Digital Filters Design for Signal and Image Processing

4.7. Linear phase filters (maximally flat delay or MFD): Bessel and Thomson filters

4.7.1. Reminders on continuous linear phase filters

In some applications, we need a filter with a transfer function whose phase varies linearly with the frequency in a specific range.

Let us assume a continuous-time filter represented by its transfer function H(j2πf) = |H(j2πf)| exp(j(f)). If we have been looking at |H(j2πf)|, we must also take the characteristics of (f) into account.

To do that, we look at an input signal that is made of two sinusoidal components:

The filtered signal then is written as:

or

Now, if we carry out a limited development of , we get:

Taking equation (4.68) into account, equation (4.67) is written as:

In order to avoid a frequency distortion, it is necessary that be cancelled. ...