Types of practical filters Filter specifications Tradeoffs...

Practical Analog Filters Overview

• Types of practical filters

• Filter specifications

• Tradeoffs

• Many examples

J. McNames Portland State University ECE 222 Practical Analog Filters Ver. 1.04 1

Ideal Filters

1Lowpass

1Highpass

1Bandpass

1Bandstop

1Notch

ω ωcωcωc

ωc1ωc1 ωc2ωc2

• There are five ideal filters

• Lowpass filters pass low frequencies: ω < ωc

• Highpass filters pass high frequencies: ω > ωc

• Bandpass filters pass a range of frequencies: ωc1 < ω < ωc2

• Bandstop filters pass two ranges: ω < ωc1 and ω > ωc2

• Notch filters pass all frequencies except ω ∼= ωc

Ideal Filters Comments

1Lowpass

1Highpass

1Bandpass

1Bandstop

1Notch

ω ωcωcωc

ωc1ωc1 ωc2ωc2

• Phase is not shown

• ωc is called the cutoff frequency

• Generally, the ideal phase is 0◦ for all frequencies

• Can not build ideal filters in practice

• Real filters appear as rounded versions of ideal filters

• Most LTI systems can be thought of as non-ideal filters

Practical Filters

• Practical filters are usually designed to meet a set of specifications

• Lowpass and highpass filters usually have the followingrequirements

– Passband range

– Stopband range

– Maximum ripple in the passband

– Minimum attenuation in the stopband

• If we know the specifications, we can ask MATLAB to generatethe filter for us

• There are four popular types of standard filters

– Butterworth

– Chebyshev Type I

– Chebyshev Type II

– Elliptic

Practical Filter Tradeoffs

• Butterworth

- Highest order H(s)+ No passband or stopband ripple

• Chebyshev Type I

+ No stopband ripple

• Chebyshev Type II

+ No passband ripple

• Elliptic

+ Lowest order H(s)- Passband and stopband ripple

Example 1: Lowpass Filter Specifications

Design a lowpass filter that meets the following specifications:

• The passband ripple is no more than 0.4455 dB(0.95 ≤ |H(jω)| ≤ 1)

• The stopband attenuation is at least 26.02 dB (|H(jω)| ≤ 0.05)

• The passband ranges from 0–450 rad/s

• The stopband ranges from 550–∞ rad/s

Plot the magnitude of the resulting transfer function on a linear-linearplot, the Bode magnitude plot, the pole-zero plot, the impulseresponse, and the step response. Try the Butterworth, Chebyshev I,Chebyshev II, and elliptic filters.

The MATLAB code is posted on the course web site.

Example 2: Butterworth Lowpass

0 100 200 300 400 500 600 700 800 900 10000

Butterworth Lowpass Filter Transfer Function Order: 21

Frequency (rad/sec)

Butterworth Lowpass Filter Transfer Function Order:21

−1000 −500 0 500

−400

−300

−200

−100

Real Axis

Butterworth Lowpass Filter Pole−Zero Plot Order:21

0 0.05 0.1 0.15 0.2 0.25−100

Time (s)

Butterworth Lowpass Filter Impulse Response Order:21

0 0.05 0.1 0.15 0.2 0.250

Time (s)

Butterworth Lowpass Filter Step Response Order:21

Example 2: Chebyshev-I Lowpass

0 100 200 300 400 500 600 700 800 900 10000

Chebyshev−I Lowpass Filter Transfer Function Order: 8

Frequency (rad/sec)

Chebyshev−I Lowpass Filter Transfer Function Order:8

−800 −600 −400 −200 0 200 400 600

−400

−300

−200

−100

Real Axis

Chebyshev−I Lowpass Filter Pole−Zero Plot Order:8

0 0.05 0.1 0.15 0.2 0.25−100

Time (s)

Chebyshev−I Lowpass Filter Impulse Response Order:8

0 0.05 0.1 0.15 0.2 0.250

Time (s)

Chebyshev−I Lowpass Filter Step Response Order:8

Example 2: Chebyshev-II Lowpass

0 100 200 300 400 500 600 700 800 900 10000

Chebyshev−II Lowpass Filter Transfer Function Order: 8

Frequency (rad/sec)

Chebyshev−II Lowpass Filter Transfer Function Order:8

−4000 −3000 −2000 −1000 0 1000 2000 3000 4000

−2000

−1000

Real Axis

Chebyshev−II Lowpass Filter Pole−Zero Plot Order:8

PolesZeros

0 0.05 0.1 0.15 0.2 0.25−200

−150

−100

Time (s)

Chebyshev−II Lowpass Filter Impulse Response Order:8

0 0.05 0.1 0.15 0.2 0.250

Time (s)

Chebyshev−II Lowpass Filter Step Response Order:8

Example 2: Elliptic Lowpass

0 100 200 300 400 500 600 700 800 900 10000

Elliptic Lowpass Filter Transfer Function Order: 5

Frequency (rad/sec)

Elliptic Lowpass Filter Transfer Function Order:5

−1200 −1000 −800 −600 −400 −200 0 200 400 600 800

−600

−400

−200

Real Axis

Elliptic Lowpass Filter Pole−Zero Plot Order:5

PolesZeros

0 0.05 0.1 0.15 0.2 0.25−100

Time (s)

Elliptic Lowpass Filter Impulse Response Order:5

0 0.05 0.1 0.15 0.2 0.250

Time (s)

Elliptic Lowpass Filter Step Response Order:5

Example 3: Highpass Filter Specifications

Design an elliptic highpass filter that meets the followingspecifications:

• The passband ranges from 550–∞ rad/s

• The stopband ranges from 0–450 rad/s

Plot the magnitude of the resulting transfer function on a linear-linearplot, the Bode magnitude plot, the pole-zero plot, the impulseresponse, and the step response.

Example 3: Elliptic Highpass

0 100 200 300 400 500 600 700 800 900 10000

Elliptic Highpass Filter Transfer Function Order: 5

Frequency (rad/sec)

Elliptic Highpass Filter Transfer Function Order:5

−1400 −1200 −1000 −800 −600 −400 −200 0 200 400

−500

Real Axis

Elliptic Highpass Filter Pole−Zero Plot Order:5

PolesZeros

0 0.05 0.1 0.15−1400

−1200

−1000

−800

−600

−400

−200

Time (s)

Elliptic Highpass Filter Impulse Response Order:5

0 0.05 0.1 0.15−0.4

−0.2

Time (s)

Elliptic Highpass Filter Step Response Order:5

Example 3: Relevant MATLAB Code

Wp = 550; % Passband endsWs = 450; % Stopband beginsRp = -20*log10(0.95); % Maximum deviation from 1 in the passband (dB)Rs = -20*log10(0.05); % Minimum attenuation in the stopband (dB)[n,wn] = ellipord(Wp,Ws,Rp,Rs,’s’);[B,A] = ellip(n,Rp,Rs,wn,’high’,’s’);

Example 4: Bandpass Filter Specifications

Design an Chebyshev type I bandpass filter that meets the followingspecifications:

• The passband ranges from 450–550 rad/s

• The stopband ranges from 0–400 rad/s and 500–∞ rad/s

Example 4: Chebyshev-I Bandpass

0 100 200 300 400 500 600 700 800 900 10000

Chebyshev−I Bandpass Filter Transfer Function Order: 8

Frequency (rad/sec)

Chebyshev−I Bandpass Filter Transfer Function Order:8

−800 −600 −400 −200 0 200 400 600 800

−500

Real Axis

Chebyshev−I Bandpass Filter Pole−Zero Plot Order:8

PolesZeros

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−40

Time (s)

Chebyshev−I Bandpass Filter Impulse Response Order:8

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−0.08

−0.06

−0.04

−0.02

Time (s)

Chebyshev−I Bandpass Filter Step Response Order:8

Wp = [450 550]; % Passband endsWs = [400 600]; % Stopband beginsRp = -20*log10(0.95); % Maximum deviation from 1 in the passband (dB)Rs = -20*log10(0.05); % Minimum attenuation in the stopband (dB)[n,wn] = cheb1ord(Wp,Ws,Rp,Rs,’s’);[B,A] = cheby1(n,Rp,wn,’s’);

Example 5: Bandstop Filter Specifications

Design an Chebyshev type II bandstop filter that meets the followingspecifications:

• The passband ranges from 0–400 rad/s and 500–∞ rad/s

• The stopband ranges from 450–550 rad/s

Example 5: Chebyshev-II Bandstop

0 100 200 300 400 500 600 700 800 900 10000

Chebyshev−II Bandstop Filter Transfer Function Order: 8

Frequency (rad/sec)

Chebyshev−II Bandstop Filter Transfer Function Order:8

−800 −600 −400 −200 0 200 400 600 800

−500

Real Axis

Chebyshev−II Bandstop Filter Pole−Zero Plot Order:8

PolesZeros

0 0.05 0.1 0.15 0.2 0.25−300

−200

−100

Time (s)

Chebyshev−II Bandstop Filter Impulse Response Order:8

0 0.05 0.1 0.15 0.2 0.250.5

Time (s)

Chebyshev−II Bandstop Filter Step Response Order:8

Wp = [400 600]; % Passband endsWs = [450 550]; % Stopband beginsRp = -20*log10(0.95); % Maximum deviation from 1 in the passband (dB)Rs = -20*log10(0.05); % Minimum attenuation in the stopband (dB)[n,wn] = cheb2ord(Wp,Ws,Rp,Rs,’s’);[B,A] = cheby2(n,Rs,wn,’stop’,’s’);

Practical Filters Summary

• In most applications, filter design begins with a set ofspecifications

• We can use design tools like MATLAB to solve for the H(s) thatmeets a set of frequency specifications

• We can then use the cascade-form of transfer function synthesis

• However, there are standard circuits for these types of filters thatare less expensive and more tolerant of errors in component values

Types of practical filters Filter specifications Tradeoffs...

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