Filters with gaussian magnitude approximation
In pulse communication systems there is a demand for filters whose impulse responses have the following properties:
- No ringing and overshoot;
- Symmetry about the time for which the response is a maximum.
A filter that satisfies the above conditions is called a Gaussian filter. The three most common filter types with widely available design tables and curves which approach the ideal Gaussian filter are:
- The gaussian magnitude filter;
- The maximally flat group delay filter;
- The equiriple group-delay filters.
Although, the delay performance of a gaussian filter is worse than the Bessel approximation, a gaussian filter has a better step response. By definition, a gaussian function has the form:
(A.3.1)
where T is the mean value and s the standard deviation. If the impulse response of a filter is of this form, then the filter will be said to be gaussian. This impulse response has no overshoot. If we denote w_{0}^{2}=2/s^{2} , then the ideal gaussian magnitude shape derived from the Fourier transform of g(t) can be written as:
(A.3.2)
The frequency w_{0} is a normalizing frequency and it can be related to the -3dB point as:
(A.3.3)
The magnitude of the gaussian transfer and the group-delay response are shown below.
Fig.A.3.1: Ideal gaussian transfer and the group delay |
When w=w_{0}, the value of the magnitude is e=2.71828 and the relative attenuation is 1Np or 8.68dB. It can be shown that a gaussian magnitude shape is unrealizable but approximations of this transfer can be obtained by using the following series expansion:
(A.3.4)
An nth-order approximation consists of the first 2n powers in the series. The attenuation of the gaussian function approximated up to n=6 is shown in fig. A.3.2.
Fig.A.3.2: Attenuation performance of gaussian approximations |
The approximation of the Gaussian function with a finite number of network elements can be made better by increasing the number of network elements. However, it is possible to approximate the gaussian function up to a certain level. Gaussian-to-6dB and gaussian-to-12 dB approximations will approximate the transfer up to -6dB point and -12dB point respectively. To show the difference between those approximations, the frequency transfer of a 5^{th} order gaussian, gaussian-to-6dB and gaussian-to-12dB transfer is depicted in fig. A.3.3. With these gaussian approximations it is possible to achieve a higher stopband attenuation with the same number of network elements at the expenses of a small decrease of performance in the time domain response and group delay.
Fig.A.3.3: Gaussian, gaussian-to-6dB and gaussian-to-12dB transfer |
The corresponding group-delay and step response of the approximations shown above are illustrated in fig.A.3.4. This explains the degradation of the group delay and correspondingly, the degradation of the step response when approximating the gaussian transfer.
Fig.A.3.4: Group delay and step response of gaussian-to-6(12) dB |