero-mean noise is drawn from the complex Gaussianwith variance σ2

noise,

σ2noise = kB TtempB , (5)

where kB is the Boltzmann constant, Ttemp is the absolutetemperature, and B is the full bandwidth. Range r and delayτ are related by

τ =2 r

c, (6)

where c is the speed of light. The typical radar estimatorattempts to estimate both the range and the amplitude. Forthe sake of discussion, we focus on range estimation. Similardevelopments can be found for amplitude estimation. A rea-sonable estimator (particularly if targets are well separated)under the assumption that Doppler shifts are unresolvable isgiven by

τ̂m = argmaxτm

∫dt z(t) s∗radar(t− τm) . (7)

We assume that we are tracking the target, and we assume theoptimistic model that we have some well understood expectedvalue of the radar return (based upon prior observations);however, there is some range fluctuation in the return due tosome underlying target process, so that the next observation isknown up to some random Gaussian process variation nτ,proc,

τ (k)m = τ (k)m,pre + nτ,proc (8)

τ (k)m,pre = f(k;Tpri,θ) .

The function f(k;Tpri,θ) is a prediction function with param-eters Tpri, which is the time between updates (pulse repetitioninterval), and θ which contains other parameters. The varianceof the process is given by

σ2τ,proc =

⟨∥∥∥τ (k)m − f(k;Tpri,θ)∥∥∥2⟩. (9)

The observed signal at the receiver z(t) at time t in thepresence of a communications signal and the radar return isgiven by

z(t) =√Pcom b scom(t) (10)

+√Pradar

N∑

m=1

am sradar(t− τm) + n(t)

D. Radar-Prediction-Suppressed Observed Signal

For the sake of the communications system, we can try tomitigate unnecessary interference by subtracting the predicted

radar return at the receiver2

z̃(t) =√Pcom b scom(t) + n(t) (11)

+√Pradar

N∑

m=1

am[sradar(t−τm)−sradar(t−τm,pre)] ,

where here we dropped the explicit indication of pulse index(k). For small delay process variation, we can replace thedifference between the waveforms at the correct and predicteddelay with a derivative,

sradar(t− τm)− sradar(t− τm,pre)= sradar(t− τm)− sradar(t− τm + nτ,proc)

≈ ∂sradar(t− τm)

∂tnτ,proc . (12)

The observed signal is then given by

z̃(t) ≈√Pcom b scom(t) + n(t)

+√Pradar

N∑

m=1

am∂sradar(t− τm)

∂tnτ,proc . (13)

From the communications receiver’s perspective, the interfer-ence plus noise is given by

nint+n ≈√Pradar

(N∑

m=1

am∂sradar(t− τm)

∂tnτ,proc

)+ n(t)

σ2int+n =

⟨‖nint+n‖2

⟩

= Pradar

(N∑

m=1

a2m (2π)2B2rms σ

2proc

)+ σ2

noise ,

(14)

where Brms is extracted by employing Parseval’s theorem[2]. The value γ is the scaling constant between B andBrms times 2π that is dependent upon the shape of the radarwaveform’s power spectral density. For a flat spectral shape,γ2 = (2π)2/12.

E. Radar Estimation Information Rate

An essential tool of this paper is to consider the estimationinformation rate (estimating delay in this case). We developthis information rate by considering the entropy of a randomparameter being estimated and the entropy of the estimationuncertainty of that parameter. As an observation, if the targetsare well separated, then each target estimation can be consid-ered an independent information channel.

1) Estimation Entropy: To find the estimation entropy, wefind the delay estimation uncertainty for each target. Forcircularly symmetric Gaussian noise, we employ the complexSlepian-Bangs formulation of the Cramer-Rao bound [2], [22].The variance of delay estimation for the mth target (ignoring

2Note: this process would theoretically remove all clutter.

inter-target interference) is given by

σ2τ ;est = Var{τ̂m} =

1

(2π)2B2rms ISNR

=σ2noise

(2π)2B2rms TB a

2m Pradar

=kB Ttemp

γ2B (TB) a2m Pradar, (15)

where ISNR = TB a2m Pradar/σ2noise indicates the integrated

SNR, and the thermal noise is given by

σ2noise = kB TtempB . (16)

Under the assumption of Gaussian estimation error, the result-ing entropy of the error is given by

hτ,est = log2[π e σ2τ,est]

= log2

[π e

kB Ttemp

γ2B (TB) a2m Pradar

]. (17)

2) Radar Random Process Entropy: The entropy of the pro-cess uncertainty plus estimation uncertainty under a Gaussianassumption for both is given by [2], [21]

hτ,rr = log2[π e (σ2

τ,proc + σ2τ,est)

]. (18)

3) Estimation Information Rate: Consequently, the mutualinformation rate in terms of bits per pulse repetition intervalTpri, which is related to the integration period T by the dutyfactor T = δ Tpri, is approximately bounded by

Rest ≤∑

m

hτ,rr − hτ,estTpri

=∑

m

δ

Tlog2

(1 +

σ2τ,proc

σ2τ,est

)

=∑

m

B log2

(1 +

σ2τ,proc γ

2B (TB) a2m Pradar

kB Ttemp

)δ/(TB)

.

(19)

It is worth noting, that by employing this estimation entropyin the rate bound, it is assumed that the estimator achieves theCramer-Rao performance. If the error variance is larger, thenthe rate bound is lowered.

III. INNER RATE BOUNDS

It would be surprising if the performance bound displayedfor the communications multiple-access scenario in Figure1 achieved the performance bounds of the joint estimationand communications problem. Here, we search for a goodachievable (inner) bounds. The fundamental system perfor-mance limit lies between these achievable bounds and theouter bounds found above. To find these inner bounds, wehypothesize an idealized receiver and determine the boundingrates. To simplify the discussion, we consider only a singletarget with delay τ and gain-propagation-cross-section producta2, and drop the explicit index to the target. For exampleσ2τ,proc → σ2

proc.

If Rest ≈ 0 is sufficiently low, then the communicationsoperates according to the bound determined by the isolatedcommunications system,

Rcom ≤ B log2

(1 +

b2 Pcom

σ2noise

)

= B log2

(1 +

b2 Pcom

kB TtempB

). (20)

If Rcom is sufficiently low for a given transmit power thenthe communications signal can be decoded and subtractedcompletely from the underlying signal, so that the radarparameters can be estimated without contamination,

Rcom ≤ B log2

[1 +

b2 Pcomσ2int+n

](21)

= B log2

[1 +

b2 Pcoma2 Pradar γ2B2 σ2

proc + kB TtempB

],

where we used Equation (14). In this regime, the correspond-ing estimation rate bound Rest is given by Equation (19).

These two vertices correspond to the points 2 (associatedwith Equation (20)) and 4 (associated with Equations (21) and(19)) in Figure 1, if R1 is interpreted as the estimation rate, andR2 is interpreted as the communications rate. An achievablerate lies within the triangle constructed by connecting astraight line between these points.

A. Water-filling

We hypothesize that we can construct tighter (larger) innerbounds than we constructed in the previous section. In thissection, we consider a water-filling approach that splits thetotal bandwidth into two sub-bands and we water fill thecommunications power between these bands. Water fillingoptimizes the power and rate allocation between multiplechannels [2], [21]. For this application, we separate the bandinto two frequency channels. One channel has only commu-nications, and the other channel is mixed-use and operates atthe SIC rate vertex define by Equations (19) and (21).

Given some α, that defines the bandwidth separation,

B = Bcom +Bmix (22)Bcom = αB

Bmix = (1− α)B ,

then we optimize β that defines the power utilization,

Pcom = Pcom,com + Pcom,mix (23)Pcom,com = β Pcom

Pcom,mix = (1− β)Pcom .

There are two effective channels

µcom =b2

kB TtempBcom

=b2

kB Ttemp αB, (24)

for the channel with only communications signal, and themixed-use channel that includes the interference to the com-munications system from the radar

µmix =b2

σ2int+n

(25)

=b2

a2 Pradar (1− α)2γ2B2 σ2proc + kB Ttemp (1− α)B

.

The communications power is split between the two channels[2], [21],

Pcom = Pcom,com + Pcom,mix

=

(αν − 1

µcom

)++

((1− α) ν − 1

µmix

)+. (26)

The critical point (the transition between using one or bothchannels for communications) occurs when

(1− α) ν − 1

µmix= 0

Pcom = αν − 1

µcom, (27)

so both channels are used if

Pcom ≥α

(1− α)µmix− 1

µcom. (28)

If the communications-only channel is used exclusively forcommunications, then Pcom = Pcom,com. If both channels areemployed for communications then

Pcom,com = αν − 1

µcom

Pcom,mix = (1− α) ν − 1

µmix, (29)

and thus when Equation (28) is satisfied

Pcom = αν − 1

µcom+ (1− α) ν − 1

µmix

ν =

(Pcom +

1

µcom+

1

µmix

). (30)

The value of power fraction β is then given by

β =Pcom,com

Pcom

=αν − 1

µcom

Pcom

=α(Pcom + 1

µcom+ 1

µmix

)− 1

µcom

Pcom

= α+1

Pcom

(α− 1

µcom+

α

µmix

);

when Pcom ≥α

(1− α)µmix− 1

µcom. (31)

The resulting communications rate bound in thecommunications-only subband is given by

Rcom,com ≤ Bcom log2

[1 +

Pcom,com b2

kB TtempBcom

]

≤ αB log2

[1 +

β Pcom b2

kB Ttemp αB

]. (32)

If Pcom < 1/µcom − 1/µmix then Rcom,mix = 0 becauseno communications power is allocated to the “mixed” usechannel, otherwise the mixed use communications rate innerbound is given by

Rcom,mix ≤ Bmix log2

[1 +

b2 Pmix

σ2int+n

]

= (1− α)B log2

[1 +

b2 (1− β)Pcom

σ2int+n

](33)

σ2int+n = a2 Pradar (1− α)2 γ2B2 σ2

proc + kB Ttemp (1− α)BThe corresponding radar estimation rate inner bound is thengiven by

Rest ≤ Bmix log2

(1 +

σ2proc γ

2Bmix (TBmix) a2 Pradar

kB Ttemp

)δ/(TBmix)

= (1− α)B log2(1 + SNRradar)δ/([1−α]TB) (34)

SNRradar =σ2proc γ

2 (1− α)B ([1− α]TB) a2 Pradar

kB Ttemp.

(35)

We assume that [1−α]TB, which is the waveform integration,is held constant as α is varied so Rest is given by

Rest ≤ (1− α)B log2

(1 +

σ2proc γ

2 (1− α)B κa2 Pradar

kB Ttemp

)δ/κ,

(36)

where waveform integration is denoted κ = (1− α)TB. Forsome very large value of α, corresponding to a very smallradar subband, the problem is no longer self consistent becauseT > Trpi.

B. Examples

In Figure 2, we display an example of inner bounds on per-formance. The parameters used in the example are displayedin Table II. It is assumed that the communications system isreceived through an antenna sidelobe, so that the radar andcommunications receive gain are not identical. In the figure,we indicate a outer bound in red. We indicate in green, thebound on successive interference cancellation (SIC), presentedin Equation (21). The best case system performance given SICis at the vertex (at the intersection of the green and red lines),which is determined by the joint solution of Equations (21) and(19). The inner bound that linearly interpolates between thisvertex and the radar-free communications bound in Equation(20) is indicated by the gray dashed line. The water-fillingbound is indicated by the blue line. The water-filling bound

is not guaranteed to be convex. The water-filling bound is notguaranteed to be greater than the linearly interpolated bound.In general, the inner bound is produced by the convex hull ofall contributing inner bounds. In the example, we see that thewater-filling bound exceeds the linearly interpolated bound.

TABLE IIPARAMETERS FOR EXAMPLE PERFORMANCE BOUND.

Parameter ValueBandwidth 5 MHz

Center Frequency 3 GHzTemperature 1000 K

Communications Range 10 kmCommunications Power 20 dBm

Communications Antenna Gain 0 dBiRadar Target Range 100 kmRadar Antenna Gain 30 dBi

Radar Power 1 kWTarget Cross Section 10 m2

Target Process Standard Deviation 100 mTime-Bandwidth Product 100

Radar duty factor 0.01

0 1000 2000 3000 40000

1¥106

2¥106

3¥106

4¥106

Estimation Rate HbêsL

DataRateHbêsL

Fig. 2. Data rate and estimation rate bounds. Outer bounds on commu-nications and radar are indicated by the red lines. Successive interferencecancellation (SIC) bound for the communications rate is indicated by thegreen dashed line. The linear interpolation between SIC vertex and the radar-free data rate bound is indicated by the gray dashed line. The water-fillinginner bound is indicated by the blue line.

IV. CONCLUSION

In this paper, we provide a novel approach for producingjoint radar and communications performance bounds. Anachievable inner bound based on a water-filling approach isdeveloped and an example is presented. This is an initialinvestigation. There are a range of potentially significantimprovements to the inner bounds, and potentially interestingextensions to the scenarios to which these bounds may beapplied.

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