The quadrature filter

Beginning with a causal response, we switched cosines and sines in the frequency domain. Here we do so again, except that we interchange the time and frequency domains, getting a more physical interpretation.

A filter that converts sines into cosines is called a ``$90^{\circ}$phase-shift filter" or a ``quadrature filter."

More specifically, if the input is $\cos (\omega t +\phi_1)$,then the output should be $\cos(\omega t + \phi_1 - \pi /2)$.An example is given in Figure 2.

 

hilb0 Figure 2 Input (top) filtered [2] with quadrature filter yields [4] phase-shifted signal (bottom).

Let U(Z) denote the Z-transform of a real signal input and Q(Z) denote a quadrature filter. Then the output signal is  

$$V(Z) \eq Q(Z) \ U(Z)$$ (5)

Let us find the numerical values of q_t. The time-derivative operation has the $90^{\circ}$ phase-shifting property we need. The trouble with a differentiator is that higher frequencies are amplified with respect to lower frequencies. Recall the FT and take its time derivative:

$$\begin{eqnarray} b(t) & \eq & \int B(\omega) e^{-i\omega t} d\omega \ \frac{db}{dt} & \eq & \int -i\omega B(\omega) e^{-i\omega t} d\omega \end{eqnarray}$$ (6) (7)

Thus we see that time differentiation corresponds to the weight factor $-i\omega$ in the frequency domain. The weight $-i\omega$ has the proper phase but the wrong amplitude. The desired weight factor is  

$$Q(\omega) = {-i\omega \over \vert\omega\vert} = -i \, {\rm sgn}\,\omega$$ (8)

where sgn is the ``signum'' or ``sign'' function. Let us transform $Q(\omega)$ into the domain of sampled time t=n:

$$\begin{eqnarray} q_n & = & \frac{1}{2\pi} \int_{-\pi}^{\pi} Q(\omega) e^{-i\omeg... ... {-2 \over \pi n} & \mbox{for $n$\space odd} \end{array} \right. \end{eqnarray}$$ (9) (10)

Examples of filtering with q_n are given in Figure 2 and 3.

Since q_n does not vanish for negative n, the quadrature filter is nonrealizable (that is, it requires future inputs to create its present output). If we were discussing signals in continuous time rather than sampled time, the filter would be of the form 1/t, a function that has a singularity at t=0 and whose integral over positive t is divergent. Convolution with the filter coefficients q_n is therefore painful because the infinite sequence drops off slowly. Convolution with the filter q_t is called ``Hilbert transformation."

 

hilb
Figure 3 A Hilbert-transform pair.