ECE 223. DT Filters. Ver. 1.03. 3. Overview of Discrete-Time Filters. • First-order
filters. • Ideal filters. • Practical filters. • Frequency-selective filter specifications.
Overview of Discrete-Time Filters
Discrete-Time Filters Overview
• First-order filters
N
• Ideal filters
ak y[n − k]
=
k=0
• Practical filters
Y (ejω )
• Frequency-selective filter specifications • Ripple versus filter order tradeoff
=
M
bk x[n − k]
k=0 M −jwk k=0 bk e X(ejω ) N −jwk a e k k=0
• Discrete-time filters are divided into two categories – Finite impulse response (FIR): h[n] = 0 for some a and b such that −∞ < a < n < b < +∞ – Infinite impulse response (IIR): not FIR
• Application example
• Filters that can be described with difference-equations – FIR: N = 0 – IIR: N > 0 • A simple FIR filter is the moving average filter • A simple IIR filter is the first-order lowpass filter Portland State University
ECE 223
DT Filters
Ver. 1.03
1
Portland State University
Example 1: First-Order Filters
ECE 223
DT Filters
Ver. 1.03
2
Ver. 1.03
4
Example 1: Workspace
Consider the following filter: y[n] − ay[n − 1] = (1 − a)x[n] 1. Solve for the filter’s transfer function 2. Find the cutoff frequency as a function of a
Portland State University
ECE 223
DT Filters
Ver. 1.03
3
Portland State University
ECE 223
DT Filters
Example 1: Ωc versus a
Example 1: H(ejω ) for various a
3.5 1 jω
|H(e )|
3
2
0.5
0
−3
1.5 ∠ H(ejω) (o)
c
ω (rad/sample)
2.5
1 0.5 0
−1
0
1
2
3
−1 0 1 Frequency (rads/sample)
2
3
50 0 −50
0
0.1
0.2
Portland State University
0.3
0.4
ECE 223
0.5 a
0.6
0.7
0.8
DT Filters
0.9
Ver. 1.03
1
−3
5
ECE 223
DT Filters
Ver. 1.03
6
Ver. 1.03
8
Example 1: MATLAB Code %function [] = FirstOrderApplied(); close all;
Intel Closing Daily Price Over 1 Year
36
−2
Portland State University
Example 1: Filtered Signals d nd x n
34
= = = =
load(’Intel.txt’); % Closing daily price length(d); d(nd:-1:1); % Reorder so first element is oldest data (0:nd-1)’; % Discrete time index
%============================================================================== % Plot the relationship between cutoff frequency and a %============================================================================== a = 0.00001:0.01:1; wc = acos((1-4*a+a.^2)./(-2*a));
32 x[n], y0.1[n], y0.6[n]
a=0.1 a=0.2 a=0.3 a=0.4 a=0.5 a=0.6 a=0.7 −2 a=0.8 a=0.9
30
figure FigureSet(1,’LTX’); h = plot(a,wc,’LineWidth’,1.0); ylabel(’\omega_c (rad/sample)’); xlabel(’a’); grid on; box off; AxisSet(8); print -depsc FOCutoff;
28 26 24 Raw signal Cutoff:0.56 Cutoff:0.11
22 20 0
50
100
150
%============================================================================== % Plot the relationship between cutoff frequency and a %============================================================================== a = 0.1:0.1:0.9; w = -pi:0.01:pi; H = zeros(length(a),length(w));
200
for cnt = 1:length(a) [h,w] = freqz(1-a(cnt),[1 -a(cnt)],w); H(cnt,:) = h;
Time (day)
Portland State University
ECE 223
DT Filters
Ver. 1.03
7
Portland State University
ECE 223
DT Filters
end;
for cnt = 2:nd, y2(cnt) = a*y2(cnt-1) + (1-a)*x(cnt); end;
figure FigureSet(1,’LTX’); subplot(2,1,1); h = plot(w,abs(H)); xlim([-pi pi]); ylim([0 1.05]); legend(’a=0.1’,’a=0.2’,’a=0.3’,’a=0.4’,’a=0.5’,’a=0.6’,’a=0.7’,’a=0.8’,’a=0.9’,2); ylabel(’|H(e^{j\omega})|’); grid on; box off; subplot(2,1,2); h = plot(w,rem(angle(H)*180/pi,180)); xlim([-pi pi]); ylim([-70 70]); ylabel(’\angle H(e^{j\omega}) (^o)’); xlabel(’Frequency (rads/sample)’); grid on; box off; AxisSet(8); print -depsc FOTransferFunctions;
figure FigureSet(1,’LTX’); h = plot(n,x,’b’,n,y1,’r’,n,y2,’g’); set(h,’LineWidth’,0.6); ylabel(’x[n], y_{0.1}[n], y_{0.6}[n]’); xlabel(’Time (day)’); title(’Intel Closing Daily Price Over 1 Year’); xlim([0 nd-1]); ylim([19 36]); box off; grid on; AxisSet(8); legend(’Raw signal’,sprintf(’Cutoff:%4.2f’,w1),sprintf(’Cutoff:%4.2f’,w2),4); print -depsc FOSignalFiltered;
%============================================================================== % Filter & Plot %============================================================================== a = 0.58; % cutoff frequency approximately 0.1 w1 = acos((1-4*a+a.^2)./(-2*a)); y1 = zeros(nd,1); for cnt = 2:nd, y1(cnt) = a*y1(cnt-1) + (1-a)*x(cnt); end; a = 0.90; % cutoff frequency approximately 0.1 w2 = acos((1-4*a+a.^2)./(-2*a)); y2 = zeros(nd,1);
Portland State University
ECE 223
DT Filters
Ver. 1.03
9
Ideal Filters Lowpass
Highpass
1 Ωc
Ωc
Bandpass
Ω
Ωc
Ω
Bandstop
1
1 Ωc1
Ωc2
Ω
Ωc1
Ωc2
DT Filters
Ver. 1.03
Ω
• MATLAB can be used to design standard frequency selective filters that meet user-specified requirements
• If we know the specifications, we can ask MATLAB to generate the filter for us
• These filters include: lowpass, highpass, bandpass, and bandstop
• There are four popular types of standard filters – Butterworth – Chebyshev Type I – Chebyshev Type II – Elliptic
• Unlike continuous-time filters, these must have cutoff frequencies that range between 0 and π
Portland State University
ECE 223
DT Filters
10
• Lowpass and highpass filters usually have the following requirements – Passband range – Stopband range – Maximum ripple in the passband – Minimum attenuation in the stopband
1
Ω
ECE 223
Practical Filters • Practical filters are usually designed to meet a set of specifications
Notch
1
Portland State University
Ver. 1.03
11
Portland State University
ECE 223
DT Filters
Ver. 1.03
12
Ripple Tradeoff Filter Butterworth Chebyshev Type I Chebyshev Type II Elliptic
Order Largest Moderate Moderate Lowest
Passband Smooth Ripple Smooth Ripple
Example 2: Lowpass Filter Specifications Design a lowpass filter that meets the following specifications:
Stopband Smooth Smooth Ripple Ripple
• The passband ripple is no more than 0.4455 dB (0.95 ≤ |H(ejω )| ≤ 1) • The stopband attenuation is at least 26.02 dB (|H(ejω )| ≤ 0.05) • The passband ranges from 0–0.2π rad/sample
• The four popular filter types differ in how they satisfy the specifications
• The stopband ranges from 0.3π–π rad/sample Plot the magnitude of the resulting transfer function on a linear-linear plot, the impulse response, and the step response. Try the Butterworth, Chebyshev I, Chebyshev II, and elliptic filters.
• In the passband and stopband, each filter is either smooth or contains ripple • Elliptic filters are also called equiripple filters and Cauer filters
Portland State University
ECE 223
DT Filters
Ver. 1.03
13
Portland State University
Example 3: Butterworth Lowpass
ECE 223
DT Filters
Ver. 1.03
14
Example 3: Butterworth Lowpass
Butterworth Lowpass Filter Transfer Function Order: 10
Butterworth Lowpass Filter Impulse Response Order:10
0.25
1 0.2 0.8
0.6
h[n]
|H(ejω)|
0.15 0.1 0.05 0.4 0 0.2 −0.05 0
0
0.5
Portland State University
1
1.5 2 Frequency (rad/sample)
ECE 223
DT Filters
2.5
−0.1
3
Ver. 1.03
15
0
10
Portland State University
20
30
40
ECE 223
50 60 Time (n)
DT Filters
70
80
90
Ver. 1.03
100
16
Example 3: Butterworth Lowpass
Example 3: Chebyshev-I Lowpass Chebyshev−I Lowpass Filter Transfer Function Order: 5
Butterworth Lowpass Filter Step Response Order:10
1.4
1 1.2 0.8
0.8
|H(ejω)|
h[n]
1
0.6
0.6
0.4 0.4 0.2 0.2 0
0
10
20
Portland State University
30
40
50 60 Time (n)
ECE 223
70
DT Filters
80
90
0
100
Ver. 1.03
17
0
0.5
Portland State University
Example 3: Chebyshev-I Lowpass
0.15
1
0.1
0.8
h[n]
h[n]
1.2
0.05
0.6
0
0.4
−0.05
0.2
0
10
Portland State University
20
30
40
ECE 223
50 60 Time (n)
DT Filters
70
ECE 223
2.5
DT Filters
3
Ver. 1.03
18
80
Chebyshev−I Lowpass Filter Step Response Order:5
1.4
0.2
−0.1
1.5 2 Frequency (rad/sample)
Example 3: Chebyshev-I Lowpass
Chebyshev−I Lowpass Filter Impulse Response Order:5
0.25
1
90
Ver. 1.03
0
100
19
0
10
Portland State University
20
30
40
ECE 223
50 60 Time (n)
70
DT Filters
80
90
Ver. 1.03
100
20
Example 3: Chebyshev-II Lowpass
Example 3: Chebyshev-II Lowpass
Chebyshev−II Lowpass Filter Transfer Function Order: 5
Chebyshev−II Lowpass Filter Impulse Response Order:5
0.25
1 0.2 0.8
0.6
h[n]
|H(ejω)|
0.15 0.1 0.05 0.4 0 0.2 −0.05 0
0
0.5
Portland State University
1
1.5 2 Frequency (rad/sample)
ECE 223
2.5
DT Filters
−0.1
3
Ver. 1.03
21
0
10
20
Portland State University
40
ECE 223
Example 3: Chebyshev-II Lowpass
50 60 Time (n)
70
80
DT Filters
90
100
Ver. 1.03
22
Example 3: Elliptic Lowpass Elliptic Lowpass Filter Transfer Function Order: 4
Chebyshev−II Lowpass Filter Step Response Order:5
1.4
30
1 1.2 0.8
0.8
|H(ejω)|
h[n]
1
0.6
0.6
0.4 0.4 0.2 0.2 0
0
10
Portland State University
20
30
40
ECE 223
50 60 Time (n)
70
DT Filters
80
90
Ver. 1.03
0
100
23
0
0.5
Portland State University
1
1.5 2 Frequency (rad/sample)
ECE 223
DT Filters
2.5
3
Ver. 1.03
24
Example 3: Elliptic Lowpass Elliptic Lowpass Filter Impulse Response Order:4
0.25 0.2
1.2
0.15
1
0.1
0.8
0.05
0.6
0
0.4
−0.05
0.2
−0.1
0
10
Portland State University
20
30
40
ECE 223
50 60 Time (n)
70
DT Filters
80
90
Ver. 1.03
0
100
25
Example 3: MATLAB Code
= = = =
0.20; 0.30; -20*log10(0.95); -20*log10(0.05);
% % % %
Passband ends Stopband begins Maximum deviation from 1 in the passband (dB) Minimum attenuation in the stopband (dB)
for cnt = 1:4, if cnt==1, [od,wn] = buttord(Wp,Ws,Rp,Rs); [B,A] = butter(od,wn); stFilter = ’Butterworth’; elseif cnt==2, [od,wn] = ellipord(Wp,Ws,Rp,Rs); [B,A] = ellip(od,Rp,Rs,wn); stFilter = ’Elliptic’; elseif cnt==3, [od,wn] = cheb1ord(Wp,Ws,Rp,Rs); [B,A] = cheby1(od,Rp,wn); stFilter = ’Chebyshev-I’; elseif cnt==4, [od,wn] = cheb2ord(Wp,Ws,Rp,Rs); [B,A] = cheby2(od,Rs,wn); stFilter = ’Chebyshev-II’; else break; end; sys = tf(B,A,-1); wp = Wp*pi; ws = Ws*pi;
Portland State University
0
10
Portland State University
20
30
40
ECE 223
50 60 Time (n)
70
80
DT Filters
90
100
Ver. 1.03
26
Ver. 1.03
28
%============================================================================== % Plot Magnitude on Linear Scale %============================================================================== ymax = 1.05; pbmax = 1; pbmin = 10^(-Rp/20); sbmax = 10^(-Rs/20); wmax = pi; w = 0:0.001:wmax; [H,w] = freqz(B,A,w); mag = abs(H); phs = angle(H); figure; FigureSet(1,’LTX’); h = patch([0 wp wp 0],[0 0 pbmin pbmin],0.5*[1 1 1]); set(h,’LineWidth’,0.0001); hold on; h = patch([0 ws ws wmax wmax 0],[pbmax pbmax sbmax sbmax ymax ymax],0.5*[1 1 1]); set(h,’LineWidth’,0.0001); h = plot(w,mag,’r’); set(h,’LineWidth’,1.0); hold off; ylim([0 ymax]); xlim([0 wmax]); grid on; ylabel(’|H(e^{j\omega})|’); title(sprintf(’%s Lowpass Filter Transfer Function Order: %d’,stFilter,od)); xlabel(’Frequency (rad/sample)’); box off; AxisSet(8); st = sprintf(’print -depsc Lowpass%s’,stFilter); eval(st);
%function [] = Lowpass(); clear all; close all; Wp Ws Rp Rs
Elliptic Lowpass Filter Step Response Order:4
1.4
h[n]
h[n]
Example 3: Elliptic Lowpass
%============================================================================== % Impulse Response %============================================================================== figure;
ECE 223
DT Filters
Ver. 1.03
27
Portland State University
ECE 223
DT Filters
FigureSet(1,’LTX’); n = 0:100; [x,t] = impulse(sys,n); h = stem(t,x,’b’); set(h(1),’MarkerFaceColor’,’b’); set(h(1),’MarkerSize’,2); ylabel(’h[n]’); xlabel(’Time (n)’); title(sprintf(’%s Lowpass Filter Impulse Response Order:%d’,stFilter,od)); box off; hold on; h = plot(xlim,[0 0],’k:’); hold off; AxisSet(8); st = sprintf(’print -depsc Lowpass%sImpulse’,stFilter); eval(st);
end;
%============================================================================== % Step Response %============================================================================== figure; FigureSet(1,’LTX’); n = 0:100; [x,t] = step(sys,n); h = stem(t,x,’b’); set(h(1),’MarkerFaceColor’,’b’); set(h(1),’MarkerSize’,2); ylabel(’h[n]’); xlabel(’Time (n)’); title(sprintf(’%s Lowpass Filter Step Response Order:%d’,stFilter,od)); box off; hold on; h = plot(xlim,[1 1],’k:’); hold off; AxisSet(8); st = sprintf(’print -depsc Lowpass%sStep’,stFilter); eval(st);
Portland State University
ECE 223
DT Filters
Ver. 1.03
29
Practical Filter Tradeoffs
Portland State University
ECE 223
DT Filters
Ver. 1.03
30
Application Example 1: Microelectrode Recording Filter
• Butterworth - Highest order H(ejω ) + No passband or stopband ripple
An engineer wishes to detect action potentials in a microelectrode recording with a simple threshold detector. The signal contains significant baseline drift. Action potentials typically last about 1 ms.
• Chebyshev Type I + No stopband ripple
• What type of filter should the engineer use?
• Chebyshev Type II + No passband ripple
• What should the cutoff frequency(ies) be?
• What should the filter specifications be?
• Elliptic + Lowest order H(ejω ) - Passband and stopband ripple
Portland State University
ECE 223
DT Filters
Ver. 1.03
31
Portland State University
ECE 223
DT Filters
Ver. 1.03
32
Application Example 1: Frequency-Selective Filters
Application Example 1: Frequency-Selective Filters 50
Microelectrode Recording FT Magnitude
0.8 0.6
40 30 20 10
0.4
0 0
0.2
500
1000
1500
2000
500 FT Magnitude
0 −0.2 −0.4
400 300 200 100
−0.6
0 0
1
1.2
1.4
1.6
Portland State University
1.8
2 2.2 Time (sec)
ECE 223
2.4
2.6
DT Filters
50
100
150
200
250
300
2.8
Ver. 1.03
33
Portland State University
Application Example 1: Frequency-Selective Filters
ECE 223
DT Filters
Ver. 1.03
34
Application Example 1: Frequency-Selective Filters 50 FT Magnitude
Microelectrode Recording 0.5 0
40 30 20 10
−0.5
0
0.2
0.4
0.6
0.8
1 1.2 Time (sec)
1.4
1.6
1.8
0
500
1000
1500
2000
0
500
1000
1500
2000
2 50 FT Magnitude Filtered
0
0.5 0 −0.5
40 30 20 10 0
0
0.2
0.4
Portland State University
0.6
0.8
1 1.2 Time (sec)
ECE 223
1.4
DT Filters
1.6
1.8
2
Ver. 1.03
35
Portland State University
ECE 223
DT Filters
Ver. 1.03
36
Application Example 1: MATLAB Code
subplot(2,1,1); h = plot(f,abs(X(k)),’r’); set(h,’LineWidth’,0.6); xlim([min(f) max(f)]); ylim([0 50]); set(gca,’XTick’,[0:500:max(f)]); box off; ylabel(’FT Magnitude’); subplot(2,1,2); h = plot(f,abs(X(k)),’r’); set(h,’LineWidth’,0.6); xlim([0 300]); ylim([0 500]); %set(gca,’XTick’,[0:500:max(f)]); box off; ylabel(’FT Magnitude’); AxisSet(6); print -depsc MERSpectralDensity;
%function [] = MER(); close all; [x,fs,nbits] = wavread(’Henderson2.wav’); x = decimate(x,2); fs = fs/2; k = round(fs*1):round(fs*3); % Look at only 5 s x = x(k); nx = length(x); figure; FigureSet(1,’LTX’); t = (k-1)/fs; h = plot(t,x,’b’); set(h,’LineWidth’,0.6); xrng = max(x)-min(x); xlim([min(t) max(t)]); ylim([min(x)-0.01*xrng max(x)+0.01*xrng]); AxisLines; xlabel(’Time (sec)’); ylabel(’’); title(’Microelectrode Recording’); box off; AxisSet(8); print -depsc MERSignal; X nX k f
= = = =
Wp = 210/(fs/2); % Passband ends Ws = 190/(fs/2); % Stopband begins Rp = -20*log10(0.95); % Maximum deviation from 1 in the passband (dB) Rs = -20*log10(0.05); % Minimum attenuation in the stopband (dB) [od,wn] = ellipord(Wp,Ws,Rp,Rs); [B,A] = ellip(od,Rp,Rs,wn,’high’); stFilter = ’Elliptic’; y = filtfilt(B,A,x);
fft(x,2^12); length(X); 1:floor((length(X)+1)/2); (k-1)*(fs)./(nX+1);
figure; FigureSet(1,’LTX’); k = 1:length(x); t = (k-1)/fs; subplot(2,1,1); t = (k-1)/fs; h = plot(t,x,’b’);
figure; FigureSet(1,’LTX’);
Portland State University
ECE 223
DT Filters
Ver. 1.03
37
set(h,’LineWidth’,0.6); xrng = max(x)-min(x); xlim([min(t) max(t)]); ylim([min(x)-0.01*xrng max(x)+0.01*xrng]); AxisLines; xlabel(’Time (sec)’); ylabel(’’); title(’Microelectrode Recording’); box off; subplot(2,1,2); h = plot(t,y,’g’); set(h,’LineWidth’,0.6); xlim([min(t) max(t)]); ylim([min(x)-0.01*xrng max(x)+0.01*xrng]); AxisLines; xlabel(’Time (sec)’); ylabel(’’); box off; AxisSet(8); print -depsc MERSignalFiltered; Y nY k f
= = = =
S
Portland State University
ECE 223
DT Filters
Ver. 1.03
38
DT Filters
Ver. 1.03
40
h = plot(f,abs(Y(k)),’g’); set(h,’LineWidth’,0.6); xlim([min(f) max(f)]); ylim([0 50]); set(gca,’XTick’,[0:500:max(f)]); box off; ylabel(’FT Magnitude Filtered’); AxisSet(6); print -depsc MERSpectralDensityFiltered;
fft(y,2^12); length(Y); 1:floor((length(Y)+1)/2); (k-1)*(fs)./(nY+1);
figure; FigureSet(1,’LTX’); subplot(2,1,1); h = plot(f,abs(X(k)),’r’); set(h,’LineWidth’,0.6); xlim([min(f) max(f)]); ylim([0 50]); set(gca,’XTick’,[0:500:max(f)]); box off; ylabel(’FT Magnitude’); subplot(2,1,2);
Portland State University
ECE 223
DT Filters
Ver. 1.03
39
Portland State University
ECE 223
