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Section: Scientific Foundations

Frame expansions

Signal representation using orthogonal basis functions (e.g., DCT, wavelet transforms) is at the heart of source coding. The key to signal compression lies in selecting a set of basis functions that compacts the signal energy over a few coefficients. Frames are generalizations of a basis for an overcomplete system, or in other words, frames represent sets of vectors that span a Hilbert space but contain more numbers of vectors than a basis. Therefore signal representations using frames are known as overcomplete frame expansions. Because of their inbuilt redundancies, such representations can be useful for providing robustness to signal transmission over error-prone communication media. Consider a signal Im9 $\#119857 $ . An overcomplete frame expansion of Im9 $\#119857 $ can be given as Im10 ${F\#119857 }$ where F is the frame operator associated with a frame Im11 ${\#934 _F\#8801 {{\#981 _i}}_{i\#8712 I}}$ , Im12 $\#981 $ 's are the frame vectors and I is the index set. The i th frame expansion coefficient of Im9 $\#119857 $ is defined as Im13 ${{(F\#119857 )}_i\#8801 {\#9001 \#981 _i,\#119857 \#9002 }}$ , for all i$ \in$I . Given the frame expansion of Im9 $\#119857 $ , it can be reconstructed using the dual frame of $ \upper_phi$F which is given as Im14 ${\mover \#934 \#732 _F\#8801 {{{(F^hF)}^{-1}\#981 _i}}_{i\#8712 I}}$ . Tight frame expansions, where the frames are self-dual, are analogous to orthogonal expansions with basis functions. Frames in finite-dimensional Hilbert spaces such as Im15 ${\#119825 }^K$ and Im16 ${\#119810 }^K$ , known as discrete frames, can be used to expand signal vectors of finite lengths. In this case, the frame operators can be looked upon as redundant block transforms whose rows are conjugate transposes of frame vectors. For a K -dimensional vector space, any set of N , N>K , vectors that spans the space constitutes a frame. Discrete tight frames can be obtained from existing orthogonal transforms such as DFT, DCT, DST, etc by selecting a subset of columns from the respective transform matrices. Oversampled filter banks can provide frame expansions in the Hilbert space of square summable sequences, i.e., Im17 ${l_2{(\#119833 )}}$ . In this case, the time-reversed and shifted versions of the impulse responses of the analysis and synthesis filter banks constitute the frame and its dual. Since overcomplete frame expansions provide redundant information, they can be used as joint source-channel codes to fight against channel degradations. In this context, the recovery of a message signal from the corrupted frame expansion coefficients can be linked to the error correction in infinite fields. For example, for discrete frame expansions, the frame operator can be looked upon as the generator matrix of a block code in the real or complex field. A parity check matrix for this code can be obtained from the singular value decomposition of the frame operator, and therefore the standard syndrome decoding algorithms can be utilized to correct coefficient errors. The structure of the parity check matrix, for example the BCH structure, can be used to characterize discrete frames. In the case of oversampled filter banks, the frame expansions can be looked upon as convolutional codes.


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