Time - Frequency Analysis of Frames *

The theory of frames is fundamental to timefrequency (TF or time-scale signal expansions like Gabor ysis of frames via two “TF frame representations” called the Weyl symbol and Wigner distribution of a frame. The T F analysis shows how a frame’s properties depend on the signal’s T F location and on certain frame parameters. 1 I N T R O D U C T I O N Linear time-frequency (TF) or time-scale signal expansions like the Gabor expansion or the wavelet transform [1]31 are often based on nonorthogonal function sets. The mat 6 ematical theory of frames [3 -[5] yields important insights well as methods for calculating the expansion coefficients. &(Et) be a Hilbert space of finite-energy signals, with dimension DX that may be 00. A set of functions Q = { g k ( t ) } with g k ( t ) E X is a frame for X if for every signal z( t ) E X expansions an d wavelet transforms. We propose a T F analinto the properties of nonort h ogonal signal expansions, as Review of Frame Theory. Let X k with 0 < Ap 5 B p < 00. Here, ( z , g k ) = t z ( t ) g i ( t ) dt is the inner product’ of z ( t ) with g k ( t ) , and = (z,z) is the energy of z( t ) . The constants Ap and B p are called f rame bounds. Frame theory now shows [3] that any signal z ( t ) E X can be expanded into the frame functions g k ( t ) as z ( t ) = c f f k g k ( t ) with f f k = ( z , 3 k ) ( 2 ) k where Here, the f rame operator G is defined as i j k ( t ) = ( G ’ g k ) ( t ) E X . (3) with the kernel G ( t , t ’ ) = x g k ( t ) g i ( t ’ ) . (4) k G is a self-adjoint, positive semidefinite, linear operator [6] that maps &(Et) into X. On X, G is positive definite and invertible, i.e., G is also an invertible mapping from X onto X. We note that (Gz)(t) = 0 for z ( t ) I X. Eq. (1) can be rewritten as Ao11~11~ 5 ( G z , z ) 5 Bp11~11~ for all z ( t ) E X. This shows that the tightest possible frame bounds (denoted A;, Bz) are given by the infimum and supremum, respectively, of the eigenvalues of G. The functions & ( t ) in (2), ( 3 ) constitute another frame ‘Funding by FWF grant P10012-OPH. ‘Integrals go from -m to 00. = { & ( t ) } for X which is called the dual frame. For the dual frame, the frame bounds are Ad = 1/Bp and Bd = 1/Ap, and the frame operator is (on X) G = G-’ . A frame Q is complete in the space X, but the frame functions g k ( t ) need not be linearly independent. A frame with linearly independent g k ( t ) (called exact frame) satisfies the biorthogonality relations ( g k , & ) = & I . A frame is called tight if Ap = B g . Here, G = Ap Px, where PX is the orthogonal projection operator on X, and & ( t ) = g k ( t ) / A p so that calculation of the dual frame is trivial. An orthonormal basis is a special case of a tight frame with Ap = B p = 1. A frame with Ap M Bp is called snug. Closer frame bounds Ap and B p entail better numerical properties of the expansion (2) and more efficient algorithms for calculating the dual frame. Indeed, (3) can be expanded as which converges faster for closer Ap, B p [3]. For snug frames, & ( t ) can hence be approximated by truncating the series (5). In particular, truncation after the n = 0 term yields g k ( t ) = C g k ( t ) and, with (2), k Motivation a n d Outline. The frame bounds Ap, B p do not show how certain parameters of a frame could be changed in order to improve the frame’s numerical properties. This information can often be obtained from the T F analysis o f f rames proposed in this paper. The T F analysis also shows how a frame’s properties depend on the T F location of the signal to be expanded; in particular, a frame may be “locally snug” in restricted T F regions. We propose two TF frame representations, the Weyl symbol and the Wagner distribution of a frame, both of which generalize the Wigner distribution of a linear signal space [7, 8 and satisfy tations are bounded in terms of &e frame bounds. Some examples show the usefulness of the T F analysis proposed. Trace, Inner Product , Energy. For use in subsequent sections, we define the trace To of a f rame B as the trace of the frame operator G, interesting properties. Local avera es of these Jr F represenR We also define the inner product of two frames Q and ‘H as A (Q, ‘H) = tr{GH} = G ( t , t‘) H * ( t , t ’ ) dtdt’ k I and the energy of a B m e Q as 0-7803-2127-8/94 $4.00 01994 IEEE 52 ~______ .in Proc. IEEE-SP Int. Sympos. Time-Frequency Time-Scale Analysis (TFTS-94), Philadelphia (PA), Oct. 1994, pp. 52–55 Copyright IEEE 1994