Engee documentation

cheby2

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Calculation of Chebyshev filter type II.

Library

EngeeDSP

Syntax

Function call

  • b, a = cheby2(n, Rs, Ws) — designs a Chebyshev type II digital low-pass filter n`of the 1st order with a normalized boundary frequency of the delay band `Ws and attenuation in the delay band Rs dB from the peak bandwidth value. Function cheby2 returns the coefficients of the numerator and denominator of the filter transfer function.

  • b, a = cheby2(n, Rs, Ws, ftype) — designs a Chebyshev type II digital filter: low-pass, high-pass, bandpass or notch filter, depending on the value of the argument ftype and the number of elements Ws. The resulting designs of the bandpass and notch filters are of the following order 2n.

  • z, p, k = cheby2(_) — designs a Chebyshev type II digital filter and returns its zeros, poles, and gain. This syntax can include any input arguments from the previous options.

  • A, B, C, D = cheby2(_) — designs a Chebyshev type II digital filter and returns matrices that define its representation in the state space.

  • _ = cheby2(_, "s") — designs an analog Chebyshev type II filter using any of the input or output arguments in the previous syntaxes.

Arguments

Input arguments

# n — filter order

+ scalar

Details

The filter order, specified as an integer scalar, is less than or equal to 500. For band-pass and notch filters n represents half of the filter order.

Data types

Float64

# Rs — attenuation in the delay band in dB

+ positive scalar

Details

The attenuation in the delay band is relative to the peak value in the passband, set as a positive scalar in dB.

If the value is expressed in linear units, you can convert it to dB using the formula Rs .

Data types

Float64

# Ws — the limit frequency of the delay band
scalar | two-element vector

Details

The boundary frequency of the delay band, defined as a scalar or a two-element vector. The boundary frequency of the delay band is the frequency at which the amplitude-frequency response (frequency response) of the filter is equal to –Rs in dB. Large delay band attenuation values Rs they lead to increased bandwidth.

  • If Ws — a scalar, then cheby2 designs a low-pass or high-pass filter with a cut-off frequency Ws.

    If Ws — two-element vector [w1 w2], where w1 < w2 Then cheby2 designs a bandpass or notch filter with a lower cut-off frequency w1 and the upper boundary frequency w2.

  • For digital filters, the frequency limits of the delay band should be in the range of 0 before 1, where 1 corresponds to the Nyquist frequency — half of the sampling frequency or rad/countdown.

    For analog filters, the boundary frequencies of the delay band must be expressed in rad/s and can take any positive value.

Data types

Float64

# ftype — filter type

+ "low" | "bandpass" | "high" | "stop"

Details

The filter type is set as:

  • "low" — low-pass filter with a boundary frequency of the delay band Ws. This value is used by default for scalar Ws;

  • "high" — high-pass filter with a boundary frequency of the delay band Ws;

  • "bandpass" — bandpass filter 2n of the order if Ws — a two-element vector. This value is used by default when Ws set as a two-element vector;

  • "stop" — notch (blocking) filter 2n of the order if Ws — a two-element vector.

Data types

String

Output arguments

# b, a are the coefficients of the transfer function

+ string vectors

Details

The coefficients of the filter transfer function are returned as row vectors. With the specified filter order n the function returns b and a with r by counting where r=n+1 for low and high pass filters and r=2*n+1 for band-pass and notch filters.

The transfer function is expressed in terms of and :

  • for digital filters

  • for analog filters

Data types

Float64

# z, p, k — zeros, poles, and gain

+ column vectors and scalar

Details

The zeros, poles, and gain of the filter are returned as two vectors-columns and a scalar. With the specified filter order n the function returns z and p with r by counting where r=n for low and high pass filters and r=2*n for band-pass and notch filters.

The transfer function is expressed in terms of , and :

  • for digital filters

  • for analog filters

Data types

Float64

# A, B, C, D — representation of the filter in the state space

+ matrices

Details

The representation of the filter in the state space, returned as matrices. If r = n for low and high pass filters and r = 2n for band-pass and notch filters, then A This is the matrix r on r, B the matrix r on 1, C the matrix 1 on r, and D — 1 on 1.

The state space matrices relate the state vector , entrance and the exit by means of systems of equations:

  • for digital filters

  • for analog filters

Data types

Float64

Algorithms

Chebyshev type II filters are monotonous in the passband and have uniform pulsation in the delay band. Type II filters do not have as fast a drop in performance as type I filters, but they do not have ripples in the bandwidth.

Chebyshev filter type II cheby2 uses a five-step algorithm:

  • Finds the poles, zeros, and gain of the analog low-pass prototype.

  • Converts poles, zeros, and gain into a state space.

  • If necessary, it uses a state space transformation to convert a low-pass filter into a high-pass filter, bandpass filter, or notch filter with the required frequency constraints.

  • To design digital filters, it converts an analog filter to a digital one by means of a bilinear frequency pre-distortion conversion. Fine frequency tuning allows analog and digital filters to have the same frequency response amplitude Ws or w1 and w2.

  • If necessary, it converts the state space filter back into a transfer function or a zero-pole-gain form.

Literature

  1. Lyons, Richard G. Understanding Digital Signal Processing. Upper Saddle River, NJ: Prentice Hall, 2004.