constant envelope waveform design for mimo radar - Semantic Scholar
CONSTANT ENVELOPE WAVEFORM DESIGN FOR MIMO RADAR S. Ahmed, J. S. Thompson and B. Mulgrew
Y. Petillot
School of Engineering University of Edinburgh Edinburgh EH8 3JL, UK
School of Engineering and Physical Sciences Heriot-Watt University Edinburgh EH14 4AS, UK
ABSTRACT A method for generating constant envelope (CE) waveforms to realise a given covariance matrix for a closely spaced MIMO radar system is proposed. In contrast to available algorithms, the technique provides closed form solutions for finding the required waveforms and suggests that waveforms can be chosen from finite alphabets such as binary-phase shift keying (BPSK) and quadrature-phase shift keying (QPSK). Gaussian random-variables (RV’s) are mapped onto CE nonGaussian RV’s using memoryless non-linear functions. The relationship between the correlation of Gaussian RV’s at the input to the nonlinear functions and non-Gaussian RV’s at their output is established. Simulation results are presented to demonstrate the effectiveness of the methodology. Index Terms— co-located antennas, constant-envelope waveforms, MIMO radar, Hermite polynomials 1. INTRODUCTION In contrast to phased-array radars, multiple-input multipleoutput (MIMO) radars allow each transmitting antenna to transmit independent waveforms, thus providing extra degrees-of-freedom (DOF) [1]. MIMO radars can be classified as either widely spaced [2] or co-located [3, 4]. In the former, the transmitting antennas are widely separated and each antenna may view a different aspect of the target. This topology can increase the spatial diversity of the system. In co-located MIMO radars, the transmitting antennas are closely spaced (on the order of half wavelength of the carrier frequency) and all the transmit antennas view the same aspect of the target. Co-located antennas cannot provide improved spatial diversity but can increase the spatial resolution of the system. Moreover, compared to phased-array radars extra DOFs of a co-located MIMO radar can provide better control of the transmit beampattern [3]. In [3–5] the covariance matrix of the waveforms for colocated MIMO radars is synthesised to achieve the desired beampattern. The work of [3] is extended in [4] to design the actual waveforms. This work is based on the assumption that This work is funded through EPSRC/DSTL joint grant scheme
978-1-4244-4296-6/10/$25.00 ©2010 IEEE
4066
signals that have CE and take on values in a finite alphabets may fail to realise a given covariance matrix. The compromise on the requirement for CE waveforms is to allow small variations in the amplitudes of the waveforms and satisfy a less restrictive condition of low peak-to-average power ratio (PAPR). To design the waveforms a cyclic algorithm [4] is used that requires a number of iterations. In this paper an alternative solution is proposed that guarantees the CE property. It is based on the principle that waveforms with specified correlation properties can be easily generated using independent Gaussian random variables and a square root of the associated covariance matrix. Memoryless nonlinearities are then used to map the Gaussian random processes to random processes with the desired density function - in this case random variables with a CE. Although the action of memoryless nonlinearities also alters the correlation between the waveforms and changes the covariance matrix, this change is well defined in terms of a series of Hermite polynomials [7] and can be inverted. Thus we can define a covariance matrix at the input to the memoryless nonlinearities that would produce the desired covariance matrix at their outputs. A similar approach was used to generate correlated Gamma processes in [6] to simulate radar sea clutter. Here we extend this work to generate a variety of CE waveforms with given cross-correlation properties. The organisation of the paper is as follows. In Section 2 the problem is formulated, Section 3 discusses the generation of CE waveforms. Simulation results are presented in Section 4, followed by conclusions in Section 5. Notation: Bold upper case letters, X, and calligraphic upper case letters, X , denote matrices while bold lower case letters, x, denote vectors. Conjugate and conjugate transposition are respectively denoted by (.)∗ and (.)H . The mth row and nth column element of a matrix X is denoted by X(m, n) and E{.} denote statistical expectation. 2. PROBLEM FORMULATION Consider a closely spaced MIMO radar with M transmit antennas. The sequence of N transmitted symbols from antenna m is denoted by the column vector xm . The (N × M ) matrix
ICASSP 2010
X = [x1 x2 · · · xM ] contains all the transmitted symbols from the M antennas. In co-located MIMO radars, for a desired beampattern a covariance matrix R of the waveforms is synthesised and then the signal waveform matrix X satisfying XH X = R is realised. Given the desired covariance matrix, N the waveform matrix X can be determined using: √ N PR1/2 (1) X = or X =
X Λ1/2 UH ,
(2)
where P is (N × M ) semi-orthonormal matrix, X is (N × M ) matrix of zero mean, unit variance columns of Gaussian RV’s, Λ is the (M × M ) matrix of eigenvalues and U is an (N × M ) matrix of the eigenvectors of R. The distribution of RV’s in the columns of X in (1) will depend on P, while the distribution in the columns of X in (2) will be Gaussian. In radar and communication systems it is often a practical requirement that the transmitted signals have a CE. However, (1) and (2) do not guarantee the CE property. 3. GENERATION OF CE RANDOM PROCESSES FROM GAUSSIAN RANDOM VARIABLES
between the Gaussian and non-Gaussian RV’s as [7] ∞ ∞ x1 √ ψ12 = f (x1 )Hn p(x1 )dx1 σ1 2 −∞ n=0 ∞ n x2 ρ12 √ · , f (x2 )Hn p(x2 )dx2 2n n! σ2 2 −∞ where
Generating non-Gaussian RV’s to realise a given covariance matrix, R, is very much difficult but generating Gaussian RV’s to realise a given covariance matrix, Rg , is straight forward. The proposed algorithm exploits this fact to convert the problem of finding non-Gaussian RV’s to realise R into finding the Gaussian RV’s to realise Rg . To establish the relationship between the Gaussian and non-Gaussian processes, a Gaussian process is mapped onto the non-Gaussian with some memoryless non-linear mapping function. Once we have the relationship between the Gaussian and non-Gaussian RV’s, the relationship between their correlation properties can be derived. Hence, Rg is obtained from R, then the matrix of Gaussian RV’s that can realise Rg can be easily determined with a de-whitening transformation. These Gaussian RV’s are converted into the desired non-Gaussian RV’s with a memoryless non-linear mapping function. The block diagram in Fig. 1 shows the process of mapping Gaussian RV’s onto non-Gaussian RV’s through a non-linear transformation. If non-Gaussian RV’s y1 = f (x1 ) and y2 = f (x2 ) are the memoryless non-linear functions of Gaussian RV’s x1 and x2 then the cross-correlation, ψ12 = E{y1 y2∗ }, between y1 and y2 can be defined as ∞ ∞ y1 y2∗ p(x1 , x2 ; ρ12 )dx1 dx2 , (3) ψ12 = −∞
Fig. 1. Nonlinear mapping of Gaussian process onto nonGaussian process.
−∞
where p(x1 , x2 ; ρ12 ) is the joint-probability-density (PDF) E{x x∗ } function of x1 and x2 , ρ12 = σ11σ22 is the cross-correlation coefficient of x1 and x2 , while σm is the square-root of the variance of xm . Hermite polynomials can be used to separate the above double integral cross-correlation relationship
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(4)
2
x 1 − m 2 p(xm ) = e 2σm 2 2πσm
is the PDF of Gaussian RV xm and 2
Hn (xm ) = (−1)n exm /2
dn −x2m /2 e . dxnm
is the Hermite polynomial of order n. Since f (x1 ) and f (x2 ) have same PDF therefore by assuming σ1 = σ2 = σ, (4) can be written in more compact form as 2 n ∞ ∞ ρ x √ (5) f (x)Hn ψ12 = p(x)dx n12 , 2 n! σ 2 −∞ n=0 where f (x) is the mapping function and p(x) is the PDF of Gaussian RV’s. In the following, some examples of generating CE RV’s from Gaussian RV’s are given. 3.1. Mapping of Gaussian RV’s onto BPSK RV’s If the BPSK process is represented by RV y and the Gaussian process by RV x then x can be mapped onto y using the following relationship y = sign(x).
(6)
If y1 = sign(x1 ), y2 = sign(x2 ) and the cross-correlation coefficient of x1 and x2 is ρ12 then the relationship between the cross-correlation of Gaussian and BPSK RV’s can be derived using (5) as [6] E{y1 y2∗ }
=
2 sin−1 (ρ12 ). π
(7)
Suppose E{y1 y2∗ } = β12 then ρ12 can be determined as π β12 . ρ12 = sin 2
To generate M CE complex waveforms with given covariance matrix R, corresponding covariance matrix Rg for Gaussian RV’s can be written as
(8)
Rg (i, j) = ln (R(i, j)) + σ 2 , i, j ∈ {1, 2, . . . M }.
Hence, if β12 is known then ρ12 can be determined using (8). The Gaussian RV’s x1 and x2 realising ρ12 can be easily determined using the de-whitening transformation in (2). Once we have x1 and x2 , the BPSK signals y1 and y2 can be generated using (6). Similarly, to generate M BPSK waveforms having covariance matrix R, using (8) the correlation matrix of unit variance M Gaussian RV’s, Rg , can be generated as π Rg (i, j) = sin R(i, j) , i, j ∈ {1, 2, . . . , M } (9) 2
Once Rg is obtained, the matrix of Gaussian RV’s, X, to realise Rg can be generated again using the de-whitening transformation in (2). Finally the CE complex RV’s to realise R can be generated by mapping X onto the CE RV’s using (10). Remark: In (13), if γ12 is negative then the crosscorrelation between the Gaussian RV’s will be complex, which require Gaussian complex RV’s. Therefore, for γ12 negative, (10) will not necessarily be CE. In Fig. 2 the cross-correlation of BPSK and CE complex RV’s with respect to ρ12 is shown. For CE complex RV’s the variance of Gaussian RV’s is 5.
where Rg should be a positive semi-definite covariance matrix for the algorithm to work. The matrix X of Gaussian RV’s to realise the correlation matrix Rg can be generated using (2). Similarly, the matrix of BPSK signals having covariance matrix R can be generated by mapping X onto the BPSK signals with the mapping function given in (6). This methodology can be extended to generate QPSK signals.
1 0.8 0.6 0.4
3.2. Mapping of Gaussian RV’s onto CE Complex RV’s
y = ejx .
0.2 ρ12
If a CE complex signal is represented by RV y and Gaussian RV by x then x can be mapped onto y using the following relationship (10)
In order to establish the relationship between the crosscorrelation of Gaussian RV’s and CE complex RV’s suppose y1 = ejx1 , y2 = ejx2 and the Gaussian RV’s x1 and x2 have cross-correlation coefficient ρ12 . The cross-correlation of RV’s y1 and y2 in terms of ρ12 can be written by assuming σ1 = σ2 = σ and using (5) as ∞ =
∞
2 dx
ρn12 . 2πσ 2 2n n! −∞ n=0 (11) jσ x √ and t = , (11) can be written as By defining u = σ√ 2 2
E{y1 y2∗ }
2
x jx − 2σ 2
e e
Hn
x √ σ 2
0
−0.2 −0.4 −0.6 −0.8 −1 −1
BPSK symbols CE complex symbols −0.5 0 0.5 Cross−correlation of CE waveforms
1
Fig. 2. The cross-correlation relationship between the mapped Gaussian and non-Gaussian process.
4. SIMULATION
In this section, we compare the beampattern obtained from the given synthesised covariance matrix, R, with the beampattern 2 ∞ n −σ2 ∞ obtained from the CE waveforms realising R. For simulation, 2 ρ12 e E{y1 y2∗ } = e−(u−t) Hn (u) du . (12) the correlation of an autoregressive process of order 1 is used n π2 n! −∞ n=0 to define the covarince matrix as
∞ z n ⎡ ⎤ ∗ z If we define γ12 = E{y1 y2 } then using e = n=0 n! and 1 γ · · · γ M−1 ∞ −(z−p)2 √ ⎢ γ 1 · · · γ M−2 ⎥ Hn (z)dz = 2n πpn [8], the cross-correlation ⎢ ⎥ −∞ e R=⎢ (14) ⎥. .. .. .. between the CE complex RV’s can be derived as ⎣ ⎦ . . ··· . 2 γ M−1 γ M−2 · · · 1 γ = eσ (ρ12 −1) . 12
The Covariance matrix R is parametrised by γ and 0 ≤ γ ≤ 1. The beampattern at location θ is defined as
Correspondingly ρ12 can be determined as ρ12
=
ln(γ12 ) + 1. σ2
(13)
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P (θ) = eH (θ)Re(θ),
5. CONCLUSION CE symbol generation algorithms to realise a given covariance matrix are proposed. The proposed algorithms provide closed form solutions for the generation of CE waveforms. This research suggests that symbols can be chosen from finite alphabets to match the desired covariance matrix.
7 6.5
P(θ)
5.5 5
4 3.5 3 2.5 2 1.5 1 0.5 0 −100
−50
0 θ
50
100
Fig. 4. Beampattern realised by BPSK symbols is compared with the beampattern obtained with the given R. versity in radars − models and detection performance,” IEEE Transaction on Signal Processing, vol. 54, no. 3, pp. 823–838, Mar. 2006. [2] A. M. Haimovich, R. S. Blum and L. J. Cimini, “MIMO radar with widely separated antennas,” IEEE Signal Processing Magazine, vol. 25, no. 1, pp. 116-129, Jan. 2008. [3] P. Stoica, J. Li, and Y. Xie, “On probing signal design for MIMO radar,” IEEE Transaction on Signal Processing, vol. 55, no. 8, pp. 4151–4161, Aug. 2007.
[5] D. R. Fuhrmann and J. S. Antonio, “Transmit beamforming for MIMO radar systems using signal crosscorrelation,” IEEE Transaction on Aerospace and Electronic Systems, vol. 44, no. 1, pp. 171–185, Jan. 2008.
Beampattern with given R Beampattern with BPSK symbols Beampattern with CE complex symbols
[6] R. J. A. Tough and K. D. Ward,“The correlation properties of gamma and other non-Gaussian processes generated by memoryless nonlinear transformation,” J. Phys. D: Appl. Phys, pp. 3075–3084, vol. 32, no. 23, Dec. 1999.
4.5 4 3.5 3 −100
Ideal beampattern R beampattern BPSK beampattern
4.5
[4] P. Stoica, J. Li, and X. Zhu, “Waveform synthesis for diversity-based transmit beampattern design,” IEEE Transaction on Signal Processing, vol. 56, no. 6, pp. 2593–2598, Jun. 2008.
7.5
6
5
P(θ)
where e(θ) = [1 ejπ sin(θ) · · · ejπ(M−1) sin(θ) ]T is the steering vector. For the first simulation, it is assumed that γ = 0.5 and M = 5 to generate the covariance matrix. The BPSK and CE complex symbols are generated with the proposed algorithms, Fig. 3 compares the beampattern of R with the beampattern of BPSK and CE complex symbols realising R. For both algorithms each antenna transmit 100 symbols and the beampattern is averaged over 100 different sets of symbols from each antenna. In the figure it can be seen that on an average CE waveforms achieve almost the same beampattern to that of obtained with R. Similarly for the second simulation to achieve the desired beampattern with M = 5 the covariance matrix is synthesised using the algorithm in [3]. Here, to realise the given covariance matrix R only BPSK symbols are determined using the corresponding proposed algorithm. Fig. 4 compares the beampattern obtained by the proposed algorithm with the beampattern obtained with the synthesised R. The number of symbols transmitted from each antenna is 100. It can be seen in the figure that the proposed algorithm achieves almost the same beampattern to that of obtained with the synthesised R. In Fig. 4 the beampattern is not the averaged one, each set of generated BPSK symbols yield close beampattern to R.
−50
0 θ
50
100
Fig. 3. Beampattern obtained by BPSK and CE complex symbols is compared with the beampattern obtained by given R. 6. REFERENCES [1] E. Fishler, A. M. Haimovich, R. S. Blum, L. J. Cimini, D. Chizhik and R. A. Valenzuela, “Spatial di-
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[7] J. Brown Junior, “On the expansion of the bivariate Gaussian probability density using results of nonlinear theory,” IEEE Transactions on Information Theory, pp. 158–159, vol. 14, no. 1, Jan. 1968. [8] A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and Series, vol. 2, USSR Academy of Sciences Moscow, Translated from the Russian by N. M. Queens, Gordon and Breach Science Publishers, 1988.
Y. Petillot
School of Engineering University of Edinburgh Edinburgh EH8 3JL, UK
School of Engineering and Physical Sciences Heriot-Watt University Edinburgh EH14 4AS, UK
ABSTRACT A method for generating constant envelope (CE) waveforms to realise a given covariance matrix for a closely spaced MIMO radar system is proposed. In contrast to available algorithms, the technique provides closed form solutions for finding the required waveforms and suggests that waveforms can be chosen from finite alphabets such as binary-phase shift keying (BPSK) and quadrature-phase shift keying (QPSK). Gaussian random-variables (RV’s) are mapped onto CE nonGaussian RV’s using memoryless non-linear functions. The relationship between the correlation of Gaussian RV’s at the input to the nonlinear functions and non-Gaussian RV’s at their output is established. Simulation results are presented to demonstrate the effectiveness of the methodology. Index Terms— co-located antennas, constant-envelope waveforms, MIMO radar, Hermite polynomials 1. INTRODUCTION In contrast to phased-array radars, multiple-input multipleoutput (MIMO) radars allow each transmitting antenna to transmit independent waveforms, thus providing extra degrees-of-freedom (DOF) [1]. MIMO radars can be classified as either widely spaced [2] or co-located [3, 4]. In the former, the transmitting antennas are widely separated and each antenna may view a different aspect of the target. This topology can increase the spatial diversity of the system. In co-located MIMO radars, the transmitting antennas are closely spaced (on the order of half wavelength of the carrier frequency) and all the transmit antennas view the same aspect of the target. Co-located antennas cannot provide improved spatial diversity but can increase the spatial resolution of the system. Moreover, compared to phased-array radars extra DOFs of a co-located MIMO radar can provide better control of the transmit beampattern [3]. In [3–5] the covariance matrix of the waveforms for colocated MIMO radars is synthesised to achieve the desired beampattern. The work of [3] is extended in [4] to design the actual waveforms. This work is based on the assumption that This work is funded through EPSRC/DSTL joint grant scheme
978-1-4244-4296-6/10/$25.00 ©2010 IEEE
4066
signals that have CE and take on values in a finite alphabets may fail to realise a given covariance matrix. The compromise on the requirement for CE waveforms is to allow small variations in the amplitudes of the waveforms and satisfy a less restrictive condition of low peak-to-average power ratio (PAPR). To design the waveforms a cyclic algorithm [4] is used that requires a number of iterations. In this paper an alternative solution is proposed that guarantees the CE property. It is based on the principle that waveforms with specified correlation properties can be easily generated using independent Gaussian random variables and a square root of the associated covariance matrix. Memoryless nonlinearities are then used to map the Gaussian random processes to random processes with the desired density function - in this case random variables with a CE. Although the action of memoryless nonlinearities also alters the correlation between the waveforms and changes the covariance matrix, this change is well defined in terms of a series of Hermite polynomials [7] and can be inverted. Thus we can define a covariance matrix at the input to the memoryless nonlinearities that would produce the desired covariance matrix at their outputs. A similar approach was used to generate correlated Gamma processes in [6] to simulate radar sea clutter. Here we extend this work to generate a variety of CE waveforms with given cross-correlation properties. The organisation of the paper is as follows. In Section 2 the problem is formulated, Section 3 discusses the generation of CE waveforms. Simulation results are presented in Section 4, followed by conclusions in Section 5. Notation: Bold upper case letters, X, and calligraphic upper case letters, X , denote matrices while bold lower case letters, x, denote vectors. Conjugate and conjugate transposition are respectively denoted by (.)∗ and (.)H . The mth row and nth column element of a matrix X is denoted by X(m, n) and E{.} denote statistical expectation. 2. PROBLEM FORMULATION Consider a closely spaced MIMO radar with M transmit antennas. The sequence of N transmitted symbols from antenna m is denoted by the column vector xm . The (N × M ) matrix
ICASSP 2010
X = [x1 x2 · · · xM ] contains all the transmitted symbols from the M antennas. In co-located MIMO radars, for a desired beampattern a covariance matrix R of the waveforms is synthesised and then the signal waveform matrix X satisfying XH X = R is realised. Given the desired covariance matrix, N the waveform matrix X can be determined using: √ N PR1/2 (1) X = or X =
X Λ1/2 UH ,
(2)
where P is (N × M ) semi-orthonormal matrix, X is (N × M ) matrix of zero mean, unit variance columns of Gaussian RV’s, Λ is the (M × M ) matrix of eigenvalues and U is an (N × M ) matrix of the eigenvectors of R. The distribution of RV’s in the columns of X in (1) will depend on P, while the distribution in the columns of X in (2) will be Gaussian. In radar and communication systems it is often a practical requirement that the transmitted signals have a CE. However, (1) and (2) do not guarantee the CE property. 3. GENERATION OF CE RANDOM PROCESSES FROM GAUSSIAN RANDOM VARIABLES
between the Gaussian and non-Gaussian RV’s as [7] ∞ ∞ x1 √ ψ12 = f (x1 )Hn p(x1 )dx1 σ1 2 −∞ n=0 ∞ n x2 ρ12 √ · , f (x2 )Hn p(x2 )dx2 2n n! σ2 2 −∞ where
Generating non-Gaussian RV’s to realise a given covariance matrix, R, is very much difficult but generating Gaussian RV’s to realise a given covariance matrix, Rg , is straight forward. The proposed algorithm exploits this fact to convert the problem of finding non-Gaussian RV’s to realise R into finding the Gaussian RV’s to realise Rg . To establish the relationship between the Gaussian and non-Gaussian processes, a Gaussian process is mapped onto the non-Gaussian with some memoryless non-linear mapping function. Once we have the relationship between the Gaussian and non-Gaussian RV’s, the relationship between their correlation properties can be derived. Hence, Rg is obtained from R, then the matrix of Gaussian RV’s that can realise Rg can be easily determined with a de-whitening transformation. These Gaussian RV’s are converted into the desired non-Gaussian RV’s with a memoryless non-linear mapping function. The block diagram in Fig. 1 shows the process of mapping Gaussian RV’s onto non-Gaussian RV’s through a non-linear transformation. If non-Gaussian RV’s y1 = f (x1 ) and y2 = f (x2 ) are the memoryless non-linear functions of Gaussian RV’s x1 and x2 then the cross-correlation, ψ12 = E{y1 y2∗ }, between y1 and y2 can be defined as ∞ ∞ y1 y2∗ p(x1 , x2 ; ρ12 )dx1 dx2 , (3) ψ12 = −∞
Fig. 1. Nonlinear mapping of Gaussian process onto nonGaussian process.
−∞
where p(x1 , x2 ; ρ12 ) is the joint-probability-density (PDF) E{x x∗ } function of x1 and x2 , ρ12 = σ11σ22 is the cross-correlation coefficient of x1 and x2 , while σm is the square-root of the variance of xm . Hermite polynomials can be used to separate the above double integral cross-correlation relationship
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(4)
2
x 1 − m 2 p(xm ) = e 2σm 2 2πσm
is the PDF of Gaussian RV xm and 2
Hn (xm ) = (−1)n exm /2
dn −x2m /2 e . dxnm
is the Hermite polynomial of order n. Since f (x1 ) and f (x2 ) have same PDF therefore by assuming σ1 = σ2 = σ, (4) can be written in more compact form as 2 n ∞ ∞ ρ x √ (5) f (x)Hn ψ12 = p(x)dx n12 , 2 n! σ 2 −∞ n=0 where f (x) is the mapping function and p(x) is the PDF of Gaussian RV’s. In the following, some examples of generating CE RV’s from Gaussian RV’s are given. 3.1. Mapping of Gaussian RV’s onto BPSK RV’s If the BPSK process is represented by RV y and the Gaussian process by RV x then x can be mapped onto y using the following relationship y = sign(x).
(6)
If y1 = sign(x1 ), y2 = sign(x2 ) and the cross-correlation coefficient of x1 and x2 is ρ12 then the relationship between the cross-correlation of Gaussian and BPSK RV’s can be derived using (5) as [6] E{y1 y2∗ }
=
2 sin−1 (ρ12 ). π
(7)
Suppose E{y1 y2∗ } = β12 then ρ12 can be determined as π β12 . ρ12 = sin 2
To generate M CE complex waveforms with given covariance matrix R, corresponding covariance matrix Rg for Gaussian RV’s can be written as
(8)
Rg (i, j) = ln (R(i, j)) + σ 2 , i, j ∈ {1, 2, . . . M }.
Hence, if β12 is known then ρ12 can be determined using (8). The Gaussian RV’s x1 and x2 realising ρ12 can be easily determined using the de-whitening transformation in (2). Once we have x1 and x2 , the BPSK signals y1 and y2 can be generated using (6). Similarly, to generate M BPSK waveforms having covariance matrix R, using (8) the correlation matrix of unit variance M Gaussian RV’s, Rg , can be generated as π Rg (i, j) = sin R(i, j) , i, j ∈ {1, 2, . . . , M } (9) 2
Once Rg is obtained, the matrix of Gaussian RV’s, X, to realise Rg can be generated again using the de-whitening transformation in (2). Finally the CE complex RV’s to realise R can be generated by mapping X onto the CE RV’s using (10). Remark: In (13), if γ12 is negative then the crosscorrelation between the Gaussian RV’s will be complex, which require Gaussian complex RV’s. Therefore, for γ12 negative, (10) will not necessarily be CE. In Fig. 2 the cross-correlation of BPSK and CE complex RV’s with respect to ρ12 is shown. For CE complex RV’s the variance of Gaussian RV’s is 5.
where Rg should be a positive semi-definite covariance matrix for the algorithm to work. The matrix X of Gaussian RV’s to realise the correlation matrix Rg can be generated using (2). Similarly, the matrix of BPSK signals having covariance matrix R can be generated by mapping X onto the BPSK signals with the mapping function given in (6). This methodology can be extended to generate QPSK signals.
1 0.8 0.6 0.4
3.2. Mapping of Gaussian RV’s onto CE Complex RV’s
y = ejx .
0.2 ρ12
If a CE complex signal is represented by RV y and Gaussian RV by x then x can be mapped onto y using the following relationship (10)
In order to establish the relationship between the crosscorrelation of Gaussian RV’s and CE complex RV’s suppose y1 = ejx1 , y2 = ejx2 and the Gaussian RV’s x1 and x2 have cross-correlation coefficient ρ12 . The cross-correlation of RV’s y1 and y2 in terms of ρ12 can be written by assuming σ1 = σ2 = σ and using (5) as ∞ =
∞
2 dx
ρn12 . 2πσ 2 2n n! −∞ n=0 (11) jσ x √ and t = , (11) can be written as By defining u = σ√ 2 2
E{y1 y2∗ }
2
x jx − 2σ 2
e e
Hn
x √ σ 2
0
−0.2 −0.4 −0.6 −0.8 −1 −1
BPSK symbols CE complex symbols −0.5 0 0.5 Cross−correlation of CE waveforms
1
Fig. 2. The cross-correlation relationship between the mapped Gaussian and non-Gaussian process.
4. SIMULATION
In this section, we compare the beampattern obtained from the given synthesised covariance matrix, R, with the beampattern 2 ∞ n −σ2 ∞ obtained from the CE waveforms realising R. For simulation, 2 ρ12 e E{y1 y2∗ } = e−(u−t) Hn (u) du . (12) the correlation of an autoregressive process of order 1 is used n π2 n! −∞ n=0 to define the covarince matrix as
∞ z n ⎡ ⎤ ∗ z If we define γ12 = E{y1 y2 } then using e = n=0 n! and 1 γ · · · γ M−1 ∞ −(z−p)2 √ ⎢ γ 1 · · · γ M−2 ⎥ Hn (z)dz = 2n πpn [8], the cross-correlation ⎢ ⎥ −∞ e R=⎢ (14) ⎥. .. .. .. between the CE complex RV’s can be derived as ⎣ ⎦ . . ··· . 2 γ M−1 γ M−2 · · · 1 γ = eσ (ρ12 −1) . 12
The Covariance matrix R is parametrised by γ and 0 ≤ γ ≤ 1. The beampattern at location θ is defined as
Correspondingly ρ12 can be determined as ρ12
=
ln(γ12 ) + 1. σ2
(13)
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P (θ) = eH (θ)Re(θ),
5. CONCLUSION CE symbol generation algorithms to realise a given covariance matrix are proposed. The proposed algorithms provide closed form solutions for the generation of CE waveforms. This research suggests that symbols can be chosen from finite alphabets to match the desired covariance matrix.
7 6.5
P(θ)
5.5 5
4 3.5 3 2.5 2 1.5 1 0.5 0 −100
−50
0 θ
50
100
Fig. 4. Beampattern realised by BPSK symbols is compared with the beampattern obtained with the given R. versity in radars − models and detection performance,” IEEE Transaction on Signal Processing, vol. 54, no. 3, pp. 823–838, Mar. 2006. [2] A. M. Haimovich, R. S. Blum and L. J. Cimini, “MIMO radar with widely separated antennas,” IEEE Signal Processing Magazine, vol. 25, no. 1, pp. 116-129, Jan. 2008. [3] P. Stoica, J. Li, and Y. Xie, “On probing signal design for MIMO radar,” IEEE Transaction on Signal Processing, vol. 55, no. 8, pp. 4151–4161, Aug. 2007.
[5] D. R. Fuhrmann and J. S. Antonio, “Transmit beamforming for MIMO radar systems using signal crosscorrelation,” IEEE Transaction on Aerospace and Electronic Systems, vol. 44, no. 1, pp. 171–185, Jan. 2008.
Beampattern with given R Beampattern with BPSK symbols Beampattern with CE complex symbols
[6] R. J. A. Tough and K. D. Ward,“The correlation properties of gamma and other non-Gaussian processes generated by memoryless nonlinear transformation,” J. Phys. D: Appl. Phys, pp. 3075–3084, vol. 32, no. 23, Dec. 1999.
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Ideal beampattern R beampattern BPSK beampattern
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[4] P. Stoica, J. Li, and X. Zhu, “Waveform synthesis for diversity-based transmit beampattern design,” IEEE Transaction on Signal Processing, vol. 56, no. 6, pp. 2593–2598, Jun. 2008.
7.5
6
5
P(θ)
where e(θ) = [1 ejπ sin(θ) · · · ejπ(M−1) sin(θ) ]T is the steering vector. For the first simulation, it is assumed that γ = 0.5 and M = 5 to generate the covariance matrix. The BPSK and CE complex symbols are generated with the proposed algorithms, Fig. 3 compares the beampattern of R with the beampattern of BPSK and CE complex symbols realising R. For both algorithms each antenna transmit 100 symbols and the beampattern is averaged over 100 different sets of symbols from each antenna. In the figure it can be seen that on an average CE waveforms achieve almost the same beampattern to that of obtained with R. Similarly for the second simulation to achieve the desired beampattern with M = 5 the covariance matrix is synthesised using the algorithm in [3]. Here, to realise the given covariance matrix R only BPSK symbols are determined using the corresponding proposed algorithm. Fig. 4 compares the beampattern obtained by the proposed algorithm with the beampattern obtained with the synthesised R. The number of symbols transmitted from each antenna is 100. It can be seen in the figure that the proposed algorithm achieves almost the same beampattern to that of obtained with the synthesised R. In Fig. 4 the beampattern is not the averaged one, each set of generated BPSK symbols yield close beampattern to R.
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0 θ
50
100
Fig. 3. Beampattern obtained by BPSK and CE complex symbols is compared with the beampattern obtained by given R. 6. REFERENCES [1] E. Fishler, A. M. Haimovich, R. S. Blum, L. J. Cimini, D. Chizhik and R. A. Valenzuela, “Spatial di-
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[7] J. Brown Junior, “On the expansion of the bivariate Gaussian probability density using results of nonlinear theory,” IEEE Transactions on Information Theory, pp. 158–159, vol. 14, no. 1, Jan. 1968. [8] A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and Series, vol. 2, USSR Academy of Sciences Moscow, Translated from the Russian by N. M. Queens, Gordon and Breach Science Publishers, 1988.
