cheby1

Calculation of a type I Chebyshev filter.

Library

EngeeDSP

Syntax

Function call

b, a = cheby1(n, Rp, Wp) — designs a Chebyshev type I digital low-pass filter n-th order with a normalized bandwidth boundary frequency Wp and the size of the irregularities in the bandwidth Rp dB. Function cheby1 returns the coefficients of the numerator and denominator of the filter transfer function.

b, a = cheby1(n, Rp, Wp, ftype) — designs a Chebyshev type I digital filter: low-pass, high-pass, bandpass or notch filter, depending on the value of the argument ftype and the number of elements Wp. The resulting designs of the bandpass and notch filters have the following order 2n.

z, p, k = cheby1(_; nout=3) — designs a type I Chebyshev digital filter and returns its zeros, poles, and gain. This syntax can include any input arguments from the previous options.

A, B, C, D = cheby1(_; nout=4) — designs a type I Chebyshev digital filter and returns matrices that define its representation in the state space.

_ = cheby1(_, "s"; nout) — designs a Type I Chebyshev analog filter using any of the input or output arguments in the previous syntaxes. Argument nout shows how many output arguments the function will return. If the argument is nout omitted, the function will calculate only for the two output arguments.

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.

Типы данных

Float64

# Rp is the size of the ripples in the bandwidth in dB

+ positive scalar

Details

The size of the ripples in the bandwidth, set as a positive scalar in dB.

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

Типы данных

Float64

# Wp — bandwidth limit frequency

+ scalar | two-element vector

Details

The bandwidth boundary frequency, specified as a scalar or two-element vector. The bandwidth cutoff frequency is the frequency at which the amplitude-frequency response of the filter is −Rp in dB. Lower frequency response ripple values in the bandwidth, Rp they lead to increased bandwidth.

If Wp — a scalar, then cheby1 designs a low-pass or high-pass filter with a cut-off frequency Wp.

If Wp — two-element vector [w1 w2], where w1 < w2 Then cheby1 designs a bandpass or notch filter with a lower cut-off frequency w1 and the upper boundary frequency w2.
For digital filters, the bandwidth boundary frequencies 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 bandwidth boundary frequencies must be expressed in rad/s and can take any positive value.

Типы данных

Float64

# ftype — filter type

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

Details

The filter type specified as:

"low" — low-pass filter with bandwidth boundary frequency Wp. This value is used by default for scalar Wp;
"high" — high-pass filter with bandwidth boundary frequency Wp;
"bandpass" — bandpass filter 2n of the order if Wp — a two-element vector. This value is used by default when Wp set as a two-element vector;
"stop" — notch (blocking) filter 2n of the order if Wp — a two-element vector.

Типы данных

String

Name-value input arguments

# nout — number of output arguments

+ 2 (by default) | 3 | 4

Details

The number of output arguments, set as 2, 3 or 4. If the argument is nout not specified, the function returns two output arguments by default.

Output arguments

# b, a are the coefficients of the transfer function

+ string vectors

Details

Coefficients of the filter transfer function 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

Типы данных

Float64

# z, p, k — zeros, poles, and gain

+ column vectors and scalar

Details

Zeros, poles, and filter gain, 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

Типы данных

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 bandpass 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

Типы данных

Float64

Examples

Chebyshev type I low-pass filter

Details

Let’s design a Chebyshev type I low-pass filter 6-th order with ripples in the bandwidth 10 dB and bandwidth cutoff frequency 300 Hz, which corresponds to 0.6π rad/count for data sampled with frequency 1000 Hz. Let’s plot the amplitude-frequency and phase characteristics. We use it to filter a random signal with a frequency 1000 counts.

import EngeeDSP.Functions: cheby1, freqz, randn, filter

fc = 300
fs = 1000

b, a = cheby1(6,10,fc/(fs/2))

freqz(b,a,out=:plot)

cheby1 1

dataIn = randn(1000,1)
dataOut = filter(b,a,dataIn)

Chebyshev Type I barrier filter

Details

Let’s design a type I Chebyshev barrier filter 3-th order with normalized boundary frequencies 0.2π and 0.6π rad/countdown and bandwidth ripples 5 dB. Let’s plot the amplitude-frequency and phase characteristics. We use it to filter a random signal.

import EngeeDSP.Functions: cheby1, freqz, randn, filter

b,a = cheby1(3,5,[0.2 0.6],"stop")

freqz(b,a,out=:plot)

cheby1 2

dataIn = randn(1000,1)
dataOut = filter(b,a,dataIn)

Chebyshev type I high-pass filter

Details

Let’s design a Chebyshev type I high-pass filter 9-th order with bandwidth cutoff frequency 300 Hz, which is for data sampled with frequency 1000 Hz, corresponds to 0.6π rad/countdown. Setting the ripple in the bandwidth 0.5 dB. Convert the zeros, poles, and gain into second-order sections. Let’s plot the amplitude-frequency and phase characteristics.

import EngeeDSP.Functions: cheby1, zp2sos

z,p,k = cheby1(9,0.5,300/500,"high"; nout=3)
sos = zp2sos(z,p,k)

freqz(sos,out=:plot)

cheby1 3

Chebyshev type I bandpass filter

Details

Let’s design a Chebyshev type I bandpass filter 10-th order with lower bandwidth 400 Hz and upper bandwidth 500 Hz. Set the ripple in the bandwidth 3 dB and sampling rate 1500 Hz. We use a representation in the state space. We transform the representation in the state space into the form of sections of the second order. Let’s plot the amplitude-frequency and phase characteristics.

import EngeeDSP.Functions: cheby1, ss2sos
fs = 1500

A,B,C,D = cheby1(10,3,[400 500]/(fs/2); nout=4)
sos = ss2sos(A,B,C,D)[1]

freqz(sos,out=:plot)

cheby1 4

Algorithms

Chebyshev type I filters have a uniform frequency response pulsation in the passband and are monotonous in the delay band. Type I filters have a faster frequency response drop than type II filters, but at the expense of a greater deviation from unity in the bandwidth.

Chebyshev filter type I cheby1 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 Wp or w1 and w2.
If necessary, it converts the state space filter back into a transfer function or a zero-pole-gain form.

Literature

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