# General transfer function for a bandpass filter

Is there a general form of transfer function (with peak frequency $\omega_m$ and quality factor $Q$) relevant for any type of bandpass filter ?

• a bandpass filter has two cutoff frequencies! May 14, 2012 at 12:00
• well I meant the peak frequency, the frequency at which the gain is maximum May 14, 2012 at 12:02
• Some bandpass filters have multiple peaks, like a Chebychev for example. May 14, 2012 at 12:04
• @snickers - even without the multiple peaks (Olin's comment) the center frequency isn't enough to know the bandwidth. May 14, 2012 at 12:06
• @snickers - example of the frequency response of a Chebychev filter: cnx.org/content/m16895/latest/c92.png May 14, 2012 at 12:09

No. Whilst a standard second-order bandpass section can be defined in this way ...

$H(s) = \dfrac{\dfrac{\omega_m}{Q}s}{s^2+\dfrac{\omega_m}{Q}s+\omega_m^2}$

... it is also possible to have a second-order bandpass filter with the same characteristic frequency and Q but with a different transfer function. This previous question which addresses a filter with a stop-band attenuation of 1 is a case-in-point.

Furthermore, higher-order filters will require more than just these two parameters to define them since there are more coefficients.

There is actually another low-entropy form presenting the transfer function in a more compact way in my opinion:

$$\H(s)=H_0\frac{1}{1+Q \left(\frac{s}{\omega_0}+\frac{\omega_0}{s}\right)}\$$

$$\H_0\$$ represents the gain at resonance. It is 20 dB in the below example:

Quote: "Is there a general form of transfer function (with peak frequency ωm and quality factor Q) relevant for any type of bandpass filter ?"

When you say "any type" - are you referring to higher order filters (n>2)?

• For a second order bandpass (lowest possible order) there is only one general form (see the formula in Mike`s answer). This form explicitely contains the midfrequency (peak) and the Q-value. Note that for this filter (n=2) the pole-quality factor Qp is identical to the filter-Q (fm/BW).

• For higher orders (n>2) different responses are possible (Cauer, Chebyshev,...) and it is not possible to derive the filter-Q (fm/BW) directly from the transfer function. Each pole pair has its own pole-Q which - of course - cannot be identical to the mentioned filter-Q.