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

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

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

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

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

Since ${\displaystyle q}$ is an analytic signal, it is either zero or complex-valued. In practice, therefore, ${\displaystyle q}$ 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, but by choosing the function ${\displaystyle f(t)}$ 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.

## Applications

### Estimation of analytic signal

Notice that the computation of an ideal analytic signal for general signals cannot be made in practice since it involves convolutions with the function

${\displaystyle {1 \over \pi t}}$

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 ${\displaystyle s(t)}$ with a quadrature filter ${\displaystyle q(t)}$. Given that ${\displaystyle q(t)}$ is designed with some care, it can be approximated by means of a filter which can be implemented in practice. The resulting function ${\displaystyle h(t)}$ is the analytic signal of ${\displaystyle f*s}$ rather than of ${\displaystyle s}$. This implies that ${\displaystyle f}$ should be chosen such that convolution by ${\displaystyle f}$ affects the signal as little as possible. Typically, ${\displaystyle f(t)}$ 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.

### Single frequency signals

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

${\displaystyle h(t)=(s*q)(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }S(u)Q(u)e^{iut}du={\frac {1}{2\pi }}\int _{-\infty }^{\infty }A\pi [\delta (u+\omega )+\delta (u-\omega )]Q(u)e^{iut}du=}$
${\displaystyle ={\frac {A}{2}}\int _{0}^{\infty }\delta (u-\omega )Q(u)e^{iut}du={\frac {A}{2}}Q(\omega )e^{i\omega t}}$
${\displaystyle |h(t)|={\frac {A}{2}}|Q(\omega )|}$

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 ${\displaystyle \omega }$ which then can be estimated.