Advantages of an orthogonal filterbank

Question

What are the advantages orthogonal filterbanks such as the DFT or DCT have over non-orthogonal filterbanks?

I have heard quite often that they have attributes that are desirable when modifying the intermediate signal, i.e. when performing quantization during audio coding.

Answer 1

Orthogonality provides an interesting backbone to the structure of the filter-banks (FB). First, from an analysis FB, the synthesis FB is very direct, so it can ease implementations. Second, the orthogonality often allows faster implementations, as there is "little redundancy" in computation. Third, orthogonality ensures that matrices are well-conditioned, this may reduce risks of error propagation (esp. with quantization). Fourth, energy is preserved, so operations performed in the dual domain can be measured (energetically) in the primal domain. Fifth: it's a great deal of simplification when you need to get tractable proofs for their design (energy compaction for instance) or performance (as for instance Gaussian noise remains Gaussian after an orthogonal transformation). Note that these five points are not completely orthogonal.

To provide a little more details:

1. Orthogonality, and more generally paraunitarity (cf. Paraunitary matrices by Barry and Ted Hurley), i.e. a square matrix $U(z)$ satisfying $U(z)U^∗(z^{−1}) = 1$, are practical, as the inverse is somehow a conjugate-transpose
2. For orthogonal bases in an $N$-dimensional space, the first orthogonal vector has about $N-1$ degrees of freedom, the second one $N-2$, etc. So orthogonal matrices have a structure (see the Orthogonal group), which can be used to factorize them, hopefully with less operations (Fast multiplication of orthogonal matrices). For instance, $f 3 \times 3$ orthogonal matrices can be represented by three Givens rotations, but other designs are possible
3. Determinants are $\pm 1$ (and of course the same of their inverses). When a matrix contains close vectors, projections split evenly of them, which has effects on 1) simple inversion 2) iterations (like in computerized tomography) where powers of matrices (like FFT) over millions of cells/pixels/voxels may diverge.
4. When energy is preserved, a structured signal (in the appropriate basis or frame) may come out of unstructured noise. This is a rationale behind "sparsifying transformations", and how thresholding values can denoise. Somehow behind Stein estimates (Stein Unbiased Risk Estimation), with orthogonaly invariant estimators, sample eigenvalues are modified (for the best) yet sample eigenvectors untouched.
5. All block orthogonal transforms are quite easy to deal with.
   • a) Overlap and windows are more involved. An early attempt is the LOT (Lapped Orthogonal Transform), and their generalizations (GenLOT - Generalized Lapped Orthogonal Transforms , GULLOT - Generalized Unequal Length Lapped Orthogonal Transforms , etc.). They can be seen as DCT (or Hadamard) matrices with additional degree-of-freedom. By decomposing them with rotations, it is possible to optimize their design by imposing constrains, like "Coding gain", "DC leakage" (Linear phase paraunitary filter bank with filters of different lengths and its application in image compression, 1999), that can be related to their performance in data compression. Trivia: read Properties of the Eigenvectors of Persymmetric Matrices with Applications to Communication Theory to see how correlation structures impose properties of eigenvectors
   • b) Some noise properties can be estimated in a data sparsifying domain. A classical example in the wavelet domain (and filter-banks generally) is the MAD (median absolute deviation) estimator of the Gaussian noise in a combination of subbands.

However, orthogonality has limits, imposes additional constraints that sometimes spoil other desirable properties. Plus, the human sensory systems care little about strict orthogonality, and you can gain degrees of freedom (even in computational efficiency) by discarding strict orthogonality. The filterbanks used in the JPEG2000 image compression standard are not orthogonal, but biorthogonal. [UPDATE] In some cases, you may want integer or dyadic rational operations (for memory or speedup), and this is rarely compatible with orthogonality. In video coding, a lot of decorrelating operations are not strictly orthogonal.

On most practical cases though, a lot of systems remain at least "close to orthogonal" in some sense.

Answer 2

Having an orthogonal basis makes finding coefficients a lot easier. It makes the matrix that needs to be inverted a diagonal matrix making inversion trivial. With the DFT it is so trivial it is implicit.