# Advantages of an orthogonal filterbank

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.

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.

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.

• Yes, those sound like exactly the points I am looking for! Could you explain/reference a little bit more, especially the "less obvious" points about error propagation, energy compaction and performance? – Nils Werner Jul 19 at 11:32
• Sure, a bit later in the week-end, because I need to select references, the topic is quite broad – Laurent Duval Jul 19 at 11:35
• Sounds good! Thanks in advance, really looking forward to hearing your insight! – Nils Werner Jul 19 at 11:36

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.

• Yes, I am aware of this. I guess I am more asking for system- and filter-theory specific advantages. – Nils Werner Jul 19 at 11:27
• @NilsWerner I had worked up a quick followup but Laurent Duval just did it better. – Cedron Dawg Jul 19 at 11:32
• @LaurentDuval No, you covered it very well, more than I was going to say, or am able to say. Worthy of my upvote. – Cedron Dawg Jul 19 at 11:42