First off, what does orthogonality mean in the context of adjacent channels in a filter bank?

Secondly, I am reviewing some code in which the dot product of the transfer functions of adjacent channels are being used to "test orthogonality". Is this a formal method with a name? How does it work?

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    $\begingroup$ Your second question sounds like a strange method to me. The magnitude of a transfer function is nonnegative, so the dot product of two vectors that contain transfer function magnitudes will only be zero (which I would interpret to indicate "orthogonal") if one of the two transfer functions is zero everywhere. $\endgroup$ – Jason R Oct 5 '12 at 13:14
  • $\begingroup$ OK, I was put off by the fact that the author of the code had called a temporary variable mag, but it turns out that it is the dot product of complex transfer functions. I have edited the question appropriately. Thanks. $\endgroup$ – learnvst Oct 5 '12 at 14:09

The mathematical definition of orthogonality between two vectors is that their dot product is zero. It just means that there is no correlation between the two- at least at that "phase". It is often the case that if you shifted one of the vectors you would get strong correlation.

Infinite vectors of different frequencies are always orthogonal, so in an ideal world the output of adjacent channels in a filter band would always be orthogonal. There are two ways, though, that the real world is not ideal.

First, time limitations can introduce non-orthogonality. The non-orthogonality is usually trivial if both of the vectors have "many" sinusoidal cycles, but can be substantial if the length is less than a cycle.

Second, non-ideal filters means that attenuated stop-band frequencies get into the filter output, which means that the adjacent channels do have frequencies in common, just at different amplitudes.

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  • $\begingroup$ perhaps taking the dot product gives an idea of the amount of overlap? $\endgroup$ – geometrikal Oct 8 '12 at 13:20
  • $\begingroup$ It measures the amount of correlation. $\endgroup$ – Jim Clay Oct 8 '12 at 13:35
  • $\begingroup$ if the dot product of the transfer functions was zero they would be orthogonal, if non-zero but small we could say they are 'practically orthogonal'? $\endgroup$ – geometrikal Oct 8 '12 at 22:48
  • $\begingroup$ Yes. In theory you can get correlation measurements that end up being 0, in practice you never do. $\endgroup$ – Jim Clay Oct 9 '12 at 13:29

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