Quadrature filter

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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.

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.

1 Applications
- 1.1 Estimation of analytic signal
- 1.2 Single frequency signals
2 See also

[edit] Applications

[edit] 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. 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.

[edit] Single frequency signals

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

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

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

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

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 $ω$ which then can be estimated.