2.1 Principle of R-ITR in anechoic chamber
In an anechoic chamber, the pulse signal is divided into multiple short pulses after ITR so that the target echo can be received when the next short pulse is transmitted. Then, the coupling can be eliminated [3]. When the ITR periods are random, the R-ITR is shown in Fig. 1.
The R-ITR control signal can be expressed as:
$$ {p}_1(t)=\sum \limits_{n\to -\infty}^{+\infty}\mathrm{rect}\left(\frac{t}{\tau_n}\right)\ast \delta \left(t-\sum \limits_k^n{T}_{s_k}\right) $$
(1)
where δ(⋅) is the impulse function and τ_{n} is the width of the nth short pulse. \( {T}_{s_k} \) is the period of the kth short pulse and k ≤ n. ∗ is the convolution operation, and rect(⋅) is:
$$ \mathrm{rect}\left(\frac{t}{\tau_n}\right)=\left\{\begin{array}{c}1,\kern0.5em \left|t/{\tau}_n\right|<0.5\\ {}0,\kern0.5em \mathrm{others}\kern1em \end{array}\right. $$
(2)
When the period of ITR and short pulse width is unchanged and it can suppose as \( {T}_{s_k}={T}_s \) and τ_{n} = τ, then we have \( {\sum}_k^n{T}_{s_k}={nT}_s \) for the nth short pulse. And the uniform ITR control signal is:
$$ {p}_2(t)=\mathrm{rect}\left(\frac{t}{\tau}\right)\ast \sum \limits_{n\to -\infty}^{+\infty}\delta \left(t-{nT}_s\right) $$
(3)
Therefore, the spectrum of p_{2}(t) can be obtained as:
$$ {P}_2(f)=\tau {f}_s\sum \limits_{n=-\infty}^{n=+\infty}\sin \mathrm{c}\left({nf}_s\tau \right)\delta \left(f-{nf}_s\right) $$
(4)
where f_{s} = 1/T_{s}, sinc(x) = sin(πx)/(πx). As the R-ITR period is random, the analytical expression of the spectrum is difficult to be obtained.
The transmitted signal is supposed as linear frequency modulation (LFM)
$$ s(t)=\mathrm{rect}\left(\frac{t}{T_p}\right)\exp \left(j2\pi \left({f}_ct+\frac{1}{2}\mu {t}^2\right)\right) $$
(5)
where T_{p} is the pulse width, f_{c} is the carrier frequency, and μ = B/T_{p} and B are the bandwidth.
Considering the ITR control signal in (3), the ITR echo can be obtained as:
$$ {\displaystyle \begin{array}{l}{s}_{r_2}(t)={p}_2(t)\cdot {s}_r(t)\\ {}\kern4em =\left(\mathrm{rect}\left(\frac{t}{\tau}\right)\ast \sum \limits_{n\to -\infty}^{+\infty}\delta \left(t-{nT}_s\right)\right){s}_r(t)\end{array}} $$
(6)
where \( {s}_r(t)=\sum \limits_{k=1}^K{\alpha}_ks\left(t-2{R}_k/C\right) \) is the target echo, K is the number of scattering centers, and α_{k} is the scattering coefficient of the kth scattering centers. R_{k} is the distance between the radar and the kth scattering center.
The de-chirp reference signal is s_{ref}(t) = s(t ‐ 2R_{ref}/C), where R_{ref} is the reference distance and C is the speed of the electromagnetic wave. Then, the difference output after de-chirping is:
$$ {\displaystyle \begin{array}{l}{s}_{f_2}(t)={p}_2(t)\cdot {s}_r(t)\cdot {s}_{\mathrm{ref}}^{\ast }(t)\\ {}=\sum \limits_{k=1}^K\left(\left(\mathrm{rect}\left(\frac{t}{\tau}\right)\ast \sum \limits_{n\to -\infty}^{+\infty}\delta \left(t-{nT}_s\right)\right){\alpha}_k\right.\mathrm{rect}\left(\frac{t-2{R}_k/C}{T_p}\right)\exp \left(-j\frac{4\pi }{C}\mu \left(t-\frac{2{R}_{\mathrm{ref}}}{C}\right){R}_{k,\Delta}\right)\\ {}\kern1em \cdot \left.\exp \left(-j\frac{4\pi }{C}{f}_c{R}_{k,\Delta}\right)\exp \left(j\frac{4\pi }{C^2}{R}_{k,\Delta}^2\right)\right)\end{array}} $$
(7)
where \( {s}_{\mathrm{ref}}^{\ast }(t) \) is the conjugation of s_{ref}(t), and R_{k, Δ} = R_{k} − R_{ref} is the distance between the scattering center and the reference point.
The HRRP of ITR echo is obtained after Fourier transform:
$$ {S}_{f_2}(f)=\tau {f}_s{T}_p\sum \limits_{k=1}^K{\alpha}_k\exp \left(-j\frac{4\pi {f}_c}{C}{R}_{k,\Delta}\right)\sum \limits_{n\to -\infty}^{n\to +\infty}\left(\sin \mathrm{c}\left({nf}_s\tau \right)\sin \mathrm{c}\left({T}_p\left(f-{nf}_s+2\frac{\mu }{C}{R}_{k,\Delta}\right)\right)\right) $$
(8)
In Eq. (8), nf_{s} denotes that different orders of fake peaks appear in HRRP. Then, the range interval is:
$$ \Delta R=\frac{Cf_s}{2\mu } $$
(9)
The constraints of ITR parameters are the same with [9]:
$$ \Big\{{\displaystyle \begin{array}{l}\tau \le \frac{2{R}_0}{C}\\ {}\tau +\frac{2\left({R}_0+L\right)}{C}\le {T}_s<\frac{C{T}_p}{2 BL}\end{array}}\operatorname{} $$
(10)
Because ΔR may be smaller than L, the real and fake peaks will be overlapped and the target peaks in the HRRP are difficult to be extracted. However, the ITR echo is sparse compared with the completed pulse; therefore, the HRRP can be reconstructed based on CS.
2.2 HRRP reconstruction based on CS
The vector of the difference-frequency output of the complete echo can be expressed as:
$$ {\mathbf{s}}_{\mathbf{f}}={\mathbf{S}}_{\mathbf{r}\mathbf{ef}}^{\ast}{\mathbf{s}}_{\mathbf{r}} $$
(11)
where s_{r} = [s_{r}(0),s_{r}(1),…,s_{r}(N-1)]^{T} is N × 1 vector of the complete echo s_{r}(t), N is the total sampling number in T_{p}, S_{ref} = diag{s_{ref}(0),s_{ref}(1),…,s_{ref}(N-1)} is the N × N diagonal matrix composed by the reference signal s_{ref}(t), and s_{f} = [s_{f}(0),s_{f}(1),…,s_{f}(N-1)]^{T} is the N × 1 vector of the difference-frequency output.
S_{f} = [S_{f} (0),S_{f} (1),…,S_{f} (N-1)]^{T} is the target HRRP which can be supposed as K-sparse according to the number of scattering centers. Then, S_{f} can be obtained after Fourier transform of (11):
$$ {\mathbf{S}}_{\mathbf{f}}={\boldsymbol{\Psi}}^{-1}{\mathbf{S}}_{\mathbf{r}\mathbf{ef}}^{\ast}{\mathbf{s}}_{\mathbf{r}} $$
(12)
where Ψ is the N × N inverse fast Fourier transform (IFFT) matrix.
Because p_{1}(t) is 1 in τ_{n} and 0 in \( {T}_{s_n} \)-τ_{n}, the rows of P = diag{p_{1}(0),p_{1}(1),…,p_{1}(N-1)} are 0 and can be eliminated. Then, the formation of P_{1} is obtained and presented in Fig. 2.
P_{1} can be expressed as:
$$ {\mathbf{P}}_1={\left(\begin{array}{cccccccc}{\mathbf{I}}_{n_{s_1}}& 0& \cdots & 0& 0& \cdots & 0& 0\\ {}0& 0& \cdots & 0& {\mathbf{I}}_{n_{s_2}}& \cdots & 0& 0\\ {}\vdots & \vdots & \cdots & \vdots & \vdots & \cdots & \vdots & \vdots \\ {}0& 0& \cdots & 0& 0& \cdots & 0& {\mathbf{I}}_{n_{s_n}^{\prime }}\end{array}\right)}_{M\times N} $$
(13)
where I_{nsn} is the n_{sn} × n_{sn} identity matrix.
Then, the ITR echo is obtained based on the measurement of s_{r}:
$$ {\mathbf{s}}_{\mathrm{ITR}}={\mathbf{P}}_1{\mathbf{s}}_{\mathbf{r}} $$
(14)
where s_{ITR} is the M × 1 vector.
As \( {\mathbf{S}}_{\mathbf{ref}}{\mathbf{S}}_{\mathbf{ref}}^{\ast}=\mathbf{I} \), Eq. (14) can be re-written as:
$$ {\mathbf{s}}_{\mathrm{ITR}}={\mathbf{P}}_1{\mathbf{S}}_{\mathbf{ref}}{\boldsymbol{\Psi} \mathbf{S}}_{\mathbf{f}} $$
(15)
After adding noise ξ, we have:
$$ {\displaystyle \begin{array}{l}{\mathbf{s}}_{\mathrm{spInter}}={\mathbf{s}}_{\mathrm{ITR}}+\boldsymbol{\upxi} \\ {}\kern4em ={\boldsymbol{\Phi} \boldsymbol{\Psi} \mathbf{S}}_{\mathbf{f}}+\boldsymbol{\upxi} \end{array}} $$
(16)
where Φ = P_{1}S_{ref} and s_{spInter} is the non-zero value of ITR echo.
The target HRRP can be reconstructed by solving the following problem:
$$ \underset{\overline{{\mathbf{S}}_{\mathbf{f}}}}{\min }{\left\Vert \overline{{\mathbf{S}}_{\mathbf{f}}}\right\Vert}_1\kern1em s.t.\kern0.5em {\left\Vert {\mathbf{s}}_{\mathrm{spInter}}-\boldsymbol{\Phi} \boldsymbol{\Psi} \overline{{\mathbf{S}}_{\mathbf{f}}}\right\Vert}_2\le \boldsymbol{\upxi} $$
(17)
where ||·||_{1} is the ℓ_{1} norm, and \( \overline{{\mathbf{S}}_{\mathbf{f}}} \) denotes the reconstructed HRRP.
Let A = ΦΨ be the sensing matrix. Ψ is the N × N the fixed IFFT matrix. Φ = P_{1}S_{ref} is the measurement matrix which can be obtained with P_{1} and S_{ref}. Because S_{ref} is composed by the reference signal s_{ref}(t), the form of A is determined by the matrix P_{1}. From Fig. 2, it can be found that different ITR period sequences can obtain different matrixes of P_{1} so that the form of A changes. Therefore, the relationship between the R-ITR period sequence and RIP conditions of A is constructed. Generally, A should satisfy the K-order RIP condition to ensure the HRRP reconstruction performance. In order to improve the RIP condition of A, the R-ITR period sequence should be optimized.
2.3 ITR period sequence optimization based on GA
Because there are hundreds of short pulses in each pulse signal, the R-ITR period sequence optimization can be converted to a multivariable optimization problem. GA is an adaptive optimization algorithm which is widely used to solve the multivariable optimization problem. The multi-point search, cross-operation, and mutation operation are adopted to eliminate the worst solution and save the optimum solution. After iteration, the optimum R-ITR period sequence can be obtained.
According to Fig. 3, the R-ITR period should be coded firstly to obtain the initialized population. Supposing the R-ITR period \( {T}_{s_n} \) is distributed in [0.5 μs, 0.8 μs] with the interval of 0.1 μs, there are four values to choose, i.e., 0.5 μs, 0.6 μs, 0.7 μs, and 0.8 μs. So, the two-bit binary can be adopted to code \( {T}_{s_n} \), i.e., 00, 01, 10, and 11. \( \mathbf{T}=\left\{{T}_{s_n,{m}^{\prime }}\right\} \) is a N′ × M′ matrix where N′ is the number of \( {T}_{s_n} \) and M′ is the number of population. If the mth R-ITR period sequence \( {T}_{s_n,{m}^{\prime }} \) is:
$$ 0.8\upmu \mathrm{s},0.6\upmu \mathrm{s},0.5\upmu \mathrm{s},0.6\upmu \mathrm{s},0.7\upmu \mathrm{s},0.7\upmu \mathrm{s} $$
(18)
The corresponding binary codes \( {Z}_{n,{m}^{\prime }} \) are obtained as:
$$ 11\ 01\ 00\ 01\ 10\ 10 $$
(19)
Generally, the sequence length of R-ITR periods is determined by the pulse width and the range of ITR periods. Therefore, the code length increases when the pulse width is large.
Researches show that the RIP of the sensing matrix equals to the eigenvalues of the Gram matrix of sensing matrix and the cross-correlation coefficient of the sensing matrix columns [11,12,13]. When the eigenvalues of the Gram matrix of sensing matrix are close to 1, the RIP property can be satisfied well. However, as the computation complexity of the Gram matrix eigenvalues is high, the cross-correlation coefficient of the sensing matrix columns is adopted to evaluate the fitness of the population.
a_{i} is supposed as the ith column of the sensing matrix A. Then, the maximum cross-correlation coefficient [11,12,13] of the sensing matrix A is defined as:
$$ \rho \left(\mathbf{A}\right)=\underset{0\le k,j<N,k\ne j}{\max}\frac{\left|{\mathbf{a}}_k^H{\mathbf{a}}_j\right|}{{\left\Vert {\mathbf{a}}_k^H\right\Vert}_2{\left\Vert {\mathbf{a}}_j\right\Vert}_2} $$
(20)
When the maximum cross-correlation coefficient is minimized, the RIP of the sensing matrix can be well satisfied. Therefore, we define the fitness function with the mth R-ITR period sequence \( {T}_{s_n,{m}^{\prime }} \) as:
$$ F\left({T}_{s_n,{m}^{\prime }}\right)=\max \frac{1}{\rho \left(\mathbf{A}\right)},\kern1.5em \mathrm{s}.\mathrm{t}.\kern1em {T}_{s_n,{m}^{\prime }}\in P $$
(21)
where P is the variation range of \( {T}_{s_n} \).
The scheme of the proposed optimization method based on GA is presented in Fig. 3.
The flow of Algorithm 1 is presented according to Fig. 3.
Algorithm 1: R-ITR periods sequence optimization. |
Input: Initial population \( \mathbf{T}=\left\{{T}_{s_n,{m}^{\prime }}\right\} \) Output: Optimized R-ITR periods sequence \( {\mathbf{T}}_{\mathrm{opt}}=\left\{{T}_{s_n,{m}^{\prime }}\right\} \) 1. Calculate \( F\left({T}_{s_n,{m}^{\prime }}\right) \) 2. Judge the fitness value 3. Code the R-ITR periods sequence to be \( {\mathbf{T}}_0={\left\{{Z}_{n,{m}^{\prime }}\right\}}_0 \) 4. Select the optimum individual \( {\mathbf{T}}_{\mathrm{s}}={\left\{{Z}_{n,{m}^{\prime }}\right\}}_{\mathrm{s}} \) 5. Cross operation and obtain \( {\mathbf{T}}_{\mathrm{c}}={\left\{{Z}_{n,{m}^{\prime }}\right\}}_{\mathrm{c}} \) 6. Mutation operation and obtain \( {\mathbf{T}}_{\mathrm{m}}={\left\{{Z}_{n,{m}^{\prime }}\right\}}_{\mathrm{m}} \) 7. Population update and obtain \( {\mathbf{T}}_{\mathrm{u}}={\left\{{Z}_{n,{m}^{\prime }}\right\}}_{\mathrm{u}} \) 8. Decode operation to obtain \( {\mathbf{T}}_{\mathrm{d}}={\left\{{T}_{s_n,{m}^{\prime }}\right\}}_{\mathrm{d}} \) and return to 1 |
By initializing the population as \( \mathbf{T}=\left\{{T}_{s_n,{m}^{\prime }}\right\} \), the proposed optimization process is expressed in detail according to Algorithm 1 and Fig. 3.
Step 1. Calculate the fitness function. The sensing matrix A = ΦΨ is constructed based on each individual \( {T}_{s_n,{m}^{\prime }} \). Then, the fitness function in Eq. (21) is calculated.
Step 2. Judge the fitness value. The fitness function is calculated for every R-ITR period sequence. If the fitness value is satisfied, the iteration stops. Otherwise, the iteration continues.
Step 3. Code operation. The length of the binary code is determined by the range of the R-ITR periods. According to the coded principle in Eqs. (18) and (19), the R-ITR period sequence can be coded as \( {\mathbf{T}}_0={\left\{{Z}_{n,{m}^{\prime }}\right\}}_0 \). M′ series R-ITR period sequences are generated in this step.
Step 4. Select the optimum individual. The roulette wheel is used to select the individual. The elitism selection is also adopted in this part to save the optimum individual {Z_{n,m′}}. After the selection operation, the population is obtained as T_{s} = {Z_{n, m′}}_{s}.
Step 5. Cross operation. The two-point crossover is utilized to conduct the cross operation. The crossover positions and individuals are randomly selected with the crossover probability of P_{c}. Then, the new individuals can be obtained. And the updated population is T_{c} = {Z_{n, m′}}_{c}.
Step 6. Mutation operation. The mutation operation is conducted based on the population obtained in step 5. The single-point mutation is performed by randomly selecting the individual and the mutation position with the mutation probability of P_{m}. Then, the population after mutation operation is T_{m} = {Z_{n, m′}}_{m}.
Step 7. Population update. The elitism selection is still used to update the population and obtain \( {\mathbf{T}}_{\mathrm{u}}={\left\{{Z}_{n,{m}^{\prime }}\right\}}_{\mathrm{u}} \).
Step 8. Decode operation. As \( {\mathbf{T}}_{\mathrm{u}}={\left\{{Z}_{n,{m}^{\prime }}\right\}}_{\mathrm{u}} \) is a binary code, T_{u} should be decoded to obtain the R-ITR periods as \( {\mathbf{T}}_{\mathrm{d}}={\left\{{T}_{s_n,{m}^{\prime }}\right\}}_{\mathrm{d}} \). Then, T_{d} is used to calculate the fitness function in step 1.