What is quadrature mirror filter (QMF) and conjugate quadrature filter (CQF)?

1. How does it work?
2. How is it designed?
3. Where is it applied?

1. The quadrature mirror filter (QMF) does not work in general. It has only two coefficients (unless the IIR filters are used).
2. The conjugate quadrature filter (CQF) can work with larger filters (usually even number of filter coefficients).

For example:

oooooo---h_1 ------- x_h_0() -------- g_1 ----
x(n)----ooooooooooooooooooooooooooooo---x(n)
oooooo---h_0 ------- x_h_1() -------- g_0 ----

Fig. Two-channel filter bank.

1. For QMF if h_0={a,b}
h_1={a,-b} is the same as h_0 but negate every other value.
g_0={a,b} is the same h_0.
g_1={-a,b} is -h_1.

2. For CQF if h_0={a,b}
h_1={b,-a} is the reversed version of h_0 and negate every other value.
g_0={b,a} is the reversed version of h_0.
g_1={-a,b} is the reversed version of h_1.