Important Filters

Chebyshev filter:

               Chebyshev filters are analog or digital filters having a steeper roll-off and more passband ripple (In type I or Chebyshev) or stopband ripple (In type II or Inverse Chebyshev) than Butterworth filters. Chebyshev filters have the property that they minimize the error between the idealized and the actual filter characteristic over the range of the filter, but with ripples in the passband. This type of filter is named so because its mathematical characteristics are derived from Chebyshev polynomials.

Butterworth filter:

                 The Butterworth filter is a type of signal processing filter designed to have a flat(i.e no ripples in the passband and rolls off towards zero in the stopband) frequency response in the passband. It is also referred to as a maximally flat magnitude filter.

                  Compared with a Chebyshev Type I/Type II filter or an elliptic filter, the Butterworth filter has a slower roll-off, and thus will require a higher order to implement a particular stopband specification, but Butterworth filters have a more linear phase response in the pass-band than Chebyshev Type I/Type II and elliptic filters can achieve.

Elliptic filter/Caur filter:

                An elliptic filter (also known as a Cauer filter) is a signal processing filter with equalized ripple (equiripple) behavior in both the passband and the stopband. The amount of ripple in each band is independently adjustable, and no other filter of equal order can have a faster transition in gain between the passband and the stopband, for the given values of ripple (whether the ripple is equalized or not). Alternatively, one may give up the ability to independently adjust the passband and stopband ripple, and instead design a filter which is maximally insensitive to component variations.

            As the ripple in the stopband approaches zero, the filter becomes a type I Chebyshev filter. As the ripple in the passband approaches zero, the filter becomes a type II Chebyshev filter and finally, as both ripple values approach zero, the filter becomes a Butterworth filter.

Gaussian filter:

                 In electronics and signal processing, a Gaussian filter is a filter whose impulse response is a Gaussian function (or an approximation to it). Gaussian filters have the properties of having no overshoot to a step function input while minimizing the rise and fall time. This behavior is closely connected to the fact that the Gaussian filter has the minimum possible group delay.Gaussian filter is considered the ideal time domain filter, just as the sinc is the ideal frequency domain filter.These properties are important in areas such as oscilloscopes and digital telecommunication systems.

           Mathematically, a Gaussian filter modifies the input signal by convolution with a Gaussian function; this transformation is also known as the Weierstrass.

All-pass filter:

               An all-pass filter is a signal processing filter that passes all frequencies equally, but changes the phase relationship between various frequencies. It does this by varying its propagation delay with frequency. Generally, the filter is described by the frequency at which the phase shift crosses 90° (i.e., when the input and output signals go into quadrature --when there is a quarter wavelength of delay between them).They are generally used to compensate for other undesired phase shifts that arise in the system, or for mixing with an unshifted version of the original to implement a notch comb filter.

The operational amplifier circuit shown in Figure 1 implements an active all-pass filter with the transfer function

                                                 

which has one pole at -1/RC and one zero at 1/RC (i.e., they are reflections of each other across the imaginary axis of the complex plane). Themagnitude and phase of H(iω) for some angular frequency ω are

                 

As expected, the filter has unity-gain magnitude for all ω. The filter introduces a different delay at each frequency and reaches input-to-output quadratureat ω=1/RC (i.e., phase shift is 90 degrees).

• At high frequencies, the capacitor is a short circuit, thereby creating a unity-gain voltage buffer (i.e., no phase shift).
• At low frequencies and DC, the capacitor is an open circuit and the circuit is an inverting amplifier (i.e., 180 degree phase shift) with unity gain.
• At the corner frequency ω=1/RC of the high-pass filter (i.e., when input frequency is 1/(2πRC)), the circuit introduces a 90 degree shift (i.e., output is in quadrature with input; it is delayed by a quarter wavelength).

In fact, the phase shift of the all-pass filter is double the phase shift of the high-pass filter at its non-inverting input.

APF Using Latice Topology:

pole zero plot of APF:(Alternate poles and zeros)