Oblique projections: Formulas, algorithms, and error bounds

IEEE Trans. Signal Process.

Dvorkind

2006

127

We treat the problem of reconstructing a signal from its non-ideal samples where the sampling and reconstruction spaces as well as the class of input signals can be arbitrary subspaces of a Hilbert space.Our formulation is general, and includes as special cases reconstruction from finitely many samples as well as uniform-sampling of continuous-time signals, which are not necessarily bandlimited. To obtain a good approximation of the signal in the reconstruction space from its samples, we suggest two design strategies that attempt to minimize the squared-norm error between the signal and its reconstruction.The approaches we propose differ in their assumptions on the input signal: If the signal is known to lie in an appropriately chosen subspace, then we propose a method that achieves the minimal squared-error.On the other hand, when the signal is not restricted, we show that the minimal-norm reconstruction cannot generally be obtained. Instead, we suggest minimizing the worst-case squared-error between the reconstructed signal, and the best possible (but usually unattainable) approximation of the signal within the reconstruction space. We demonstrate both theoretically and through simulations that the suggested methods can outperform the consistent reconstruction approach previously proposed for this problem.

“…The oblique projection 1 [17] onto W along S ⊥ is denoted by E WS ⊥ , and is defined as the unique operator satisfying E WS ⊥ w = w for any w ∈ W;…”

Section: Sampling Formulationmentioning

confidence: 99%

A minimum squared-error framework for generalized sampling

IEEE Trans. Signal Process.

Dvorkind

2006

127

“…Theorem 1 below asserts that (2.2) is satisfied for all f ∈ H with W ∩ S ⊥ = {0} if and only if G = W H S * is an oblique 2 projection [16,2,20] with R(G) = W and N (G) = S ⊥ , denoted by E WS ⊥ . The oblique projection E WS ⊥ is defined as the unique operator satisfying Proof.…”

Section: Consistency Conditionmentioning

confidence: 99%

Sampling with Arbitrary Sampling and Reconstruction Spaces and Oblique Dual Frame Vectors

Journal of Fourier Analysis and Applications

2003

183

229

“…Thus, which in general is not equal to , except in the special case that is orthogonal to , , in which case . We now show that (12) implies that is the oblique projection of onto along , denoted by 1 [5]. To this end we first show that (13) From the definition of an oblique projection, is the unique operator satisfying for any for any…”

Section: Methods First We Note That Ifmentioning

confidence: 89%

“…Thus, we choose to minimize the least-squares error (3) In [3, p. 241] it is claimed that the minimizing is (4) It is important to note that when the vectors are linearly dependent, given by (4) is not the only vector that minimizes (3). Indeed, let (5) where is any vector in the null space of . Then since , and , so that also minimizes (3).…”

Section: Minimum-norm Least-squares Approximationmentioning

confidence: 99%

On geometric properties of the decorrelator

2002

IEEE Commun. Lett.

Abstract-In this letter we discuss geometric properties of the multiuser decorrelator receiver. In particular we show that the prevalent geometric interpretation of the correlating vectors comprising the decorrelator receiver in terms of orthogonal projections of the signature vectors is incorrect. The correct interpretation established in this letter is that the correlating vectors are oblique projections of the signature vectors onto appropriate spaces. Furthermore, we show that each branch of the decorrelator consists of an oblique projection onto the space spanned by the corresponding user's signature vector along the space spanned by the interferers, followed by a correlator with correlating vector equal to the corresponding user's signature vector.