## 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 . An overcomplete frame expansion of
can be given as where F is the frame
operator associated with a frame , 's are the frame vectors and I is the index set.
The i th frame expansion coefficient of is defined as
, for all iI . Given the frame expansion of , it can be reconstructed
using the dual frame of _{F} which is given as . 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 and , 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., .
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.