Assuming the time *t*_{
a
} it is observed that received SAR signal **s**(*t*_{
r
}*, t*_{
a
} ) is to be measured at the discrete fast-time sample *t*_{
r
}*, t*_{
r
} = 0,1,...,*M*-1 and *t*_{
a
} is the discrete aperture (slow-time) samples, *t*_{
a
} = 0,1,...,*N*-1. It is SAR echo signal **s**_{echo}(*t*_{
r
}*, t*_{
a
} ) contaminated by RFI signal **s**_{RFI}(*t*_{
r
}*, t*_{
a
} ) and system noise **s**_{Noise}(*t*_{
r
}*, t*_{
a
} ):

\mathbf{s}\left({t}_{r},{t}_{a}\right)={\mathbf{s}}_{\text{echo}}\left({t}_{r},{t}_{a}\right)+{\mathbf{s}}_{\text{RFI}}\left({t}_{r},{t}_{a}\right)+{\mathbf{s}}_{\text{Noise}}\left({t}_{r},{t}_{a}\right)

(1)

Generally, for distributed targets (e.g., in real scenario) **s**_{echo}(*t*_{
r
}*, t*_{
a
} ) resembles Additive White Gaussian Noise (AWGN), and **s**_{Noise}(*t*_{
r
}*, t*_{
a
} ) generated by system is AWGN, too. RFI signal has comparatively narrow band, and can be considered but not limit to complex exponential signal. Previous researches [1–6] pointed that the power of RFI signals is tens of dBs greater than the power of the SAR echoes. So, **s**_{echo}(*t*_{
r
}*, t*_{
a
} ) and **s**_{Noise}(*t*_{
r
}*, t*_{
a
} ) can be together considered as wideband background noise which is independent from the RFI source, denoted as **s**_{WB}(*t*_{
r
}*, t*_{
a
} ) = **s**_{echo}(*t*_{
r
}*, t*_{
a
} )+**s**_{Noise}(*t*_{
r
}*, t*_{
a
} ).

For a given *t*_{
a
} , **s**(*t*_{
r
}*, t*_{
a
} ) is a one-dimensional time series, denoted as **s** = (*s*[1], *s*[2],...,*s*[*M*]) for simplicity. Normally, we remove mean value from the original data vector ahead. According to the first step of SSA algorithm [7], an *embedding* step, we choose a window length *L* and an embedded dimension *K* to construct *K* = *M*-*L*+1 lagged vectors:

{\mathbf{s}}_{k}={\left(s\left[k-1+L-1\right],\dots ,s\left[k-1\right]\right)}^{\text{T}}={\mathbf{s}}_{\text{RFI},k}+{\mathbf{s}}_{\text{WB},k},\phantom{\rule{1em}{0ex}}k=1,...,K,

(2)

where superscript 'T' denotes transpose. Then, an *L* × *K* matrix with Toeplitz structure is composed:

\mathbf{S}=\left({\mathbf{s}}_{1},{\mathbf{s}}_{2},\dots ,{\mathbf{s}}_{k}\right)=\left(\begin{array}{cccc}\hfill s\left[L-1\right]\hfill & \hfill s\left[L\right]\hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill s\left[M-1\right]\hfill \\ \hfill s\left[L-2\right]\hfill & \hfill s\left[L-1\right]\hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill s\left[M-2\right]\hfill \\ \hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \vdots \hfill \\ \hfill s\left[0\right]\hfill & \hfill s\left[1\right]\hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill s\left[M-L\right]\hfill \end{array}\right),\phantom{\rule{1em}{0ex}}k=1,...,K,

(3)

Note that **G** = **SS**^{H} is an *L* × *L* symmetric and positive semi-definite (SPSD) matrix, where superscript 'H' denotes conjugate transpose. We may express it in spectral decomposition form as **G** = **UΛU**^{H}, where **U** is an orthogonal matrix whose columns are the eigenvectors **u**_{1},...,**u**_{L} of **G**, and **Λ** = diag(*λ*_{1},*λ*_{2,}....,*λ*_{
L
} ) is a diagonal matrix whose entries on diagonal are ordered eigenvalues *λ*_{1} ≥ *λ*_{2} ≥...,≥*λ*_{
L
} of **G**. Based on Equation (1), we suppose that the *r* leading eigenpairs {\left\{\left({\lambda}_{i},{\mathbf{u}}_{i}\right)\right\}}_{1}^{r} which should be separated from the other *L*-*r* eigenpairs to reconstruct a 'clean' data matrix without 'noise' are related to RFI subspace. We know that if the signal and noise subspace can be sufficiently separated, it implies that the noise has to be white and zero-mean. In the preceding paragraph, we have expatiated on the precondition of the proposed algorithm: radar echoes from distributed targets have White Gaussian Noise characters, and the mean value has been removed from the original record ahead. Then, the reconstructed RFI clustering could be obtained via

{\stackrel{\u0303}{\mathbf{S}}}_{RFI}=\mathbf{UP}{\mathbf{U}}^{\text{H}}\mathbf{S}=\mathbf{U}\left(\mathbf{P}{\mathbf{U}}^{\text{H}}\mathbf{S}\right)=\mathbf{UQ}

(4)

where **P** is a diagonal selected matrix, with the *i* th diagonal entry satisfies *P*_{
ii
} = 1 if the *i* th row of **Q** = **PU**^{H}**S** is selected whereas *P*_{
ii
} = 0 it will be discard. Thus, we select the first *r* = rank(**G**) corresponding to the dimension of RFI subspace entries, {\left\{{P}_{ii}=1\right\}}_{i=1}^{r}.

Note that, there may be varying number of RFI sources in the region where carrier platform transits. Namely, the dimension *r* of the RFI subspace is mutative in the aperture synthetic duration and needs to be ascertained. Historically, multiple methods are proposed for how to select the rank *r*, such as AIC-type (AIC, Akaike Information Criterion and MDL, Minimum Description Length [18]) which are based on information theoretic criteria or ESPRIT-type (ESTER, ESTimation Error [19] and SAMOS, Subspace-based Automatic Model Order Selection [20]) based on the shift invariance equation. Besides that, the method we adopt here is based on the eigenvalue distribution of the approximated sample covariance matrix [21]. It uses several results from random matrix theory which provides a set of remarkable tools for rank estimation and just has progressed substantially over the last 10 years. In a word, this method takes use of the theories developed in [22–24], and consequently determines a specified eigenvalue threshold. Here, we just give the rank estimation formula:

r=\text{rank}\left(G\right)=\text{arg}\underset{j}{\text{min}}\left\{{\lambda}_{j}\le {\sigma}_{j}^{2}\left(\mu \left(L,K-j\right)+\tau \left(\varsigma \right)\u2022\delta \left(L,K-j\right)\right)\right\}-1\phantom{\rule{1em}{0ex}}1\le j\le \text{min}\left(L,K\right),

(5)

where {\sigma}_{j}^{2} is the noise level whose calculation procedure is also in [21], *τ*(*ς*) is the corresponding value computed by inversion of the Tracy-Widom distribution [21, 23], *ς* indicates that the eigenvalue significance level is normally a small positive number, in our raw data experiment, we set *ς* = 0.05. Besides, *μ*(*L, K*-*j*) and *δ*(*L, K*-*j*) are center and scaling quantities, respectively:

\mu \left(L,K-j\right)={\left(\sqrt{L}+\sqrt{K-j}\right)}^{2}

(6)

\delta \left(L,K-j\right)=\left(\sqrt{L}+\sqrt{K-j}\right){\left(\frac{1}{\sqrt{L}}+\frac{1}{\sqrt{K-j}}\right)}^{\frac{1}{3}}

(7)

Now, recall Equation (4), the *i* th row of **Q** is denoted as {\mathbf{q}}_{i}={P}_{ii}{\mathbf{u}}_{i}^{\text{H}}\mathbf{S}, *i* = 1,...,*L*, then (4) can be written as

{\stackrel{\u0303}{\mathbf{S}}}_{\text{RFI}}={\mathbf{u}}_{1}\left({P}_{11}{\mathbf{u}}_{1}^{\text{H}}\mathbf{S}\right)+{\mathbf{u}}_{2}\left({P}_{2}{\mathbf{u}}_{2}^{\text{H}}\mathbf{S}\right)+\cdots +{\mathbf{u}}_{r}\left(\text{P}{}_{rr}{\mathbf{u}}_{r}^{\text{H}}\mathbf{S}\right)={\mathbf{u}}_{1}{\mathbf{q}}_{1}+{\mathbf{u}}_{2}{\mathbf{q}}_{2}+\cdots +{\mathbf{u}}_{r}{\mathbf{q}}_{r}=\sum _{i=1}^{r}{\stackrel{\u0303}{\mathbf{S}}}_{\text{RFI},i}

(8)

Interestingly, the 1 × *L* row vector {\mathbf{q}}_{i}={P}_{ii}{\mathbf{u}}_{i}^{\text{H}}\mathbf{S} can be considered as a filtered version of the original data sequence [8, 9]. Each sample of **q**_{
i
} can be written as follows

{q}_{i}\left[n\right]={P}_{ii}\sum _{j=1}^{L}{u}_{j,i}s\left[n-j+1\right],\phantom{\rule{1em}{0ex}}\left(L-1\right)\le n<M,

(9)

There are *K* = *M*-(*L*-1) samples drawn from the time series *q*_{
i
} [*n*] in **q** _{
i
}. Equation (6) is the convolution sum of the eigenvector **u**_{
i
} and the (*n* - *K* + 2)th original lagged data vector. Thus, the entries of eigenvector **u**_{
i
} correspond to the coefficients of an FIR filter which is namely the *analysis* filter. Introducing *z*-transform Z\left(\u2022\right), we can get the transfer function H_{
i
}(*z*) of the analysis filter

\text{H}{}_{i}\left(z\right)=\frac{{Q}_{i}\left(z\right)}{S\left(z\right)}={P}_{ii}\sum _{j=1}^{L}{u}_{j,i}{z}^{-\left(j-1\right)}={P}_{ii}\left({u}_{1,i}+{u}_{2,i}z+\cdots +{u}_{K,i}{z}^{-\left(L-1\right)}\right)

(10)

where {Q}_{i}\left(z\right)=Z\left({q}_{i}\left[n\right]\right), S\left(z\right){z}^{-\left(j-1\right)}=Z\left(s\left[n-j+1\right]\right), and *z* is a complex value.

In the sequel we will perform *re-embedding* step on the reconstructed RFI-embedded matrix {\stackrel{\u0303}{\mathbf{S}}}_{\text{RFI}} to recover it in a one-dimensional time series again. However, {\stackrel{\u0303}{\mathbf{S}}}_{\text{RFI}} has not kept Toeplitz structure anymore. To substitute all entries on each diagonal with their respective average value, we obtain a new Toeplitz matrix. All of the averaging values comprise the final pure RFI signals [7]. We give an example of {\stackrel{\u0303}{\mathbf{S}}}_{\text{RFI},i}={\mathbf{u}}_{i}{\mathbf{q}}_{i}, the procedure can be described as follows.

{\stackrel{\u0303}{\mathbf{S}}}_{\text{RFI},i}={\mathbf{u}}_{i}{\mathbf{q}}_{i}\text{=}\left(\begin{array}{cccc}\hfill {u}_{1,i}{q}_{i}\left[L-1\right]\hfill & \hfill {u}_{1,i}{q}_{i}\left[L\right]\hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill {u}_{1,i}{q}_{i}\left[M-1\right]\hfill \\ \hfill {u}_{2,i}{q}_{i}\left[L-1\right]\hfill & \hfill {u}_{2,i}{q}_{i}\left[L\right]\hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill {u}_{2,i}{q}_{i}\left[M-1\right]\hfill \\ \hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \vdots \hfill \\ \hfill {u}_{r,i}{q}_{i}\left[L-1\right]\hfill & \hfill {u}_{r,i}{q}_{i}\left[L\right]\hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill {u}_{r,i}{q}_{i}\left[M-1\right]\hfill \end{array}\right)

(11)

The diagonal averaging process of this rank one matrix is given by

{\stackrel{\u0303}{s}}_{i}\left[n\right]=\frac{1}{{N}_{d}}\sum _{j=a}^{b}{u}_{j,i}{q}_{i}\left[n+j-1\right],\phantom{\rule{1em}{0ex}}i=1,\dots ,M,

(12)

where *N*_{
d
} is the number of current diagonal entries, *a* and *b* depend on the location of current diagonal. For lower left corner of the matrix in Equation (11), *N*_{
d
} = *n* + 1, *a* = *L*-*N*_{
d
}*, b* = *L*; otherwise, for upper right corner of the same matrix, *N*_{
d
} = *M-n, a* = 1, *b* = *L*-*N*_{
d
} . Obviously, Equation (12) can also be seen as a convolution sum that can be interpreted as an anti-causal FIR filter. Accordingly, both of the above two cases can be unified by formally setting *q*_{
i
} [*n*] = 0, 0 ≤ *n* ≤ *L*-2, or *L* ≤ *n* ≤ *M* + *L*-2. Therefore, the synthesis transfer function in the steady-state case is given by

\text{F}{}_{i}\left(z\right)=\frac{{\stackrel{\u0303}{S}}_{i}\left(z\right)}{{Q}_{i}\left(z\right)}=\frac{1}{L}\sum _{j=1}^{L}{u}_{j,i}{z}^{\left(j-1\right)}=\frac{1}{L}\left({u}_{1,i}+{u}_{2,i}z+\cdots +{u}_{L,i}{z}^{\left(L-1\right)}\right)

(13)

where {\stackrel{\u0303}{S}}_{i}\left(z\right)=Z\left({\stackrel{\u0303}{s}}_{i}\left[n\right]\right), {Q}_{i}\left(z\right){z}^{\left(j-1\right)}=Z\left({q}_{i}\left[n+j-1\right]\right).

Consequently, the whole SSA procedure which contains projection, reconstruction, and diagonal averaging can be described by a global transfer function below

\text{T}{}_{i}\left(z\right)=\text{H}{}_{i}\left(z\right)\text{F}{}_{i}\left(z\right)=\frac{{Q}_{i}\left(z\right)}{S\left(z\right)}\u2022\frac{{\stackrel{\u0303}{S}}_{i}\left(z\right)}{{Q}_{i}\left(z\right)}=\frac{{\stackrel{\u0303}{S}}_{i}\left(z\right)}{S\left(z\right)}=\sum _{j=-\left(L-1\right)}^{L-1}{t}_{j,i}{z}^{j}

(14)

By substituting (10) and (11) into (14), we can see that *t*_{
j, i
} = *t*_{
-j, i
}*, j* = 1,...,*L*-1. Let *z* = *e*^{jω} , \text{j}=\sqrt{-1}, the related frequency response arises via

\text{T}{}_{i}\left({e}^{\text{j}\omega}\right)={t}_{0,i}+2\sum _{j=1}^{L-1}{t}_{j,i}\text{cos}\left(j\omega \right)

(15)

This is a zero-phase filter whose output signal is always in phase with the input signal. So, the proposed RFI suppression algorithm is a phase preserving method. Otherwise, it can be observed that |H _{
i
} (*e*^{jω} )| and |F _{
i
} (*e*^{jω} )| only differ in a scaling factor and the sign of the power of the complex exponential argument. Hence, T _{
i
} (*e*^{jω} ) resembles |H _{
i
} (*e*^{jω} )| in shape and its value can be given by |T _{
i
} (*e*^{jω} )| = |H _{
i
} (*e*^{jω} )|^{2}/*L*. The pattern of the global FIR filter group is shown in Figure 1.

Besides, once alternative embedding procedure is adopted--the trajectory matrix **S** has Hankel structure but not Toeplitz structure (see [8, 9] for example), the properties of H _{
i
} (*e*^{jω} ) and F _{
i
} (*e*^{jω} ) will interchange, i.e., the former corresponding to an anti-casual filter whereas the latter a casual filter.