Chebyshev filter | EPFL Graph Search

Chebyshev filters are analog or digital filters that have a steeper roll-off than Butterworth filters, and have either passband ripple (type I) or stopband ripple (type II). Chebyshev filters have the property that they minimize the error between the idealized and the actual filter characteristic over the operating frequency range of the filter, but they achieve this with ripples in the passband. This type of filter is named after Pafnuty Chebyshev because its mathematical characteristics are derived from Chebyshev polynomials. Type I Chebyshev filters are usually referred to as "Chebyshev filters", while type II filters are usually called "inverse Chebyshev filters". Because of the passband ripple inherent in Chebyshev filters, filters with a smoother response in the passband but a more irregular response in the stopband are preferred for certain applications. Type I Chebyshev filters are the most common types of Chebyshev filters. The gain (or amplitude) response, , as a function of angular frequency of the th-order low-pass filter is equal to the absolute value of the transfer function evaluated at : where is the ripple factor, is the cutoff frequency and is a Chebyshev polynomial of the th order. The passband exhibits equiripple behavior, with the ripple determined by the ripple factor . In the passband, the Chebyshev polynomial alternates between -1 and 1 so the filter gain alternate between maxima at and minima at . The ripple factor ε is thus related to the passband ripple δ in decibels by: At the cutoff frequency the gain again has the value but continues to drop into the stopband as the frequency increases. This behavior is shown in the diagram on the right. The common practice of defining the cutoff frequency at −3 dB is usually not applied to Chebyshev filters; instead the cutoff is taken as the point at which the gain falls to the value of the ripple for the final time. The 3 dB frequency is related to by: The order of a Chebyshev filter is equal to the number of reactive components (for example, inductors) needed to realize the filter using analog electronics.

Adaptive optics in speckle interferometry and in super-resolution

Phanindra Narayan Gundu

This thesis describes innovative techniques for reducing speckle noise and improving the intensity profile of the speckle correlation fringes. The methods are based on reducing the range of the modulation intensity values of the speckle interference pattern. After the fringe pattern is corrected adaptively at each pixel, a simple morphological filtering of the fringes is sufficient to obtain smoothed fringes. The concepts are presented both analytically and by simulation by using computer-generated speckle patterns and experimental verifications are performed wherever possible. A new generalized method for designing continuous amplitude-only pupil filters for transverse superresolution using a nonlinear programming method is also presented. The thesis emphasises the principal advantage of amplitude-only filters over their phase-only counterparts, that the side lobe intensities can be highly reduced along with the spot size. A quantitative comparison with continuous phase-only filters as well as the two-zone binary phase filter is shown with respect to spot size and ratio of side lobe to central peak intensity. The work is extended to combine the advantages of amplitude and phase filters in one complex filter that performs better than either phase or amplitude filters designed so far. The performance here refers to having a smaller spot size along with higher peak to side lobe intensity ratio. Using numerical simulation the limitations of phase and amplitude filters are assessed. The experimental verification of the designed combination filter is performed with two LCD spatial light modulators used for displaying separately the phase and amplitude part of the filter. Results obtained from this setup confirm the simulation. Finally, the effect of optical superresolution on speckle correlations is studied. Simulations reveal that using a lateral super-resolution pupil filter more than twice the out of plane correlation length of the clear pupil can be achieved. This means that the measurement range in speckle correlation measurements doubles. To verify the correlation length an experiment is performed using a liquid crystal (LCD) spatial light modulator as a programmable superresolution filter. The results corroborate the simulation.

EPFL2007

Adaptive optics in speckle interferometry and in super-resolution

Phanindra Narayan Gundu

EPFL2007