# What bandpass filter design will yield the shortest impulse response?

Designing a simple 2nd order IIR bandpass Buterworth filter with a centre frequency of 500 Hz and a 1 octave bandwidth gives me the following frequency response ...

Now, if I take the impulse response and normalise it and convert it to dB, we can observe the decay of the impulse response.

The decay of the impulse response is approximately linear with time when plotted on this scale, allowing us to define a decay time statistic (just like in room acoustics where you can define reverb time). For the impulse response of this filter to drop below 30 dB, it takes about 11 ms.

We are trying to minimize this decay time keeping the following constant:

• -3 dB bandwidth
• Filter order

I am happy to accept (within limits) passband and stopband ripple, and/or a compromise on the steepness of the transition band to achieve this. Can anyone suggest a method for filtering with the shortest possible impulse response duration as defined above?

• Please include sampling frequency, to give those 11ms some meaning. – Juancho May 31 '12 at 14:13
• Poles in the filter will yield exponentially-decaying terms in the impulse response, which when plotted on a log scale gives a linear decay, as you showed. The rate of decay is related to the poles' distance to the unit circle; the closer they are, the slower the decay. The steepness of the transition band is also related to how close the poles are to the unit circle. I don't know of any design techniques off hand that would allow you to prioritize this particular characteristic. – Jason R May 31 '12 at 14:16
• @ Juancho Sample rate was omitted as I thought it was completely irrelevant: using 5 kHz or 500 kHz does not change the decay rate of the impulse response. I am targeting 44.1 kHz if you are curious. Thanks for looking :) – learnvst May 31 '12 at 14:19
• @JimClay yes you can I'm sure, but I want to keep computational cost very low. To efficiently use Fir I'd need to use an fft based technique, and this would introduce latency to the algorithm while the FFT buffer is filled with samples. Yes/no? – learnvst May 31 '12 at 21:28
• @JimClay why do the laws of physics always halt my plans for world domination! Grumble grumble – learnvst Jun 1 '12 at 14:51