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Universidad San Francisco de Quito
February 10, 2014
Ronny RamírezDeber 2
3.1
1.
%%%3.1function [X k] = dtft(x,n,w)k=w/pi;W=exp(-1j*pi/500).^(n'*k); X=x*W;endw=-pi:pi/500:pi;
%1n1=-10:10;x1=0.6.^abs(n1);[X1 k] = dtft(x1,n1,w);
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El ángulo varía signi�cativamente cuando la frecuencia se incrementa de [0, π]
2.
%2n2=0:20;x2=n2.*0.9.^n2;[X2 k] = dtft(x2,n2,w);
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3.
%3n3=0:50;x3=cos(0.5*pi*n3)+1j*sin(0.5*pi*n3);[X3 k] = dtft(x3,n3,w);
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Se puede observar que la magnitud se incrementa mientras que la fase decrece cuando se varia la frecuencia.
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%4n4=0:7;x4=[4 3 2 1 1 2 3 4];[X4 k] = dtft(x4,n4,w);
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Se puede observar que el ángulo varía levemente cuando se incrementa la frecuencia.
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%5n5=0:7;x5=[4 3 2 1 -1 -2 -3 -4];[X5 k] = dtft(x5,n5,w);
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Se puede observar que el ángulo da un salto cuando frecuencia es igual a cero.
3.3
w=0:pi/500:pi;k=w/pi;
1
%1X1=0.25*(exp(1j*2.*w)+exp(1j.*w)+1./(1-exp(1j.*w)));
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2
%2 X2=(1-(0.6.*exp(1j.*w)).^10)./(1-0.6.*exp(1j.*w))-1+(1-(0.6.*exp(-1j.*w)).^10)./(1-0.6.*exp(-1j.*w));
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3
%3 X3=1j*(1.11.*exp(1j.*w))./(1-0.9.*exp(-1j.*w)).^2-4.12.*exp(1j*3.*w)./(1-0.9.*exp(-1j.*w));
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%4 X4=1j*0.512-exp(-1j.*w)./(1-0.8.*exp(-1j.*w)).^2+4.*exp(-1j*2.*w)./(1-0.8.*exp(-1j.*w));
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%5X5=4./(1+(0.7.*exp(-1j.*w)).^2);
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3.4
M1=10;M2=25;M3=50;M4=101;w=-pi:pi/500:pi;
n1=0:M1-1;R1=ones(1,length(n1));C1=0.5*(1-cos((2*pi*n1)/(M1-1)));T1=1-abs(M1-1-2*n1)/(M1+1);H1=0.54-0.46*cos((2*pi*n1)/(M1-1));
[Rf1 k] = dtft(R1,n1,w);
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[Cf1 k] = dtft(C1,n1,w);
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[Tf1 k] = dtft(T1,n1,w);
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[Hf1 k] = dtft(H1,n1,w);
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n2=0:M2-1;R2=ones(1,length(n2));C2=0.5*(1-cos((2*pi*n2)/(M2-1)));T2=1-abs(M2-1-2*n2)/(M2+1);H2=0.54-0.46*cos((2*pi*n2)/(M2-1));[Rf2 k] = dtft(R2,n2,w);
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[Cf2 k] = dtft(C2,n2,w);
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[Tf2 k] = dtft(T2,n2,w);
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[Hf2 k] = dtft(H2,n2,w);
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n3=0:M1-1;R3=ones(1,length(n3));C3=0.5*(1-cos((2*pi*n3)/(M3-1)));T3=1-abs(M3-1-2*n3)/(M3+1);H3=0.54-0.46*cos((2*pi*n3)/(M3-1));[Rf3 k] = dtft(R3,n3,w);
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[Cf3 k] = dtft(C3,n3,w);
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[Tf3 k] = dtft(T3,n3,w);
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[Hf3 k] = dtft(H3,n3,w);
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n4=0:M4-1;R4=ones(1,length(n4));C4=0.5*(1-cos((2*pi*n4)/(M4-1)));T4=1-abs(M4-1-2*n4)/(M4+1);H4=0.54-0.46*cos((2*pi*n4)/(M4-1));[Rf4 k] = dtft(R4,n4,w);
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[Cf4 k] = dtft(C4,n4,w);
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[Tf4 k] = dtft(T4,n4,w);
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[Hf4 k] = dtft(H4,n4,w);
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3.6 Entregado a mano
3.8
%3.8n=0:19;x=exp(1j*0.1*pi.*n);w=-pi:pi/500:pi;xr=real(x);xi=imag(x);xc=conj(x);[X k] = dtft(x,n,w);[Xr k] = dtft(xr,n,w);[Xi k] = dtft(xi,n,w);[Xc k] = dtft(xc,n,w);Xe=1/2*(X+Xc);Xo=1/2*(X-Xc);subplot(2,1,1)plot(k,abs(Xr-Xe)) title('DTFT');ylabel('Magnitud')subplot(2,1,2)plot(k,abs(1j*Xi-Xo))ylabel('Magnitud');xlabel('frecuencia (rad/s)')
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3.10
%3.10M=10;n=0:M-1;T10=1-abs(M-1-2*n)/(M+1);x=T10;w=-pi:pi/500:pi;[X k] = dtft(x,n,w);
1
%1X1=conj(X);
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2
%2X2=X-X.*exp(1j.*w*10);
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3
%3X3=X.*X1;
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4
%4[X4 k4]=sigshift(X,k,pi);
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5
%5xcos=cos(0.1*pi.*n);[Xcos k] = dtft(xcos,n,w);X5=conv(X,Xcos);k5=-1:1/1000:1;
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3.14
1 Entregado a mano.
2
%2N=41;a=20;wc=0.5*pi;n=0:N-1;h=wc/pi*sinc(wc.*(n-a)/pi);w=-pi:pi/500:pi;[H k] = dtft(h,n,w);
3
%3wd=-wc:pi/500:wc;Hd=[zeros(1,length(-pi:pi/500:-wc)-1)exp(-1j*a.*wd)zeros(1,length(wc:pi/500:pi)-1)];
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3.15
1 Entregado a mano
2
%2N=31;a=15;wc=0.5*pi;n=0:N-1;h=sinc(n-a)-sin(wc.*(n-a))./(pi.*(n-a));w=-pi:pi/500:pi;[H k] = dtft(h,n,w);
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%3wd1=-pi:pi/500:-wc;wd2=wc:pi/500:pi;Hd=[exp(-1j*a.*wd1)zeros(1,length(-wc:pi/500:wc)-2)exp(-1j*a.*wd2)];
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3.16
function [H] = freqresp(b,a,w)M=size(b)-1;N=size(a)-1;S1=0;S2=0;for m=0:MS1=S1+b(m+1).*exp(-1j.*w*m);endfor l=1:NS2=S2+a(l+1).*exp(-1j.*w*l);endH=S1./(1+S2);end
3.18
%%%3.18n=0:200;b=[1 0 1 0 1 0 1];a=[1 0 -0.81 0 -0.656 0 -0.531];
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1
%1x1=5+10*(-1).^n;y1=�lter(b,a,x1);
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%2x2=1+cos(0.5*pi.*n+pi/2);y2=�lter(b,a,x2);
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3
%3x3=2*sin(pi.*n/4)+3*cos(3*pi.*n/4);y3=�lter(b,a,x3);
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4
%4x4=0;for k=0:5;x4=x4+(k+1)*cos(pi*k.*n/4);endy4=�lter(b,a,x4);
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5
%5x5=cos(pi.*n);y5=�lter(b,a,x5);
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3.19
1
%1t1=0:0.0001:0.1;x1=sin(1000*pi.*t1);
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%2t2=0:0.001:0.1;x2=sin(1000*pi.*t2);
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3
%3t3=0:0.01:0.1;x3=sin(1000*pi.*t3);
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