Transcript ee221_Opt

Circuits II
EE221
Optimization of Filter Parameters
Instructor: Kevin D. Donohue
Iteration, Objective/Error Functions,
Visualization
Matlab and Parameter Optimization:
Equations should be developed that relate design parameters to the
error/objective criteria. If optimal parameters can be solved for
directly, the problem is finished. Unfortunately in many cases, the large
number of parameters or the nonlinear relationship between them, makes
a direct (closed-form) solution difficult or impossible.
The performance equations can, however, be examined through
programs such as Matlab, and optimization performed with a creative
strategy for iterating through guesses at critical performance
parameters until the error reaches a practical minimum.
Matlab functions: Available Matlab functions can be combined
together in a program to create a custom function for optimization.
Example: Create a Matlab function that evaluates transfer function
samples of a first-order low-pass filter over an array of frequency axis
points (in Hz), for a given cutoff frequency (in Hz), and the gain at DC (in
dB).
Function for Use in Iteration Process
1) Use a text editor to create a file lpf1.m (Matlab program files always end in .m).
2) Then enter the following text.
function h = lpf1(f,fc,gdb)
%
This function evaluates complex points of a first
%
order low-pass filter with cut-off frequency fc
%
in Hertz, and a gain in dB at DC of gdb. The
%
function syntax is:
%
%
tflp = lpf1(f, fc, gdb)
%
%
where f is an array of points in Hertz where the
%
function is eavluated at, and tfpl is the array of
%
complex evaluation points.
s = j*2*pi*f; %
compute complex frequencies along j omega axis.
gdc = 10^(gdb/20); %
convert gain in dB to linear gain.
h = gdc./(s/(2*pi*fc) + 1 ); % Evaluate TF point by point
3) Save the file in your “working directory” and execute in Matlab just like any other
function. If you type help lpf1 in Matlab, the first sequence of comments will print to
the screen.
Function for Use in Iteration Process
4) All variables defined inside the function will be undefined when the
function is through (local variables). Create an analogous function for a
high-pass filter:
1) Create file with a text editor called hpf1.m and enter:
function h = hpf1(f,fc,gdb)
%
This function evaluates complex points of a first
%
order high-pass filter with cut-off frequency fc
%
in Hertz, and a gain in dB at infinity of gdb.
%
The function syntax is:
%
%
tfhp = hpf1(f, fc, gdb)
%
%
where f is an array of points in Hertz where the
%
function is evaluated at, and tfhp is the array of
%
complex evaluation points.
%
s = j*2*pi*f; %
compute complex frequencies along j omega axis.
ginf=10^(gdb/20); %
convert gain in dB to linear gain.
h = ginf*(s/(2*pi*fc))./(s/(2*pi*fc) + 1 ); % Evaluate TF point by point
Script for Optimization
Matlab Scripts: Matlab functions can be combined together to create a
series of commands to be executed.
Example: Create a Matlab script to examine the mean and variance error
between a cascaded low-pass and high-pass filter, and a band-pass-like
transfer function with a flat 10 dB passband extending from 5 kHz to 50
kHz with a 20 dB per decade roll-off at either end of the passband. The
gain in the passband is 10 dB. Only consider the error from 500 Hz to 500
kHz.
1)Strategy: Since the major portion of overlap between the passbands of
the LPF and HPF will exist between the cut-off frequencies, set the gains
equal to about 5.6 dB each. Now only consider minimizing the error with
respect to 2 parameters - the cut-off frequencies of the LPF and the HPF.
The gain can be adjusted later if our assumption on the passband gain is
not accurate.
2) Create a file in a text editor called ex6.m, and enter:
f =logspace(log10(500),log10(500e3), 512);
% define axis for evaluation
%
Create target function (Described in Example)
ht = zeros(1,512);
% Set up array to fill with function points
Script for Optimization
dB
% Loop through each linear section piece-wise, to create target shape
for k=1:512
% For points less than 5kHz create a line on a log scale
if( f(k) < 5e3) ,
ht(k) = -10 + 20*log10(f(k)/500);
% For points greater than 5kHz and less than 50kHz
Target TF
10
elseif(f(k) < 5e4),
ht(k) = 10;
5
% for points greater than 50kHz create a line on a
0
% log scale
else
-5
ht(k) = 10 - 20*log10(f(k)/5e4);
-10
end
end
-15
10
10
10
10
% Note the above code has nothing to do with your project
Hertz
% It should not be seen in any of your codes
figure(1)
semilogx(f,ht)
% Check to see if it did what we wanted it to
grid
title('Target TF')
xlabel('Hertz')
ylabel('dB')
pause
%
Script will pause until the keyboard is struck
% Set up nested loops to vary the cutoff frequencies for HPF and LPF
% Try a neighborhood of frequencies around f= 5 kHz and 50 kHz.
fhc = linspace(3500,7500,25);
% Select a set of high-pass cut-offs
flc = linspace(35000, 55000, 25); % Select a set of low-pass cut-offs
2
3
4
5
6
10
Script for Optimization
% Set up arrays to store computed error terms
em = zeros(25,25); % mean error (bias)
ev = zeros(25,25); % variance error (consistency of fit)
% Fix Gain at 5.6 dB => 10^(5.6/20) = 1.9055 and optimize for low-pass
% and high-pass cutoff
% Note, this may not be the best gain, but is just an initial guess.
for k=1:25,
for n=1:25,
h = lpf1(f,flc(k),5.6).*hpf1(f,fhc(n),5.6); % Compute Resultant TF
% Compute errors
em(k,n) = mean(ht-20*log10(abs(h)));
ev(k,n) = std(ht-20*log10(abs(h))); %This is actually square root of the
% variance, which is just as good in terms of finding a minimum point.
end
end
[k,n] = find(abs(em) == min(min(abs(em))))%find mean error index close to 0
% look at optimal TF
h = lpf1(f,flc(k),5.6).*hpf1(f,fhc(n),5.6); % Compute Resultant TF
% points at optimum
semilogx(f,20*log10(abs(h)),'g')
hold on
semilogx(f, ht,':r')
hold off
title('Compare target with design TF for minimum mean error')
xlabel('Hertz')
Script for Optimization
% To get a better view, try an intensity plot of the same data:
figure(3) % Plot Mean Error
imagesc(fhc, flc, abs(em))
colormap('gray')
colorbar
title('mean error')
xlabel(' High-pass cut-off')
ylabel(' Low-pass cut-off')
figure(4) % Plot Variance (standard deviation) Error
imagesc(fhc, flc, ev)
colormap('gray')
colorbar
xlabel(' High-pass cut-off')
ylabel(' Low-pass cut-off')
title('standard deviation error')
pause
3) Save file and type ex6 at Matlab prompt. These commands will
execute and when finished, all variables defined in the script will still
be available in the Matlab workspace.
Script for Optimization
ylabel('dB')
flcmm = flc(k)
% Convert index to frequency value
fhcmm = fhc(n)
% Convert index to frequency value
pause
[k,n] = find(ev == min(min(ev))) % find index of minimum variance error
% Compute Resultant TF points at optimum
h = lpf1(f,flc(k),5.6).*hpf1(f,fhc(n),5.6)
figure(2)
semilogx(f,20*log10(abs(h)),'g')
hold on
semilogx(f, ht,':r')
hold off
title('Compare target with design TF for minimum variance error')
xlabel('Hertz')
ylabel('dB')
flcsd = flc(k)
% Convert index to frequency value
fhcsd = fhc(n)
% Convert index to frequency value
pause
figure(3)
mesh(fhc, flc, abs(em))
%
Look at graphic distribution of error.
title('mean error')
figure(4)
mesh(fhc, flc, ev)
title('standard deviation error')
pause
Script for Optimization
% Changing the color map to something other than gray may help in
% observing changes in levels. Use the 'jet' color scheme To get a
% better view, try an intensity plot of the same data:
figure(5)
imagesc(fhc, flc, abs(em))
colormap('jet')
colorbar
title('mean error')
xlabel(' High-pass cut-off')
ylabel(' Low-pass cut-off')
figure(6)
imagesc(fhc, flc, ev)
colormap('jet')
colorbar
xlabel(' High-pass cut-off')
ylabel(' Low-pass cut-off')
title('standard deviation error')
3) Save file and type ex6 at Matlab prompt. These commands will
execute and
when finished, all variables defined in the script will
still be available in the Matlab workspace.
Optimization Results
4) The results of this program should yield a variance error at k=3 and n=16, which
corresponds to cutoffs at 6000 Hz and 41667 Hz. Further improvement can be obtained by
fixing the cutoffs and varying the gain. The following is the output of the previous script:
Mean Error
10
dB
5
0
-5
-10
-15
2
10
3
4
10
5
6
10
10
10
Hertz
Compare target with design TF for minimum variance error
15
10
5
dB
k=
12
n=
8
flcmm =
4.4167e+004
fhcmm =
4.6667e+003
Variance Error
k=
9
n=
16
flcsd =
4.1667e+004
fhcsd =
6000
Compare target with design TF for minimum mean error
15
0
-5
-10
-15
2
10
3
10
4
10
Hertz
5
10
6
10
Optimization Iterated Error Surfaces
(mesh)
Optimization Iterated Error Surfaces (Intensity
Image)
Result for color map ‘gray’
Optimization Iterated Error Surfaces (Intensity
Image)
Result for color map ‘jet’
4
4
standard deviation error
x 10
mean error
x 10
3.5
2.4
3.5
2.2
2
2.2
1.8
1.6
1.4
1.2
4.5
1
Low-pass cut-off
Low-pass cut-off
2
4
4
1.8
1.6
4.5
1.4
0.8
0.6
5
1.2
5
1
0.4
0.2
5.5
3500
4000
4500
5000 5500 6000
High-pass cut-off
6500
7000
7500
0.8
5.5
3500
4000
4500
5000 5500 6000
High-pass cut-off
6500
7000
7500
Parameter Optimization BPF Example:
Example: Create a Matlab script that evaluates matching a desired
transfer function shape by adding 3 second-order band-pass filters and
varies the mid frequency bandwidth to variance error. The test points
are at 100, 500, 1000, 2000, 3000, and 5000 Hz and correspond to TF
magnitudes of 0, 6, 10, 2, 6, and -1 dB.
See script opt1.m on course web site
http://www.engr.uky.edu/~donohue/ee221/mfile.html
Since shape is the criterion, only need to evaluate variance error:
>> errv(k) = std(ht-tfg)^2; % Compute error for this bandwidth where ht is the
vector of test point magnitudes for the target TF and tgf is the vector of corresponding
points evaluated on the kth synthesized filter
Parameter Optimization BPF Example:
Target TF and Initial Guess
Desired Circuit TF (black) and Test Circuit (red), bandwidth on middle filter = 600 Hz
12
dB
% Initial guess for parameters of 3 BPFs in parallel
10
% BPF1
8
f01 = 1000; % Center Frequency Hz
6
b1 = 600; % Bandwidth Hz
g1 = 10; % Gain at center frequency in dB
4
% BPF2
2
f02 = 3000; % Center Frequency Hz
0
b2 = 2000; % Bandwidth Hz
g2 = 6; % Gain at center frequency in dB
-2
10
% BPF3
% The third BPF helps fit the shape at the low end
% to slow down the roll off from the middle BPF
f03 = 300; % Center Frequency Hz
b3 = 500; % Bandwidth Hz
g3 = 4; % Gain at center frequency in dB
2
3
10
Hz
4
10
Parameter Optimization BPF Example:
With all other parameters fixed, vary bandwidth of middle BPF
(BPF1) from 200 to 2000 Hz in linear steps of 10. Various iterations:
Desired Circuit TF (black) and Test Circuit (red), bandwidth on middle filter = 450 Hz
12
Desired Circuit TF (black) and Test Circuit (red), bandwidth on middle filter = 800 Hz
12
8
8
6
6
dB
10
dB
10
4
4
2
2
0
0
-2
2
10
3
10
Hz
4
10
Desired Circuit TF (black) and Test Circuit (red), bandwidth on middle filter = 1100 Hz
12
-2
2
10
4
10
Desired Circuit TF (black) and Test Circuit (red), bandwidth on middle filter = 1750 Hz
12
10
10
8
8
6
6
dB
dB
3
10
Hz
4
4
2
2
0
0
-2
2
10
3
10
Hz
4
10
-2
2
10
3
10
Hz
4
10
Parameter Optimization BPF Example
Iteration corresponding to minimum variance error.
Error as a function of filter parameter, with minimum error at 810 Hz
Desired Circuit TF (black) and Test Circuit (red), bandwidth on middle filter = 810 Hz
12
8
7.5
10
7
8
6
6
dB
Variance error
6.5
5.5
4
5
2
4.5
X: 810
Y: 3.76
4
3.5
200
400
600
800
1000
1200
1400
Bandwidth of Filter 1 in Hz
0
1600
1800
2000
-2
2
10
3
10
Hz
4
10