magnitude-phase representation of fourier transform
DESCRIPTION
Magnitude-phase representation of Fourier Transform. Magnitude-phase representation of frequency response of LTI systems. Ignore reflection at both interfaces. Propagation constant. Is linear with . gives the group delay t 0. represents the group velocity. Is linear with . - PowerPoint PPT PresentationTRANSCRIPT
16.362 Signal and System I • Magnitude-phase representation of Fourier Transform
][nx
jejXjX )()( )(tx
jjj eeXeX )()(
jejHjH )()( )(th
• Magnitude-phase representation of frequency response of LTI systems
)(ty)(tx )(th
jejHjH )()(
16.362 Signal and System I
10 / nc
Lt
tjetx 1)(1 ?)(1 ty
Ignore reflection at both interfaces
)()( 0111 ttxAty
0111)( tjeAjH
Ln
j
nf
Lj
nc
Lj
eA
eA
eAjH
1
11
11
/
2
1
/1
/11)(
Propagation constant1
1 /
2
nk
16.362 Signal and System I
10 / nc
Lt
tjetx 1)(1 )()( 0111 ttxAty
0111)( tjeAjH
tjetx 2)(2
0212 )( tjeAjH
)()( 0212 ttxAty
jejHjH )()(
Is linear with . d
d gives the group delay t0.
Ld
d1
represents the group velocity.
16.362 Signal and System I
10 / nc
Lt
tjetx 1)(1 )()( 0111 ttxAty
0111)( tjeAjH
jkLejHjH )()(
Is linear with . d
d gives the group delay t0.
represents the group velocity. Ld
d1
1
ddk
jkLejHjH )()(
represents the group velocity.
16.362 Signal and System I 1
ddk represents the group velocity.
Free space
cf
fk
00
22
cd
dk
1
16.362 Signal and System I
Example #1
All pass filter
3
1
)()(i
i jHjH
)()(2
)(1
)()(2
)(1
2
2
22
2
2txtx
dt
dtx
dt
dtyty
dt
dty
dt
d
i
i
ii
i
i
iii
iiii
jj
jjjH
/2/1
/2/1)( 2
2
jiii
jiii
iii
iiii
e
e
j
jjH
222
222
2
2
/2/1
/2/1
/2/1
/2/1)(
2)/(1
)/(2arctan
i
ii
2)()( jii ejHjH
1)( jH i
16.362 Signal and System I
Example #1 All pass filter
3
1
)()(i
i jHjH iii
iiii
jj
jjjH
/2/1
/2/1)( 2
2
2)/(1
)/(2arctan
i
ii 2)()( jii ejHjH 1)( jH i
f 2f1 = 50 Hz
f2 = 150 Hz
f3 = 300 Hz d
dg
16.362 Signal and System I
Example #1 All pass filter
3
1
)()(i
i jHjH
2)/(1
)/(2arctan
i
iii
2)()( jii ejHjH 1)( jH i
f1 = 50 Hz f2 = 150 Hz f3 = 300 Hz
)(1 jH )(2 jH )(3 jH
1)()(3
1
i
i jHjH
3
1ii
16.362 Signal and System I
1)( jH i
3
1ii
d
dg
)(th
16.362 Signal and System I 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 transformt = 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);endfigure(4)plot(t,real(h));zoom on;
16.362 Signal and System I • Log-magnitude and phase plot
)(log20 10 jH i
dB scale, I, V
10% -20dB 90% loss
1% -40dB 99% loss
P10log10
dB scale for power, P, intensity
10% -10dB 90% loss
1% -20dB 99% loss
f (Hz)50% -3dB 50% loss
16.362 Signal and System I • Time-domain properties of ideal frequency-selective filter
c
1
c
1
)( jH
3
)( jeH
c
1
c
)( jH
c
1
c
)( jH
#1
#2
#3
)(th
][nh
)( th