Tue Mar 19 13:24:42 EDT 2013

Complex 1-poles

Something that has puzzled me for a while:

* The ideas of "low pass" (LP) and "high pass" (HP) filter are not
  meanigful for a discrete, complex filter.

The ideas of LP,HP,BP,LP stem from real, continuous filters.  They
refer to the behavior of the transfer function at zero, infinity and a
reference frequency f.

      | 0  f  inf
  LP  | 1  ?  0
  HP  | 0  ?  1
  BP  | 0  1  0
  LP  | 1  0  1

This has to do with the symmetry of the transfer functions.

For continuous filters, transfer functions are defined on the real
line, which have 2 special points apart from a current reference
point: zero and infinity.

For a complex discrete filter, no special points exist, other than
"not the point of interest".  This is due to the symmetry: transfer
functions, which is apparent by them being defined on the complex unit
circle (z-transform).

Looking at the symmetry, there are only two "kinds" of discrete
filters: "band pass" (BP) and "band stop" (BS).

    f=f0  f!-f0
BS   0     >0
BP   1     <1

* The ideas of LP/HP can be introduced into the discrete world by
  - Sticking to real signals.  This creates 2 special points: DC, NY
  - Defining a mapping from continuous to discrete.

Sticking to real signals creates two special points where the real
axis intersects the unit circle: DC (the direct current) and NY
(Nyquist frequency).

By itself, DC and NY are 2 points that are not essentially different.
However, they can be distinguished by relating discrete and continuous
transfer functions.

Impulse invariance:
  - DC  0 (but also 2NY, 4NY, ...)
  - NY  not in any way special.  infinity is not mapped

Bilinear transform:  
  - DC  0
  - NY  infinity

Once this mapping is in place, the ideas of HP and LP can be made to
carry over.

So it is important when talking about LP and HP filters in a discrete
sense to also take into account how these ideas are defined,
i.e. through impulse invariance or the bilinear transform.

Using the impulse invariance transform which is useful for musical
applications, it might be useful to still define properties in terms
of behaviour at infinity, even if that doesn't make sense in
actuality, as filter transfer functions will alias.