# Quadrature filter

In signal processing, a **quadrature filter** [math]\displaystyle{ q(t) }[/math] is the analytic representation of the impulse response [math]\displaystyle{ f(t) }[/math] of a real-valued filter:

- [math]\displaystyle{ q(t) = f_{a}(t) = \left(\delta(t) + j\delta(jt) \right) * f(t) }[/math]

If the quadrature filter [math]\displaystyle{ q(t) }[/math] is applied to a signal [math]\displaystyle{ s(t) }[/math], the result is

- [math]\displaystyle{ h(t) = (q * s)(t) = \left(\delta(t) + j\delta(jt)\right) * f(t) * s(t) }[/math]

which implies that [math]\displaystyle{ h(t) }[/math] is the analytic representation of [math]\displaystyle{ (f * s)(t) }[/math].

Since [math]\displaystyle{ q }[/math] is an analytic signal, it is either zero or complex-valued. In practice, therefore, [math]\displaystyle{ q }[/math] 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. It has single sided support, but by choosing the (analog) function [math]\displaystyle{ f(t) }[/math] carefully, it is possible to design quadrature filters which are localized such that they can be approximated by means of functions of finite support. A digital realization without feedback (FIR) has finite support.

## Applications

This construction will simply assemble an analytic signal with a starting point to finally create a causal signal with finite energy. The two Delta Distributions will perform this operation. This will impose an additional constraint on the filter.

### Single frequency signals

For single frequency signals (in practice narrow bandwidth signals) with frequency [math]\displaystyle{ \omega }[/math] the *magnitude* of the response of a quadrature filter equals the signal's amplitude *A* times the frequency function of the filter at frequency [math]\displaystyle{ \omega }[/math].

- [math]\displaystyle{ h(t) = (s * q)(t) = \frac{1}{\pi} \int_{0}^{\infty} S(u) Q(u) e^{i u t} du = \frac{1}{\pi} \int_{0}^{\infty} A \pi \delta(u - \omega) Q(u) e^{i u t} du = }[/math]

- [math]\displaystyle{ = A \int_{0}^{\infty} \delta(u - \omega) Q(u) e^{i u t} du = A Q(\omega) e^{i \omega t} }[/math]

- [math]\displaystyle{ |h(t)| = A |Q(\omega)| }[/math]

This property can be useful when the signal *s* is a narrow-bandwidth signal of unknown frequency. By choosing a suitable frequency function *Q* of the filter, we may generate known functions of the unknown frequency [math]\displaystyle{ \omega }[/math] which then can be estimated.

## See also

Original source: https://en.wikipedia.org/wiki/Quadrature filter.
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