Sparsity-based MIMO Noise Radar for Multiple Target ... - IEEE Xplore
2012 IEEE 7th Sensor Array and Multichannel Signal Processing Workshop (SAM)
Sparsity-based MIMO Noise Radar for Multiple Target Estimation Sandeep Gogineni, Student Member, IEEE, and Arye Nehorai∗ , Fellow, IEEE Preston M. Green Department of Electrical and Systems Engineering Washington University in St. Louis One Brookings Drive, St. Louis, MO 63130, USA Email: {sgogineni,nehorai}@ese.wustl.edu Phone: 314-935-7520 Fax: 314-935-7500 Abstract—We solve a multiple moving-target estimation problem using a colocated multiple-input multiple-output (MIMO) radar system. Each antenna of the radar array transmits noise waveforms to achieve high resolution. These waveforms are further covered by codes that are inspired from code division multiple access (CDMA) to exploit code diversity. We formulate the measurement model using a sparse representation in an appropriate basis to estimate the unknown target parameters (delays, Dopplers) using support recovery algorithms. We demonstrate the performance of the proposed system using numerical simulations.
waveforms for each transmitter, to obtain good resolution during estimation [9]. These noise waveforms are very important in radar applications since they provide a low probability of intercept (LPI) [10]. Recently, there has been some active research on using noise radar in MIMO configuration [11], [12]. In [11], beam forming application of MIMO noise radar has been studied. In [12], the authors exploit spatial diversity by considering the antennas to be distributed over a circle. However, we use linear colocated arrays with known inter-element spacings. While using UWB waveforms, the target can no longer be treated as point like because of the enhanced resolution offered by these waveforms. It is represented by a series of individual point scatterers along with the corresponding impulse response. Therefore, the target has a frequency dependent response given by the Fourier transform of this impulse response. This information about the target cannot be obtained while using conventional norrowband waveforms. Noise waveforms have a flat frequency spectrum, and may not always provide the best match to the target response. Further, even though different transmitters emit independent noise sequences, the cross correlation between them is small but not exactly zero even for zero-lag. Note that orthogonality of the waveforms is a desired quality for MIMO radar. We will overcome these problems by covering the noise waveforms by codes that are inspired from code division multiple access (CDMA) [13], [14]. Since these codes have a nonflat frequency response, target responses vary for different codes. Therefore, we will completely exploit the code diversity by selecting waveforms that match the target responses (see Section V). Further, we will allocate different orthogonal codes for each transmitter to ensure zero cross correlation at zerolag. This is similar to the idea of using orthogonal codes for different users in a CDMA based communication system to avoid crosstalk between the users. In addition, we consider multiple moving targets as opposed to single stationary target considered in [12]. Due to the recent advances in hardware, computationally intense problems are being solved with ease. As a result, sparse modeling and reconstruction have been applied to many fields such as engineering and medicine [15]–[17]. In the
I. I NTRODUCTION Multiple input multiple output (MIMO) radar offers additional degrees of freedom by employing multiple transmit and receive antennas [1]–[5]. These degrees of freedom can be exploited in different ways. Firstly, the antennas can be placed wide apart to obtain spatially diverse looks of the target. This configuration is called distributed MIMO radar (or MIMO radar with widely separated antennas) [1], [2], [6]. Though this configuration provides spatial diversity, we cannot easily perform coherent processing because it is very difficult to achieve phase synchronization between the antennas. The degradation in the performance of coherent distributed MIMO radar in the presence of phase synchronization mismatch was shown in [7]. Secondly, we can place the multiple antennas in an array, and transmit different waveforms from each element of the array, thereby exploiting the waveform diversity. This configuration is called colocated MIMO radar [5]. Coherent processing is easily possible while using this configuration since phase synchronization is easy to achieve. In this paper, when we refer to MIMO radar, it stands for colocated MIMO radar. The transmit waveforms can be chosen in many different ways to provide diversity gain. For example, one can obtain frequency diversity by transmitting waveforms of different sub-carrier frequencies from each array element. In [8], the authors exploit this idea to implement an OFDM MIMO radar system. In this paper, we choose ultra wideband (UWB) noise This work was supported by ONR Grant N000140810849, NSF Grant CCF-1014908, and AFOSR Grant FA9550-11-1-0210. ∗Corresponding author
978-1-4673-1071-0/12/$31.00 ©2012 IEEE
33
last few years, many radar problems have been solved using sparse modeling [18]–[19]. This has been made possible by the discretization of the target delay-Doppler space. In this paper, we represent the measurements of our MIMO noise radar system using a sparse model in an appropriate basis. Therefore, we reduce the estimation problem to a sparse recovery problem. II. S IGNAL M ODEL We assume the radar operates in monostatic configuration. Let MT and MR denote the number of transmitters and receivers, respectively. These antennas are arranged in colocated arrays with inter-element spacings given by dT and dR , respectively. We assume the targets are located in the far-field. Let the antenna arrays make an angle θ with the group of targets (see Fig. 1). Since the targets are far away from the radar arrays, the angular separation between them with respect to the arrays will be negligible. Therefore, we chose the same angle θ for all the targets.
space and reflect off the surface of the target before reaching the receive array. In this process, the waveform is attenuated and delay-Doppler shifted. These signals can resolve the paths emanating from different scattering centers of the target. Let there be a total of C scattering centers with ac being the th attenuation corresponding to the c scattering center. We assume C is known. The received signal at each receiver is a linear combination of the target-reflected waveforms from all the transmitters. Therefore, the demodulated received signal at the k th receiver can be expressed as [21]
yk (t)
≈
MT C −1
ν ν a 1 + si 1+ (t − τ − c Δt) fc fc c
i=1 c =0
ν
× ej2πνt ej2π(1+ fc )f (γi+k) + ek (t), where τ and ν denote the delay and Doppler shift, respectively. ek (t) denotes the additive noise at the k th receiver. fc is the carrier frequency and
dĂƌŐĞƚƐ ͘
͘ ͘ ͘
ͬ
Transmit/Receive antenna array used in monostatic configuration.
First, we shall present the measurement model for a single target and later extend it to multiple targets. Let the ith transmitter emit the noise waveform ui (t) covered by the code wi (t). We construct these waveforms and codes as ui (t) =
C−1
ui,c rec(t − cΔt),
wi (t) =
C−1
(2)
rec(t) =
1, if 0 < t < Δt, 0, otherwise.
(3)
(5)
III. S PARSE R EPRESENTATION In this section, we will express the measurement model presented in the previous section using sparse signal representation. We discretize the target delay-Doppler space into V grid points. Only R of them correspond to the actual targets. Let (τv , νv ) represent the delay and Doppler corresponding to the v th grid point. Define νv νv ψi,k,c (n, v) = 1 + si 1+ (nTS − τv − c Δt) fc fc νv × ej2πνv t ej2π(1+ fc )f (γi+k) .
c=0
where
=
(4)
where TS denotes the sampling interval.
(1)
wi,c rec(t − cΔt),
γ
dR sin(θ) , λ dT , dR
k
c=0
and
=
where λ = fcv+ν , v is the speed of wave propagation in the medium. So far, we assumed a single target. Now, consider R targets in the scene illuminated by the radar. Let τ r , ν r , and arc denote the delay, Doppler, and attenuations corresponding to the rth target, respectively. After sampling, we can express the measurements as MT C R −1 νr r ac 1 + yk (n) ≈ fc i=1 c =0 r=1 r ν r × si 1+ (nTS − τ − c Δt) fc r νr × ej2πν t ej2π(1+ fc )f (γi+k) + e (t),
WůĂŶĞ tĂǀĞ
Fig. 1.
f
Δt denotes the chip interval. All the covering waveforms are mutually orthogonal [13]. Historically, Walsh functions have been used as covering waveforms in IS-95 systems [20]. In this paper, we will use Walsh functions of order 64. We express the ith transmitted waveform as si (t) = ui (t)wi (t). These waveforms travel in
Let there be N samples in a processing interval. Then, we N stack {ψi,k,c (n, v)}n=1 into a column vector ψ i,k,c (v). Next, MR we arrange ψ i,k,c (v) k=1 into a longer column vector
34
rT
, . . . , a
rT
T
. Further, we define a sparse vector a(v) r a , if (τv , νv ) = (τ r , ν r ), (6) a(v) = 0, otherwise.
Arranging the vectors a(v) corresponding to all the grid points, we obtain a V MT C dimensional block-sparse vector T T T . (7) a = a(1) , . . . , a(V ) Stack the measurements and the noise samples at each receiver, we obtain yk ek
T
= [yk (1), . . . , yk (N )] , = [ek (1), . . . , ek (N )]T .
(8) (9)
Additionally, we arrange =
e
=
T T y 1 , . . . , y TMR ,
T T e1 , . . . , eTMR .
(10) (11)
Therefore, the measurement model y = Ψa + e.
9
8 7 6
10
20
30
40
50
60
10
20
30
40
50
60
10
20
30 40 Walsh code index
50
60
8 7 6
4 3.5 3 2.5
(12)
IV. S PARSE R ECONSTRUCTION
Fig. 2. 2 norms of the target returns as a function of the Walsh code index.
We observed from the previous section that the non-zero entries of the sparse vector appear in blocks. Therefore, we will exploit the block-sparsity while recovering the sparse vector a. More specifically, we will use the block orthogonal matching pursuit (BOMP) algorithm for sparse recovery [22]. We initialize the residual vector r(0) = y and the estimate vector a(0) = 0. In each subsequent iteration, we project the residual from the previous iteration onto all the blocks of columns of Ψ and pick the block that gives the largest projection energy. Let v (k ) be the block selected in the k th iteration. Further, define V (k ) as the set containing the indices of all the blocks selected in the previous iterations. We update the corresponding entries of the estimate vector by solving argmin{a(v)} (k ) y − Ψ(v)a(v). (13) v∈V
9
Target 3 responses
y
V. N UMERICAL S IMULATIONS We consider an x-band radar system with carrier frequency of 10GHz. Further, we assume the signal bandwidth to be 2GHz. Therefore, the chip interval Δt = 0.5nS. Let the number of chips in a pulse C = 6400. We consider illumination of the target scene using a single pulse. We simulate a radar system consisting of 3 transmit antennas and 3 receive antennas. Choose θ = 30◦ and dT = dR = λ2 . Further, assume there are 3 targets in the illuminated area. Considering C = 3, we define the attenuations corresponding to the 1 2 T T targets a = [0.5, 0.2, 0.8] , a = [0.3, 0.6, 0.7] , and 3 T a = [0.2, 0.4, 0.1] . Target 1 responses
a
Assume that we terminate the algorithm after K iterations. Therefore, a(K ) denotes the final estimate of the sparse vector a containing information about the unknown target parameters.
Target 2 responses
ψ i,c (v). Each of these columns forms a basis function in our sparse representation. We stack these columns corresponding to different transmitters and paths into a block of columns Ψ(v). Each block corresponds to a different grid point. We have V such blocks that we can concatenate to obtain the dictionary matrix Ψ. Now, we arrange the attenuations arc corresponding r dimensional vector a = paths
T into a C to r different r r a0 , . . . , aC −1 . Since a is independent of the transmitter index, we define a MT C dimensional vector ar =
8.5
Minimum target response
8
7.5
7
6.5
6
v∈V (k )
Denote a(k ) as the updated estimate vector. Next, we update the residue as Ψa(k ) (v). (14) r(k ) = y −
10
20
30 40 Walsh code index
50
60
Fig. 3. 2 norm of the minimum target returns as a function of the Walsh code index.
As mentioned earlier, we generate Walsh codes of the order 64. Assuming we have accurate estimates of the attenuations
v∈V (k )
35
used numerical simulations to demonstrate the performance of the proposed system. In future work, we will analyze the performance in the presence of jamming signals. We will investigate the robustness of the system to modeling errors. Further, we will validate our results using real radar data.
from the previous processing intervals, we will select the Walsh codes for the current iteration. We compute the 2 norm of the signals reflected from the targets. In Fig. 2, we plot these norms as a function of the Walsh code indices. We notice that the signal returns are highly dependent on the choice of Walsh codes. Therefore, we need to match the codes to the target responses in order to improve the system performance. Since, we have multiple targets, the codes giving the optimal response for one target need not provide the optimal response for the other targets. Therefore, we select the code that maximizes the minimum target returns. We plot the minimum target returns in Fig. 3. Note hear that the weakest target is not fixed apriori. It varies depending on the choice of the code. We observe that the codes with indices 33, 49, and 17 are best matched to the target responses. We assign these codes to the three transmit antennas respectively.
R EFERENCES
2
1.5
1
0.5
0 20 15
5 4
10
3 5
Delay
Fig. 4.
2 1
−6
x 10
Stretch factor
Reconstructed sparse vector using optimal Walsh codes.
We discretize the target delay-Doppler space. Assume the true target delays lie in the interval [2nS, 20nS]. The spacing between these grid points is 2nS. Further, assume that the stretch factor νfvc (due to the Doppler) lies in the interval [0.000001, 0.0000055]. Let the delays corresponding to the three targets be 12nS, 18nS, and 6nS respectively. Similarly, let the corresponding stretch factors be 0.000004, 0.000002, and 0.000003 respectively. In Fig. 4, we plot the reconstructed vector at an SNR of −19.8983dB. We can clearly observe that the parameters of the three targets have been accurately reconstructed. The three largest peaks correspond to the three targets. VI. C ONCLUDING R EMARKS We used a colocated MIMO noise radar system to solve the multiple target estimation problem. We covered the transmitted noise waveforms using Walsh codes to ensure orthogonality between different transmitters and to match the transmitted waveforms to the target responses. We developed a signal model using sparse signal representation. We used a sparse reconstruction algorithm to estimate the target parameters. We
36
[1] A. M. Haimovich, R. S. Blum, and L. J. Cimini, “MIMO radar with widely separated antennas,” IEEE Signal Process. Mag., vol. 25, pp. 116–129, Jan. 2008. [2] S. Gogineni and A. Nehorai, “Polarimetric MIMO radar with distributed antennas for target detection,” IEEE Trans. Signal Process., vol. 58, pp. 1689–1697, Mar. 2010. [3] J. Li and P. Stoica, MIMO radar signal processing. Hoboken, NJ: John Wiley & Sons, Inc., 2009. [4] A. Hassanien and S. A. Vorobyov, “Transmit/receive beamforming for MIMO radar with colocated antennas,” in IEEE International Conference on Acoustics, Speech and Signal Processing, Taipei, Apr. 2009, pp. 2089–2092. [5] J. Li and P. Stoica, “MIMO radar with colocated antennas,” IEEE Signal Process. Mag., vol. 24, pp. 106–114, Sep. 2007. [6] S. Gogineni and A. Nehorai, “Target estimation using sparse modeling for distributed MIMO radar,” IEEE Trans. Signal Process., vol. 59, pp. 5315–5325, Nov. 2011. [7] M. Akcakaya and A. Nehorai, “MIMO radar detection and adaptive design under a phase synchronization mismatch,” IEEE Trans. Signal Process., vol. 58, pp. 4994–5005, Oct. 2010. [8] S. Sen and A. Nehorai, “OFDM MIMO radar with mutual-information waveform design for low-grazing angle tracking,” IEEE Trans. Signal Process., vol. 58, pp. 3152–3162, Jun. 2010. [9] R. M. Narayanan, Y. Xu, P. D. Hoffmeyer, and J. O. Curtis, “Design, performance, and applications of a coherent ultra-wideband random noise radar,” Opt. Eng., vol. 37, pp. 1855–1869, 1998. [10] L. Turner, “The evolution of featureless waveforms for LPI communications,” in Proc. IEEE NAECON, Dayton, OH, May 1991, pp. 1325–1331. [11] D. A. Gray and A. Capria, “MIMO noise radar - matched filters and coarrays,” in IEEE Radar Conference, Rome, Italy, May 2008, pp. 1–6. [12] W.-J. Chen and R. M. Narayanan, “Antenna placement for minimizing target localization error in UWB MIMO noise radar,” IEEE Antennas and Wireless Propagation Letters, vol. 10, pp. 135–138, 2011. [13] J. S. Lee and L. E. Miller, CDMA Systems Engineering Handbook. Artech House Publishers Mobile Communications Series, 1998. [14] A. J. Viterbi, CDMA: Principles of Spread Spectrum Communication. Addison-Wesley Wireless Communications Series, 1995. [15] J. Trzasko, C. Haider, and A. Manduca, “Practical nonconvex compressive sensing reconstruction of highly-accelerated 3D parallel MR angiograms,” in IEEE International Symposium on Biomedical Imaging: From Nano to Micro, Boston, MA, Jun. 2009, pp. 274–277. [16] R. Chartrand, “Fast algorithms for nonconvex compressive sensing: MRI reconstruction from very few data,” in IEEE International Symposium on Biomedical Imaging: From Nano to Micro, Boston, MA, Jun. 2009, pp. 262–265. [17] M. Herman and T. Strohmer, “Compressed sensing radar,” in IEEE Radar Conference, Rome, May 2008, pp. 1–6. [18] R. Baraniuk and P. Steeghs, “Compressive radar imaging,” in IEEE Radar Conference, Boston, MA, Apr. 2007, pp. 128–133. [19] S. Gogineni and A. Nehorai, “Adaptive design for distributed MIMO radar using sparse modeling,” in Proc. Int. Waveform Diversity and Design (WDD) Conf., Niagara Falls, Canada, Aug. 2010, pp. 23–27. [20] “Mobile station-base station compatibility standard for dual-mode wideband spread spectrum cellular system,” TIA/EIA Interim Standard 95 (IS-95) (amended as IS-95A in May 1995), Washington: Telecommunications Industry Association, July 1993. [21] A. Rihaczek, “Delay-Doppler ambiguity functions for wideband signals,” IEEE Trans. Aerosp. Electron. Syst., vol. 3, pp. 705–711, Jul. 1967. [22] Y. C. Eldar, P. Kuppinger, and H. Bolcskei, “Block-sparse signals: Uncertainty relations and efficient recovery,” IEEE Trans. Signal Process., vol. 58, pp. 3042–3054, Jun. 2010.
Sparsity-based MIMO Noise Radar for Multiple Target Estimation Sandeep Gogineni, Student Member, IEEE, and Arye Nehorai∗ , Fellow, IEEE Preston M. Green Department of Electrical and Systems Engineering Washington University in St. Louis One Brookings Drive, St. Louis, MO 63130, USA Email: {sgogineni,nehorai}@ese.wustl.edu Phone: 314-935-7520 Fax: 314-935-7500 Abstract—We solve a multiple moving-target estimation problem using a colocated multiple-input multiple-output (MIMO) radar system. Each antenna of the radar array transmits noise waveforms to achieve high resolution. These waveforms are further covered by codes that are inspired from code division multiple access (CDMA) to exploit code diversity. We formulate the measurement model using a sparse representation in an appropriate basis to estimate the unknown target parameters (delays, Dopplers) using support recovery algorithms. We demonstrate the performance of the proposed system using numerical simulations.
waveforms for each transmitter, to obtain good resolution during estimation [9]. These noise waveforms are very important in radar applications since they provide a low probability of intercept (LPI) [10]. Recently, there has been some active research on using noise radar in MIMO configuration [11], [12]. In [11], beam forming application of MIMO noise radar has been studied. In [12], the authors exploit spatial diversity by considering the antennas to be distributed over a circle. However, we use linear colocated arrays with known inter-element spacings. While using UWB waveforms, the target can no longer be treated as point like because of the enhanced resolution offered by these waveforms. It is represented by a series of individual point scatterers along with the corresponding impulse response. Therefore, the target has a frequency dependent response given by the Fourier transform of this impulse response. This information about the target cannot be obtained while using conventional norrowband waveforms. Noise waveforms have a flat frequency spectrum, and may not always provide the best match to the target response. Further, even though different transmitters emit independent noise sequences, the cross correlation between them is small but not exactly zero even for zero-lag. Note that orthogonality of the waveforms is a desired quality for MIMO radar. We will overcome these problems by covering the noise waveforms by codes that are inspired from code division multiple access (CDMA) [13], [14]. Since these codes have a nonflat frequency response, target responses vary for different codes. Therefore, we will completely exploit the code diversity by selecting waveforms that match the target responses (see Section V). Further, we will allocate different orthogonal codes for each transmitter to ensure zero cross correlation at zerolag. This is similar to the idea of using orthogonal codes for different users in a CDMA based communication system to avoid crosstalk between the users. In addition, we consider multiple moving targets as opposed to single stationary target considered in [12]. Due to the recent advances in hardware, computationally intense problems are being solved with ease. As a result, sparse modeling and reconstruction have been applied to many fields such as engineering and medicine [15]–[17]. In the
I. I NTRODUCTION Multiple input multiple output (MIMO) radar offers additional degrees of freedom by employing multiple transmit and receive antennas [1]–[5]. These degrees of freedom can be exploited in different ways. Firstly, the antennas can be placed wide apart to obtain spatially diverse looks of the target. This configuration is called distributed MIMO radar (or MIMO radar with widely separated antennas) [1], [2], [6]. Though this configuration provides spatial diversity, we cannot easily perform coherent processing because it is very difficult to achieve phase synchronization between the antennas. The degradation in the performance of coherent distributed MIMO radar in the presence of phase synchronization mismatch was shown in [7]. Secondly, we can place the multiple antennas in an array, and transmit different waveforms from each element of the array, thereby exploiting the waveform diversity. This configuration is called colocated MIMO radar [5]. Coherent processing is easily possible while using this configuration since phase synchronization is easy to achieve. In this paper, when we refer to MIMO radar, it stands for colocated MIMO radar. The transmit waveforms can be chosen in many different ways to provide diversity gain. For example, one can obtain frequency diversity by transmitting waveforms of different sub-carrier frequencies from each array element. In [8], the authors exploit this idea to implement an OFDM MIMO radar system. In this paper, we choose ultra wideband (UWB) noise This work was supported by ONR Grant N000140810849, NSF Grant CCF-1014908, and AFOSR Grant FA9550-11-1-0210. ∗Corresponding author
978-1-4673-1071-0/12/$31.00 ©2012 IEEE
33
last few years, many radar problems have been solved using sparse modeling [18]–[19]. This has been made possible by the discretization of the target delay-Doppler space. In this paper, we represent the measurements of our MIMO noise radar system using a sparse model in an appropriate basis. Therefore, we reduce the estimation problem to a sparse recovery problem. II. S IGNAL M ODEL We assume the radar operates in monostatic configuration. Let MT and MR denote the number of transmitters and receivers, respectively. These antennas are arranged in colocated arrays with inter-element spacings given by dT and dR , respectively. We assume the targets are located in the far-field. Let the antenna arrays make an angle θ with the group of targets (see Fig. 1). Since the targets are far away from the radar arrays, the angular separation between them with respect to the arrays will be negligible. Therefore, we chose the same angle θ for all the targets.
space and reflect off the surface of the target before reaching the receive array. In this process, the waveform is attenuated and delay-Doppler shifted. These signals can resolve the paths emanating from different scattering centers of the target. Let there be a total of C scattering centers with ac being the th attenuation corresponding to the c scattering center. We assume C is known. The received signal at each receiver is a linear combination of the target-reflected waveforms from all the transmitters. Therefore, the demodulated received signal at the k th receiver can be expressed as [21]
yk (t)
≈
MT C −1
ν ν a 1 + si 1+ (t − τ − c Δt) fc fc c
i=1 c =0
ν
× ej2πνt ej2π(1+ fc )f (γi+k) + ek (t), where τ and ν denote the delay and Doppler shift, respectively. ek (t) denotes the additive noise at the k th receiver. fc is the carrier frequency and
dĂƌŐĞƚƐ ͘
͘ ͘ ͘
ͬ
Transmit/Receive antenna array used in monostatic configuration.
First, we shall present the measurement model for a single target and later extend it to multiple targets. Let the ith transmitter emit the noise waveform ui (t) covered by the code wi (t). We construct these waveforms and codes as ui (t) =
C−1
ui,c rec(t − cΔt),
wi (t) =
C−1
(2)
rec(t) =
1, if 0 < t < Δt, 0, otherwise.
(3)
(5)
III. S PARSE R EPRESENTATION In this section, we will express the measurement model presented in the previous section using sparse signal representation. We discretize the target delay-Doppler space into V grid points. Only R of them correspond to the actual targets. Let (τv , νv ) represent the delay and Doppler corresponding to the v th grid point. Define νv νv ψi,k,c (n, v) = 1 + si 1+ (nTS − τv − c Δt) fc fc νv × ej2πνv t ej2π(1+ fc )f (γi+k) .
c=0
where
=
(4)
where TS denotes the sampling interval.
(1)
wi,c rec(t − cΔt),
γ
dR sin(θ) , λ dT , dR
k
c=0
and
=
where λ = fcv+ν , v is the speed of wave propagation in the medium. So far, we assumed a single target. Now, consider R targets in the scene illuminated by the radar. Let τ r , ν r , and arc denote the delay, Doppler, and attenuations corresponding to the rth target, respectively. After sampling, we can express the measurements as MT C R −1 νr r ac 1 + yk (n) ≈ fc i=1 c =0 r=1 r ν r × si 1+ (nTS − τ − c Δt) fc r νr × ej2πν t ej2π(1+ fc )f (γi+k) + e (t),
WůĂŶĞ tĂǀĞ
Fig. 1.
f
Δt denotes the chip interval. All the covering waveforms are mutually orthogonal [13]. Historically, Walsh functions have been used as covering waveforms in IS-95 systems [20]. In this paper, we will use Walsh functions of order 64. We express the ith transmitted waveform as si (t) = ui (t)wi (t). These waveforms travel in
Let there be N samples in a processing interval. Then, we N stack {ψi,k,c (n, v)}n=1 into a column vector ψ i,k,c (v). Next, MR we arrange ψ i,k,c (v) k=1 into a longer column vector
34
rT
, . . . , a
rT
T
. Further, we define a sparse vector a(v) r a , if (τv , νv ) = (τ r , ν r ), (6) a(v) = 0, otherwise.
Arranging the vectors a(v) corresponding to all the grid points, we obtain a V MT C dimensional block-sparse vector T T T . (7) a = a(1) , . . . , a(V ) Stack the measurements and the noise samples at each receiver, we obtain yk ek
T
= [yk (1), . . . , yk (N )] , = [ek (1), . . . , ek (N )]T .
(8) (9)
Additionally, we arrange =
e
=
T T y 1 , . . . , y TMR ,
T T e1 , . . . , eTMR .
(10) (11)
Therefore, the measurement model y = Ψa + e.
9
8 7 6
10
20
30
40
50
60
10
20
30
40
50
60
10
20
30 40 Walsh code index
50
60
8 7 6
4 3.5 3 2.5
(12)
IV. S PARSE R ECONSTRUCTION
Fig. 2. 2 norms of the target returns as a function of the Walsh code index.
We observed from the previous section that the non-zero entries of the sparse vector appear in blocks. Therefore, we will exploit the block-sparsity while recovering the sparse vector a. More specifically, we will use the block orthogonal matching pursuit (BOMP) algorithm for sparse recovery [22]. We initialize the residual vector r(0) = y and the estimate vector a(0) = 0. In each subsequent iteration, we project the residual from the previous iteration onto all the blocks of columns of Ψ and pick the block that gives the largest projection energy. Let v (k ) be the block selected in the k th iteration. Further, define V (k ) as the set containing the indices of all the blocks selected in the previous iterations. We update the corresponding entries of the estimate vector by solving argmin{a(v)} (k ) y − Ψ(v)a(v). (13) v∈V
9
Target 3 responses
y
V. N UMERICAL S IMULATIONS We consider an x-band radar system with carrier frequency of 10GHz. Further, we assume the signal bandwidth to be 2GHz. Therefore, the chip interval Δt = 0.5nS. Let the number of chips in a pulse C = 6400. We consider illumination of the target scene using a single pulse. We simulate a radar system consisting of 3 transmit antennas and 3 receive antennas. Choose θ = 30◦ and dT = dR = λ2 . Further, assume there are 3 targets in the illuminated area. Considering C = 3, we define the attenuations corresponding to the 1 2 T T targets a = [0.5, 0.2, 0.8] , a = [0.3, 0.6, 0.7] , and 3 T a = [0.2, 0.4, 0.1] . Target 1 responses
a
Assume that we terminate the algorithm after K iterations. Therefore, a(K ) denotes the final estimate of the sparse vector a containing information about the unknown target parameters.
Target 2 responses
ψ i,c (v). Each of these columns forms a basis function in our sparse representation. We stack these columns corresponding to different transmitters and paths into a block of columns Ψ(v). Each block corresponds to a different grid point. We have V such blocks that we can concatenate to obtain the dictionary matrix Ψ. Now, we arrange the attenuations arc corresponding r dimensional vector a = paths
T into a C to r different r r a0 , . . . , aC −1 . Since a is independent of the transmitter index, we define a MT C dimensional vector ar =
8.5
Minimum target response
8
7.5
7
6.5
6
v∈V (k )
Denote a(k ) as the updated estimate vector. Next, we update the residue as Ψa(k ) (v). (14) r(k ) = y −
10
20
30 40 Walsh code index
50
60
Fig. 3. 2 norm of the minimum target returns as a function of the Walsh code index.
As mentioned earlier, we generate Walsh codes of the order 64. Assuming we have accurate estimates of the attenuations
v∈V (k )
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used numerical simulations to demonstrate the performance of the proposed system. In future work, we will analyze the performance in the presence of jamming signals. We will investigate the robustness of the system to modeling errors. Further, we will validate our results using real radar data.
from the previous processing intervals, we will select the Walsh codes for the current iteration. We compute the 2 norm of the signals reflected from the targets. In Fig. 2, we plot these norms as a function of the Walsh code indices. We notice that the signal returns are highly dependent on the choice of Walsh codes. Therefore, we need to match the codes to the target responses in order to improve the system performance. Since, we have multiple targets, the codes giving the optimal response for one target need not provide the optimal response for the other targets. Therefore, we select the code that maximizes the minimum target returns. We plot the minimum target returns in Fig. 3. Note hear that the weakest target is not fixed apriori. It varies depending on the choice of the code. We observe that the codes with indices 33, 49, and 17 are best matched to the target responses. We assign these codes to the three transmit antennas respectively.
R EFERENCES
2
1.5
1
0.5
0 20 15
5 4
10
3 5
Delay
Fig. 4.
2 1
−6
x 10
Stretch factor
Reconstructed sparse vector using optimal Walsh codes.
We discretize the target delay-Doppler space. Assume the true target delays lie in the interval [2nS, 20nS]. The spacing between these grid points is 2nS. Further, assume that the stretch factor νfvc (due to the Doppler) lies in the interval [0.000001, 0.0000055]. Let the delays corresponding to the three targets be 12nS, 18nS, and 6nS respectively. Similarly, let the corresponding stretch factors be 0.000004, 0.000002, and 0.000003 respectively. In Fig. 4, we plot the reconstructed vector at an SNR of −19.8983dB. We can clearly observe that the parameters of the three targets have been accurately reconstructed. The three largest peaks correspond to the three targets. VI. C ONCLUDING R EMARKS We used a colocated MIMO noise radar system to solve the multiple target estimation problem. We covered the transmitted noise waveforms using Walsh codes to ensure orthogonality between different transmitters and to match the transmitted waveforms to the target responses. We developed a signal model using sparse signal representation. We used a sparse reconstruction algorithm to estimate the target parameters. We
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