Chebyshev filter

The Chebyshev filter offers higher attenuation and a steeper roll-off near the cutoff frequency than the Butterworth filter. There is a tradeoff to achieve the higher attenuation. The cost of utilizing a Chebyshev filter is higher values of Q, which leads to difficulties in hardware realization, and nonlinear phase characteristics, which can result in difficulties in predicting circuit performance. 

Active filters utilize an operational amplifier as part of the circuit and allow the design of any RLC filter without the need of an inductance. Filters of this type include Buttersworth and Chebyshev filters. 

To investigate the efficiency of the DRP-FDTD method, its phase error per wavelength for a Chebyshev filtering process is examined in Figure 2.11(a). As can be observed,... 

FIGURE 2.11 (a) Phase error per wavelength of Chebyshev filtering for the 3-D DRP schemes with different pc. (b) Maximum phase error over the largest wavelength via diverse Ap and pc = 0.15... 

Using Chebyshev-Filtered Subspace Iteration and Windowing Methods to Solve the Kohn-Sham Problem... 

Using Chebyshev-Filtered Subspace Iteration and Windowing Methods.. 

After self-consistency is reached, the Chebyshev filtered subspace includes the eigensubspace corresponding to occupied states. ExpUcit eigenvectors can be readily obtained by a Rayleigh-Ritz refinement [40] (also called subspace rotation) step. 

The Chebyshev filtering in Step 3 costs 0(s N/p) flops. The discretized Hamiltonian is sparse and each matrix-vector product on one processor costs 0 N/p) flops. Step 3 requires m s matrix-vector products, at a total cost of 0(s m N/ p) where the degree w of the polynomial is small (typically between 8 and 20). 

Algorithm 6.3 Chebyshev-filtered Subspace (CheFS) method ... 

The first step diagonalization by the block Chebyshev-Davidson method, together with the Chebyshev-filtered subspace method (Algorithm 6.3), enabled us to... 

A simple rule of thumb which allows one estimation of the required order for a given filtering need has already been presented. However, it should be stressed that this is only approximate and that the precise roll-off behavior of a filter can be improved through various methods, such as, for example, introducing ripple into the filter band. The Chebyshev filter is an example of this, and of the three designs, it presents the best roll-off i.e., for a given filter order the transition band is shortest in this design. 

Passband ripple The passband of a filter should ideally be flat. As we have seen, the Chebyshev filter sacrifices a flat passband for faster roll-off. The Butterworth filter, on the other hand, has been designed to have the shortest possible transition band while still maintaining a flat passband i.e., there is no ripple. The Bessel filter also has a flat passband however, it has the worst roll-off of the three designs. 

The principal advantage of the Butterworth alignment is the flatness of the ampHtude response within the pass band. The principal disadvantage is that the rolloff is not as steep in the vicinity of cutoff as that of some other filter designs of the same complexity (number of poles). The Chebyshev filter has a frequency response with a steeper rolloff in the vicinity of cutoff than that of the Butterworth filter, but ripples are present in the pass band. The Chebyshev filter is sometimes called an equiripple filter because the ripples have the same peak-to-peak amplitude throughout the pass band. 

The mathematicalbasis for frequency mapping is omitted here however, the technique involves replacing the LP transfer function variable (s or ) with the mapped variable, as presented in Table 4.13. The mapped amplitude response for a Butterworth filter or a Chebyshev filter is obtained by performing the appropriate operation from column two of Table 4.13 on Eq. (4.60) or (4.61), respectively. The mapped phase response... 

FIGURE 8.107 Magnitude responses of low-pass analog filters (a) Butterworth filter, (b) Chebyshev filter, (c) inverse Chebyshev filter. 

The Nth-order Chebyshev filter has a transfer function given by... 

For inverse Chebyshev filters, the equiripples are inside the stop band, as opposed to the pass band in the case of Chebyshev filters. The magnitude response square of the inverse Chebyshev filter is... 

In comparison, the Butterworth filter requires a higher order than both types of Chebyshev filters to satisfy the same specification. There is another type of filters called elliptic filters (Cauer filters) that have equiripples in the pass band as well as in the stop band. Because of the lengthy expressions, this type of filters is not given here (see the references). The Butterworth filter and the inverse Chebyshev filter have better (closer to linear) phase characteristics in the pass band than Chebyshev and elliptic filters. Elliptic filters require a smaller order than Chebyshev filters to satisfy the same specification. 

Chebyshev filter as the prototype filter. Following the same design process as the first example, we have... 

Obtain the analog LPF transfer function. From Eq. (8.39) and with N = 4, we find the transfer function of the analog Chebyshev filter as... 

After initial testing on small systems, Chelikowsky s group extended their real-space code (now called PARSEC) for a wide range of challenging applications.The applications include quantum dots, semiconductors, nanowires, spin polarization, and molecular dynamics to determine photoelectron spectra, metal clusters, and time-dependent DFT (TDDFT) calculations for excited-state properties. PARSEC calculations have been performed on systems with more than 10,000 atoms. The PARSEC code does not utilize MG methods but rather employs Chebyshev-filtered subspace acceleration and other efficient techniques during the iterative solution process. When possible, symmetries may be exploited to reduce the numbers of atoms treated explicitly. 

We complete this section with a listing of other algorithmic developments in real-space electronic structure. As mentioned above, the PARSEC code has incorporated alternative techniques for accelerating the solution of the eigenvalue problem based on Chebyshev-filtered subspace methods, thus circumventing the need for multiscale methods. Jordan and Mazziotti have developed new spectral difference methods for real-space electronic structure that can yield the same accuracies as the FD representation with... 

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