Bessel filter: explicit calculus of the order of the filter

Question

As it is well known Butterworth and Chebyshev filters have explicit ways of calculating the order, derived by its polynomials.

For instance, for Butterworth filters:

$$ n \geq \displaystyle \frac{\log\left[\varepsilon^{-2}\left(10^{0.1A_S}-1\right)\right]}{ 2 \log \Omega_S} $$

and for Chebyshev filters

$$ n \geq \displaystyle \frac{\cosh^{-1}\left[\varepsilon^{-2}\left(10^{0.1A_S}-1\right)\right]}{\cosh^{-1} \Omega_S}$$

where, for both cases, \$ \varepsilon= \sqrt{10^{0.1A_P}-1} \$. To briefly explain the nomenclature used here: \$ A_P \$ is the passband ripple, \$ A_S \$ is the stopband attenuation and \$\Omega_S\$ is the normalized stopband frequency (the filters are normalized to a passband frequency of 1).

I was wondering if a similar approach can be taken for Bessel filters. My attempt was to consider the attenuation characteristic of the filters as:

$$A(\Omega)=10 \log \left[ \varepsilon^2C_n^2(\Omega) \right]$$

where \$C_n\$ is the \$n\$th order Bessel polynomial. I have taken a look at the Wikipedia for the Bessel polynomials, but they seem to have a complicated explicit formula, instead of a logarithm or hyperbolic cosine. Any idea on how can I predict the necessary order for a Bessel filter?

Answer 1

I think that for Bessel filters a slightly different approach is necessary.

Why does one decide to use the Bessel approximation? Answer: Because the timely properties (delay) play an important role and not only the frequency properties (3dB cut-off frequency wc). Therefore, another "frequency limit" wo is defined by the basic group delay at very low frequencies tau(w=0)=tau_o:

wo=1/tau_o.

Example: For the 2nd order Bessel low pass we have wo=wpQp=wp/SQRT(3).

Thus it can be shown that for w=wc the group delay is then tau(w=wc)=0.923 * tau_o (with 7.7% deviation from tau_o). For the 2nd order Bessel function the relation wc/wo=1.3616 is also valid.

For higher filter orders (n>2) an approximation can be given:

wc * tau_o=wc/wo~SQRT[ln2 * (2n-1)]

The maximum error is less than 5% and decreases with increasing degree n. This can be used to estimate the minimum required filter order:

n_min=(1/2)+(wc/wo)²/(2 * ln2).

The value for the basic group delay tau_o=1/wo must then be determined from the permissible deviation (in %) between tau_o and the actual delay at the frequency limit w=wc (tables available).