The last comment I made was regarding those perfect rectangular filters that we always describe with such reverence. But as I said, they don’t exist. Taking a receiver as an example, the filter amplitude versus frequency characteristics are measured by sweeping a narrowband signal through the passband frequency range of the circuit. Typically the passband width is given at the – 6 dB and – 60 dB points, and a prefect rectangular filer would have the same bandwidth at both points. This filter would define a 1:1 shape factor. A close examination of the amplitude versus frequency data on typical filters will show that different circuit types have different shape factors.
Even though the 6 dB (or 3 dB) narrowband bandwidths of various circuits could all be equal, their circuit response shape might be square, trapezoidal, triangular, cosine, Gaussian, or whatever; and the shape determines the broadband response. This makes the broadband bandwidth (also known as the impulse bandwidth) wider than the narrowband bandwidth. A lot of filters are Gaussian-like and a good rule of thumb for a Gaussian filter is BB BW ≈ 1.5 * NB BW. Expressed in dB this is approximately the – 7dB bandwidth. There are a number of ways to measure the impulse bandwidth, and either ANSI/IEEE Std 376-1975 or SAE ARP 1267 can be used. I feel that the procedure from ARP 1267 is easier to read. As an alternative, just plug your trusty calibrated impulse generator into the front end—through an attenuator, of course—and measure it directly. This provides the factor necessary to correct for the broadband signal energy filling the entire passband of the filter.
While I’m on my high horse about bandwidths, let me tell you about one of my peeves. I have many. This one is not one of my pets, but it does annoy me never the less. The EMC community frequently uses graphical Fourier analysis to convert signals from the time domain to the frequency domain. The approach is fast and easy. First, we make a simplifying assumption that the pulse is trapezoidal with amplitude A, pulse width D, and rise-time tr. Then, we translate that to an amplitude versus frequency envelope in the frequency domain, stating that the amplitude will be no greater than 2AD from DC to F1= 1/(pi D), decrease at a 20 dB/decade rate from F1= 1/(pi D) to F2= 1/(pi tr), and then decrease at a 40 dB/decade rate above F2= 1/(pi tr).
The frequency F2 defines the bandwidth (BW) of the trapezoidal pulse which is related to the rise-time giving BW = 0.318 / tr. I have often made that statement and, then, had someone tell me: “Oh no! That’s not the BW! The BW = 0.35 / tr.” I admit that this relationship is a very popular definition of pulse bandwidth; but in a real circuit, the broadband BW is a function of the filter shape (read that as Q) and the 0.35 / tr relationship only applies to a 1st order RC low pass filter. Here’s a table that provides a comparison of the step rise-time bandwidth of different circuit types.
Pure trapezoidal wave* BW = 1 / (π * tr ) ≈ 0.318 / tr
Pure Gaussian BW ≈ 0.332 / tr
10- 90% Gaussian BW ≈ 0.34 / tr
RC LPF (1st order) BW ≈ 0.35 / tr
Bessel-Thompson BW ≈ 0.36 / tr
Butterworth BW ≈ 0.49 / tr
Chebychev BW ≈ 0.60 / tr
* This is a mathematical construct and not related to circuit characteristics
