Tuning stepsize between convergence rate and steady state error level or stability is a problem in some subspace tracking schemes. Methods in DPM and OJA class may show sparks in their steady state error sometimes, even with a rather small stepsize. By a study on the schemes' updating formula, it is found that the update only happens in a specific plane but not all the subspace basis. Through an analysis on relationship between the vectors in that plane, an amendment as needed is made on the algorithm routine to fix the problem by constricting the stepsize at every update step. The simulation confirms elimination of the sparks.
Tracking a subspace is Estimating a projection matrix onto a space or a basis for the space, from a random vector sequence observed by a sensor array. It is a powerful tool in some signal processing fields such as: telecommunication, radar, sonar and navigation, serves as a measure of adaptive filter, DOA estimation, or interference mitigation. Subspace tracking methodology is classified in into two categories: the first is estimating the space where the signal is generated from, the second one is to find orthogonal complement of that space. The former is known as a principal subspace (PS, PSA) tracker or signal subspace tracker, the later is often referred to as minor subspace (MS, MSA) tracker or a noise subspace tracker. For our earlier works on MUSIC, we are used to the term signal subspace track or noise subspace track.
N.L. Owsley developed the first algorithm for subspace tracking in [1]. Assuming the problem’s dimension is N, the rank of the subspace we are interested in is L .Usually L « N. Complex of this solution proportion to N 2 L,or O(N 2 L) namely. Many schemes with less compute complex were developed after then. An excellent survey paper [20] outlined almost all of achievements on this topic before 1990, which cost O(N 2 L) or O(NL 2 ) operations. Algorithms with O(NL) operations were developed after it. The new class is called as Fast Subspace Tracking method. Surveys on fast subspace schemes are presented in [3 pp30-43] or [19, pp221-270].
x k is an N-dim observer vector from an N-element sensor array, as (1),
Where i a is N-dim vector with unit length, independent to each other, representing the manifold of one of the arriving
which spans a subspace same as the space spanned by V; the noise subspace tracking is searching a ( ) W k which spans a subspace as orthogonal compensate of span(V). From now, we may use the name of the basis as the name of the space, such as space W or space W(k) to span(W(k)), if there is no confusion.
One of criterion on subspace problem is the distance between the spaces. Majority of those solutions [2][3][4][5][6][7][8][9][10][11] use the projection error power as (2) for signal subspace tracking or (3) for noise subspace tracking.
DPM [21] class algorithm is started from optimization the coordinate length of input vector’s image projected onto the subspace as (5) , while Oja class scheme optimize the length of the projection image as (6).
The final routines of DPM or OJA type approaches are very similar. A typical routine of DPM class scheme is similar to (7).
means orthonormalization operation.
Plus sign stands for signal subspace tracking, Minus sign is for noise subspace tracking.
The typical routine for an Oja scheme only replaces ( )
in the temporary general basis ( ) T k update in (7). The variety of DPM might include FDPM, FRANS, HFRANS [6], MFPDM [7] and a version of SOOJA [8]. The branches of Oja include OOJA [9], OOJAH [9], FOOJA [10], original version of SOOJA [11].
Some unreasonable random sparks were observed when we apply these schemes, especially when the noise subspace tracking was applied. It happens to DPM class schemes under noise subspace tracking and all variety of Oja methods.
In this paper, we present the geometric relationship among x(k), y(k), the old last estimated space W(k-1) and the next estimated space W(k) by update equations analysis. An Amendment as needed on the schemes is made to fix the sparks problem by the applying a limiter on stepsize at every update step. The presented simulation confirmed the amendment.
Considering the subspace tracking problem, the new arriving x(k) is projected onto the last estimated space W(k-1) to get a y(k) in space W(k-1). Vector t(k) is the y(k)’s projection image in x(k)’s orthogonal complement as (8).
The relation between those vectors and the last estimated space W(k-1) shows in Fig. 1 with an omission of time index k. We omit the time index k in some of following equations if there is no confusion.
Assuming there is a vector set with L elements including y direction served as a orthonormal basis set for space W(k-1), the basis is noted by ( / , ) y y COM , where COM is an L-1 dim subspace of space ( 1) W k , and
) is another orthonormal basis for the same space. Both ( / , ) y y COM and W(k-1) are orthonormal base set for the same subspace. Therefore an orthonormal matrix Q exists and meet (9,10);
H W k to (9), it is easy to find the first row of Q is 0
( 1)
( )
For (11) x is orthogonal to COM, therefore any linear combination of x and y is orthogonal to COM too, so is p.
Before the orthnorm( ) in ( 7), ( ) k T is a general basis of the newly estimated subspace. Right multiply the update equation of ( )
vector set for the newly estimated subspace as
For Oja schemes:
( 1) , ( ) ( ) ,
.
Or for DPM class:
After the multiplication, all newly estimated subspaces in Oja or DPM class schemes have a basis in term of COM and a linear combinations of x and y. By c
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