Even and Odd Filters and Quadrature

Question

I am a mathematician trying to understand some signal processing terms. My first question regards the terms 'even' and 'odd' filters. What does this mean? Does it reference the fact that upon a Fourier transform, there is a cosine and a sine function, which in turn are even and odd? If not, a link to a reliable source would be appreciated.

Second question: What is a quadrature? I have heard the term used for the phase shift between two Fourier transforms, the method for dividing a frame into four parts, and taking a gradient-like subtraction from vertical, horizontal, and both diagonal directions. Is it all of these at once? Also, a link to a reliable source would be great.

Answer 1

An even function is symmetric:

$$f(x) = f(-x)$$

An odd function is anti-symmetric:

$$-f(x) = f(-x)$$

A signal in quadrature with another has all its Fourier series components phase-shifted by 90 degrees. The two signals are thus orthogonal.

To get the quadrature signal we apply the Hilbert transform (or Riesz transform in more than 1 dimension)

$$g(x) = \pm \mathcal{H}[f](x)$$

A cosine wave is even and symmetric. A sine wave is odd and anti-symmetric. Therefore if you apply the Hilbert transform to an even signal (cosine Fourier series components) you get an odd signal (sine Fourier series components). If you apply the Hilbert transform again you get the negative original signal. Apply again you get the negative of the original Hilbert transformed signal. Apply again you get the original signal. This is where the "quad" part comes in.

This is useful in signal analysis. Since the Hilbert transform only phase-shifts the Fourier series components, the energy of the signal remains constant. We can thus reconstruct the original signal (e.g. wavelet denoising). Also, given an even filter that responds to peaks and troughs, we can create an odd filter that responds to edges. The pair of filters is called a quadrature filter pair and allows the analysis of signal features. See the analytic signal for more details.

Answer 2

For the 1st part of your question:

Some definitions of even and odd functions here:

https://www.mathsisfun.com/algebra/functions-odd-even.html

and how this applies to the sine and cosine functions:

http://math2.org/math/algebra/functions/sincos/properties.htm

Answer 3

To answer your "What is quadrature?" question, perhaps the discussion at: http://www.dsprelated.com/blogimages/RickLyons/Quad%20Signals%20Tutorial-Lyons.pdf would be of some use to you.