Quadrature Filter

In signal processing, a quadrature filter is the analytic representation of the impulse response of a real-valued filter:

$q(t) = f_{a}(t) = \left(\delta(t) + {i \over \pi t} \right) * f(t)$

If the quadrature filter is applied to a signal, the result is

$h(t) = (q * s)(t) = \left(\delta(t) + {i \over \pi t} \right) * f(t) * s(t)$

which implies that is the analytic representation of .

Since is an analytic signal, it is either zero or complex-valued. In practice, therefore, is often implemented as two real-valued filters, which correspond to the real and imaginary parts of the filter, respectively.

An ideal quadrature filter cannot have a finite support, but by choosing the function carefully, it is possible to design quadrature filters which are localized such that they can be approximated reasonably well by means of functions of finite support.