Time-frequency analysis of frames

F. Hlawatsch, H. Bolcskei
Proceedings of IEEE-SP International Symposium on Time- Frequency and Time-Scale Analysis  
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. 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
more » ... unction sets. The mat 6ematical 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 frame bounds. Frame theory now shows [3] that any signal
doi:10.1109/tfsa.1994.467365 fatcat:hi6o6c4ypnhgrdoayemb62v5ji