How does linear phase shift of bessel filter avoid distortion of nonsinusoidal inputs?

How does linear phase shift help here? I'm familiar with some fourier series but I don't get my textbook explanation. Any help?

A linear phase shift (as produced by a Bessel filter) means a constant time delay for all the main frequency components of a signal. So, for instance, if the filter were a 2nd order low-pass type having a 3.01 dB cut-off point at (say) 12.64 kHz: -

• 159.2 Hz would see a phase shift of about 0.99 degrees
• 1.592 kHz would see a phase shift of about 9.86 degrees
• 7.977 kHz would see a phase shift of about 49.02 degrees

Then, if you factored in the time shift you would get: -

• 159.2 Hz has a time shift of 17.27 us
• 1.592 kHz has a time shift of 17.20 us
• 7.977 kHz has a time shift of 17.07 us

In other words, a 2nd order Bessel filter is good for constant delay up to around 50% of the cut-off frequency AND, that will largely maintain good shape with maybe about 0.5% overshoot in the time domain from a step change.

You can use this on-line tool to check out the numbers. Manuipulate the R value to give a \$\zeta\$ of 0.86. I used R = 172 ohms knowing that a 2nd order Bessel filter has a damping ratio of about 0.86.

When we require that a frequency-dependent block (filter) should only delay any signal by the time T - without changing the form of the signal the output-to-input relation is:

v2(t)=v1(t-T) .

Applying the LAPLACE transformation to this equation the ratio (transfer function) is:

V2(s)/V1(s)=H(s)=exp(-sT).

That means: (1) Magnitude |H(w)|=1 and (2) phase function phi(w)=-wT.

Hence, we require a phase function that is a LINEAR function of the frequency.

Definition: The group delay D is defined as D=-d(phi)/d(w). We see that for a linear phase function the group delay is D=constant.

However, it is to be mentioned that a real lowpass can never realize a transfer function H(s)=exp(-sT). Hence, we need an approximation "as good as possible" within a limited frequency band. This requirement leads to the well-known Thomson-Bessel approximation.

In most cases, the cut-off frequency for a Bessel filter is defined in the time domain (and NOT in the frequency domain, as it is normal for all other filter types). The cut-off (end of the passband) is defined at a frequency where the deviation of the (nearly constant) group delay D has a certain percentage of the value for low frequencies - depending on the particular application.

Remember: The BUTTERWORTH approximation has a "maximally flat magnitude" within the passband - the BESSEL approximation is derived from a "maximally flat group delay" requirement.

Consider a cable. That has constant time delay. This is the same as linear phase with frequency. Does a cable distort your signals?