144 Chapter 7 referred to as staggered tuning. It is impossible to have a first-order bandpass or notch (band-reject) response. 7.8 Filter Cutoff Frequencies and Characteristic Frequencies The cutoff frequency of a lowpass or highpass filter is a measure of where in the spectrum its roll-off starts. In many cases it is defined as the frequency at which the gain is down by -3 dB (= 1/2). The word "cutoff" is perhaps unfortunate because it seems to imply a frequency response that drops suddenly, like falling off a cliff. So-called "brickwall" filters with very fast roll-offs do exist, but the filters used in active crossover design are rarely higher than fourth-order and the start of the roll-off is actually quite gentle. However, nobody's going to change the word now. Filter types like the Butterworth and Bessel characteristics have their cutoff frequencies defined at the -3 dB point, whereas the Linkwitz-Riley filter may be defined at the -3 dB or the -6 dB point. To an extent the choice of attenuation is arbitrary; it would be quite possible to design these filters with cutoff defined at -4.5 dB or whatever, but there would be no point at all in doing so. For the Butterworth filter in particular, using -3 dB makes the mathematics very simple. Other filters like the Chebyshev type, which has amplitude ripples in the passband, have their cutoff defined as the point where the gain passes through 0 dB for the last time as the frequency increases (in the lowpass case) and the steady roll-off begins. The characteristic frequency of a lowpass or highpass filter is not at all the same thing as its cutoff frequency, and the two terms should not be confused. Taking lowpass filters as an example, the amplitude response off all second-order filters will eventually become a 12 dB/ octave straight line heading downwards. If you extend that line upwards and to the left it will eventually cut the 0 dB gain line, and the point where it cuts it is the characteristic frequency. You can design a Bessel filter and a Chebshev filter so that their responses converge on the same 12 dB/octave line and so have the same characteristic frequency, but they will have different cutoff frequencies. Characteristic frequencies are not widely used in crossover design but they are useful when dealing with state variable filters; this is discussed in an excellent article by Ramkumar Ramaswamy [6]. This book uses cutoff frequencies exclusively. 7.9 First-Order Filters A first-order filter is just a single RC time-constant, with an ultimate roll-off of 6 dB/octave. Some writers seem to regard first-order filters as having a fixed Q of 0.5, though quite how this might be helpful is unclear. This is absolutely NOT the same thing as a second-order