Band pass filter = Band stop filter if you switch the cutoff frequencies?

Question

The normalized frequency response of a N points bandpass digital filter is:

$\frac{sin(\omega_{c2}(n-M))}{\pi(n-M)}-\frac{sin(\omega_{c1}(n-M))}{\pi(n-M)}$ where $\omega_{c1}$ is the normalized lower cutoff angular frequency , where $\omega_{c2}$ is the normalized upper cutoff angular frequency and $M = floor(N/2)$.

However the normalized frequency response of a N points bandstop digital filter is:

$\frac{sin(\omega_{c1}(n-M))}{\pi(n-M)}-\frac{sin(\omega_{c2}(n-M))}{\pi(n-M)}$ where $\omega_{c1}$ is the normalized lower cutoff angular frequency and where $\omega_{c2}$ is the normalized upper cutoff angular frequency .

But obviously $H_{bp}(e^{j\omega}) =-H_{bs}(e^{j\omega})$ Does this mean that a band pass filter has the same frequency response to a band stop filter if you switch the angular cutoff frequencies?

And what implications does this have in digital filter design?

Answer 1

The impulse response of the ideal bandstop filter given in your question isn't correct. Think about it: inverting the sign of the impulse response of a bandpass filter is just another bandpass filter with an extra phase shift of $\pi$.

In the frequency domain it's easy to see that the ideal response of a (zero phase) bandstop filter is just $1$ minus the ideal response of a (zero phase) bandpass filter (assuming unity gain of both filters in their passbands):

$$H_{BS}(e^{j\omega})=1-H_{BP}(e^{j\omega})$$

In the time domain this is equivalent to

$$h_{BS}[n]=\delta[n]-h_{BP}[n]$$

The impulse at the origin makes an important difference: it switches passbands and stopbands, which is exactly what we need when transforming a bandpass filter to a bandstop filter.

Note that all these filters are zero phase filters. You can simply shift them by an integer $M$ (as in the formulas in your question), but don't forget to also shift the impulse.

Answer 2

But obviously $H_{bp}(e^{j\omega}) =-H_{bs}(e^{j\omega})$.

Nope. What gives a bandpass or bandstop filter its name is the amplitude response. Changing the phase by 180$^\circ$ doesn't change the amplitude response by one whit.

You need to look at some transfer functions of bandpass and bandstop filters. They're quite different from each other. Using the $z$ domain really obfuscates this, so I'm going to demonstrate this in the Laplace domain. Consider two filters, of the same bandwidth:

$$\begin{align} H_{bp}(s) &= \frac{\frac{\omega_0}{Q}s}{s^2 + \frac{\omega_0}{Q}s + \omega_0^2} \\ H_{bs}(s) &= \frac{s^2 + \omega_0^2}{s^2 + \frac{\omega_0}{Q}s + \omega_0^2} \end{align}$$

$H_{bs} \ne -H_{bp}$ here, and that is never the case. Bandpass and bandstop filters are different, but related beasts.

It happens in this case that $H_{bs} = 1 - H_{bp}$ -- that's not the case in general, but it's far more evocative of the fundamental relationship between the two. $\left | H_{bs} \right | = 1 - \left | H_{bp} \right |$ is, possibly, a more useful generalization to think about bandpass and bandstop filters, but won't help you much in constructing them.