The VolumeCorrelation Subspace Detector
Abstract
Detecting the presence of subspace signals with unknown clutter (or interference) is a widely known difficult problem encountered in various signal processing applications. Traditional methods fails to solve this problem because they require knowledge of clutter subspace, which has to be learned or estimated beforehand. In this paper, we propose a novel detector, named volumecorrelation subspace detector, that can detect signal from clutter without any knowledge of clutter subspace. This detector effectively makes use of the hidden geometrical connection between the known target signal subspace to be detected and the subspace constructed from sampled data to ascertain the existence of target signal. It is derived based upon a mathematical tool, which basically calculates volume of parallelotope in highdimensional linear space. Theoretical analysis show that while the proposed detector is detecting the known target signal, the unknown clutter signal can be explored and eliminated simultaneously. This advantage is called "detecting while learning", and implies perfect performance of this detector in the clutter environment. Numerical simulation validated our conclusion.
I Introduction
We consider the following problem widely existing in communication, radar, sonar and other fields of signal detection and processing: How to effectively detect a target signal buried in clutter lying in an UNKNOWN low rank subspace and random noise. The noticeable point here is that the clutter subspace is unknown at the receiver. Without what is called training data, we could not sample directly in this clutter subspace and get enough information to eliminate its influence on target signal detection.
As a matter of fact, detecting target signal in certain signal subspace has been considered by several researchers and various schemes has been proposed. Denote by Hilbert space, which is the basic signal space, then the problem of detecting signal in subspace could be formulated roughly as follows.
Problem 1
Suppose be the known target signal subspace with dimension ,
(1) 
where are KNOWN signal vectors. Given the sampled data contaminated by Gaussian random noise, could we determine whether or not lies in ?
The wellknown projection method has long been regarded as the basic step for these kinds of detectors [1, 2]. Indeed, to solve Problem 1, we firstly project the sampled data onto subspace , then we use the energy detector to make the decision. Actually, the conventional optimal detector for single known signal vector, i.e., the matched filter, is also a special kind of this projection based detector. Because the known signal vector indeed represents a onedimensional subspace, and the matched filter can be regarded as essentially sequential projections of received data on the timeshift versions of target signal. On the other hand, least square estimation and its alternatives also involve projection of raw data on the subspace spanned by several prescribed signal vectors. The optimality of linear least square method as the linear estimator could be sufficiently guaranteed by GaussMarkov theorem in statistical inference. It is clear that detection, being the counterpart of estimation, of signal in known subspace is closely contacted with projection operation in linear space. In fact, more complicated problem involving clutter or interference has been solved by projectionbased methods, such problem is formulated as follows.
Problem 2
In addition to , suppose be the KNOWN clutter subspace with dimension .
(2) 
where are KNOWN clutter (interference) vectors. Given the sampled data , could we determine whether or not lies in but not entirely in from the contamination of random noise? In other words, whether satisfies
(3) 
or
(4) 
where and is the Gaussian random noise.
Unlike Problem 1, Problem 2 can not be solved by the simple projection and energy detector mentioned above. The reason is that clutter subspace is not necessarily orthogonal to signal subspace in general (they are even alike in hostile environment). In practical scenario the power of clutter is often much higher than that of target signal. So it is hard to tell the existence of signal component from sample data only based on the argument that the energy of its projection on is relatively large. The most remarkable approach to the problem is the Matched Subspace Detector firstly proposed by L. Scharf et.al. [3]. It is essentially a "twofolds" projection: Firstly, sampled data was projected on orthogonal complement of to eliminate the influence of clutter thoroughly; Secondly, the result of first projection was further projected onto the part of that is orthogonal to , and then an energy detector was applied to infer the existence of signal component in . Denoting and by the projection operators on subspace and its orthogonal complement respectively, the matched subspace detector could be written as
(5) 
Although tremendous variations and applications of matched subspace detector has appeared [4][5][6][7][8], it should be noticed that the key precondition for the success of matched subspace detector is the clutter subspace must be KNOWN beforehand. It is seldom satisfied in practice, especially in radar, reconnaissance, mobile communication and underwater signal processing. Hence the problem we encounter is actually like this:
Problem 3
Let be KNOWN target signal subspace and be UNKNOWN clutter subspace, given the sampled data contaminated by Guassian noise, if we assume , could we determine whether contains signal that lies in or entirely in ? In other words, whether satisfies
(6) 
or
(7) 
where and is the Gaussian random noise.
Because of the unknownness of clutter subspace, projectionbased detectors can not be constructed explicitly. However, the structure of clutter subspace could be explored by successively sampling in it to gain the information of the basis for subspace. Because the generic property of randomly sampling in linear space ensures the linear independence of sampled vectors, the clutter subspace could be ’reconstructed’ by multiple samples. But it should be noted that, the information of target signal is mixed intimately with the clutter in the samples, like the case shown in (6). In this case, it is impossible to separated the ’pure’ clutter signal from target signal so that the basis for clutter subspace could be extracted alone. In other words, with multiple sampled vectors of that satisfies the generic property, we cannot determine whether the sample subspace these span is or just , this issue is the core difficulty for detecting target signal against structured deterministic clutter.
It is interesting to make a comparison of our problem to the problem of detection in random noise without clutter. Both two detection problems have similar formulations. In fact, there are two hypothesis, and , which are
(8) 
The only difference is that in traditional detection problems appears to be only random noise and is described by a certain probability distribution. On the contrary, in our problem includes both random noise and a vector lying within certain deterministic and unknown lowdimensional subspace. Detectors based on Likelihood Ratio Test have been proved to be optimal under the probabilistic assumption of and projection is the natural consequence of likelihood ratio test in the case of Gaussian distributed noise. However, it can’t be applied directly in our problem for the deterministic and unknown clutter subspace structure. Hence, more effective detector which could take fully advantage of the geometrical properties of subspace must be designed to overcome the obstacle we are facing.
In this paper a novel detector for target signal buried in structured lowdimensional clutter was given. The main idea of our detector is that the geometrical characteristic of sampled data could be utilized in solving detection problem. Here the volume, a common concept for geometrical objects, was suitably defined for basis for subspaces (more concretely, the parallelotope with its edges being the basis vectors of this subspace). It is intuitive that the ’volume’ of basis for lowdimensional subspaces in highdimensional linear space is zero. So the judgment of whether or not the sampled vectors span a subspace that contains the target signal could be transformed naturally to the calculation of ’volume’ of a parallelotope built by sampled vectors together with basis for target subspace. If the ’volume’ is zero, then the conclusion can be drawn that the sample subspace contains the target signal subspace, otherwise the target vector must lie outside the sample subspace. Thus the volumebased subspace detector, instead of projectionbased ones, can be used to cope with the problem of detecting target signal lying in known subspace under clutter background with unknown subspace structure.
Throughout this paper, we use small bold letters to denote vectors, capital bold letters to denote matrices(or subspaces); we use and to denote the norm of the matrix and vector , and to denote the Frobenius norm of matrix . The dimensional identity matrix is denoted by . represents the linear subspace spanned by column vectors of the matrix . denotes the direct sum of subspaces and . In addition, and denotes the probability and expectation respectively.
The remainder of this paper is organized as follows: Some preliminary backgrounds on the geometrical concepts for linear subspaces such as principal angles and volumes were summarized in section II. Then the Volumebased Subspace Detector was introduced and demonstrated in detail in section III. In section IV, theoretical analysis on the performance of our volumebased subspace detector was given. The potential application and future work on volumebased subspace detector was discussed in final section.
Ii Preliminary Background
In this section, some important concepts of linear space geometry were reviewed concisely. Although these results are fundamental for a deep understanding of linear subspace, these concepts rarely appear in common textbooks of linear algebra. Only necessary material for our discussion was put forward for the space limitation. For details, please see [9] and reference therein.
Iia Principal Angles between Subspaces
The concept of principal angles [10] is the natural generalization of that of angles between two vectors. Principal angles can be used to formulate the relationship between two subspaces.
Definition 1
For two linear subspaces and , with dimensions . Take , then the principal angles between and are defined by
where .
As an important concept of linear space geometry, the principal angles are widely applied in scientific and engineering fields. For instance, the geodesic distance which is the key metric measure on Grassmann manifold, as well as numerous kinds of distance measures, is defined using the principal angles [9][11][12], such as

Chordal Distance or Projection distance

BinetCauchy Distance

Procrustes Distance
Moreover, the volume of subspace which was used in this paper to construct our subspace detector is also closely related to the principal angles.
IiB The volume of a matrix
The definition of volume for certain geometrical object is ambiguous without the stipulation of dimension. For example, the volume of a parallelogram on a plane is the absolute value of cross product of its two adjacent sides. This is its twodimensional volume. On the contrary, when the parallelogram is regarded as the threedimensional body, its threedimensional volume is definitely zero. It means that the volume value of an object depends on the dimension of space it lies in. For a square matrix , the dimensional volume of the parallelotope spanned by column vectors of is wellknown to be the absolute value of determinant of , which is the product of all the eigenvalues of . When is rectangular, the concept of volume could be generalized naturally. Suppose a matrix with column vectors, and , its dimensional volume is defined as [13]
(9) 
where are the singular values of . If is of full column rank, its dimensional volume can be written equivalently as [13][10]
(10) 
The following simple lemma is widely useful in application of volume for subspaces. It means that the dimensional volume of a matrix with a rank less than is definitely zero.
Lemma 1
Suppose be a group of vectors in Hilbert space and , then
(11) 
The dimensional volume provides a kind of measure for separation between two linear subspaces. For the dimensional Hilbert space and its two subspaces and with dimensions , denote their bases matrices by and , we define the volumebased correlation as
(12) 
where means putting columns of matrices and together.
The volumebased correlation is closely related to the principal angles between subspaces, i.e., according to [10],
(13) 
where are the principal angles of subspace and .
It can be seen intuitively from (13) that the volumebased correlation can actually play the role of distance measure between subspaces and . When and have vectors in common, i.e., , we have . On the other side, when is orthogonal to , we have , in other words, . Although volumebased correlation and the conventional correlation in statistics satisfy CauchySchwarz inequality alike, they are essentially different, because the volumebased correlation isn’t an inner product operation induced from some kind of distance in linear space of subspaces (more formally, Grassmann manifold). But it does not matter for the following discussion. The volumebased correlation will be seen as a generalized distance measure for convenience that plays a key role in our proposed subspace detector.
Iii The Volumebased Correlation Subspace Detector without random noise
In order to fully convey the geometrical intuition about our subspace detector, in this section, we temporarily assume the noise component is not present, i,e., in Problem 3. We introduce a novel detector called volumebased correlation subspace detector, or VC subspace detector for short, that can detect subspace signals buried in unknown, but usually highpower clutter. The main characteristic of the detector is that, it can exploit the geometric relation between the subspaces extracted from the sampled data and the target signal subspace, then it will eliminate the influence of clutter subspace gradually through the process of target detection.
Iiia Main Idea
As we have mentioned, the unknown clutter with an unknown subspace structure is the primary obstacle for efficient detection of target signal. To reach the purposes, the designers of detector must find a way to clarify the intrinsic construction of clutter subspace. Just as most of the traditional approaches for background learning, in our method, multiple samples are used to explore the clutter subspace. The following observation is the inspiration about the exploration of clutter subspace.

Suppose be a dimensional Hilbert space, and be randomly sampled vectors in , then in the generic situation, we have
(14) In other words, are linearly independent.

In the case of , then in the generic situation, we have
(15) In a word, are linearly dependent, but span the whole space .
Let be unknown clutter subspace with unknown dimension , be known target subspace with dimension , we assume throughout this paper. Denote by the randomly sampled data satisfying the generic property mentioned above. The critical issue must be concerned with is that the sampled data might contain both clutter and target components in general, that is,
(16) 
Only with these , it is impossible to separate the clutter and target signal apart. According to the generic property of random sampling, we have
(17) 
Even if the dimension was given virtually, the sample subspace still could not be regarded as the clutter subspace , for the existence of target signal component . How could the sampled data be mined effectively to get knowledge of the clutter subspace?
We will show step by step that, the volumebased correlation between subspaces is helpful for us to eliminate the impact of mixing of clutter and target signal.
Let be the known basis vectors of . It has been mentioned that in the generic scenario of random sampling, different sampled from are linearly independent. In other words, innovative directions of basis vectors in are revealed continually along with the sampling process. Combined with the known basis for signal subspace , we have
(18) 
When we already have samples, actually has been explored by successive sampling thoroughly, in other words, the projection of sample subspace on has already constitute a complete basis for . The interesting point is, further sampling can not modify the intrinsic dimension of the subspace constituted by both the sampled data and basis for signal subspace, that is,
(19) 
The whole process about change of dimensions could be illustrated more clearly from the viewpoint of volume. In particular, the inspection of volume is the core idea of our proposed detector.
Firstly, when both the signal and clutter are present, i.e., we have
(20) 
the magic will happen for the next dimension, i.e., when there are sample vectors, there will be
(21) 
while on the other hand,
(22) 
The reason is because, sample vectors in this scenario have not spanned the entire subspace according to the previous statement of randomly sampling; but can span , in another word,
(23) 
On the other hand, if the sample data only contains pure clutter, i.e., , we obtain
(24) 
and
(25) 
(21), (22), (24) and (25) indicates that, is the critical number of samples for detection of target signal in the background of clutter with unknown subspace structure, i.e., the "breakpoint". The knack of detection in this noiseless situation is, sampling continually, computing the volume of parallelotope spanned by all the sample vectors and known basis of target subspace at various dimensions and inspect the change of results. Once the volume vanishes, it means the number of samples reaches the critical point. Then the process of sampling should be stopped and the volume of sample vector themselves is calculated. The decision can be made based on whether the result is zero, i.e., whether (22) or (25).
IiiB The Volumebased Correlation Subspace Detector on raw data
Following the above thinking, an informal formulation of our detector in noiseless scenario, can be given below.
Detector 0.

Initial Step :
Obtain as the basis vectors of signal subspace . Let the initial matrix of sample data be . Index is set to . Set two thresholds and at appropriate values.

Step 1 :
Get the new sample , let , then compute
(26) build the test quantity as
(27) if , goto Step 2; else let , goto Step 1,

Step 2 :
Compute
(28) build the test quantity as
(29) if , then concludes the nonexistence of target signal; otherwise the conclusion is converse.
It should be noted that the decision on existence of target can’t be drawn by only examining the test quantity in (27), because the volume of will also be zero when and , leading naturally to zero value of . That is to say, will become zero even when the target doesn’t exist, because the dimension of matrix is larger than the dimension of clutter subspace. So additionally checking the volume of sample subspace is definitely necessary.
As we have said, when there is no target signal, we have
(30)  
(31) 
for , but on the other hand, when target does exist, we have
(32)  
(33) 
if , and will become zero only when . So as a whole, when only clutter exists, both and will vanish simultaneously for the same . While if target exists, they will become zero one after another as the increase of . This is the essential sign of presence of target and the key point in our detector, and that’s the reason why the detector relies on a joint test of (27) and (29).
IiiC The Volumebased Correlation Subspace Detector on orthogonalized data
The forementioned detector is impractical, because there will be numerical stability problem when we are calculating volume of matrices with large dimensions in practical situation. Hence the procedure of Orthogonalization will be introduced into our detector. The advantages of orthogonalization include reducing the procedure of volume calculation and threshold testing from two steps to one, and improving the numerical stability dramatically.
Specifically, let be the matrix whose columns are the orthonormal basis for target signal subspace , which could be obtained offline. The sample data taken from the subspace (or ) could be orthogonalized and normalized. We denote the result as matrix . Then the test quantity (27) in VC subspace detector, that is, the volume correlation between subspaces and can be written as
(34) 
because of the fact that
(35) 
we have
(36) 
The matrix in (34) can be prepared in advance. and matrix can be generated recursively as
(37) 
It should be noted that the twostep test in our previous VC subspace detector could be reduced to just one with the help of orthogonalization. In fact, if there is no target signal in received data, we have
(38) 
Contrast to the case without orthogonalization, the volume of will not become zero when . Because until now, the innovative vector could be linearly expressed by columns of . So if
we have
(39) 
hence
(40) 
Similarly, when there exists target signal, we also have
(41) 
As we can see, the volume of will never become zero.
When we are considering the other test quantity, according to (13), when there is no target signal, we have
(42) 
because .
On the other hand, when there exists target signal, according to the analysis in subsection A, it is obvious that
(43) 
because . As a result, the detector only needs a test on (34), because it is not possible that the volume of equals zero whether or not the target presents in received data, orthogonalization eliminates completely the possibility of rank deficiency of matrix . There is no need to check the value of alone in VC subspace detector at all.
The detector could be adapted as follows,
Detector 1.

Initial Step :
Obtain as the orthonormal basis vectors of target subspace and denote it by . Let the initial matrix of sample data be . Index is set to . Set two threshold and at appropriate values.

Step 1 :
Get the new sample , let
for ; while we let otherwise.
Then we compute the test quantity as
(44) if , concludes the existence of target signal and exits;

Step 2 :
if , then set and go to step 1; otherwise concludes the nonexistence of target signal and exits.
There are some further remarks deserve being mentioned explicitly.

Remark 1. The core of the proposed noiseless VC subspace detector is a volumebased test, the final decision is made based on the test of (44), which is actually the reciprocal of the volume correlation between the target signal subspace and the sample subspace. According to the previous analysis, when target exists in sampled data, will reach infinity for , while on the other hand, will remains a finite value if only clutter exists. The breakpoint of dimension for volume computation could be discovered AUTOMATICALLY, which is the indication of unknown dimension of clutter subspace. The reasons can be described briefly as that results of volumebased correlations is independent of the intrinsic structure of subspaces, and depends only on the dimensions and mutual relationship of the subspaces. So VC subspace detector focuses on the evolution of values of volumebased correlations along with increase of dimensions for volume computation only, regardless of the basis structure of clutter subspace. Concerning with the unknown characteristic of clutter subspace, VC subspace detector could be listed among the blind detecting methods. It is important to note that although matched subspace detector also detect signals from clutters, these two detection methods are essentially different. Because the prerequisite for matched subspace detector includes the detailed information on clutter subspace. However, it is needless for VC subspace detector.

Remark 2. It should be emphasized that the most remarkable advantage of VC subspace detector is its feature of "Detecting while Learning". To be specific, the detection could be completed without separated sessions for background learning with VC subspace detector. As is well known, background learning is very popular in adaptive processing for radar, communication and other signal processing problems. Channel equalization in communication transmission, CFAR (Constant False Alarm Rate) operation in radar detection and estimation of covariance matrices for clutter echoes in STAP (SpaceTime Adaptive Processing) all belong to sessions of background learning. There are double common defects for all these schemes. The first is that the efficacy of estimating clutter background might be influenced heavily by existence of target signal, which is referred to as target leakage in literatures; the second is the nonhomogeneousness widely existed in clutter environment which easily leads to mismatch of the consequence of learning with the actual clutter scenario at the target location. Nevertheless, VC subspace detector stands far away from these trouble because the process of background process is accomplished implicitly and simultaneously with the detection operation. While the raw data are being sampled and put into the detector sequentially, the volumebased correlations are examined and tested constantly until the breakpoint is reached. The information of clutter subspace is being learned in the form of volumes of basis for sample subspace. At the breakpoint, background learning ends while simultaneously the decision on the existence of target is made naturally. There is no need for extra effort of background learning. The learning and detection procedures are merged perfectly in VC subspace detector. We call this interesting property "Detecting while Learning".

Remark 3. The acceptance criterion of VC subspace detector for the hypothesis on existence of target signal is whether or not certain volumebased correlations are zero. It is generally believed that testing some quantities to be zero is impractical. Lots of statistical methods, such as MUSIC and other subspaceclass algorithms use reciprocal to transform the value near zero to be relatively large. Hence the difference between negligible results could be sharpen and the power of detection is greatly strengthened. VC subspace detector is not an exception.

Remark 4. In noisy situation, the detection problem becomes more complicated. Random noise will disturb our judgement and must be eliminated to assure the quality of detector. There are plenty of mature techniques for extracting the informative subspaces from noise, such as eigendecomposition based filtering and subspace tracking, for us to choose as the preprocessing steps of VC subspace detector. Some detailed discussion on these issues will be given in section IV.
IiiD Theoretical property of the proposed detector in noiseless situation
We will show theoretically that our volumebased correlation subspace detector can totally eliminate the influence of the clutter and detect the target signal in known target subspace effectively. Besides, an interesting monotonicity property for volumesbased correlation values of subspaces with different dimension will also be given, which provide a theoretical insight of our subspace detector.
Theorem 1
Let be the dimensional Hilbert space, and be the subspace of corresponding to target and clutter respectively. , , , Suppose be randomly sampled data either containing both target and clutter,
(45) 
or only containing clutter
(46) 
where and . Let
(47) 
and and be orthogonal matrix with columns being the basis vectors of and , then we have the following monotone property
(48) 
where
(49) 
The monotone property formulated in Theorem 1 might be useful when VC subspace detector was used in practical scenario. It guarantees that the test quantity of our detector will increase continuously with the dimension of sample subspace until the breakpoint is reached. So the breakpoint should be found easily without the annoying fluctuation of test quantity.
Theorem 2
Under the same assumption of theorem 1, the sufficient and necessary condition for existence of target signal in sample subspace is there is an integer such that
(50) 
Theorem 1 and 2 theoretically guarantee the behavior of our volumebased detector, as is shown in figure 1, the volume quantity will gradually drop as the increase of samples, it will drop to zero at index if and only if target signal exists; and remain nonzero when only clutter exists. The index in Theorem 2 is exactly the dimension of clutter subspace when the target signal is presented in sample subspace. Concerning with the monotone property in Theorem 1, is the smallest index for sample subspace to satisfy (50). Such could be called the critical point or phase transition point, for it indicates the essential change of volume correlation between sample subspace and target signal subspace. The reason for this change must be rank deficiency of direct sum of sample subspace and target subspace, which actually indicates the existence of target components in sample subspace. The behavior of our VC detector described in theorem 1 and 2 is illustrated in figure 1, in which we randomly generate two subspaces as and , and plot the variation of volumebased correlation with increase of in Detector 1.
Iv The VolumeCorrelation Subspace Detector in noisy environment
In this section, we will modify our VC subspace detector to make it be more suitable for noisy environment. Then some asymptotic result on the performance of VC subspace detector will be given.
Iva Main Idea
The main problem here is the sample subspace has been contaminated by random noise and can not be used directly to compute the volume correlation in VC subspace detector. Therefore the noise must be cleared in advance. For most statistical signal processing algorithms concerned with subspaces, such as MUSIC, ESPRIT and so on, the target signal and random noise are separated into signal subspaces and noise subspaces by eigendecomposition of correlation matrices firstly for further treatment. It implies the natural strategy of denoising for subspacebased signal processing. That is extracting signal subspaces for followup analysis and discarding noise subspaces simply for the purpose of noise elimination. In our case, the theme of VC subspace detector is detecting target signal from known target subspace lying in unknown clutter subspace. The main obstacle for our detector is how to accomplish subspace detection under the background of deterministic but sealed clutter. Hence random noise isn’t the critical factor in the detection and will be treated concisely.
To be specific, let be dimensional Hilbert space, and be target and clutter subspaces of respectively, and , , , the signal model for our detector is
where is the noise vector usually assumed to be white and Gaussian distributed with mean 0 and variance , and represents the signal components being sampled randomly from or just . The deterministic vector can be randomized and written as
where or , and is a random vector with finite second order moments. Then the correlation matrix of the sample data is
(51) 
without loss of generality, is assumed to be full rank, we denote the column rank of by , then the eigenvalues of could be listed as
(52) 
and the corresponding eigenvectors are
Denote , . It is clear that
(53) 
when both signal and clutter are present, and
(54) 
when the sampled data contains "pure" clutter. and are commonly called signal subspace and noise subspace. Therefore could be used as proxy of (or ) and the main idea in previous section is workable as well in the noisy environment.
IvB VC Subspace Detector in noisy environment
The VC subspace detector could be extended to noisy scenario as follows:
Detector 2.

Initial Step :
Denote the received data by . Obtain as the orthonormal basis vectors of target subspace and denote it by . Let the sample covariance matrix be . Index is set to . Set two thresholds and at appropriate values.

Step 1 :
Get the new sample , compute the covariance matrix as
(55) Assume the eigenvalues of be
(56) and the corresponding eigenvectors be
(57) determine the dimension of signal subspace , and construct the estimated sample subspace as
(58) 
Step 2 :
Compute the test quantity as
(59) if , concludes the existence of target signal and exits;

Step 3 :
if , then set and go to step 1; otherwise concludes the nonexistence of target signal and exits.
As described above, the subspace built from eigenvectors corresponding to large eigenvalues of covariance matrix of sampled data was taken to be the sample subspace in VC subspace detector. It is because the true covariance matrices can’t be obtained straight from sample data such that the sample covariance matrices were calculated via (55) instead. The dimension of signalplusclutter subspace (or clutter subspace), i.e., in (56) actually needs to be estimated. There are various methods can be used, from the conventional AIC or MDL and their variations [14, 15], to the newest Bayesian Information Criterion (BIC [16], GBIC [17]), Random Matrix Theory (RMT, [18]) and Entropy Estimation of Eigenvalues (EEE, [19]), etc. Since they all belong to another domain of research, we will not discuss this topic in detail here. In the following analysis, we just assume the dimension , which is related with and , is accurately known or estimated. With the eigendecomposition method, the accuracy about this approximation of subspace (or ) had been studied extensively [20][21][22] and the feasibility of had been proved asymptotically. Hence we use it in VC subspace detector as the substitution of sample subspace when noise is presented. We can expect the proposed VC subspace detector will asymptotically approximate the VC subspace detector in noiseless scenario, and this expectation is validated by the theory in next section.
IvC Property of Detector
To avoid the vagueness brought by asymptotic conclusion of the performance of VC subspace detector in the noisy background, we give some nonasymptotic analysis on the capability of our detector with knowledge of random matrices and concentration inequalities. Denote the received data by , and denote the correlation matrix by as in (51). Suppose the eigenvalues of be as (56) and its eigenvectors be as (57), We have
Theorem 3
Let be dimensional Hilbert space, and be target and clutter subspaces of respectively, , , ,
where is the sample data, , is Gaussian white noise.
If the target signal presents in sample data, then for any and , if
(60) 
then there exists a constant , such that
(61) 
holds with probability
(62) 
On the contrary, in the case of nontarget, for any and , when
(63) 
we have