Types of practical filters Filter specifications Tradeoffs...
Transcript of Types of practical filters Filter specifications Tradeoffs...
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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
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Ideal Filters
1Lowpass
1Highpass
1Bandpass
1Bandstop
1Notch
ω
ω
ω
ω
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
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Ideal Filters Comments
1Lowpass
1Highpass
1Bandpass
1Bandstop
1Notch
ω
ω
ω
ω
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
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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
J. McNames Portland State University ECE 222 Practical Analog Filters Ver. 1.04 4
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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
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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.
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Example 2: Butterworth Lowpass
0 100 200 300 400 500 600 700 800 900 10000
0.2
0.4
0.6
0.8
1
|H(j
ω)|
Butterworth Lowpass Filter Transfer Function Order: 21
Frequency (rad/sec)
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Example 2: Butterworth Lowpass
102
103
−30
−25
−20
−15
−10
−5
0
|H(j
ω)|
(dB
)
Butterworth Lowpass Filter Transfer Function Order:21
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Example 2: Butterworth Lowpass
−1000 −500 0 500
−400
−300
−200
−100
0
100
200
300
400
Imag
inar
y A
xis
Real Axis
Butterworth Lowpass Filter Pole−Zero Plot Order:21
Poles
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Example 2: Butterworth Lowpass
0 0.05 0.1 0.15 0.2 0.25−100
−50
0
50
100
150
h(t)
Time (s)
Butterworth Lowpass Filter Impulse Response Order:21
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Example 2: Butterworth Lowpass
0 0.05 0.1 0.15 0.2 0.250
0.2
0.4
0.6
0.8
1
1.2
1.4
h(t)
Time (s)
Butterworth Lowpass Filter Step Response Order:21
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Example 2: Chebyshev-I Lowpass
0 100 200 300 400 500 600 700 800 900 10000
0.2
0.4
0.6
0.8
1
|H(j
ω)|
Chebyshev−I Lowpass Filter Transfer Function Order: 8
Frequency (rad/sec)
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Example 2: Chebyshev-I Lowpass
102
103
−30
−25
−20
−15
−10
−5
0
|H(j
ω)|
(dB
)
Chebyshev−I Lowpass Filter Transfer Function Order:8
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Example 2: Chebyshev-I Lowpass
−800 −600 −400 −200 0 200 400 600
−400
−300
−200
−100
0
100
200
300
400
Imag
inar
y A
xis
Real Axis
Chebyshev−I Lowpass Filter Pole−Zero Plot Order:8
Poles
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Example 2: Chebyshev-I Lowpass
0 0.05 0.1 0.15 0.2 0.25−100
−50
0
50
100
150
h(t)
Time (s)
Chebyshev−I Lowpass Filter Impulse Response Order:8
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Example 2: Chebyshev-I Lowpass
0 0.05 0.1 0.15 0.2 0.250
0.2
0.4
0.6
0.8
1
1.2
1.4
h(t)
Time (s)
Chebyshev−I Lowpass Filter Step Response Order:8
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Example 2: Chebyshev-II Lowpass
0 100 200 300 400 500 600 700 800 900 10000
0.2
0.4
0.6
0.8
1
|H(j
ω)|
Chebyshev−II Lowpass Filter Transfer Function Order: 8
Frequency (rad/sec)
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Example 2: Chebyshev-II Lowpass
102
103
−30
−25
−20
−15
−10
−5
0
|H(j
ω)|
(dB
)
Chebyshev−II Lowpass Filter Transfer Function Order:8
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Example 2: Chebyshev-II Lowpass
−4000 −3000 −2000 −1000 0 1000 2000 3000 4000
−2000
−1000
0
1000
2000
Imag
inar
y A
xis
Real Axis
Chebyshev−II Lowpass Filter Pole−Zero Plot Order:8
PolesZeros
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Example 2: Chebyshev-II Lowpass
0 0.05 0.1 0.15 0.2 0.25−200
−150
−100
−50
0
50
100
150
h(t)
Time (s)
Chebyshev−II Lowpass Filter Impulse Response Order:8
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Example 2: Chebyshev-II Lowpass
0 0.05 0.1 0.15 0.2 0.250
0.2
0.4
0.6
0.8
1
1.2
1.4
h(t)
Time (s)
Chebyshev−II Lowpass Filter Step Response Order:8
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Example 2: Elliptic Lowpass
0 100 200 300 400 500 600 700 800 900 10000
0.2
0.4
0.6
0.8
1
|H(j
ω)|
Elliptic Lowpass Filter Transfer Function Order: 5
Frequency (rad/sec)
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Example 2: Elliptic Lowpass
102
103
−30
−25
−20
−15
−10
−5
0
|H(j
ω)|
(dB
)
Elliptic Lowpass Filter Transfer Function Order:5
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Example 2: Elliptic Lowpass
−1200 −1000 −800 −600 −400 −200 0 200 400 600 800
−600
−400
−200
0
200
400
600
Imag
inar
y A
xis
Real Axis
Elliptic Lowpass Filter Pole−Zero Plot Order:5
PolesZeros
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Example 2: Elliptic Lowpass
0 0.05 0.1 0.15 0.2 0.25−100
−50
0
50
100
150
h(t)
Time (s)
Elliptic Lowpass Filter Impulse Response Order:5
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Example 2: Elliptic Lowpass
0 0.05 0.1 0.15 0.2 0.250
0.2
0.4
0.6
0.8
1
1.2
1.4
h(t)
Time (s)
Elliptic Lowpass Filter Step Response Order:5
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Example 3: Highpass Filter Specifications
Design an elliptic highpass filter that meets the followingspecifications:
• 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 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.
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Example 3: Elliptic Highpass
0 100 200 300 400 500 600 700 800 900 10000
0.2
0.4
0.6
0.8
1
|H(j
ω)|
Elliptic Highpass Filter Transfer Function Order: 5
Frequency (rad/sec)
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Example 3: Elliptic Highpass
102
103
−30
−25
−20
−15
−10
−5
0
|H(j
ω)|
(dB
)
Elliptic Highpass Filter Transfer Function Order:5
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Example 3: Elliptic Highpass
−1400 −1200 −1000 −800 −600 −400 −200 0 200 400
−500
0
500
Imag
inar
y A
xis
Real Axis
Elliptic Highpass Filter Pole−Zero Plot Order:5
PolesZeros
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Example 3: Elliptic Highpass
0 0.05 0.1 0.15−1400
−1200
−1000
−800
−600
−400
−200
0
200
h(t)
Time (s)
Elliptic Highpass Filter Impulse Response Order:5
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Example 3: Elliptic Highpass
0 0.05 0.1 0.15−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
h(t)
Time (s)
Elliptic Highpass Filter Step Response Order:5
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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’);
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Example 4: Bandpass Filter Specifications
Design an Chebyshev type I bandpass filter that meets the followingspecifications:
• 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 450–550 rad/s
• The stopband ranges from 0–400 rad/s and 500–∞ 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.
J. McNames Portland State University ECE 222 Practical Analog Filters Ver. 1.04 34
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Example 4: Chebyshev-I Bandpass
0 100 200 300 400 500 600 700 800 900 10000
0.2
0.4
0.6
0.8
1
|H(j
ω)|
Chebyshev−I Bandpass Filter Transfer Function Order: 8
Frequency (rad/sec)
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Example 4: Chebyshev-I Bandpass
102
103
−30
−25
−20
−15
−10
−5
0
|H(j
ω)|
(dB
)
Chebyshev−I Bandpass Filter Transfer Function Order:8
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Example 4: Chebyshev-I Bandpass
−800 −600 −400 −200 0 200 400 600 800
−500
0
500
Imag
inar
y A
xis
Real Axis
Chebyshev−I Bandpass Filter Pole−Zero Plot Order:8
PolesZeros
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Example 4: Chebyshev-I Bandpass
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−40
−30
−20
−10
0
10
20
30
40
h(t)
Time (s)
Chebyshev−I Bandpass Filter Impulse Response Order:8
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Example 4: Chebyshev-I Bandpass
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
0
0.02
0.04
0.06
0.08
h(t)
Time (s)
Chebyshev−I Bandpass Filter Step Response Order:8
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Example 4: Relevant MATLAB Code
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’);
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Example 5: Bandstop Filter Specifications
Design an Chebyshev type II bandstop filter that meets the followingspecifications:
• 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 0–400 rad/s and 500–∞ rad/s
• The stopband ranges from 450–550 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.
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Example 5: Chebyshev-II Bandstop
0 100 200 300 400 500 600 700 800 900 10000
0.2
0.4
0.6
0.8
1
|H(j
ω)|
Chebyshev−II Bandstop Filter Transfer Function Order: 8
Frequency (rad/sec)
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Example 5: Chebyshev-II Bandstop
102
103
−30
−25
−20
−15
−10
−5
0
|H(j
ω)|
(dB
)
Chebyshev−II Bandstop Filter Transfer Function Order:8
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Example 5: Chebyshev-II Bandstop
−800 −600 −400 −200 0 200 400 600 800
−500
0
500
Imag
inar
y A
xis
Real Axis
Chebyshev−II Bandstop Filter Pole−Zero Plot Order:8
PolesZeros
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Example 5: Chebyshev-II Bandstop
0 0.05 0.1 0.15 0.2 0.25−300
−200
−100
0
100
200
h(t)
Time (s)
Chebyshev−II Bandstop Filter Impulse Response Order:8
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Example 5: Chebyshev-II Bandstop
0 0.05 0.1 0.15 0.2 0.250.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
h(t)
Time (s)
Chebyshev−II Bandstop Filter Step Response Order:8
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Example 5: Relevant MATLAB Code
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’);
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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
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