Quadrature filter
From Wikipedia, the free encyclopedia
In signal processing, a quadrature filter q(t) is the analytic signal of a real-valued filter f(t):
If the quadrature filter q(t) is applied to a signal s(t), the result is
which implies that h(t) is the analytic signal of (f * s)(t).
Since q is an analytic signal, it is either zero or complex-valued. In practice, therefore, q 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.
On the other hand, by choosing the function f(t) 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.
[edit] Applications
Notice that the computation of an ideal analytic signal for general signals cannot be made in practice since it involves convolutions with the function
which is difficult to approximate as a filter which is either causal or of finite support, or both. According to the above result, however, it is possible to obtain an analytic signal by convolving the signal s(t) with a quadrature filter q(t). Given that q(t) is designed with some care, it can be approximated by means of a filter which can be implemented in practice. On the other hand, the resulting function h(t) is now the analytic signal of f * s rather than of s. This implies that f should be chosen such that convolution by f affects the signal as little as possible. Typically, f(t) is a band-pass filter, removing low and high frequencies, but allowing frequencies within a range which includes the interesting components of the signal to pass.
