</refsection>
<refsection>
<title>Examples</title>
- <programlisting role="example"><![CDATA[
-
+ <programlisting role="example"><![CDATA[
//Evaluate magnitude response of continuous-time system
hs=analpf(4,'cheb1',[.1 0],5)
fr=0:.1:15;
hf=freq(hs(2),hs(3),%i*fr);
hm=abs(hf);
plot(fr,hm)
-
- ]]></programlisting>
+ ]]></programlisting>
</refsection>
<refsection>
<title>Authors</title>
<refsection>
<title>Examples</title>
- <programlisting role="example"><![CDATA[
- Hlp=iir(3,'lp','ellip',[0.1 0],[.08 .03]);
- pl=roots(Hlp.den);
- zr=roots(Hlp.num);
- gn=coeff(Hlp.num,degree(Hlp.num))/coeff(Hlp.den,degree(Hlp.den));
- z=poly(0,'z');
- a=0.3;
- num=z-a;
- den=1-a*z;
- [npl,nzr,ngn] = bilt(pl,zr,gn,num,den)
+ <programlisting role="example"><![CDATA[
+Hlp=iir(3,'lp','ellip',[0.1 0],[.08 .03]);
+pl=roots(Hlp.den);
+zr=roots(Hlp.num);
+gn=coeff(Hlp.num,degree(Hlp.num))/coeff(Hlp.den,degree(Hlp.den));
+z=poly(0,'z');
+a=0.3;
+num=z-a;
+den=1-a*z;
+[npl,nzr,ngn] = bilt(pl,zr,gn,num,den)
- Hlpt=ngn*poly(nzr,'z','r')/poly(npl,'z','r')
- //comparison with horner
- horner(Hlp,num/den)
- ]]></programlisting>
+Hlpt=ngn*poly(nzr,'z','r')/poly(npl,'z','r')
+
+//comparison with horner
+horner(Hlp,num/den)
+ ]]></programlisting>
</refsection>
<refsection>
<title>Authors</title>
</refsection>
<refsection>
<title>Examples</title>
- <programlisting role="example"><![CDATA[
-
+ <programlisting role="example"><![CDATA[
//squared magnitude response of Butterworth filter
h=buttmag(13,300,1:1000);
mag=20*log(h)'/log(10);
plot2d((1:1000)',mag,[2],"011"," ",[0,-180,1000,20])
-
- ]]></programlisting>
+ ]]></programlisting>
</refsection>
<refsection>
<title>Authors</title>
</refsection>
<refsection>
<title>Examples</title>
- <programlisting role="example"><![CDATA[
-
+ <programlisting role="example"><![CDATA[
x=[1,2,3;4,5,6;7,8,9;10,11,12]
cels=casc(x,'z')
-
- ]]></programlisting>
+ ]]></programlisting>
</refsection>
</refentry>
</refsection>
<refsection>
<title>Examples</title>
- <programlisting role="example"><![CDATA[
-
+ <programlisting role="example"><![CDATA[
w=0.1:0.1:5;mag=1+abs(sin(w));
fresp=cepstrum(w,mag);
plot2d([w',w'],[mag(:),abs(fresp)])
-
- ]]></programlisting>
+ ]]></programlisting>
</refsection>
<refsection>
<title>See Also</title>
</refsection>
<refsection>
<title>Examples</title>
- <programlisting role="example"><![CDATA[
-
+ <programlisting role="example"><![CDATA[
//Chebyshev; ripple in the passband
n=13;epsilon=0.2;omegac=3;sample=0:0.05:10;
h=cheb1mag(n,omegac,epsilon,sample);
plot2d(sample,h)
xtitle('','frequencies','magnitude')
-
- ]]></programlisting>
+ ]]></programlisting>
</refsection>
<refsection>
<title>See Also</title>
</refsection>
<refsection>
<title>Examples</title>
- <programlisting role="example"><![CDATA[
-
+ <programlisting role="example"><![CDATA[
//Chebyshev; ripple in the stopband
n=10;omegar=6;A=1/0.2;sample=0.0001:0.05:10;
h2=cheb2mag(n,omegar,A,sample);
plot(sample,log(h2)/log(10),'frequencies','magnitude in dB')
+
//Plotting of frequency edges
minval=(-maxi(-log(h2)))/log(10);
plot2d([omegar;omegar],[minval;0],[2],"000");
+
//Computation of the attenuation in dB at the stopband edge
attenuation=-log(A*A)/log(10);
plot2d(sample',attenuation*ones(sample)',[5],"000")
-
- ]]></programlisting>
+ ]]></programlisting>
</refsection>
<refsection>
<title>See Also</title>
</refsection>
<refsection>
<title>Examples</title>
- <programlisting role="example"><![CDATA[
-
+ <programlisting role="example"><![CDATA[
chepol(4,'x')
-
- ]]></programlisting>
+ ]]></programlisting>
</refsection>
<refsection>
<title>Authors</title>
<refsection>
<title>Examples</title>
- <programlisting role="example">
-
+ <programlisting role="example"><![CDATA[
x=1:3;
h1=[1,0,0,0,0];h2=[0,1,0,0,0];h3=[0,0,1,0,0];
x1=convol(h1,x),x2=convol(h2,x),x3=convol(h3,x),
p1=poly(x,'x','coeff')
p2=poly(h1+h2+h3,'x','coeff')
p1*p2
-
- </programlisting>
+ ]]></programlisting>
</refsection>
<refsection>
<para>
Computes
</para>
- <programlisting><![CDATA[
-
- n - m
- ====
- \ 1
- cov(m) = > (x(k) - xmean) (y(m+k) - ymean) * ---
- / n
- ====
- k = 1
-
- ]]></programlisting>
+ <programlisting role = ""><![CDATA[
+ n - m
+ ====
+ \ 1
+cov(m) = > (x(k) - xmean) (y(m+k) - ymean) * ---
+ / n
+ ====
+ k = 1
+ ]]></programlisting>
<para>
for m=0,..,<literal>nlag-1</literal> and two vectors <literal>x=[x(1),..,x(n)]</literal>
<literal>y=[y(1),..,y(n)]</literal>
<varlistentry>
<term>Updating method</term>
<listitem>
- <programlisting><![CDATA[
-
- [w,xu]=corr('updt',x1,[y1],w0)
- [w,xu]=corr('updt',x2,[y2],w,xu)
- ...
- wk=corr('updt',xk,[yk],w,xu)
-
- ]]></programlisting>
+ <programlisting role = ""><![CDATA[
+[w,xu]=corr('updt',x1,[y1],w0)
+[w,xu]=corr('updt',x2,[y2],w,xu)
+ ...
+wk=corr('updt',xk,[yk],w,xu)
+ ]]></programlisting>
<para>
With this calling sequence the calculation is updated at each
call to <literal>corr</literal>.</para>
- <programlisting><![CDATA[
-
- w0 = 0*ones(1,2*nlags);
- nlags = power of 2.
-
- ]]></programlisting>
+ <programlisting role = ""><![CDATA[
+w0 = 0*ones(1,2*nlags);
+nlags = power of 2.
+ ]]></programlisting>
<para><literal>x1,x2,...</literal> are parts of <literal>x</literal> such that
<literal>x=[x1,x2,...]</literal> and sizes of <literal>xi</literal> a power of
2. To get <literal>nlags</literal> coefficients a final fft must be
</refsection>
<refsection>
<title>Examples</title>
- <programlisting role="example"><![CDATA[
-
+ <programlisting role="example"><![CDATA[
x=%pi/10:%pi/10:102.4*%pi;
rand('seed');rand('normal');
y=[.8*sin(x)+.8*sin(2*x)+rand(x);.8*sin(x)+.8*sin(1.99*x)+rand(x)];
c=matrix(c,2,128);cov=[];
for j=1:64,cov=[cov;c(:,(j-1)*2+1:2*j)];end;
rand('unif')
-//
+
rand('normal');x=rand(1,256);y=-x;
deff('[z]=xx(inc,is)','z=x(is:is+inc-1)');
deff('[z]=yy(inc,is)','z=y(is:is+inc-1)');
zz=real(fft(w2,1))/256;c6=zz(1:32);
norm(c6-c2,1)
rand('unif')
+
// test for Fortran or C external
//
deff('[y]=xmacro(sec,ist)','y=sin(ist:(ist+sec-1))');
[cc,mm]=corr('fft',xmacro,ymacro,100,2^3);
[cc2,mm2]=corr('fft','corexx','corexy',100,2^3);
[maxi(abs(cc-cc1)),maxi(abs(mm-mm1)),maxi(abs(cc-cc2)),maxi(abs(mm-mm2))]
-
-
- ]]></programlisting>
+ ]]></programlisting>
</refsection>
<refsection>
<title>See Also</title>
<refsection>
<title>Examples</title>
- <programlisting role="example">
-
+ <programlisting role="example"><![CDATA[
rand('normal');rand('seed',0);
x=rand(1:1024-33+1);
//make low-pass filter with eqfir
sm=cspect(100,200,'tr',y);
smsize=maxi(size(sm));fr=(1:smsize)/smsize;
plot(fr,log(sm))
-
- </programlisting>
+ ]]></programlisting>
</refsection>
<refsection>
</refsection>
<refsection>
<title>Examples</title>
- <programlisting role="example"><![CDATA[
-
+ <programlisting role="example"><![CDATA[
a=.7*exp(%i*%pi/6);
[ffr,bds]=xgetech(); //preserve current context
rect=[-1.2,-1.2*sqrt(2),1.2,1.2*sqrt(2)];
zr=real(z);zi=imag(z);
plot2d(zr',zi',[5],"000")
xsetech(ffr,bds); //restore context
-
- ]]></programlisting>
+ ]]></programlisting>
</refsection>
<refsection>
<title>Authors</title>
</refsection>
<refsection>
<title>Examples</title>
- <programlisting role="example"><![CDATA[
+ <programlisting role="example"><![CDATA[
// example #1
t = linspace(0,16*%pi,1000)';
x = -20 + t + 0.3*sin(0.5*t) + sin(t) + 2*sin(2*t) + 0.5*sin(3*t);
plot2d(t,[x y],style=[2 5])
legend(["before detrend","after detrend"]);
xgrid()
- ]]></programlisting>
+ ]]></programlisting>
</refsection>
<refsection>
<title>Authors</title>
</refsection>
<refsection>
<title>Examples</title>
- <programlisting role="example"><![CDATA[
-
+ <programlisting role="example"><![CDATA[
n=8;omega = exp(-2*%pi*%i/n);
j=0:n-1;F=omega.^(j'*j); //Fourier matrix
x=1:8;x=x(:);
inv(F)*x
fft(x,1)
dft(x,1)
-
- ]]></programlisting>
+ ]]></programlisting>
</refsection>
<refsection>
<title>See Also</title>
</refsection>
<refsection>
<title>Examples</title>
- <programlisting role="example"><![CDATA[
-
+ <programlisting role="example"><![CDATA[
deff('[alpha,BeTa]=alpha_beta(n,m,m1)',...
'if 2*int(n/2)==n then, BeTa=K1; else, BeTa=0;end;...
alpha=%k(1-m1)/%k(1-m);')
//Now we map the positive real axis into the contour...
z=alpha*%asn(sample/omegac,m)+Beta*ones(sample);
plot(sample,ell1mag(epsilon,m1,z))
-
- ]]></programlisting>
+ ]]></programlisting>
</refsection>
<refsection>
<title>See Also</title>
</refsection>
<refsection>
<title>Examples</title>
- <programlisting role="example"><![CDATA[
-
+ <programlisting role="example"><![CDATA[
hn=eqfir(33,[0 .2;.25 .35;.4 .5],[0 1 0],[1 1 1]);
[hm,fr]=frmag(hn,256);
plot(fr,hm),
-
- ]]></programlisting>
+ ]]></programlisting>
</refsection>
<refsection>
<title>Authors</title>
<refsection>
<title>Examples</title>
- <programlisting role="example">
+ <programlisting role="example"><![CDATA[
[cells,fact,zzeros,zpoles]=eqiir('lp','ellip',[2*%pi/10,4*%pi/10],0.02,0.001)
h=fact*poly(zzeros,'z')/poly(zpoles,'z')
- </programlisting>
+ ]]></programlisting>
</refsection>
<refsection>
filter model.
The algorithm tries to compute the solution P as the growing limit of a
sequence of matrices Pn such that</para>
- <programlisting><![CDATA[
-
+ <programlisting role = ""><![CDATA[
-1
Pn+1=F*Pn*F'+(G-F*Pn*h')*(R0-H*Pn*H') *(G'-H*Pn*F')
-1
P0=G*R0 *G'
-
- ]]></programlisting>
+ ]]></programlisting>
<para>
Note that this method may not converge,especially when F has poles
near the unit circle. Use preferably the srfaur function.</para>
<literal>j1</literal> lies in <literal>1:dim(1),</literal> <literal>j2</literal> in
<literal>1:dim(2),...</literal> one gets a p-variate FFT by calling p times
<literal>fft</literal> as follows</para>
- <programlisting><![CDATA[incrk=1; x=a; for k=1:p x=fft(x ,-1,dim(k),incrk)
- incrk=incrk*dim(k) end]]></programlisting>
+ <programlisting role = ""><![CDATA[
+incrk=1; x=a;
+for k=1:p
+ x=fft(x ,-1,dim(k),incrk)
+ incrk=incrk*dim(k)
+end
+ ]]></programlisting>
<para>where <literal>dimk</literal> is the dimension of the current variable
w.r.t which one is integrating and <literal>incrk</literal> is the increment
which separates two successive <literal>jk</literal> elements in
</refsection>
<refsection>
<title>Examples</title>
- <programlisting role="example"><![CDATA[
-
- //Comparison with explicit formula
- //----------------------------------
- a=[1;2;3];n=size(a,'*');
- norm(1/n*exp(2*%i*%pi*(0:n-1)'.*.(0:n-1)/n)*a -fft(a,1))
- norm(exp(-2*%i*%pi*(0:n-1)'.*.(0:n-1)/n)*a -fft(a,-1))
+ <programlisting role="example"><![CDATA[
+//Comparison with explicit formula
+//----------------------------------
+a=[1;2;3];n=size(a,'*');
+norm(1/n*exp(2*%i*%pi*(0:n-1)'.*.(0:n-1)/n)*a -fft(a,1))
+norm(exp(-2*%i*%pi*(0:n-1)'.*.(0:n-1)/n)*a -fft(a,-1))
- //Frequency components of a signal
- //----------------------------------
- // build a noides signal sampled at 1000hz containing to pure frequencies
- // at 50 and 70 Hz
- sample_rate=1000;
- t = 0:1/sample_rate:0.6;
- N=size(t,'*'); //number of samples
- s=sin(2*%pi*50*t)+sin(2*%pi*70*t+%pi/4)+grand(1,N,'nor',0,1);
+//Frequency components of a signal
+//----------------------------------
+// build a noides signal sampled at 1000hz containing to pure frequencies
+// at 50 and 70 Hz
+sample_rate=1000;
+t = 0:1/sample_rate:0.6;
+N=size(t,'*'); //number of samples
+s=sin(2*%pi*50*t)+sin(2*%pi*70*t+%pi/4)+grand(1,N,'nor',0,1);
- y=fft(s);
- //the fft response is symetric we retain only the first N/2 points
- f=sample_rate*(0:(N/2))/N; //associated frequency vector
- n=size(f,'*')
- clf()
- plot2d(f,abs(y(1:n)))
-
- ]]></programlisting>
+y=fft(s);
+//the fft response is symetric we retain only the first N/2 points
+f=sample_rate*(0:(N/2))/N; //associated frequency vector
+n=size(f,'*')
+clf()
+plot2d(f,abs(y(1:n)))
+ ]]></programlisting>
</refsection>
<refsection>
<title>See Also</title>
</refsection>
<refsection>
<title>Description</title>
- <programlisting><![CDATA[This functions performs the two-dimension discrete Fourier
- transform.]]></programlisting>
+ <programlisting role = ""><![CDATA[
+This functions performs the two-dimension discrete Fourier transform.
+ ]]></programlisting>
<para><literal>y=fft2(x)</literal>y and x have the same size</para>
<para><literal>y=fft2(x,m,n):</literal> If <literal>m</literal> (respectively
<literal>n</literal>) is less than the rows number (respectively columns) of
</refsection>
<refsection>
<title>Examples</title>
- <programlisting role="example"><![CDATA[
+ <programlisting role="example"><![CDATA[
//Comparison with explicit formula
a=[1 2 3 ;4 5 6 ;7 8 9 ;10 11 12]
m=size(a,1)
n=size(a,2)
+
// fourier transform along the rows
for i=1:n
a1(:,i)=exp(-2*%i*%pi*(0:m-1)'.*.(0:m-1)/m)*a(:,i)
end
+
// fourier transform along the columns
for j=1:m
a2temp=exp(-2*%i*%pi*(0:n-1)'.*.(0:n-1)/n)*(a1(j,:)).'
a2(j,:)=a2temp.'
end
norm(a2-fft2(a))
-
-
-
- ]]></programlisting>
+ ]]></programlisting>
</refsection>
<refsection>
<title>See Also</title>
<para>
If <literal>x</literal> is an <literal>m</literal> by <literal>n</literal> matrix <literal>y</literal> is the matrix
<literal>x([m/2+1:n,1:m/2],[n/2+1:n,1:n/2])</literal>.</para>
- <programlisting><![CDATA[
-
+ <programlisting role = ""><![CDATA[
[x11 x12] [x22 x21]
x=[ ] gives y=[ ]
[x21 x22] [x12 x11]
-
- ]]></programlisting>
+ ]]></programlisting>
<para><literal>y= fftshift(x,n)</literal> make the swap only along the <literal>n</literal>th dimension</para>
</refsection>
<refsection>
<title>Examples</title>
- <programlisting role="example"><![CDATA[
-
+ <programlisting role="example"><![CDATA[
//make a signal
t=0:0.1:1000;
x=3*sin(t)+8*sin(3*t)+0.5*sin(5*t)+3*rand(t);
//compute the fft
y=fft(x,-1);
+
//display
xbasc();
subplot(2,1,1);plot2d(abs(y))
0.5*sin(5*t')*sin(5*t)+3*rand(t')*rand(t);
//compute the fft
y=fft(x,-1);
+
//display
xbasc();
xset('colormap',hotcolormap(256))
subplot(2,1,1);Matplot(abs(y))
subplot(2,1,2);Matplot(fftshift(abs(y)))
-
-
- ]]></programlisting>
+ ]]></programlisting>
</refsection>
<refsection>
<title>See Also</title>
</refsection>
<refsection>
<title>Examples</title>
- <programlisting role="example"><![CDATA[
-
+ <programlisting role="example"><![CDATA[
plot(filt_sinc(100,0.1))
- ]]></programlisting>
+ ]]></programlisting>
</refsection>
<refsection>
<title>See Also</title>
</para>
</refsection>
<refsection>
- <title>Examples</title>
- <programlisting role="example"><![CDATA[
-
- ]]></programlisting>
- </refsection>
- <refsection>
<title>References</title>
<para>
Oppenheim, A. V. and R.W. Schafer. Discrete-Time Signal Processing, Englewood Cliffs, NJ: Prentice-Hall, 1989, pp. 311-312.
</refsection>
<refsection>
<title>Examples</title>
- <programlisting role="example"><![CDATA[
-
+ <programlisting role="example"><![CDATA[
w=0.01:0.01:2;s=poly(0,'s');
G=syslin('c',2*(s^2+0.1*s+2), (s^2+s+1)*(s^2+0.3*s+1));
fresp=repfreq(G,w);
Gid=frfit(w,fresp,4);
frespfit=repfreq(Gid,w);
bode(w,[fresp;frespfit])
-
- ]]></programlisting>
+ ]]></programlisting>
</refsection>
<refsection>
<title>See Also</title>
<refsection>
<title>Examples</title>
- <programlisting role="example">
- hz=iir(3,'bp','cheb1',[.15 .25],[.08 .03]);
- [hzm,fr]=frmag(hz,256);
- plot(fr,hzm)
- hz=iir(3,'bp','ellip',[.15 .25],[.08 .03]);
- [hzm,fr]=frmag(hz,256);
- plot(fr,hzm,'r')
- </programlisting>
+ <programlisting role="example"><![CDATA[
+hz=iir(3,'bp','cheb1',[.15 .25],[.08 .03]);
+[hzm,fr]=frmag(hz,256);
+plot(fr,hzm)
+hz=iir(3,'bp','ellip',[.15 .25],[.08 .03]);
+[hzm,fr]=frmag(hz,256);
+plot(fr,hzm,'r')
+ ]]></programlisting>
</refsection>
<refsection>
</refsection>
<refsection>
<title>Examples</title>
- <programlisting role="example"><![CDATA[
-
+ <programlisting role="example"><![CDATA[
//
//Example of how to use the fsfirlin macro for the design
//of an FIR filter by a frequency sampling technique.
pas=1/prod(size(hst1))*.5;
fg=0:pas:.5;//normalized frequencies grid
plot2d([1 1].*.fg(1:257)',[hst1' hst2']);
+
// 2nd example
hd=[0*ones(1,15) ones(1,10) 0*ones(1,39)];//desired samples
hst1=fsfirlin(hd,1);//filter with no sample in the transition
n=prod(size(hst1))
plot(fg(1:n),hst1);
plot2d(fg(1:n)',hst2',[3],"000");
-
- ]]></programlisting>
+ ]]></programlisting>
</refsection>
<refsection>
<title>See Also</title>
</refsection>
<refsection>
<title>Examples</title>
- <programlisting role="example"><![CDATA[
-
+ <programlisting role="example"><![CDATA[
z=poly(0,'z');
h=z/(z-.5);
[tg,fr]=group(100,h);
plot(fr,tg)
-
- ]]></programlisting>
+ ]]></programlisting>
</refsection>
<refsection>
<title>Authors</title>
</refsection>
<refsection>
<title>Examples</title>
- <programlisting role="example"><![CDATA[
-
+ <programlisting role="example"><![CDATA[
//Example of how to use the hank macro for
//building a Hankel matrix from multidimensional
//data (covariance or Markov parameters e.g.)
//being the transposition in scilab)
//
//we take here d=2 and n=64
-//
+
c=rand(2,2*64)
-//
+
//generate the hankel matrix H (with 4 bloc-rows and 5 bloc-columns)
//from the data in c
-//
+
H=hank(4,5,c);
-//
-
- ]]></programlisting>
+ ]]></programlisting>
</refsection>
<refsection>
<title>See Also</title>
</simplelist>
</refsection>
<refsection><title>Examples</title>
- <programlisting role="example"><![CDATA[
-
- plot(hilb(51))
-
- ]]></programlisting></refsection>
+ <programlisting role="example"><![CDATA[
+plot(hilb(51))
+ ]]></programlisting></refsection>
<refsection>
<title>Authors</title>
<para>C. B. </para></refsection>
</simplelist>
</refsection>
<refsection><title>Examples</title>
- <programlisting role="example"><![CDATA[
-
-
-
- //compare the discrete-time analytic signal imaginary part of the impulse real signal
- // with the FIR approximation of the Hilbert transform filter
- m=25;
- n=2*m+1;
- y=hilbert(eye(n,1));
- h=hilb(n)';
- h=[h((m+1):$);h(1:m)];
- plot([imag(y) h])
-
- ]]></programlisting></refsection>
+ <programlisting role="example"><![CDATA[
+//compare the discrete-time analytic signal imaginary part of the impulse real signal
+// with the FIR approximation of the Hilbert transform filter
+m=25;
+n=2*m+1;
+y=hilbert(eye(n,1));
+h=hilb(n)';
+h=[h((m+1):$);h(1:m)];
+plot([imag(y) h])
+ ]]></programlisting></refsection>
<refsection>
<title>Authors</title>
<para>C. B. </para></refsection>
<para>And then, the filter can be represented in bode plot.</para>
- <programlisting>
+ <programlisting role = ""><![CDATA[
// analog elliptic (Bessel), order 2, cutoff 1 Hz
-Epsilon = 3; // ripple of filter in pass band (0<epsilon<1)
-A = 60; // attenuation of filter in stop band (A>1)
+Epsilon = 3; // ripple of filter in pass band (0<epsilon<1)
+A = 60; // attenuation of filter in stop band (A<1)
OmegaC = 10; // pass band cut-off frequency in Hertz
OmegaR = 50; // stop band cut-off frequency in Hertz
// Plot the resulting filter
bode(elatf,0.01,100);
title('Analog Elliptic filter');
- </programlisting>
+ ]]></programlisting>
<para>Bode plot is only suited for analog filters.</para>
<para>For our demonstration, we will use the <link
linkend="iir">iir</link> function.</para>
- <programlisting role="example">
+ <programlisting role="example"><![CDATA[
Order = 2; // The order of the filter
Fs = 1000; // The sampling frequency
Fcutoff = 40; // The cutoff frequency
subplot(2,1,2);
plot2d(Fs*frq,phi_repf);
xtitle('Obtained Frequency Response (Phase in degree)');
- </programlisting>
+ ]]></programlisting>
<para>Here is the representation of the digital elliptic filter.</para>
<para>In the following examples, we will filter a gaussian noise.</para>
- <programlisting>
+ <programlisting role = ""><![CDATA[
rand('normal');
Input = rand(1,1000); // Produce a random gaussian noise
t = 1:1000;
subplot(2,1,2);
plot(t,y);
xtitle('The filtered gaussian noise','t','y');
- </programlisting>
+ ]]></programlisting>
<para>Here is the representation of the signal before and after
filtering.</para>
<para>First, we try using the <link linkend="dscr">dscr</link> + <link
linkend="flts">flts</link> functions.</para>
- <programlisting>
+ <programlisting role = ""><![CDATA[
rand('normal');
Input = rand(1,1000); // Produce a random gaussian noise
n = 1:1000; // The sample index
subplot(2,1,2);
plot(n,y);
xtitle('The filtered gaussian noise','n','y');
- </programlisting>
+ ]]></programlisting>
<para>Here is the representation of the signal before and after filtering
using the <link linkend="dscr">dscr</link> + <link
<para>Next, we use the <link linkend="csim">csim</link> function.</para>
- <programlisting>
+ <programlisting role = ""><![CDATA[
rand('normal');
Input = rand(1,1000); // Produce a random gaussian noise
t = 1:1000;
subplot(2,1,2);
plot(t,y);
xtitle('The filtered gaussian noise','t','y');
- </programlisting>
+ ]]></programlisting>
<para>Here is the representation of the signal before and after filtering
using the <link linkend="csim">csim</link> approach.</para>
<refsection>
<title>Examples</title>
- <programlisting role="example">
-
+ <programlisting role="example"><![CDATA[
hz=iir(3,'bp','ellip',[.15 .25],[.08 .03]);
[hzm,fr]=frmag(hz,256);
plot2d(fr',hzm')
xtitle('Discrete IIR filter band pass 0.15<fr<0.25 ',' ',' ');
q=poly(0,'q'); //to express the result in terms of the delay operator q=z^-1
hzd=horner(hz,1/q)
-
- </programlisting>
+ ]]></programlisting>
</refsection>
<refsection>
<refsection>
<title>Examples</title>
- <programlisting role="example">
-
+ <programlisting role="example"><![CDATA[
//We use the 'levin' macro for solving the normal equations
//on two examples: a one-dimensional and a two-dimensional process.
//We need the covariance sequence of the stochastic process.
//We generate the process defined by two sinusoids (1Hz and 2 Hz)
//in additive Gaussian noise (this is the observed process);
//the simulated process is sampled at 10 Hz (step 0.1 in t, underafter).
-//
+
t1=0:.1:100;rand('normal');
y1=sin(2*%pi*t1)+sin(2*%pi*2*t1);y1=y1+rand(y1);plot(t1,y1);
-//
+
//covariance of y1
-//
+
nlag=128;
c1=corr(y1,nlag);
c1=c1';//c1 needs to be given columnwise (see the section PARAMETERS of this help)
-//
+
//compute the filter for a maximum order of n=10
//la is a list-type variable each element of which
//containing the filters of order ranging from 1 to n; (try varying n)
//in the d-dimensional case this is a matrix polynomial (square, d X d)
//sig gives, the same way, the mean-square error
-//
+
n=15;
[la1,sig1]=levin(n,c1);
-//
+
//verify that the roots of 'la' contain the
//frequency spectrum of the observed process y
//(remember that y is sampled -in our example
//the original frequencies (1Hz and 2 Hz) through
//the log and correct scaling by the frequency sampling)
//we verify this for each filter order
-//
+
for i=1:n, s1=roots(la1(i));s1=log(s1)/2/%pi/.1;
-//
+
//now we get the estimated poles (sorted, positive ones only !)
-//
+
s1=sort(imag(s1));s1=s1(1:i/2);end;
-//
+
//the last two frequencies are the ones really present in the observed
-//process ---> the others are "artifacts" coming from the used model size.
+//process ---> the others are "artifacts" coming from the used model size.
//This is related to the rather difficult problem of order estimation.
//
//2) A 2-dimensional process
// |y_1| y_1=sin(2*Pi*t)+sin(2*Pi*2*t)+Gaussian noise
// y=| | with :
// |y_2| y_2=sin(2*Pi*3*t)+sin(2*Pi*4*t)+Gaussian noise
-//
-//
+
d=2;dt=0.1;
nlag=64;
t2=0:2*%pi*dt:100;
c2=matrix(c2,2,128);cov=[];
for j=1:64,cov=[cov;c2(:,(j-1)*d+1:j*d)];end;//covar. columnwise
c2=cov;
-//
+
//in the multidimensional case, we have to compute the
//roots of the determinant of the matrix polynomial
-//(easy in the 2-dimensional case but tricky if d>=3 !).
+//(easy in the 2-dimensional case but tricky if d>=3 !).
//We just do that here for the maximum desired
//filter order (n); mp is the matrix polynomial of degree n
-//
+
[la2,sig2]=levin(n,c2);
mp=la2(n);determinant=mp(1,1)*mp(2,2)-mp(1,2)*mp(2,1);
s2=roots(determinant);s2=log(s2)/2/%pi/0.1;//same trick as above for 1D process
s2=sort(imag(s2));s2=s2(1:d*n/2);//just the positive ones !
-//
+
//There the order estimation problem is seen to be much more difficult !
//many artifacts ! The 4 frequencies are in the estimated spectrum
//but beneath many non relevant others.
-//
-
- </programlisting>
+ ]]></programlisting>
</refsection>
<refsection>
along its first dimension, two points along its second
dimension and three points along its third dimension the row
vector is arranged as follows</para>
- <programlisting><![CDATA[
-
- x=[x(1,1,1),x(2,1,1),x(3,1,1),
- x(1,2,1),x(2,2,1),x(3,2,1),
- x(1,1,2),x(2,1,2),x(3,1,2),
- x(1,2,2),x(2,2,2),x(3,2,2),
- x(1,1,3),x(2,1,3),x(3,1,3),
- x(1,2,3),x(2,2,3),x(3,2,3)]
-
- ]]></programlisting>
+ <programlisting role = ""><![CDATA[
+x=[x(1,1,1),x(2,1,1),x(3,1,1),
+ x(1,2,1),x(2,2,1),x(3,2,1),
+ x(1,1,2),x(2,1,2),x(3,1,2),
+ x(1,2,2),x(2,2,2),x(3,2,2),
+ x(1,1,3),x(2,1,3),x(3,1,3),
+ x(1,2,3),x(2,2,3),x(3,2,3)]
+ ]]></programlisting>
<para>
and the <literal>dim</literal> vector is: <literal>dim=[3,2,3]</literal></para>
</refsection>
</refsection>
<refsection>
<title>Examples</title>
- <programlisting role="example"><![CDATA[
-
+ <programlisting role="example"><![CDATA[
w=0.01:0.01:2;s=poly(0,'s');
G=syslin('c',2*(s^2+0.1*s+2),(s^2+s+1)*(s^2+0.3*s+1)); // syslin('c',Num,Den);
fresp=repfreq(G,w);
Gid=mrfit(w,mag,4);
frespfit=repfreq(Gid,w);
plot2d([w',w'],[mag(:),abs(frespfit(:))])
-
- ]]></programlisting>
+ ]]></programlisting>
</refsection>
<refsection>
<title>See Also</title>
</refsection>
<refsection>
<title>Examples</title>
- <programlisting role="example"><![CDATA[
-
+ <programlisting role="example"><![CDATA[
m=0.8;z=%asn(1/sqrt(m),m);K=real(z);Ktilde=imag(z);
x2max=1/sqrt(m);
x1=0:0.05:1;x2=1:((x2max-1)/20):x2max;x3=x2max:0.05:10;
y=%asn(x,m);
rect=[0,-Ktilde,1.1*K,2*Ktilde];
plot2d(real(y)',imag(y)',1,'011',' ',rect)
-//
+
deff('y=f(t)','y=1/sqrt((1-t^2)*(1-m*t^2))');
intg(0,0.9,f)-%asn(0.9,m) //Works for real case only!
-
- ]]></programlisting>
+ ]]></programlisting>
</refsection>
<refsection>
<title>Authors</title>
</refsection>
<refsection>
<title>Examples</title>
- <programlisting role="example"><![CDATA[
-
+ <programlisting role="example"><![CDATA[
m=0.4;
%asn(1,m)
%k(m)
-
- ]]></programlisting>
+ ]]></programlisting>
</refsection>
<refsection>
<title>See Also</title>
</refsection>
<refsection>
<title>Examples</title>
- <programlisting role="example"><![CDATA[
-
+ <programlisting role="example"><![CDATA[
m=0.36;
K=%k(m);
P=4*K; //Real period
z1=%sn(%i*ima_val1,m);z2=%sn(%i*ima_val2,m);
plot2d([ima_val1',ima_val2'],[imag(z1)',imag(z2)']);
xgrid(3)
-
- ]]></programlisting>
+ ]]></programlisting>
</refsection>
<refsection>
<title>See Also</title>
</refsection>
<refsection>
<title>Examples</title>
- <programlisting role="example"><![CDATA[
-
-//
+ <programlisting role="example"><![CDATA[
//This example may usefully be compared with the results from
//the 'levin' macro (see the corresponding help and example)
//
//We consider the process defined by two sinusoids (1Hz and 2 Hz)
//in additive Gaussian noise (this is the observation);
//the simulated process is sampled at 10 Hz.
-//
+
t=0:.1:100;rand('normal');
y=sin(2*%pi*t)+sin(2*%pi*2*t);y=y+rand(y);plot(t,y)
-//
+
//covariance of y
-//
+
nlag=128;
c=corr(y,nlag);
-//
+
//hankel matrix from the covariance sequence
//(we can choose to take more information from covariance
//by taking greater n and m; try it to compare the results !
-//
+
n=20;m=20;
h=hank(n,m,c);
-//
+
//compute the Markov representation (mh,mf,mg)
//We just take here a state dimension equal to 4 :
//this is the rather difficult problem of estimating the order !
//Try varying ns !
//(the observation dimension is here equal to one)
+
ns=4;
[mh,mf,mg]=phc(h,1,ns);
-//
+
//verify that the spectrum of mf contains the
//frequency spectrum of the observed process y
//(remember that y is sampled -in our example
//at 10Hz (T=0.1s) so that we need
//to retrieve the original frequencies through the log
//and correct scaling by the frequency sampling)
-//
+
s=spec(mf);s=log(s);
s=s/2/%pi/.1;
-//
+
//now we get the estimated spectrum
imag(s),
-//
-
- ]]></programlisting>
+ ]]></programlisting>
</refsection>
<refsection>
<title>See Also</title>
<refsection>
<title>Examples</title>
- <programlisting role="example">
-
- rand('normal');rand('seed',0);
- x=rand(1:1024-33+1);
- //make low-pass filter with eqfir
- nf=33;bedge=[0 .1;.125 .5];des=[1 0];wate=[1 1];
- h=eqfir(nf,bedge,des,wate);
- //filter white data to obtain colored data
- h1=[h 0*ones(1:maxi(size(x))-1)];
- x1=[x 0*ones(1:maxi(size(h))-1)];
- hf=fft(h1,-1); xf=fft(x1,-1);y=real(fft(hf.*xf,1));
-
- //plot magnitude of filter
- h2=[h 0*ones(1:968)];hf2=fft(h2,-1);hf2=real(hf2.*conj(hf2));
- hsize=maxi(size(hf2));fr=(1:hsize)/hsize;plot(fr,log(hf2));
- //pspect example
- sm=pspect(100,200,'tr',y);smsize=maxi(size(sm));fr=(1:smsize)/smsize;
- plot(fr,log(sm));
- rand('unif');
-
- </programlisting>
+ <programlisting role="example"><![CDATA[
+rand('normal');rand('seed',0);
+x=rand(1:1024-33+1);
+
+//make low-pass filter with eqfir
+nf=33;bedge=[0 .1;.125 .5];des=[1 0];wate=[1 1];
+h=eqfir(nf,bedge,des,wate);
+
+//filter white data to obtain colored data
+h1=[h 0*ones(1:maxi(size(x))-1)];
+x1=[x 0*ones(1:maxi(size(h))-1)];
+hf=fft(h1,-1); xf=fft(x1,-1);y=real(fft(hf.*xf,1));
+
+//plot magnitude of filter
+h2=[h 0*ones(1:968)];hf2=fft(h2,-1);hf2=real(hf2.*conj(hf2));
+hsize=maxi(size(hf2));fr=(1:hsize)/hsize;plot(fr,log(hf2));
+
+//pspect example
+sm=pspect(100,200,'tr',y);smsize=maxi(size(sm));fr=(1:smsize)/smsize;
+plot(fr,log(sm));
+rand('unif');
+ ]]></programlisting>
</refsection>
<refsection>
<para>Minimax approximation of a frequency domain magnitude response. The
approximation takes the form</para>
- <programlisting> h = sum[a(i)*cos(weight)], i=1:n</programlisting>
+ <programlisting role = ""><![CDATA[
+h = sum[a(i)*cos(weight)], i=1:n
+ ]]></programlisting>
<para>An FIR, linear-phase filter can be obtained from the the output of
<literal>remez</literal> by using the following commands:</para>
- <programlisting> hn(1:nc-1)=an(nc:-1:2)/2;
- hn(nc)=an(1);
- hn(nc+1:2*nc-1)=an(2:nc)/2; </programlisting>
+ <programlisting role = ""><![CDATA[
+hn(1:nc-1)=an(nc:-1:2)/2;
+hn(nc)=an(1);
+hn(nc+1:2*nc-1)=an(2:nc)/2;
+ ]]></programlisting>
<para>This function is mainly intended to be called by the <link
linkend="remezb">remezb function</link>.</para>
<refsection>
<title>Examples</title>
- <programlisting role="example"> nc=21;
- ngrid=nc*250;
- fgrid=.5*(0:(ngrid-1))/(ngrid-1);
- mag(1:ngrid/2)=ones(1:ngrid/2);
- mag(ngrid/2+1:ngrid)=0*ones(1:ngrid/2);
- weight=ones(fgrid);
- guess=round(1:ngrid/nc:ngrid);
- guess(nc+1)=ngrid;
- guess(nc+2)=ngrid;
- an=remez(guess,mag,fgrid,weight);</programlisting>
+ <programlisting role="example"><![CDATA[
+nc=21;
+ngrid=nc*250;
+fgrid=.5*(0:(ngrid-1))/(ngrid-1);
+mag(1:ngrid/2)=ones(1:ngrid/2);
+mag(ngrid/2+1:ngrid)=0*ones(1:ngrid/2);
+weight=ones(fgrid);
+guess=round(1:ngrid/nc:ngrid);
+guess(nc+1)=ngrid;
+guess(nc+2)=ngrid;
+an=remez(guess,mag,fgrid,weight);
+ ]]></programlisting>
</refsection>
<refsection>
<member><link linkend="eqfir">eqfir</link></member>
</simplelist>
</refsection>
-</refentry>
\ No newline at end of file
+</refentry>
for n=0,1,...,nc. An FIR, linear-phase filter
can be obtained from the the output of the function
by using the following commands</para>
- <programlisting><![CDATA[
-
- hn(1:nc-1)=an(nc:-1:2)/2;
- hn(nc)=an(1);
- hn(nc+1:2*nc-1)=an(2:nc)/2;
-
- ]]></programlisting>
+ <programlisting role = ""><![CDATA[
+hn(1:nc-1)=an(nc:-1:2)/2;
+hn(nc)=an(1);
+hn(nc+1:2*nc-1)=an(2:nc)/2;
+ ]]></programlisting>
</refsection>
<refsection>
<title>Examples</title>
- <programlisting role="example"><![CDATA[
-
+ <programlisting role="example"><![CDATA[
// Choose the number of cosine functions and create a dense grid
// in [0,.24) and [.26,.5)
nc=21;ngrid=nc*16;
fg=.24*(0:ngrid/2-1)/(ngrid/2-1);
fg(ngrid/2+1:ngrid)=fg(1:ngrid/2)+.26*ones(1:ngrid/2);
+
// Specify a low pass filter magnitude for the desired response
ds(1:ngrid/2)=ones(1:ngrid/2);
ds(ngrid/2+1:ngrid)=zeros(1:ngrid/2);
+
// Specify a uniform weighting function
wt=ones(fg);
+
// Run remezb
an=remezb(nc,fg,ds,wt)
+
// Make a linear phase FIR filter
hn(1:nc-1)=an(nc:-1:2)/2;
hn(nc)=an(1);
hn(nc+1:2*nc-1)=an(2:nc)/2;
+
// Plot the filter's magnitude response
plot(.5*(0:255)/256,frmag(hn,256));
-//////////////
+
// Choose the number of cosine functions and create a dense grid in [0,.5)
nc=21; ngrid=nc*16;
fg=.5*(0:(ngrid-1))/ngrid;
+
// Specify a triangular shaped magnitude for the desired response
ds(1:ngrid/2)=(0:ngrid/2-1)/(ngrid/2-1);
ds(ngrid/2+1:ngrid)=ds(ngrid/2:-1:1);
+
// Specify a uniform weighting function
wt=ones(fg);
+
// Run remezb
an=remezb(nc,fg,ds,wt)
+
// Make a linear phase FIR filter
hn(1:nc-1)=an(nc:-1:2)/2;
hn(nc)=an(1);
hn(nc+1:2*nc-1)=an(2:nc)/2;
+
// Plot the filter's magnitude response
plot(.5*(0:255)/256,frmag(hn,256));
-
- ]]></programlisting>
+ ]]></programlisting>
</refsection>
<refsection>
<title>See Also</title>
<para>
Applicable values for the first call:
</para>
- <programlisting><![CDATA[
-
+ <programlisting role = ""><![CDATA[
theta=phi=psi=l=0*ones(1,3*n). p=eye(3*n,3*n)
-
- ]]></programlisting>
+ ]]></programlisting>
</listitem>
</varlistentry>
<varlistentry>
<listitem>
<para>optional argument (forgetting constant) choosed close to 1 as convergence occur:</para>
<para><literal>lambda=[lambda0,alfa,beta]</literal> evolves according to :</para>
- <programlisting><![CDATA[
-
+ <programlisting role = ""><![CDATA[
lambda(t)=alfa*lambda(t-1)+beta
-
- ]]></programlisting>
+ ]]></programlisting>
<para>
with <literal>lambda(0)=lambda0</literal></para>
</listitem>
<listitem>
<para>contraction factor to be chosen close to 1 as convergence occurs.</para>
<para><literal>k=[k0,mu,nu]</literal> evolves according to:</para>
- <programlisting><![CDATA[
-
+ <programlisting role = ""><![CDATA[
k(t)=mu*k(t-1)+nu
-
- ]]></programlisting>
+ ]]></programlisting>
<para>
with <literal>k(0)=k0</literal>.</para>
</listitem>
Uses Ljung-Soderstrom recursive prediction error method.
Model considered is the following:
</para>
- <programlisting><![CDATA[
-
+ <programlisting role = ""><![CDATA[
y(t)+a(1)*y(t-1)+...+a(n)*y(t-n)=
b(1)*u(t-1)+...+b(n)*u(t-n)+e(t)+c(1)*e(t-1)+...+c(n)*e(t-n)
-
- ]]></programlisting>
+ ]]></programlisting>
<para>
</para>
<para>
</refsection>
<refsection>
<title>Examples</title>
- <programlisting role="example"><![CDATA[
-
+ <programlisting role="example"><![CDATA[
plot(sincd(10,1))
-
- ]]></programlisting>
+ ]]></programlisting>
</refsection>
<refsection>
<title>Authors</title>
</refsection>
<refsection>
<title>Examples</title>
- <programlisting role="example"><![CDATA[
-
+ <programlisting role="example"><![CDATA[
//GENERATE SIGNAL
x=%pi/10:%pi/10:102.4*%pi;
rand('seed',0);rand('normal');
y=[1;1]*sin(x)+[sin(2*x);sin(1.9*x)]+rand(2,1024);
+
//COMPUTE CORRELATIONS
c=[];for j=1:2,for k=1:2,c=[c;corr(y(k,:),y(j,:),64)];end;end
c=matrix(c,2,128);
+
//FINDING H,F,G with 6 states
hk=hank(20,20,c);
[H,F,G]=phc(hk,2,6);
+
//SOLVING RICCATI EQN
r0=c(1:2,1:2);
[P,s,t,l,Rt,Tt]=srfaur(H,F,G,r0,200);
+
//Make covariance matrix exactly symetric
Rt=(Rt+Rt')/2
-
- ]]></programlisting>
+ ]]></programlisting>
</refsection>
<refsection>
<title>See Also</title>
<refsection>
<title>Examples</title>
- <programlisting role="example">
- [fact,b2,b1,b0,c1,c0,zzeros,zpoles]=syredi(1,4,[2*%pi/10,4*%pi/10,0,0],0.02,0.001);
- h=fact*(b0+b1*%z+b2*%z^2)./(c0+c1*%z+%z^2)
- </programlisting>
+ <programlisting role="example"><![CDATA[
+[fact,b2,b1,b0,c1,c0,zzeros,zpoles]=syredi(1,4,[2*%pi/10,4*%pi/10,0,0],0.02,0.001);
+h=fact*(b0+b1*%z+b2*%z^2)./(c0+c1*%z+%z^2)
+ ]]></programlisting>
</refsection>
<refsection>
<member><link linkend="eqiir">eqiir</link></member>
</simplelist>
</refsection>
-</refentry>
\ No newline at end of file
+</refentry>
define system function which generates the next
observation given the old state.
System recursively calculated</para>
- <programlisting><![CDATA[
-
- x1=f*x0+g*u
- y=h*x0+v
-
- ]]></programlisting>
+ <programlisting role = ""><![CDATA[
+x1=f*x0+g*u
+y=h*x0+v
+ ]]></programlisting>
<para>
where <literal>u</literal> is distributed <literal>N(0,q)</literal>
and <literal>v</literal> is distribute <literal>N(0,r)</literal>.</para>
</refsection>
<refsection>
<title>Examples</title>
- <programlisting role="example"><![CDATA[
-
+ <programlisting role="example"><![CDATA[
clf()
Hlp=iir(3,'lp','ellip',[0.1 0],[.08 .03]);
Hbp=trans(Hlp,'bp',[0.01 0.1]);
clf();gainplot([Hlp;Hbp;Hsb],1d-3,0.48);
l=legend(['original low pass';'band pass';'stop band']);
l.legend_location="in_lower_left";
- ]]></programlisting>
+ ]]></programlisting>
</refsection>
<refsection>
<title>Authors</title>
</refsection>
<refsection>
<title>Examples</title>
- <programlisting role="example"><![CDATA[
-
+ <programlisting role="example"><![CDATA[
// Hamming window
clf()
N=64;
n=256;[W,fr]=frmag(w,n);
plot2d(fr,20*log10(W),style=color('blue'))
set(gca(),'grid',[1 1]*color('gray'))
-
- ]]></programlisting>
+ ]]></programlisting>
</refsection>
<refsection>
<title>See Also</title>
<title>Description</title>
<para>
Hz = yulewalk(N,frq,mag) finds the N-th order iir filter</para>
- <programlisting><![CDATA[
-
+ <programlisting role = ""><![CDATA[
n-1 n-2
B(z) b(1)z + b(2)z + .... + b(n)
H(z)= ---- = ---------------------------------
n-1 n-2
A(z) z + a(2)z + .... + a(n)
-
- ]]></programlisting>
+ ]]></programlisting>
<para>
which matches the magnitude frequency response given by vectors frq and mag.
Vectors frq and mag specify the frequency and magnitude of the desired
</refsection>
<refsection>
<title>Examples</title>
- <programlisting role="example"><![CDATA[
-
+ <programlisting role="example"><![CDATA[
f=[0,0.4,0.4,0.6,0.6,1];H=[0,0,1,1,0,0];Hz=yulewalk(8,f,H);
fs=1000;fhz = f*fs/2;
xbasc(0);xset('window',0);plot2d(fhz',H');
[frq,repf]=repfreq(Hz,0:0.001:0.5);
xbasc(1);xset('window',1);plot2d(fs*frq',abs(repf'));
xtitle('Obtained Frequency Response (Magnitude)')
-
- ]]></programlisting>
+ ]]></programlisting>
</refsection>
</refentry>
<title>Description</title>
<para>
Poles of a Type 1 Chebyshev analog filter. The transfer function is given by :</para>
- <programlisting><![CDATA[
-
- H(s)=gain/poly(poles,'s')
-
- ]]></programlisting>
+ <programlisting role = ""><![CDATA[
+H(s)=gain/poly(poles,'s')
+ ]]></programlisting>
</refsection>
<refsection>
<title>Authors</title>
<para>
Poles and zeros of a type 2 Chebyshev analog filter
gain is the gain of the filter</para>
- <programlisting><![CDATA[
-
+ <programlisting role = ""><![CDATA[
H(s)=gain*poly(zeros,'s')/poly(poles,'s')
-
- ]]></programlisting>
+ ]]></programlisting>
</refsection>
<refsection>
<title>Authors</title>
projects / scilab.git / commitdiff