Quadrature filter

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

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

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

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

which implies that $h(t)$ is the analytic representation 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, but 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.

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

${1 \over \pi t}$

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 $s(t)$ 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. The resulting function $h(t)$ is 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.

Single frequency signals

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

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

$= \frac{A}{2} \int_{0}^{\infty} \delta(u - \omega) Q(u) e^{i u t} du = \frac{A}{2} Q(\omega) e^{i \omega t}$

$|h(t)| = \frac{A}{2} |Q(\omega)|$

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 $\omega$ which then can be estimated.

See also

• Analytic signal
• Hilbert transform