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16.362 Signal and System I Magnitude-phase representation of Fourier Transform Magnitude-phase representation of frequency response of LTI systems

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16.362 Signal and System I Ignore reflection at both interfaces Propagation constant

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16.362 Signal and System I Is linear with . gives the group delay t 0. represents the group velocity.

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16.362 Signal and System I Is linear with . gives the group delay t 0. represents the group velocity.

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16.362 Signal and System I represents the group velocity. Free space

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16.362 Signal and System I Example #1 All pass filter

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16.362 Signal and System I Example #1 All pass filter f 1 = 50 Hz f 2 = 150 Hz f 3 = 300 Hz

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16.362 Signal and System I Example #1 All pass filter f 1 = 50 Hzf 2 = 150 Hz f 3 = 300 Hz

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16.362 Signal and System I

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clear; clf; f1 = 50; w1 = f1*2*pi; f2 = 150; w2 = f2*2*pi; f3 = 300; w3 = f3*2*pi; ks1 = 0.066; ks2 = 0.033; ks3 = 0.058; od = 0.0001*2*pi; omega = 0:od:400*2*pi; HW1 =(1+(j*omega./w1).^2-2*j*ks1*(omega./w1)); HW1 = HW1./(1+(j*omega./w1).^2+2*j*ks1*(omega./w1)); HW2 =(1+(j*omega./w2).^2-2*j*ks2*(omega./w2)); HW2 = HW2./(1+(j*omega./w2).^2+2*j*ks2*(omega./w2)); HW3 =(1+(j*omega./w3).^2-2*j*ks3*(omega./w3)); HW3 = HW3./( 1+(j*omega./w3).^2+2*j*ks3*(omega./w3)); HW = HW1.*HW2.*HW3; plot(omega/(2*pi),abs(HW)); zoom on; figure(2) AH = unwrap(angle(HW)); plot(omega/(2*pi),AH); zoom on; DH = -diff(AH); DH = DH./od; figure(3) plot(omega(2:length(omega))/(2*pi), DH); zoom on; % inverse Fourier transform t = 0:0.01:0.2; for n=1:length(t) h(n) = 0; for m=1:length(omega) h(n) = h(n)+HW(m)*exp(j*omega(m)*t(n))*od; end h(n) = h(n)./(2*pi); end figure(4) plot(t,real(h)); zoom on;

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16.362 Signal and System I Log-magnitude and phase plot dB scale, I, V 10%-20dB90% loss 1%-40dB99% loss dB scale for power, P, intensity 10%-10dB90% loss 1%-20dB99% loss f (Hz) 50%-3dB50% loss

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16.362 Signal and System I Time-domain properties of ideal frequency-selective filter 1 1 1 1 #1 #2 #3

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>> n=-50:50; wc=0.4*pi; >> hn=sinc(n*wc/pi)*wc/pi; >> stem(n,hn); title('Ideal LPF with cutoff frequency = 0.4 \pi')

>> n=-50:50; wc=0.4*pi; >> hn=sinc(n*wc/pi)*wc/pi; >> stem(n,hn); title('Ideal LPF with cutoff frequency = 0.4 \pi')

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