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If the quadrature filter is applied to a signal , the result is
which implies that is the analytic representation of .
Since is an analytic signal, it is either zero or complex-valued. In practice, therefore, 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 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.
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
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 with a quadrature filter . Given that is designed with some care, it can be approximated by means of a filter which can be implemented in practice. The resulting function is the analytic signal of rather than of . This implies that should be chosen such that convolution by affects the signal as little as possible. Typically, 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 the magnitude of the response of a quadrature filter equals the signal's amplitude A times the frequency function of the filter at frequency .
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 which then can be estimated.