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?


1 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:


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).


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