# Orthogonal Radiation Field Construction for Microwave Staring Correlated Imaging

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2 Liu and Wang the radiation field samples. Then the ideal ORF samples are approximated by properly designing the transmitting signals. Suppose that each signal is made up of N known frequencies, and the signal is determined when the complex amplitude of each frequency is obtained. The details of this method are described in Section 3. Finally, when the approximated ORF samples are applied in MSCI, high resolution imaging results with good robustness to noise are obtained. The rest of this paper is organized as follows. In Section, the imaging model of MSCI is established. The details of the ORF construction are presented in Section 3. In Section, the numerical simulations are presented, and we make a concluding remark in Section to summarize this paper.. MICROWAVE STARING CORRELATED IMAGING MODEL The geometry of MSCI system is illustrated in Figure. There is an array of M transmitting elements and a receiving element. Let (x, y, z) be Cartesian coordinates with the origin O. The array aperture is labeled as D, and the -D imaging region is labeled as S. r m, r and r is the position vector of m-th transmitting element, the receiving element and an arbitrary point within S, respectively. Figure. The geometry of MSCI. The radiation field E i (r,t) generated by the M transmitting elements can be expressed as [] M E i (r,t) = π r r A ( m r ) f m (t r ) r dr () c m= D m where c denotes the speed of light, D m the sub-aperture of the m-th transmitting element, f m (t) the signal of the m-th transmitting element, and A m (r ) the amplitude of the source in the sub-aperture D m. The received echo signal s r (t) of the target in the imaging region can be expressed as follows s r (t) = S π r r E i r r (r,t c ) σ(r)dr + n (t) ()

4 Liu and Wang There are many kinds of -D orthogonal basis functions. The general form of a group of orthogonal basis functions can be expressed as {Ψ (r), Ψ (r),...,ψ k (r),...}, r S (9) where Ψ k (r) is considered as the ideal ORF sample that E r (r,t k ) is going to approximate. Assuming that E r (r,t k )=Ψ k (r), substitute Eq. () to Eq. (3), then Ψ k (r) can be expressed as Ψ k (r) = (π) r r M m= D m r r A ( m r ) f m (t k r r c r r )dr () c Assume that F m (ω) is the Fourier transform result of f m (t), and it is known that the spectrum range of this signal is ω L to ω H.Thusf m (t) can be expressed as f m (t) = ωh F m (ω) e jωt dω () π ω L Further, the spectral domain is discretized as {ω,ω,...,ω n,...,ω N },andω n is ω n = ω L +(n ) Δω () where Δω =(ω H ω L )/N.Thusf m (t) can be written as f m (t) = Δω π N F m (ω n ) e jωnt (3) n= According to Eq. (3), if we calculate the value of F m (ω n ), the signal f m (t) can be determined. Substituting Eq. (3) to Eq. (), we can obtain where Φ m,n (r,t k )= Ψ k (r) = Δω 3π 3 r r M m= n= D m N Φ m,n (r,t k )F m (ω n ) () ( r r A ( m r ) e jωn ) t k r r r r c c dr () In Eq. (), r is discretized as {r, r,...,r q,...,r Q }. And the radiation field values at these points are arranged into a vector as e rk =[Ψ k (r ), Ψ k (r ),...,Ψ k (r q ),...,Ψ k (r Q )] T () Therefore, Eq. () can be described as the following matrix equation e rk = Φ k F (7) where, Φ, (r,t k ) Φ, (r,t k )... Φ,N (r,t k )... Φ M, (r,t k ) Φ M, (r,t k )... Φ M,N (r,t k ) Φ, (r,t k ) Φ, (r,t k )... Φ,N (r,t k )... Φ M, (r,t k ) Φ M, (r,t k )... Φ M,N (r,t k ) Φ k = () Φ, (r Q,t k ) Φ, (r Q,t k )... Φ,N (r Q,t k )... Φ M, (r Q,t k ) Φ M, (r Q,t k )... Φ M,N (r Q,t k ) F =[ F (ω ) F (ω )... F (ω N )... F M (ω ) F M (ω )... F M (ω N ) ] T (9) With known e rk and Φ k, the result of spectrum vector F is achieved by utilizing least squares (LS) method as F = ( Φ T k Φ k) Φ T k e rk () When the spectrum vector F is known, the signal of each transmitting element can be obtained by Eq. (3). And the approximate ORF sample E r (r,t k ) can be calculated by Eq. (3) or ().

5 Progress In Electromagnetics Research M, Vol. 7, 7 3. NUMERICAL SIMULATIONS In this section, numerical experiments are performed to validate the performance of the proposed scheme. Assume that the uniform spacing -D array with spacing d = m is located on the XOY plane (a) (b) (c) (d) (e) (f) Figure. Radiation ﬁeld samples approximate to diﬀerent orthogonal basis functions. (a) Orthogonal basis (i = 3, j = ). (b) Radiation ﬁeld (i = 3, j = ). (c) Orthogonal basis (i = 3, j = 3). (d) Radiation ﬁeld (i = 3, j = 3). (e) Orthogonal basis (i =, j = ). (f) Radiation ﬁeld (i =, j = ).

6 Liu and Wang The receiver is located at the origin O. The distance between the array center and the imaging region center is m. In the experiments, the imaging region is m m... Orthogonal Radiation Field Approximation The -D orthogonal basis functions are chosen as ( iπx Ψ i,j (x, y) =sin + π ) ( jπx cos + π ) L x L y where x [ L x /,L x /], y [ L y /,L y /], i =,,...,I, j =,,,...,J. According to the simulation parameters, L x = L y =m. The bandwidth of the signals is limited to GHz, which has the center frequency of GHz. frequency points with uniform spacing are used to synthesize the signals. Therefore, the vector F has (M N = ) unknowns. Here, the imaging region S is discretized into () (a) (b) Figure 3. Imaging target and imaging result based on orthogonal radiation field. (a) Imaging target model. (b) Imaging result with no noise (a) (b)

7 Progress In Electromagnetics Research M, Vol. 7, (c) (d) Figure. Imaging results with different SNRs based on orthogonal radiation field. (a) Imaging result with SNR = db. (b) Imaging result with SNR = db. (c) Imaging result with SNR = db. (d) Imaging result with SNR = db. (Q = ) cells. It means that an ORF sample provides spatial sampling values, which is much bigger than the required number to resolve the unknowns in F. The results of the constructed ORF samples are shown in Figure. Figure shows that the ORF samples can be constructed using the method proposed in this paper. However, with the increasing of i and j, and the orthogonal basis function becomes more and more complex as shown in Figures (a), (c), (e). Figures (b), (d), (f) show that the error of the ORF samples constructed by designing the transmitting signals increases as well. The root mean square error (RMSE) is used to describe the error between the constructed ORF samples and the orthogonal basis functions. And the errors of results shown in Figures (b), (d), (f) are.,., and.9 respectively.... RMSE.... Figure. RMSE versus SNR SNR/dB

8 Liu and Wang.. Imaging Based on Orthogonal Radiation Field The target model is shown in Figure 3(a). Here the imaging region S is discretized into cells to reconstruct the target. Due to the limitation of simulation parameters, such as the bandwidth, array size and number of transmitting elements, (I =,J = ) ORF samples are constructed to use in the imaging experiment. Utilizing the pseudo-inverse method, the imaging result with no noise is shown in Figure 3(b). We can see from the imaging result that the shape of the target is successfully reconstructed. According to the traditional radar resolution formula Δρ = Rλ/D, itisabout7.m under the parameters in the experiment. Therefore, the imaging result based on ORF has achieved a high resolution reconstruction. Furthermore, considering the additive noise, the imaging results with different SNRs are shown in Figure. It can be seen that with the increase of SNR, the imaging quality becomes better. Moreover, a good imaging performance can be achieved when SNR = db. It shows that the imaging based on the ORF has better robustness to noise. Figure shows that the imaging error varies with SNR. With the increase of SNR, the imaging error decreases rapidly at the beginning, and then it tends to be constant (a) (b) Figure. The ordinary temporal-spatial independent radiation field samples. (a) Sample. (b) Sample (a) (b)

9 Progress In Electromagnetics Research M, Vol. 7, (c) (d) Figure 7. Imaging results with different SNRs based on the ordinary temporal-spatial independent radiation field. (a) Imaging result with SNR = db. (b) Imaging result with SNR = db. (c) Imaging result with SNR = db. (d) Imaging result with SNR = db. when the SNR is greater than db. It means that the effect of noise can be ignored when the SNR is greater than db..3. Comparison In this section, with the same simulation parameters, an imaging experiment based on the ordinary temporal-spatial independent radiation field is performed to compare with the proposed method. Herein, the signals are chosen to be bandpass white Gaussian noise. Then, the ordinary temporal-spatial independent radiation field samples can be derived from Eq. (3), as shown in Figure. Figure 7 shows.. Current method Proposed method..3 RMSE.3... Figure. Imaging errors comparison SNR/dB

11 Progress In Electromagnetics Research M, Vol. 7, Hunt, J., J. Gollub, T. Driscoll, et al., Metamaterial microwave holographic imaging system, JOSA A, Vol. 3, No., 9 9,.. Watts, C. M., D. Shrekenhamer, J. Montoya, et al., Terahertz compressive imaging with metamaterial spatial light modulators, Nature Photonics, Vol., No., 9,.

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