Narrow-band filters

It seems we can represent a sinusoid by Z-transforms by setting a pole on the unit circle. Taking $Z_p=e^{i\omega_0}$, we have the filter  

$$B(Z) \eq {1 \over 1 - Z/Z_0} \eq {1 \over 1 - Ze^{-i\omega_0}} \eq 1 + Ze^{-i\omega_0} + Z^2e^{-i2\omega_0} + \cdots$$ (25)

The signal b_t seems to be the complex exponential $e^{-i\omega_0t}$,but it is not quite that because b_t is ``turned on'' at t = 0, whereas $e^{-i\omega_0t}$ is nonzero at negative time.

Now, how can we make a real-valued sinusoid starting at t=0? Just as with zeros, we need to complement the pole at $+\omega_p$by one at $-\omega_p$.The resulting signal b_t is shown on the left in Figure 7. On the right is a graphical attempt to plot the impulse function of dividing by zero at $\omega = \omega_0$.

 

sinus
Figure 7 A pole on the real axis (and its mate at negative frequency) gives an impulse function at that frequency and a sinusoidal function in time.

Next, let us look at a damped case like leaky integration. Let $Z_p = e^{i\omega_0}/\rho$ and $\vert\rho \vert<1$.Then $1/Z_p = \rho e^{-i\omega_0}$.Define

$$\begin{eqnarray} B(Z) &= & {1 \over {A(Z)}} \eq {1 \over {1 - Z / Z_p}} \eq 1 + ... ... 1 + Z \rho e^{-i\omega_0} + Z^2 \rho^2 e^{-i2\omega_0} + \cdots\end{eqnarray}$$ (26) (27)

The signal b_t is zero before t = 0 and is $\rho^t e^{-i\omega_0t}$ after t = 0. It is a damped sinusoidal function with amplitude decreasing with time as $\rho^t$.We can readily recognize this as an exponential decay

$$\rho^t \eq e^{t \log \rho} \quad \approx \quad e^{-t(1-\rho )}$$ (28)

where the approximation is best for values of $\rho$ near unity.

The wavelet b_t is complex. To have a real-valued time signal, we need another pole at the negative frequency, say $\overline{Z_p}$.So the composite denominator is  

$$A(Z) \eq \left( 1- {Z \over Z_p} \right) \ \left( 1- {Z \ov... ...verline{Z}_p} \right) \eq 1-Z \rho 2\cos \omega_0 + \rho^2 Z^2$$ (29)

Multiplying the two poles together as we did for roots results in the plots of 1/A(Z) in Figure 8.

 

dsinus
Figure 8 A damped sinusoidal function of time transforms to a pole near the real $\omega$-axis, i.e., just outside the unit circle in the Z-plane.

Notice the ``p'' in the figure. It indicates the location of the pole Z<sub>p</sub> but is shown in the $\omega_0$-plane, where $Z_p=e^{i\omega_0}$.Pushing the ``p'' left and right will lower and raise the resonant frequency. Pushing it down and up will raise and lower the duration of the resonance.

EXERCISES:

1. How far from the unit circle are the poles of 1/(1 - .1Z + .9Z²)? What is the decay time of the filter and its resonant frequency?